-------------------------------------------------------------------------------------------------------------------------------
      name:  <unnamed>
       log:  c:\acdbookrevision\stata_final_programs_2013\racd03.txt
  log type:  text
 opened on:  14 Jan 2013, 20:38:41

. 
. ********** OVERVIEW OF racd03.do **********
. 
. * STATA Program 
. * copyright C 2013 by A. Colin Cameron and Pravin K. Trivedi 
. * used for "Regression Analyis of Count Data" SECOND EDITION
. * by A. Colin Cameron and Pravin K. Trivedi (2013)
. * Cambridge University Press
. 
. * Chapter 3
. *   3.2 POISSON REGRESSION WITH VARIOUS STANDARD ERRORS
. *   3.3 NEGATIVE BINOMIAL WITH VARIOUS STANDARD ERRORS
. *   3.4 OVERDISPERSION TESTS
. *   3.5 MARGINAL EFFECTS AFTER POISSON
. *   3.7 OTHER MODELS
. * The bootstraps are commented out to speed up execution time.
. 
. * To run you need file
. *   racd03data.dta
. * The included output also inlcued output from user-written Stata addon countfit
. 
. ********** SETUP **********
. 
. set more off

. version 12

. clear all

. * set linesize 82
. set scheme s1mono  /* Graphics scheme */

.  
. ********** DATA DESCRIPTION
. 
. * The data set racd3data.dta is the same data as originally used in
. * (1) A.C. Cameron and P.K. Trivedi (1986), "Econometric Models Based on
. * Count Data: Comparisons and Applications of  Some Estimators and Tests",
. * Journal of Applied Econometrics, Vol. 1, pp. 29-54.
. * and in other papers.
. 
. * This data is not a representative sample of Australians as it oversamples
. * young and old. In particular, use of health services may be overstated.
. * This is because while the original sample of 40,650 individuals
. * from the 1977-78 Australian Health Survey is representative,
. * the sample used here is restricted to single people over 18 years of age.
.  
. * See the R.E.Stud. (1988, pp.85-106) section 3 for more detailed
. * discussion of the data than that given in the RACD book.
. * Also see racd03makedata.do for further details 
. 
. ********** 3.2 READ DATA AND SUMMARIZE 
. 
. use racd03data.dta, clear

. 
. *** TABLE 3.1: FREQUENCIES
. 
. * Tabulate counts of doctor visits
. tabulate DVISITS

  Number of |
 doctor (or |
specialist) |
  visits in |
     past 2 |
      weeks |      Freq.     Percent        Cum.
------------+-----------------------------------
          0 |      4,141       79.79       79.79
          1 |        782       15.07       94.86
          2 |        174        3.35       98.21
          3 |         30        0.58       98.79
          4 |         24        0.46       99.25
          5 |          9        0.17       99.42
          6 |         12        0.23       99.65
          7 |         12        0.23       99.88
          8 |          5        0.10       99.98
          9 |          1        0.02      100.00
------------+-----------------------------------
      Total |      5,190      100.00

. 
. *** TABLE 3.2: VARIABLE DEFINITIONS AND SUMMARY STATISTICS
. 
. * Variable descriptions and summary statistics
. describe

Contains data from racd03data.dta
  obs:         5,190                          
 vars:            20                          21 Jun 2011 11:27
 size:       415,200                          
-------------------------------------------------------------------------------------------------------------------------------
              storage  display     value
variable name   type   format      label      variable label
-------------------------------------------------------------------------------------------------------------------------------
SEX             float  %9.0g                  Equals 1 if female
AGE             float  %9.0g                  Age in years divided by 100 (midpoint of 10 age groups)
AGESQ           float  %9.0g                  AGE squared
INCOME          float  %9.0g                  Annual income in tens of thousands of dollars
LEVYPLUS        float  %9.0g                  Equals if private insurance
FREEPOOR        float  %9.0g                  Equals 1 if free government insurance due to low income
FREEREPA        float  %9.0g                  Equals 1 if free government insurance due to old-age, disability or veteran stat
ILLNESS         float  %9.0g                  Number of illnesses in past 2 weeks
ACTDAYS         float  %9.0g                  Number of days of reduced activity in past two weeks due to illness or injury
HSCORE          float  %9.0g                  General health questionnaire score using Goldberg's method (High score bad hlth)
CHCOND1         float  %9.0g                  Equals 1 if chronic condition(s) but not limited in activity, 0 other
CHCOND2         float  %9.0g                  Equals 1 if chronic condition(s) and limited in activity, 0 otherwise
DVISITS         float  %9.0g                  Number of doctor (or specialist) visits in past 2 weeks
NONDOCCO        float  %9.0g                  Number of consultations with non-doctor health professionals
HOSPADMI        float  %9.0g                  Number of hospital admissions in the past 12 months
HOSPDAYS        float  %9.0g                  Number of nights in a hospital, etc. during most recent admission (mid-range)
MEDICINE        float  %9.0g                  Number of prescribed/ nonprescribed medications in past 2 days
PRESCRIB        float  %9.0g                  Number of prescribed medications used in past 2 days
NONPRESC        float  %9.0g                  Number of nonprescribed medications used in past 2 days
CONSTANT        float  %9.0g                  
-------------------------------------------------------------------------------------------------------------------------------
Sorted by:  

. summarize

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
         SEX |      5190    .5206166    .4996229          0          1
         AGE |      5190    .4063854    .2047818        .19        .72
       AGESQ |      5190    .2070766    .1856365      .0361      .5184
      INCOME |      5190    .5831599    .3689067          0        1.5
    LEVYPLUS |      5190    .4427746    .4967623          0          1
-------------+--------------------------------------------------------
    FREEPOOR |      5190    .0427746     .202368          0          1
    FREEREPA |      5190    .2102119    .4074983          0          1
     ILLNESS |      5190    1.431985    1.384152          0          5
     ACTDAYS |      5190    .8618497    2.887628          0         14
      HSCORE |      5190    1.217534    2.124266          0         12
-------------+--------------------------------------------------------
     CHCOND1 |      5190    .4030829    .4905644          0          1
     CHCOND2 |      5190    .1165703    .3209385          0          1
     DVISITS |      5190    .3017341    .7981338          0          9
    NONDOCCO |      5190    .2146435    .9652756          0         11
    HOSPADMI |      5190    .1736031    .5075236          0          5
-------------+--------------------------------------------------------
    HOSPDAYS |      5190    1.333719    6.120081          0         80
    MEDICINE |      5190    1.218304    1.556643          0          8
    PRESCRIB |      5190    .8626204    1.415375          0          8
    NONPRESC |      5190     .355684     .712389          0          8
    CONSTANT |      5190           1           0          1          1

. 
. * Global for the regressors
. global XLIST SEX AGE AGESQ INCOME LEVYPLUS FREEPOOR FREEREPA ILLNESS ///
>    ACTDAYS HSCORE CHCOND1 CHCOND2

. 
. ********** 3.2 POISSON REGRESSION WITH VARIOUS STANDARD ERRORS
. 
. *** POISSON MLE and QMLE
. 
. * Poisson Robust standard errors
. poisson DVISITS $XLIST, vce(robust) 

Iteration 0:   log pseudolikelihood = -4923.1976  
Iteration 1:   log pseudolikelihood = -3890.2934  
Iteration 2:   log pseudolikelihood = -3356.8559  
Iteration 3:   log pseudolikelihood = -3355.5431  
Iteration 4:   log pseudolikelihood = -3355.5413  
Iteration 5:   log pseudolikelihood = -3355.5413  

Poisson regression                                Number of obs   =       5190
                                                  Wald chi2(12)   =     964.02
                                                  Prob > chi2     =     0.0000
Log pseudolikelihood = -3355.5413                 Pseudo R2       =     0.1576

------------------------------------------------------------------------------
             |               Robust
     DVISITS |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |    .156882   .0792209     1.98   0.048     .0016118    .3121522
         AGE |   1.056299   1.364474     0.77   0.439    -1.618021     3.73062
       AGESQ |  -.8487041   1.459683    -0.58   0.561    -3.709631    2.012223
      INCOME |  -.2053206   .1292572    -1.59   0.112      -.45866    .0480188
    LEVYPLUS |   .1231854   .0951652     1.29   0.196    -.0633348    .3097057
    FREEPOOR |  -.4400609   .2900225    -1.52   0.129    -1.008494    .1283726
    FREEREPA |   .0797984   .1257953     0.63   0.526    -.1667558    .3263527
     ILLNESS |   .1869484   .0239387     7.81   0.000     .1400295    .2338674
     ACTDAYS |   .1268465   .0077698    16.33   0.000     .1116179     .142075
      HSCORE |    .030081   .0142359     2.11   0.035     .0021791    .0579829
     CHCOND1 |   .1140853   .0908541     1.26   0.209    -.0639854    .2921561
     CHCOND2 |   .1411583   .1227226     1.15   0.250    -.0993737    .3816902
       _cons |  -2.223848   .2544567    -8.74   0.000    -2.722574   -1.725122
------------------------------------------------------------------------------

. estimates store PRobust

. 
. * The following GLM command gives the same
. glm DVISITS $XLIST, family(poisson) link(log) vce(robust) 

Iteration 0:   log pseudolikelihood = -3652.9281  
Iteration 1:   log pseudolikelihood = -3358.3637  
Iteration 2:   log pseudolikelihood = -3355.5441  
Iteration 3:   log pseudolikelihood = -3355.5413  
Iteration 4:   log pseudolikelihood = -3355.5413  

Generalized linear models                          No. of obs      =      5190
Optimization     : ML                              Residual df     =      5177
                                                   Scale parameter =         1
Deviance         =  4379.515095                    (1/df) Deviance =  .8459562
Pearson          =  6873.982244                    (1/df) Pearson  =  1.327793

Variance function: V(u) = u                        [Poisson]
Link function    : g(u) = ln(u)                    [Log]

                                                   AIC             =  1.298089
Log pseudolikelihood = -3355.541345                BIC             = -39907.07

------------------------------------------------------------------------------
             |               Robust
     DVISITS |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |    .156882   .0792209     1.98   0.048     .0016118    .3121522
         AGE |   1.056299   1.364474     0.77   0.439    -1.618021     3.73062
       AGESQ |  -.8487041   1.459683    -0.58   0.561    -3.709631    2.012223
      INCOME |  -.2053206   .1292572    -1.59   0.112      -.45866    .0480188
    LEVYPLUS |   .1231854   .0951652     1.29   0.196    -.0633348    .3097057
    FREEPOOR |  -.4400609   .2900225    -1.52   0.129    -1.008494    .1283726
    FREEREPA |   .0797984   .1257953     0.63   0.526    -.1667558    .3263527
     ILLNESS |   .1869484   .0239387     7.81   0.000     .1400295    .2338674
     ACTDAYS |   .1268465   .0077698    16.33   0.000     .1116179     .142075
      HSCORE |    .030081   .0142359     2.11   0.035     .0021791    .0579829
     CHCOND1 |   .1140853   .0908541     1.26   0.209    -.0639854    .2921561
     CHCOND2 |   .1411583   .1227226     1.15   0.250    -.0993737    .3816902
       _cons |  -2.223848   .2544567    -8.74   0.000    -2.722574   -1.725122
------------------------------------------------------------------------------

. 
. * Poisson Bootstrap is asymptotically equivalent
. * Comment out to save time
. * poisson DVISITS $XLIST, vce(boot, reps(400) seed(10101)) 
. * estimates store PBoot
. 
. * Poisson Default ml standard errors (Same as vce(oim))
. poisson DVISITS $XLIST

Iteration 0:   log likelihood = -4923.1976  
Iteration 1:   log likelihood = -3890.2934  
Iteration 2:   log likelihood = -3356.8559  
Iteration 3:   log likelihood = -3355.5431  
Iteration 4:   log likelihood = -3355.5413  
Iteration 5:   log likelihood = -3355.5413  

Poisson regression                                Number of obs   =       5190
                                                  LR chi2(12)     =    1255.31
                                                  Prob > chi2     =     0.0000
Log likelihood = -3355.5413                       Pseudo R2       =     0.1576

------------------------------------------------------------------------------
     DVISITS |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |    .156882   .0561368     2.79   0.005     .0468558    .2669081
         AGE |   1.056299   1.000781     1.06   0.291    -.9051946    3.017794
       AGESQ |  -.8487041   1.077785    -0.79   0.431    -2.961123    1.263715
      INCOME |  -.2053206   .0883793    -2.32   0.020    -.3785409   -.0321003
    LEVYPLUS |   .1231854   .0716398     1.72   0.086    -.0172261    .2635969
    FREEPOOR |  -.4400609   .1798115    -2.45   0.014     -.792485   -.0876369
    FREEREPA |   .0797984   .0920603     0.87   0.386    -.1006364    .2602333
     ILLNESS |   .1869484   .0182805    10.23   0.000     .1511192    .2227776
     ACTDAYS |   .1268465    .005034    25.20   0.000     .1169801    .1367129
      HSCORE |    .030081   .0100994     2.98   0.003     .0102866    .0498754
     CHCOND1 |   .1140853   .0666396     1.71   0.087    -.0165258    .2446964
     CHCOND2 |   .1411583   .0831451     1.70   0.090    -.0218032    .3041197
       _cons |  -2.223848   .1898161   -11.72   0.000    -2.595881   -1.851815
------------------------------------------------------------------------------

. estimates store PMLHess

. 
. * The following GLM command gives the same
. glm DVISITS $XLIST, family(poisson) link(log)

Iteration 0:   log likelihood = -3652.9281  
Iteration 1:   log likelihood = -3358.3637  
Iteration 2:   log likelihood = -3355.5441  
Iteration 3:   log likelihood = -3355.5413  
Iteration 4:   log likelihood = -3355.5413  

Generalized linear models                          No. of obs      =      5190
Optimization     : ML                              Residual df     =      5177
                                                   Scale parameter =         1
Deviance         =  4379.515095                    (1/df) Deviance =  .8459562
Pearson          =  6873.982244                    (1/df) Pearson  =  1.327793

Variance function: V(u) = u                        [Poisson]
Link function    : g(u) = ln(u)                    [Log]

                                                   AIC             =  1.298089
Log likelihood   = -3355.541345                    BIC             = -39907.07

------------------------------------------------------------------------------
             |                 OIM
     DVISITS |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |    .156882   .0561368     2.79   0.005     .0468558    .2669081
         AGE |   1.056299   1.000781     1.06   0.291    -.9051946    3.017794
       AGESQ |  -.8487041   1.077785    -0.79   0.431    -2.961123    1.263715
      INCOME |  -.2053206   .0883793    -2.32   0.020    -.3785409   -.0321003
    LEVYPLUS |   .1231854   .0716398     1.72   0.086    -.0172261    .2635969
    FREEPOOR |  -.4400609   .1798115    -2.45   0.014     -.792485   -.0876369
    FREEREPA |   .0797984   .0920603     0.87   0.386    -.1006364    .2602333
     ILLNESS |   .1869484   .0182805    10.23   0.000     .1511192    .2227776
     ACTDAYS |   .1268465    .005034    25.20   0.000     .1169801    .1367129
      HSCORE |    .030081   .0100994     2.98   0.003     .0102866    .0498754
     CHCOND1 |   .1140853   .0666396     1.71   0.087    -.0165258    .2446964
     CHCOND2 |   .1411583   .0831451     1.70   0.090    -.0218032    .3041197
       _cons |  -2.223848   .1898161   -11.72   0.000    -2.595881   -1.851815
------------------------------------------------------------------------------

. 
. * Poisson OPG standard errors
. poisson DVISITS $XLIST, vce(opg) 

Iteration 0:   log likelihood = -4923.1976  
Iteration 1:   log likelihood = -3890.2934  
Iteration 2:   log likelihood = -3356.8559  
Iteration 3:   log likelihood = -3355.5431  
Iteration 4:   log likelihood = -3355.5413  
Iteration 5:   log likelihood = -3355.5413  

Poisson regression                                Number of obs   =       5190
                                                  LR chi2(12)     =    1255.31
                                                  Prob > chi2     =     0.0000
Log likelihood = -3355.5413                       Pseudo R2       =     0.1576

------------------------------------------------------------------------------
             |                 OPG
     DVISITS |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |    .156882   .0406153     3.86   0.000     .0772774    .2364865
         AGE |   1.056299   .7498656     1.41   0.159    -.4134101    2.526009
       AGESQ |  -.8487041   .8092148    -1.05   0.294    -2.434736    .7373278
      INCOME |  -.2053206    .061921    -3.32   0.001    -.3266834   -.0839578
    LEVYPLUS |   .1231854   .0560472     2.20   0.028     .0133349    .2330359
    FREEPOOR |  -.4400609   .1163511    -3.78   0.000     -.668105   -.2120169
    FREEREPA |   .0797984   .0700594     1.14   0.255    -.0575154    .2171123
     ILLNESS |   .1869484   .0141893    13.18   0.000     .1591378     .214759
     ACTDAYS |   .1268465   .0035073    36.17   0.000     .1199722    .1337207
      HSCORE |    .030081   .0073544     4.09   0.000     .0156666    .0444954
     CHCOND1 |   .1140853   .0514849     2.22   0.027     .0131767    .2149939
     CHCOND2 |   .1411583    .058631     2.41   0.016     .0262436     .256073
       _cons |  -2.223848   .1443307   -15.41   0.000    -2.506731   -1.940965
------------------------------------------------------------------------------

. estimates store PMLOPG

. 
. * The following GLM command gives the same
. glm DVISITS $XLIST, family(poisson) link(log) vce(opg) 

Iteration 0:   log likelihood = -3652.9281  
Iteration 1:   log likelihood = -3358.3637  
Iteration 2:   log likelihood = -3355.5441  
Iteration 3:   log likelihood = -3355.5413  
Iteration 4:   log likelihood = -3355.5413  

Generalized linear models                          No. of obs      =      5190
Optimization     : ML                              Residual df     =      5177
                                                   Scale parameter =         1
Deviance         =  4379.515095                    (1/df) Deviance =  .8459562
Pearson          =  6873.982244                    (1/df) Pearson  =  1.327793

Variance function: V(u) = u                        [Poisson]
Link function    : g(u) = ln(u)                    [Log]

                                                   AIC             =  1.298089
Log likelihood   = -3355.541345                    BIC             = -39907.07

------------------------------------------------------------------------------
             |                 OPG
     DVISITS |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |    .156882   .0406114     3.86   0.000      .077285    .2364789
         AGE |   1.056299   .7497934     1.41   0.159    -.4132685    2.525867
       AGESQ |  -.8487041   .8091368    -1.05   0.294    -2.434583     .737175
      INCOME |  -.2053206    .061915    -3.32   0.001    -.3266717   -.0839694
    LEVYPLUS |   .1231854   .0560418     2.20   0.028     .0133455    .2330254
    FREEPOOR |  -.4400609   .1163399    -3.78   0.000     -.668083   -.2120388
    FREEREPA |   .0797984   .0700526     1.14   0.255    -.0575022    .2170991
     ILLNESS |   .1869484    .014188    13.18   0.000     .1591405    .2147563
     ACTDAYS |   .1268465    .003507    36.17   0.000     .1199729    .1337201
      HSCORE |    .030081   .0073537     4.09   0.000      .015668     .044494
     CHCOND1 |   .1140853     .05148     2.22   0.027     .0131864    .2149842
     CHCOND2 |   .1411583   .0586254     2.41   0.016     .0262547    .2560619
       _cons |  -2.223848   .1443168   -15.41   0.000    -2.506704   -1.940993
------------------------------------------------------------------------------

. 
. * Poisson NB1 standard errors
. glm DVISITS $XLIST, family(poisson) link(log) scale(x2)

Iteration 0:   log likelihood = -3652.9281  
Iteration 1:   log likelihood = -3358.3637  
Iteration 2:   log likelihood = -3355.5441  
Iteration 3:   log likelihood = -3355.5413  
Iteration 4:   log likelihood = -3355.5413  

Generalized linear models                          No. of obs      =      5190
Optimization     : ML                              Residual df     =      5177
                                                   Scale parameter =         1
Deviance         =  4379.515095                    (1/df) Deviance =  .8459562
Pearson          =  6873.982244                    (1/df) Pearson  =  1.327793

Variance function: V(u) = u                        [Poisson]
Link function    : g(u) = ln(u)                    [Log]

                                                   AIC             =  1.298089
Log likelihood   = -3355.541345                    BIC             = -39907.07

------------------------------------------------------------------------------
             |                 OIM
     DVISITS |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |    .156882   .0646864     2.43   0.015      .030099     .283665
         AGE |   1.056299   1.153198     0.92   0.360    -1.203928    3.316527
       AGESQ |  -.8487041    1.24193    -0.68   0.494    -3.282842    1.585434
      INCOME |  -.2053206   .1018394    -2.02   0.044    -.4049221   -.0057191
    LEVYPLUS |   .1231854   .0825505     1.49   0.136    -.0386106    .2849814
    FREEPOOR |  -.4400609   .2071966    -2.12   0.034    -.8461587   -.0339631
    FREEREPA |   .0797984   .1060809     0.75   0.452    -.1281164    .2877133
     ILLNESS |   .1869484   .0210647     8.87   0.000     .1456625    .2282344
     ACTDAYS |   .1268465   .0058006    21.87   0.000     .1154774    .1382155
      HSCORE |    .030081   .0116375     2.58   0.010     .0072719    .0528901
     CHCOND1 |   .1140853   .0767887     1.49   0.137    -.0364177    .2645883
     CHCOND2 |   .1411583    .095808     1.47   0.141     -.046622    .3289385
       _cons |  -2.223848   .2187249   -10.17   0.000    -2.652541   -1.795155
------------------------------------------------------------------------------
(Standard errors scaled using square root of Pearson X2-based dispersion.)

. estimates store PNB1

. 
. * Poisson NB2 standard errors
. quietly poisson DVISITS $XLIST 

. matrix InvHessian = e(V)

. matrix b = e(b)

. scalar Nobs = e(N)

. scalar k = e(k)

. predict mu, n

. generate terminsum = ((DVISITS - mu)^2 - mu) / (mu^2)

. quietly summarize terminsum

. scalar alphanb2 = r(sum) / (Nobs-k)

. display "alpha for NB2 : " alphanb2
alpha for NB2 : .28644145

. generate NB2weight = mu + alphanb2*mu^2

. matrix accum Vmiddle = $XLIST [pweight = NB2weight]
(obs=1932)

. matrix VNB2 = InvHessian*Vmiddle*InvHessian

. ereturn post b VNB2

. 
. ** TABLE 3.3: POISSON PMLE WITH NB2 STANDARD ERRORS
. 
. ereturn display
------------------------------------------------------------------------------
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
DVISITS      |
         SEX |    .156882   .0618302     2.54   0.011      .035697    .2780669
         AGE |   1.056299   1.111559     0.95   0.342    -1.122316    3.234915
       AGESQ |  -.8487041    1.21038    -0.70   0.483    -3.221006    1.523598
      INCOME |  -.2053206   .0960251    -2.14   0.033    -.3935264   -.0171148
    LEVYPLUS |   .1231854   .0765315     1.61   0.107    -.0268135    .2731843
    FREEPOOR |  -.4400609   .1876806    -2.34   0.019    -.8079082   -.0722136
    FREEREPA |   .0797984   .1021836     0.78   0.435    -.1204778    .2800747
     ILLNESS |   .1869484   .0206126     9.07   0.000     .1465484    .2273485
     ACTDAYS |   .1268465   .0058892    21.54   0.000     .1153038    .1383892
      HSCORE |    .030081   .0117875     2.55   0.011     .0069779    .0531841
     CHCOND1 |   .1140853   .0709159     1.61   0.108    -.0249073    .2530779
     CHCOND2 |   .1411583   .0922214     1.53   0.126    -.0395924     .321909
       _cons |  -2.223848   .2069494   -10.75   0.000    -2.629462   -1.818235
------------------------------------------------------------------------------

. 
. *** TABLE 3.3: POISSON PMLE WITH DIFFERENT STANDARD ERRORS 
. 
. * Most of Table 3.3 (except PNB2 given just above and PBoot )
. estimates table PRobust PMLHess PMLOPG PNB1, b(%9.3f) se(%9.3f) t(%9.2f)

--------------------------------------------------------------
    Variable |  PRobust     PMLHess     PMLOPG       PNB1     
-------------+------------------------------------------------
         SEX |     0.157       0.157       0.157       0.157  
             |     0.079       0.056       0.041       0.065  
             |      1.98        2.79        3.86        2.43  
         AGE |     1.056       1.056       1.056       1.056  
             |     1.364       1.001       0.750       1.153  
             |      0.77        1.06        1.41        0.92  
       AGESQ |    -0.849      -0.849      -0.849      -0.849  
             |     1.460       1.078       0.809       1.242  
             |     -0.58       -0.79       -1.05       -0.68  
      INCOME |    -0.205      -0.205      -0.205      -0.205  
             |     0.129       0.088       0.062       0.102  
             |     -1.59       -2.32       -3.32       -2.02  
    LEVYPLUS |     0.123       0.123       0.123       0.123  
             |     0.095       0.072       0.056       0.083  
             |      1.29        1.72        2.20        1.49  
    FREEPOOR |    -0.440      -0.440      -0.440      -0.440  
             |     0.290       0.180       0.116       0.207  
             |     -1.52       -2.45       -3.78       -2.12  
    FREEREPA |     0.080       0.080       0.080       0.080  
             |     0.126       0.092       0.070       0.106  
             |      0.63        0.87        1.14        0.75  
     ILLNESS |     0.187       0.187       0.187       0.187  
             |     0.024       0.018       0.014       0.021  
             |      7.81       10.23       13.18        8.87  
     ACTDAYS |     0.127       0.127       0.127       0.127  
             |     0.008       0.005       0.004       0.006  
             |     16.33       25.20       36.17       21.87  
      HSCORE |     0.030       0.030       0.030       0.030  
             |     0.014       0.010       0.007       0.012  
             |      2.11        2.98        4.09        2.58  
     CHCOND1 |     0.114       0.114       0.114       0.114  
             |     0.091       0.067       0.051       0.077  
             |      1.26        1.71        2.22        1.49  
     CHCOND2 |     0.141       0.141       0.141       0.141  
             |     0.123       0.083       0.059       0.096  
             |      1.15        1.70        2.41        1.47  
       _cons |    -2.224      -2.224      -2.224      -2.224  
             |     0.254       0.190       0.144       0.219  
             |     -8.74      -11.72      -15.41      -10.17  
--------------------------------------------------------------
                                                legend: b/se/t

. * estimates table PRobust PMLHess PMLOPG PNB1 PBoot, b(%9.3f) se(%9.3f) t(%9.2f)
. 
. /* Jackknife takes a long time so commented out. Results are as follows.
> poisson DVISITS $XLIST, vce(jackknife)
> Poisson regression                              Number of obs      =      5190
>                                                 Replications       =      5190
>                                                 F(  12,   5189)    =     77.11
>                                                 Prob > F           =    0.0000
> Log likelihood = -3355.5413                     Pseudo R2          =    0.1576
> ------------------------------------------------------------------------------
>              |              Jackknife
>      DVISITS |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
> -------------+----------------------------------------------------------------
>          SEX |    .156882    .080145     1.96   0.050     -.000236    .3139999
>          AGE |   1.056299   1.380652     0.77   0.444    -1.650361     3.76296
>        AGESQ |  -.8487041   1.477986    -0.57   0.566    -3.746179    2.048771
>       INCOME |  -.2053206   .1309306    -1.57   0.117    -.4619997    .0513585
>     LEVYPLUS |   .1231854   .0961254     1.28   0.200    -.0652609    .3116318
>     FREEPOOR |  -.4400609   .3139898    -1.40   0.161    -1.055613    .1754914
>     FREEREPA |   .0797984   .1273904     0.63   0.531    -.1699404    .3295373
>      ILLNESS |   .1869484   .0242433     7.71   0.000     .1394213    .2344755
>      ACTDAYS |   .1268465   .0078955    16.07   0.000     .1113679     .142325
>       HSCORE |    .030081   .0144887     2.08   0.038     .0016771     .058485
>      CHCOND1 |   .1140853   .0917735     1.24   0.214    -.0658294        .294
>      CHCOND2 |   .1411583   .1241703     1.14   0.256    -.1022679    .3845844
>        _cons |  -2.223848   .2569408    -8.66   0.000     -2.72756   -1.720136
> ------------------------------------------------------------------------------
> */
. 
. * Poisson estimated using Stata ml command 
. program lfpois
  1.   version 11
  2.   args lnf theta1                  // theta1=x'b, lnf=lnf(y)
  3.   tempvar lnyfact mu
  4.   local y "$ML_y1"                 // Define y so program more readable
  5.   generate double `lnyfact' = lnfactorial(`y')
  6.   generate double `mu'      = exp(`theta1')
  7.   quietly replace `lnf'     = -`mu' + `y'*`theta1' - `lnyfact'
  8. end

. ml model lf lfpois (DVISITS = $XLIST), vce(robust)

. ml maximize 

initial:       log pseudolikelihood = -5730.7989
alternative:   log pseudolikelihood =  -4471.693
rescale:       log pseudolikelihood = -4016.0932
Iteration 0:   log pseudolikelihood = -4016.0932  
Iteration 1:   log pseudolikelihood =   -3485.72  
Iteration 2:   log pseudolikelihood = -3356.3275  
Iteration 3:   log pseudolikelihood = -3355.5416  
Iteration 4:   log pseudolikelihood = -3355.5413  

                                                  Number of obs   =       5190
                                                  Wald chi2(12)   =     964.02
Log pseudolikelihood = -3355.5413                 Prob > chi2     =     0.0000

------------------------------------------------------------------------------
             |               Robust
     DVISITS |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |    .156882   .0792209     1.98   0.048     .0016118    .3121522
         AGE |     1.0563   1.364474     0.77   0.439     -1.61802     3.73062
       AGESQ |  -.8487049   1.459683    -0.58   0.561    -3.709631    2.012222
      INCOME |  -.2053207   .1292571    -1.59   0.112      -.45866    .0480186
    LEVYPLUS |   .1231855   .0951651     1.29   0.196    -.0633347    .3097058
    FREEPOOR |  -.4400608   .2900224    -1.52   0.129    -1.008494    .1283726
    FREEREPA |   .0797985   .1257953     0.63   0.526    -.1667557    .3263527
     ILLNESS |   .1869483   .0239387     7.81   0.000     .1400294    .2338673
     ACTDAYS |   .1268465   .0077698    16.33   0.000     .1116179     .142075
      HSCORE |    .030081   .0142359     2.11   0.035     .0021792    .0579829
     CHCOND1 |   .1140855   .0908541     1.26   0.209    -.0639852    .2921562
     CHCOND2 |   .1411585   .1227226     1.15   0.250    -.0993734    .3816904
       _cons |  -2.223848   .2544567    -8.74   0.000    -2.722574   -1.725122
------------------------------------------------------------------------------

. 
. /* Following not run to save time but cited in discussion of Table 3.3
>    Output is given for the first bootstrap
> * Two checks: 
> * (1) correct standard errors if DVISITS_se observed Coef. 
> *     is close to DVISITS Bootstrap Std. Error
> * (2) variablity of the s.e. is DVISITS_se Bootstrap Std. Error
> * Poisson Robust sandwich se's
> bootstrap _b _se, reps(400) seed(10101): poisson DVISITS $XLIST, vce(robust)
> Bootstrap results                               Number of obs      =      5190
>                                                 Replications       =       400
> ------------------------------------------------------------------------------
>              |   Observed   Bootstrap                         Normal-based
>              |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
> -------------+----------------------------------------------------------------
> DVISITS      |
>          SEX |    .156882    .076327     2.06   0.040     .0072839      .30648
>          AGE |   1.056299   1.390803     0.76   0.448    -1.669625    3.782224
>        AGESQ |  -.8487041   1.476844    -0.57   0.566    -3.743264    2.045856
>       INCOME |  -.2053206    .128924    -1.59   0.111    -.4580069    .0473657
>     LEVYPLUS |   .1231854   .0998184     1.23   0.217    -.0724551    .3188259
>     FREEPOOR |  -.4400609   .2932651    -1.50   0.133     -1.01485    .1347281
>     FREEREPA |   .0797984    .131406     0.61   0.544    -.1777525    .3373494
>      ILLNESS |   .1869484   .0243491     7.68   0.000     .1392252    .2346717
>      ACTDAYS |   .1268465   .0079706    15.91   0.000     .1112243    .1424686
>       HSCORE |    .030081   .0138043     2.18   0.029     .0030251    .0571369
>      CHCOND1 |   .1140853   .0869783     1.31   0.190    -.0563889    .2845595
>      CHCOND2 |   .1411583   .1198889     1.18   0.239    -.0938196    .3761362
>        _cons |  -2.223848   .2705066    -8.22   0.000    -2.754031   -1.693665
> -------------+----------------------------------------------------------------
> DVISITS_se   |
>          SEX |   .0792209   .0031799    24.91   0.000     .0729885    .0854534
>          AGE |   1.364474   .0778062    17.54   0.000     1.211977    1.516972
>        AGESQ |   1.459683   .0716192    20.38   0.000     1.319312    1.600054
>       INCOME |   .1292572   .0097382    13.27   0.000     .1101707    .1483436
>     LEVYPLUS |   .0951652   .0047033    20.23   0.000     .0859468    .1043835
>     FREEPOOR |   .2900225   .0384038     7.55   0.000     .2147525    .3652924
>     FREEREPA |   .1257953   .0067341    18.68   0.000     .1125967    .1389939
>      ILLNESS |   .0239387   .0014318    16.72   0.000     .0211323     .026745
>      ACTDAYS |   .0077698   .0004421    17.58   0.000     .0069034    .0086362
>       HSCORE |   .0142359   .0008156    17.45   0.000     .0126373    .0158345
>      CHCOND1 |   .0908541   .0040289    22.55   0.000     .0829577    .0987505
>      CHCOND2 |   .1227226   .0066592    18.43   0.000     .1096708    .1357744
>        _cons |   .2544567    .011415    22.29   0.000     .2320838    .2768297
> ------------------------------------------------------------------------------
> * Poisson NB1 se's (assumes variance multiple of the mean)
> bootstrap _b _se, reps(400) seed(10101): glm DVISITS $XLIST, family(poisson) link(log)
> */
. 
. ********** 3.3 NEGATIVE BINOMIAL WITH VARIOUS STANDARD ERRORS
. 
. *** NB2 MLE
. 
. * Negbin2 MLE with default standard errors
. nbreg DVISITS $XLIST, dispersion(mean) 

Fitting Poisson model:

Iteration 0:   log likelihood = -4923.1976  
Iteration 1:   log likelihood = -3890.2934  
Iteration 2:   log likelihood = -3356.8559  
Iteration 3:   log likelihood = -3355.5431  
Iteration 4:   log likelihood = -3355.5413  
Iteration 5:   log likelihood = -3355.5413  

Fitting constant-only model:

Iteration 0:   log likelihood = -3657.9344  
Iteration 1:   log likelihood = -3589.7157  
Iteration 2:   log likelihood = -3586.0033  
Iteration 3:   log likelihood = -3585.9916  
Iteration 4:   log likelihood = -3585.9916  

Fitting full model:

Iteration 0:   log likelihood = -3301.0284  
Iteration 1:   log likelihood = -3204.6437  
Iteration 2:   log likelihood = -3198.7794  
Iteration 3:   log likelihood = -3198.7438  
Iteration 4:   log likelihood = -3198.7438  

Negative binomial regression                      Number of obs   =       5190
                                                  LR chi2(12)     =     774.50
Dispersion     = mean                             Prob > chi2     =     0.0000
Log likelihood = -3198.7438                       Pseudo R2       =     0.1080

------------------------------------------------------------------------------
     DVISITS |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |   .2166435   .0693873     3.12   0.002     .0806469    .3526402
         AGE |  -.2161581   1.281016    -0.17   0.866    -2.726903    2.294587
       AGESQ |    .609158   1.406185     0.43   0.665    -2.146914     3.36523
      INCOME |  -.1422016   .1081902    -1.31   0.189    -.3542504    .0698472
    LEVYPLUS |   .1180641   .0855379     1.38   0.168    -.0495872    .2857153
    FREEPOOR |   -.496611   .2068902    -2.40   0.016    -.9021084   -.0911135
    FREEREPA |   .1449816   .1169489     1.24   0.215     -.084234    .3741971
     ILLNESS |   .2143414   .0242276     8.85   0.000     .1668561    .2618266
     ACTDAYS |   .1437537    .007814    18.40   0.000     .1284385    .1590689
      HSCORE |   .0380601   .0137992     2.76   0.006     .0110142     .065106
     CHCOND1 |    .099355   .0787023     1.26   0.207    -.0548987    .2536087
     CHCOND2 |    .190327   .1044087     1.82   0.068    -.0143102    .3949642
       _cons |  -2.190007   .2335801    -9.38   0.000    -2.647815   -1.732198
-------------+----------------------------------------------------------------
    /lnalpha |   .0742145   .0956442                     -.1132447    .2616737
-------------+----------------------------------------------------------------
       alpha |   1.077038   .1030124                      .8929321    1.299103
------------------------------------------------------------------------------
Likelihood-ratio test of alpha=0:  chibar2(01) =  313.60 Prob>=chibar2 = 0.000

. estimates store NB2MLHess

. 
. * Negbin2 MLE with robust standard errors
. nbreg DVISITS $XLIST, dispersion(mean) vce(robust)

Fitting Poisson model:

Iteration 0:   log pseudolikelihood = -4923.1976  
Iteration 1:   log pseudolikelihood = -3890.2934  
Iteration 2:   log pseudolikelihood = -3356.8559  
Iteration 3:   log pseudolikelihood = -3355.5431  
Iteration 4:   log pseudolikelihood = -3355.5413  
Iteration 5:   log pseudolikelihood = -3355.5413  

Fitting constant-only model:

Iteration 0:   log pseudolikelihood = -3657.9344  
Iteration 1:   log pseudolikelihood = -3589.7157  
Iteration 2:   log pseudolikelihood = -3586.0033  
Iteration 3:   log pseudolikelihood = -3585.9916  
Iteration 4:   log pseudolikelihood = -3585.9916  

Fitting full model:

Iteration 0:   log pseudolikelihood = -3301.0284  
Iteration 1:   log pseudolikelihood = -3204.6437  
Iteration 2:   log pseudolikelihood = -3198.7794  
Iteration 3:   log pseudolikelihood = -3198.7438  
Iteration 4:   log pseudolikelihood = -3198.7438  

Negative binomial regression                      Number of obs   =       5190
Dispersion           = mean                       Wald chi2(12)   =     859.34
Log pseudolikelihood = -3198.7438                 Prob > chi2     =     0.0000

------------------------------------------------------------------------------
             |               Robust
     DVISITS |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |   .2166435   .0744094     2.91   0.004     .0708037    .3624833
         AGE |  -.2161581   1.366925    -0.16   0.874    -2.895282    2.462966
       AGESQ |    .609158   1.473207     0.41   0.679    -2.278275    3.496591
      INCOME |  -.1422016   .1221171    -1.16   0.244    -.3815467    .0971435
    LEVYPLUS |   .1180641   .0914323     1.29   0.197      -.06114    .2972681
    FREEPOOR |   -.496611   .2541404    -1.95   0.051     -.994717     .001495
    FREEREPA |   .1449816   .1213457     1.19   0.232    -.0928516    .3828147
     ILLNESS |   .2143414   .0236518     9.06   0.000     .1679846    .2606981
     ACTDAYS |   .1437537   .0087244    16.48   0.000     .1266542    .1608532
      HSCORE |   .0380601   .0137203     2.77   0.006     .0111689    .0649514
     CHCOND1 |    .099355   .0832814     1.19   0.233    -.0638736    .2625835
     CHCOND2 |    .190327   .1170918     1.63   0.104    -.0391687    .4198228
       _cons |  -2.190007   .2493685    -8.78   0.000     -2.67876   -1.701253
-------------+----------------------------------------------------------------
    /lnalpha |   .0742145   .1081377                     -.1377315    .2861604
-------------+----------------------------------------------------------------
       alpha |   1.077038   .1164684                      .8713326    1.331306
------------------------------------------------------------------------------

. estimates store NB2Robust

. 
. * Negbin2 MLE with OPG standard errors
. nbreg DVISITS $XLIST, dispersion(mean) vce(opg)

Fitting Poisson model:

Iteration 0:   log likelihood = -4923.1976  
Iteration 1:   log likelihood = -3890.2934  
Iteration 2:   log likelihood = -3356.8559  
Iteration 3:   log likelihood = -3355.5431  
Iteration 4:   log likelihood = -3355.5413  
Iteration 5:   log likelihood = -3355.5413  

Fitting constant-only model:

Iteration 0:   log likelihood = -3657.9344  
Iteration 1:   log likelihood = -3589.7157  
Iteration 2:   log likelihood = -3586.0033  
Iteration 3:   log likelihood = -3585.9916  
Iteration 4:   log likelihood = -3585.9916  

Fitting full model:

Iteration 0:   log likelihood = -3301.0284  
Iteration 1:   log likelihood = -3204.6437  
Iteration 2:   log likelihood = -3198.7794  
Iteration 3:   log likelihood = -3198.7438  
Iteration 4:   log likelihood = -3198.7438  

Negative binomial regression                      Number of obs   =       5190
                                                  LR chi2(12)     =     774.50
Dispersion     = mean                             Prob > chi2     =     0.0000
Log likelihood = -3198.7438                       Pseudo R2       =     0.1080

------------------------------------------------------------------------------
             |                 OPG
     DVISITS |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |   .2166435   .0656017     3.30   0.001     .0880665    .3452206
         AGE |  -.2161581    1.23418    -0.18   0.861    -2.635107     2.20279
       AGESQ |    .609158   1.380479     0.44   0.659    -2.096531    3.314847
      INCOME |  -.1422016   .0976648    -1.46   0.145     -.333621    .0492178
    LEVYPLUS |   .1180641   .0848782     1.39   0.164    -.0482943    .2844224
    FREEPOOR |   -.496611   .1750976    -2.84   0.005    -.8397959   -.1534261
    FREEREPA |   .1449816    .117429     1.23   0.217    -.0851751    .3751383
     ILLNESS |   .2143414   .0257348     8.33   0.000      .163902    .2647807
     ACTDAYS |   .1437537   .0074716    19.24   0.000     .1291096    .1583978
      HSCORE |   .0380601   .0142574     2.67   0.008     .0101161    .0660042
     CHCOND1 |    .099355   .0766507     1.30   0.195    -.0508777    .2495876
     CHCOND2 |    .190327   .0948246     2.01   0.045     .0044742    .3761799
       _cons |  -2.190007   .2224622    -9.84   0.000    -2.626025   -1.753989
-------------+----------------------------------------------------------------
    /lnalpha |   .0742145   .0914208                     -.1049669    .2533959
-------------+----------------------------------------------------------------
       alpha |   1.077038   .0984636                      .9003543    1.288393
------------------------------------------------------------------------------
Likelihood-ratio test of alpha=0:  chibar2(01) =  313.60 Prob>=chibar2 = 0.000

. estimates store NB2MLOPG

. 
. * Negbin2 MLE with OIM standard errors
. nbreg DVISITS $XLIST, dispersion(mean) vce(oim)

Fitting Poisson model:

Iteration 0:   log likelihood = -4923.1976  
Iteration 1:   log likelihood = -3890.2934  
Iteration 2:   log likelihood = -3356.8559  
Iteration 3:   log likelihood = -3355.5431  
Iteration 4:   log likelihood = -3355.5413  
Iteration 5:   log likelihood = -3355.5413  

Fitting constant-only model:

Iteration 0:   log likelihood = -3657.9344  
Iteration 1:   log likelihood = -3589.7157  
Iteration 2:   log likelihood = -3586.0033  
Iteration 3:   log likelihood = -3585.9916  
Iteration 4:   log likelihood = -3585.9916  

Fitting full model:

Iteration 0:   log likelihood = -3301.0284  
Iteration 1:   log likelihood = -3204.6437  
Iteration 2:   log likelihood = -3198.7794  
Iteration 3:   log likelihood = -3198.7438  
Iteration 4:   log likelihood = -3198.7438  

Negative binomial regression                      Number of obs   =       5190
                                                  LR chi2(12)     =     774.50
Dispersion     = mean                             Prob > chi2     =     0.0000
Log likelihood = -3198.7438                       Pseudo R2       =     0.1080

------------------------------------------------------------------------------
     DVISITS |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |   .2166435   .0693873     3.12   0.002     .0806469    .3526402
         AGE |  -.2161581   1.281016    -0.17   0.866    -2.726903    2.294587
       AGESQ |    .609158   1.406185     0.43   0.665    -2.146914     3.36523
      INCOME |  -.1422016   .1081902    -1.31   0.189    -.3542504    .0698472
    LEVYPLUS |   .1180641   .0855379     1.38   0.168    -.0495872    .2857153
    FREEPOOR |   -.496611   .2068902    -2.40   0.016    -.9021084   -.0911135
    FREEREPA |   .1449816   .1169489     1.24   0.215     -.084234    .3741971
     ILLNESS |   .2143414   .0242276     8.85   0.000     .1668561    .2618266
     ACTDAYS |   .1437537    .007814    18.40   0.000     .1284385    .1590689
      HSCORE |   .0380601   .0137992     2.76   0.006     .0110142     .065106
     CHCOND1 |    .099355   .0787023     1.26   0.207    -.0548987    .2536087
     CHCOND2 |    .190327   .1044087     1.82   0.068    -.0143102    .3949642
       _cons |  -2.190007   .2335801    -9.38   0.000    -2.647815   -1.732198
-------------+----------------------------------------------------------------
    /lnalpha |   .0742145   .0956442                     -.1132447    .2616737
-------------+----------------------------------------------------------------
       alpha |   1.077038   .1030124                      .8929321    1.299103
------------------------------------------------------------------------------
Likelihood-ratio test of alpha=0:  chibar2(01) =  313.60 Prob>=chibar2 = 0.000

. estimates store NB2MLOIM

. 
. * Negbin2 MLE with bootstrap standard errors
. * nbreg DVISITS $XLIST, dispersion(mean) vce(boot, reps(400) seed(10101) nodots) 
. * estimates store NB2Boot
. 
. * Negbin2 MLE with four different ways to estimate standard errors 
. estimates table NB2Robust NB2MLHess NB2MLOPG NB2MLOIM, b(%9.3f) se

--------------------------------------------------------------
    Variable | NB2Robust   NB2MLHess   NB2MLOPG    NB2MLOIM   
-------------+------------------------------------------------
DVISITS      |
         SEX |     0.217       0.217       0.217       0.217  
             |     0.074       0.069       0.066       0.069  
         AGE |    -0.216      -0.216      -0.216      -0.216  
             |     1.367       1.281       1.234       1.281  
       AGESQ |     0.609       0.609       0.609       0.609  
             |     1.473       1.406       1.380       1.406  
      INCOME |    -0.142      -0.142      -0.142      -0.142  
             |     0.122       0.108       0.098       0.108  
    LEVYPLUS |     0.118       0.118       0.118       0.118  
             |     0.091       0.086       0.085       0.086  
    FREEPOOR |    -0.497      -0.497      -0.497      -0.497  
             |     0.254       0.207       0.175       0.207  
    FREEREPA |     0.145       0.145       0.145       0.145  
             |     0.121       0.117       0.117       0.117  
     ILLNESS |     0.214       0.214       0.214       0.214  
             |     0.024       0.024       0.026       0.024  
     ACTDAYS |     0.144       0.144       0.144       0.144  
             |     0.009       0.008       0.007       0.008  
      HSCORE |     0.038       0.038       0.038       0.038  
             |     0.014       0.014       0.014       0.014  
     CHCOND1 |     0.099       0.099       0.099       0.099  
             |     0.083       0.079       0.077       0.079  
     CHCOND2 |     0.190       0.190       0.190       0.190  
             |     0.117       0.104       0.095       0.104  
       _cons |    -2.190      -2.190      -2.190      -2.190  
             |     0.249       0.234       0.222       0.234  
-------------+------------------------------------------------
lnalpha      |
       _cons |     0.074       0.074       0.074       0.074  
             |     0.108       0.096       0.091       0.096  
--------------------------------------------------------------
                                                  legend: b/se

. 
. * ASIDE: Negbin2 ML estimated using Stata ml command 
. program lfnb2
  1.   version 11
  2.   args lnf theta1 a               // theta1=x'b, a=alpha, lnf=lnf(y)
  3.   tempvar mu
  4.   local y $ML_y1                  // Define y so program more readable
  5.   generate double `mu'  = exp(`theta1')
  6.   quietly replace `lnf' = lngamma(`y'+(1/`a')) - lngamma((1/`a'))  ///
>                -  lnfactorial(`y') - (`y'+(1/`a'))*ln(1+`a'*`mu')  ///
>                + `y'*ln(`a') + `y'*ln(`mu') 
  7. end

. ml model lf lfnb2 (DVISITS = $XLIST) ()

. ml maximize

initial:       log likelihood =     -<inf>  (could not be evaluated)
feasible:      log likelihood = -6593.6234
rescale:       log likelihood = -5730.7989
rescale eq:    log likelihood = -4016.0931
Iteration 0:   log likelihood = -4016.0931  (not concave)
Iteration 1:   log likelihood = -3910.7963  (not concave)
Iteration 2:   log likelihood = -3910.7905  (not concave)
Iteration 3:   log likelihood = -3672.4833  (not concave)
Iteration 4:   log likelihood = -3630.4662  (not concave)
Iteration 5:   log likelihood = -3491.1024  (not concave)
Iteration 6:   log likelihood = -3330.1697  
Iteration 7:   log likelihood = -3266.3185  
Iteration 8:   log likelihood = -3206.4172  
Iteration 9:   log likelihood = -3198.8164  
Iteration 10:  log likelihood = -3198.7439  
Iteration 11:  log likelihood = -3198.7438  

                                                  Number of obs   =       5190
                                                  Wald chi2(12)   =     706.67
Log likelihood = -3198.7438                       Prob > chi2     =     0.0000

------------------------------------------------------------------------------
     DVISITS |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
eq1          |
         SEX |   .2166435   .0693873     3.12   0.002     .0806469    .3526402
         AGE |  -.2161583   1.281016    -0.17   0.866    -2.726903    2.294586
       AGESQ |   .6091583   1.406185     0.43   0.665    -2.146913    3.365229
      INCOME |  -.1422016   .1081901    -1.31   0.189    -.3542504    .0698472
    LEVYPLUS |    .118064   .0855379     1.38   0.168    -.0495872    .2857153
    FREEPOOR |   -.496611   .2068902    -2.40   0.016    -.9021084   -.0911136
    FREEREPA |   .1449816   .1169488     1.24   0.215     -.084234    .3741971
     ILLNESS |   .2143413   .0242276     8.85   0.000     .1668561    .2618266
     ACTDAYS |   .1437537    .007814    18.40   0.000     .1284386    .1590689
      HSCORE |   .0380601   .0137992     2.76   0.006     .0110142     .065106
     CHCOND1 |    .099355   .0787023     1.26   0.207    -.0548987    .2536087
     CHCOND2 |    .190327   .1044086     1.82   0.068    -.0143102    .3949642
       _cons |  -2.190007   .2335801    -9.38   0.000    -2.647815   -1.732198
-------------+----------------------------------------------------------------
eq2          |
       _cons |   1.077037   .1030123    10.46   0.000     .8751365    1.278937
------------------------------------------------------------------------------

. 
. *** NB1 MLE
. 
. * Negbin1 MLE with default standard errors
. nbreg DVISITS $XLIST, dispersion(constant) 

Fitting Poisson model:

Iteration 0:   log likelihood = -4923.1976  
Iteration 1:   log likelihood = -3890.2934  
Iteration 2:   log likelihood = -3356.8559  
Iteration 3:   log likelihood = -3355.5431  
Iteration 4:   log likelihood = -3355.5413  
Iteration 5:   log likelihood = -3355.5413  

Fitting constant-only model:

Iteration 0:   log likelihood = -3591.3465  
Iteration 1:   log likelihood = -3586.0046  
Iteration 2:   log likelihood = -3585.9916  
Iteration 3:   log likelihood = -3585.9916  

Fitting full model:

Iteration 0:   log likelihood = -3276.9999  
Iteration 1:   log likelihood = -3229.3162  
Iteration 2:   log likelihood = -3226.8706  
Iteration 3:   log likelihood =  -3226.859  
Iteration 4:   log likelihood =  -3226.859  

Negative binomial regression                      Number of obs   =       5190
                                                  LR chi2(12)     =     718.27
Dispersion     = constant                         Prob > chi2     =     0.0000
Log likelihood = -3226.859                        Pseudo R2       =     0.1001

------------------------------------------------------------------------------
     DVISITS |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |   .1638501   .0652316     2.51   0.012     .0359984    .2917018
         AGE |   .2789488   1.160849     0.24   0.810    -1.996273     2.55417
       AGESQ |   .0205551   1.250843     0.02   0.987    -2.431052    2.472162
      INCOME |  -.1345763   .1022989    -1.32   0.188    -.3350785     .065926
    LEVYPLUS |   .2124024    .083491     2.54   0.011      .048763    .3760418
    FREEPOOR |  -.5375816   .2283607    -2.35   0.019    -.9851604   -.0900028
    FREEREPA |   .2081484   .1072239     1.94   0.052    -.0020065    .4183033
     ILLNESS |   .1958316   .0210602     9.30   0.000     .1545543    .2371088
     ACTDAYS |   .1123215   .0062328    18.02   0.000     .1001054    .1245376
      HSCORE |   .0357503    .011798     3.03   0.002     .0126267    .0588739
     CHCOND1 |     .13255   .0765784     1.73   0.083    -.0175408    .2826409
     CHCOND2 |   .1741327   .0969892     1.80   0.073    -.0159627     .364228
       _cons |  -2.201653     .22038    -9.99   0.000     -2.63359   -1.769716
-------------+----------------------------------------------------------------
    /lndelta |  -.7869279   .1036767                     -.9901306   -.5837253
-------------+----------------------------------------------------------------
       delta |   .4552412   .0471979                      .3715282    .5578165
------------------------------------------------------------------------------
Likelihood-ratio test of delta=0:  chibar2(01) =  257.36 Prob>=chibar2 = 0.000

. estimates store NB1MLHess

. 
. * Negbin1 MLE with robust standard errors
. nbreg DVISITS $XLIST, dispersion(constant) vce(robust)

Fitting Poisson model:

Iteration 0:   log pseudolikelihood = -4923.1976  
Iteration 1:   log pseudolikelihood = -3890.2934  
Iteration 2:   log pseudolikelihood = -3356.8559  
Iteration 3:   log pseudolikelihood = -3355.5431  
Iteration 4:   log pseudolikelihood = -3355.5413  
Iteration 5:   log pseudolikelihood = -3355.5413  

Fitting constant-only model:

Iteration 0:   log pseudolikelihood = -3591.3465  
Iteration 1:   log pseudolikelihood = -3586.0046  
Iteration 2:   log pseudolikelihood = -3585.9916  
Iteration 3:   log pseudolikelihood = -3585.9916  

Fitting full model:

Iteration 0:   log pseudolikelihood = -3276.9999  
Iteration 1:   log pseudolikelihood = -3229.3162  
Iteration 2:   log pseudolikelihood = -3226.8706  
Iteration 3:   log pseudolikelihood =  -3226.859  
Iteration 4:   log pseudolikelihood =  -3226.859  

Negative binomial regression                      Number of obs   =       5190
Dispersion           = constant                   Wald chi2(12)   =     871.27
Log pseudolikelihood = -3226.859                  Prob > chi2     =     0.0000

------------------------------------------------------------------------------
             |               Robust
     DVISITS |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |   .1638501   .0712176     2.30   0.021     .0242662     .303434
         AGE |   .2789488   1.208079     0.23   0.817    -2.088842     2.64674
       AGESQ |   .0205551   1.315093     0.02   0.988    -2.556981    2.598091
      INCOME |  -.1345763   .1101629    -1.22   0.222    -.3504915     .081339
    LEVYPLUS |   .2124024   .0843392     2.52   0.012     .0471006    .3777042
    FREEPOOR |  -.5375816   .2537494    -2.12   0.034    -1.034921   -.0402418
    FREEREPA |   .2081484   .1131071     1.84   0.066    -.0135375    .4298344
     ILLNESS |   .1958316   .0218244     8.97   0.000     .1530564    .2386067
     ACTDAYS |   .1123215   .0073409    15.30   0.000     .0979336    .1267093
      HSCORE |   .0357503   .0134405     2.66   0.008     .0094074    .0620933
     CHCOND1 |     .13255   .0797999     1.66   0.097    -.0238548    .2889549
     CHCOND2 |   .1741327   .1073244     1.62   0.105    -.0362193    .3844846
       _cons |  -2.201653     .22829    -9.64   0.000    -2.649093   -1.754213
-------------+----------------------------------------------------------------
    /lndelta |  -.7869279    .125188                     -1.032292   -.5415639
-------------+----------------------------------------------------------------
       delta |   .4552412   .0569908                      .3561896    .5818376
------------------------------------------------------------------------------

. estimates store NB1Robust

. 
. * Negbin1 MLE with OPG standard errors
. nbreg DVISITS $XLIST, dispersion(mean) vce(opg)

Fitting Poisson model:

Iteration 0:   log likelihood = -4923.1976  
Iteration 1:   log likelihood = -3890.2934  
Iteration 2:   log likelihood = -3356.8559  
Iteration 3:   log likelihood = -3355.5431  
Iteration 4:   log likelihood = -3355.5413  
Iteration 5:   log likelihood = -3355.5413  

Fitting constant-only model:

Iteration 0:   log likelihood = -3657.9344  
Iteration 1:   log likelihood = -3589.7157  
Iteration 2:   log likelihood = -3586.0033  
Iteration 3:   log likelihood = -3585.9916  
Iteration 4:   log likelihood = -3585.9916  

Fitting full model:

Iteration 0:   log likelihood = -3301.0284  
Iteration 1:   log likelihood = -3204.6437  
Iteration 2:   log likelihood = -3198.7794  
Iteration 3:   log likelihood = -3198.7438  
Iteration 4:   log likelihood = -3198.7438  

Negative binomial regression                      Number of obs   =       5190
                                                  LR chi2(12)     =     774.50
Dispersion     = mean                             Prob > chi2     =     0.0000
Log likelihood = -3198.7438                       Pseudo R2       =     0.1080

------------------------------------------------------------------------------
             |                 OPG
     DVISITS |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |   .2166435   .0656017     3.30   0.001     .0880665    .3452206
         AGE |  -.2161581    1.23418    -0.18   0.861    -2.635107     2.20279
       AGESQ |    .609158   1.380479     0.44   0.659    -2.096531    3.314847
      INCOME |  -.1422016   .0976648    -1.46   0.145     -.333621    .0492178
    LEVYPLUS |   .1180641   .0848782     1.39   0.164    -.0482943    .2844224
    FREEPOOR |   -.496611   .1750976    -2.84   0.005    -.8397959   -.1534261
    FREEREPA |   .1449816    .117429     1.23   0.217    -.0851751    .3751383
     ILLNESS |   .2143414   .0257348     8.33   0.000      .163902    .2647807
     ACTDAYS |   .1437537   .0074716    19.24   0.000     .1291096    .1583978
      HSCORE |   .0380601   .0142574     2.67   0.008     .0101161    .0660042
     CHCOND1 |    .099355   .0766507     1.30   0.195    -.0508777    .2495876
     CHCOND2 |    .190327   .0948246     2.01   0.045     .0044742    .3761799
       _cons |  -2.190007   .2224622    -9.84   0.000    -2.626025   -1.753989
-------------+----------------------------------------------------------------
    /lnalpha |   .0742145   .0914208                     -.1049669    .2533959
-------------+----------------------------------------------------------------
       alpha |   1.077038   .0984636                      .9003543    1.288393
------------------------------------------------------------------------------
Likelihood-ratio test of alpha=0:  chibar2(01) =  313.60 Prob>=chibar2 = 0.000

. estimates store NB1MLOPG

. 
. * Negbin1 MLE with OIM standard errors
. nbreg DVISITS $XLIST, dispersion(constant) vce(oim)

Fitting Poisson model:

Iteration 0:   log likelihood = -4923.1976  
Iteration 1:   log likelihood = -3890.2934  
Iteration 2:   log likelihood = -3356.8559  
Iteration 3:   log likelihood = -3355.5431  
Iteration 4:   log likelihood = -3355.5413  
Iteration 5:   log likelihood = -3355.5413  

Fitting constant-only model:

Iteration 0:   log likelihood = -3591.3465  
Iteration 1:   log likelihood = -3586.0046  
Iteration 2:   log likelihood = -3585.9916  
Iteration 3:   log likelihood = -3585.9916  

Fitting full model:

Iteration 0:   log likelihood = -3276.9999  
Iteration 1:   log likelihood = -3229.3162  
Iteration 2:   log likelihood = -3226.8706  
Iteration 3:   log likelihood =  -3226.859  
Iteration 4:   log likelihood =  -3226.859  

Negative binomial regression                      Number of obs   =       5190
                                                  LR chi2(12)     =     718.27
Dispersion     = constant                         Prob > chi2     =     0.0000
Log likelihood = -3226.859                        Pseudo R2       =     0.1001

------------------------------------------------------------------------------
     DVISITS |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |   .1638501   .0652316     2.51   0.012     .0359984    .2917018
         AGE |   .2789488   1.160849     0.24   0.810    -1.996273     2.55417
       AGESQ |   .0205551   1.250843     0.02   0.987    -2.431052    2.472162
      INCOME |  -.1345763   .1022989    -1.32   0.188    -.3350785     .065926
    LEVYPLUS |   .2124024    .083491     2.54   0.011      .048763    .3760418
    FREEPOOR |  -.5375816   .2283607    -2.35   0.019    -.9851604   -.0900028
    FREEREPA |   .2081484   .1072239     1.94   0.052    -.0020065    .4183033
     ILLNESS |   .1958316   .0210602     9.30   0.000     .1545543    .2371088
     ACTDAYS |   .1123215   .0062328    18.02   0.000     .1001054    .1245376
      HSCORE |   .0357503    .011798     3.03   0.002     .0126267    .0588739
     CHCOND1 |     .13255   .0765784     1.73   0.083    -.0175408    .2826409
     CHCOND2 |   .1741327   .0969892     1.80   0.073    -.0159627     .364228
       _cons |  -2.201653     .22038    -9.99   0.000     -2.63359   -1.769716
-------------+----------------------------------------------------------------
    /lndelta |  -.7869279   .1036767                     -.9901306   -.5837253
-------------+----------------------------------------------------------------
       delta |   .4552412   .0471979                      .3715282    .5578165
------------------------------------------------------------------------------
Likelihood-ratio test of delta=0:  chibar2(01) =  257.36 Prob>=chibar2 = 0.000

. estimates store NB1MLOIM

. 
. * Negbin1 MLE with bootstrap standard errors
. * nbreg DVISITS $XLIST, dispersion(mean) vce(boot, reps(400) seed(10101) nodots) 
. * estimates store NB1Boot
. 
. * Negbin2 MLE with four different ways to estimate standard errors 
. estimates table NB1Robust NB1MLHess NB1MLOPG NB1MLOIM, b(%9.3f) se

--------------------------------------------------------------
    Variable | NB1Robust   NB1MLHess   NB1MLOPG    NB1MLOIM   
-------------+------------------------------------------------
DVISITS      |
         SEX |     0.164       0.164       0.217       0.164  
             |     0.071       0.065       0.066       0.065  
         AGE |     0.279       0.279      -0.216       0.279  
             |     1.208       1.161       1.234       1.161  
       AGESQ |     0.021       0.021       0.609       0.021  
             |     1.315       1.251       1.380       1.251  
      INCOME |    -0.135      -0.135      -0.142      -0.135  
             |     0.110       0.102       0.098       0.102  
    LEVYPLUS |     0.212       0.212       0.118       0.212  
             |     0.084       0.083       0.085       0.083  
    FREEPOOR |    -0.538      -0.538      -0.497      -0.538  
             |     0.254       0.228       0.175       0.228  
    FREEREPA |     0.208       0.208       0.145       0.208  
             |     0.113       0.107       0.117       0.107  
     ILLNESS |     0.196       0.196       0.214       0.196  
             |     0.022       0.021       0.026       0.021  
     ACTDAYS |     0.112       0.112       0.144       0.112  
             |     0.007       0.006       0.007       0.006  
      HSCORE |     0.036       0.036       0.038       0.036  
             |     0.013       0.012       0.014       0.012  
     CHCOND1 |     0.133       0.133       0.099       0.133  
             |     0.080       0.077       0.077       0.077  
     CHCOND2 |     0.174       0.174       0.190       0.174  
             |     0.107       0.097       0.095       0.097  
       _cons |    -2.202      -2.202      -2.190      -2.202  
             |     0.228       0.220       0.222       0.220  
-------------+------------------------------------------------
lndelta      |
       _cons |    -0.787      -0.787                  -0.787  
             |     0.125       0.104                   0.104  
-------------+------------------------------------------------
lnalpha      |
       _cons |                             0.074              
             |                             0.091              
--------------------------------------------------------------
                                                  legend: b/se

. 
. *** NB2 QGPMLE
. 
. * Negbin2 QGPMLE estimated using glm (with default log link)
. * Use alpha found earlier
. display "alpha for NB2 : " alphanb2
alpha for NB2 : .28644145

. global aglm = alphanb2

. 
. * Negbin2 QGPMLE with Hessian standard errors
. glm DVISITS $XLIST, family(nbinomial $aglm)

Iteration 0:   log likelihood = -3311.6315  
Iteration 1:   log likelihood = -3254.2516  
Iteration 2:   log likelihood = -3254.1099  
Iteration 3:   log likelihood = -3254.1099  

Generalized linear models                          No. of obs      =      5190
Optimization     : ML                              Residual df     =      5177
                                                   Scale parameter =         1
Deviance         =  3809.779195                    (1/df) Deviance =  .7359048
Pearson          =  6232.843038                    (1/df) Pearson  =  1.203949

Variance function: V(u) = u+(.2864)u^2             [Neg. Binomial]
Link function    : g(u) = ln(u)                    [Log]

                                                   AIC             =  1.259002
Log likelihood   = -3254.109874                    BIC             = -40476.81

------------------------------------------------------------------------------
             |                 OIM
     DVISITS |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |   .1878597   .0607905     3.09   0.002     .0687125    .3070069
         AGE |   .5112983   1.101468     0.46   0.643    -1.647539    2.670136
       AGESQ |  -.2272432   1.197255    -0.19   0.849    -2.573821    2.119334
      INCOME |  -.1737962   .0952024    -1.83   0.068    -.3603894     .012797
    LEVYPLUS |   .1128012   .0762651     1.48   0.139    -.0366756     .262278
    FREEPOOR |  -.4605906   .1877967    -2.45   0.014    -.8286654   -.0925158
    FREEREPA |   .1003907   .1009725     0.99   0.320    -.0975118    .2982932
     ILLNESS |   .1980457   .0203329     9.74   0.000      .158194    .2378974
     ACTDAYS |   .1324919   .0058202    22.76   0.000     .1210845    .1438992
      HSCORE |   .0337695   .0114535     2.95   0.003      .011321     .056218
     CHCOND1 |   .1038871   .0705455     1.47   0.141    -.0343796    .2421538
     CHCOND2 |   .1590955    .090503     1.76   0.079    -.0182872    .3364783
       _cons |  -2.202967   .2050591   -10.74   0.000    -2.604875   -1.801059
------------------------------------------------------------------------------

. estimates store NB2QGPH

. 
. * Negbin2 QGPMLE with robust standard errors
. glm DVISITS $XLIST, family(nbinomial $aglm) vce(robust)

Iteration 0:   log pseudolikelihood = -3311.6315  
Iteration 1:   log pseudolikelihood = -3254.2516  
Iteration 2:   log pseudolikelihood = -3254.1099  
Iteration 3:   log pseudolikelihood = -3254.1099  

Generalized linear models                          No. of obs      =      5190
Optimization     : ML                              Residual df     =      5177
                                                   Scale parameter =         1
Deviance         =  3809.779195                    (1/df) Deviance =  .7359048
Pearson          =  6232.843038                    (1/df) Pearson  =  1.203949

Variance function: V(u) = u+(.2864)u^2             [Neg. Binomial]
Link function    : g(u) = ln(u)                    [Log]

                                                   AIC             =  1.259002
Log pseudolikelihood = -3254.109874                BIC             = -40476.81

------------------------------------------------------------------------------
             |               Robust
     DVISITS |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |   .1878597   .0760759     2.47   0.014     .0387536    .3369658
         AGE |   .5112983   1.362334     0.38   0.707    -2.158826    3.181423
       AGESQ |  -.2272432   1.458772    -0.16   0.876    -3.086385    2.631898
      INCOME |  -.1737962   .1258583    -1.38   0.167     -.420474    .0728815
    LEVYPLUS |   .1128012   .0938866     1.20   0.230    -.0712133    .2968156
    FREEPOOR |  -.4605906   .2760557    -1.67   0.095     -1.00165    .0804687
    FREEREPA |   .1003907   .1239757     0.81   0.418    -.1425971    .3433785
     ILLNESS |   .1980457   .0235064     8.43   0.000     .1519741    .2441174
     ACTDAYS |   .1324919   .0078727    16.83   0.000     .1170617    .1479221
      HSCORE |   .0337695   .0138376     2.44   0.015     .0066484    .0608907
     CHCOND1 |   .1038871   .0873308     1.19   0.234    -.0672781    .2750523
     CHCOND2 |   .1590955   .1201493     1.32   0.185    -.0763928    .3945839
       _cons |  -2.202967   .2518036    -8.75   0.000    -2.696493   -1.709441
------------------------------------------------------------------------------

. estimates store NB2QGPMLE

. 
. * Following with canonical link does not converge
. * glm DVISITS $XLIST, family(nbinomial 1) link(nbinomial) difficult
. 
. * ASIDE: Negbin2 QGPMLE using method ML (should be same as glm)
. global invaglm = 1/$aglm

. program glmnb2
  1.   version 11
  2.   args lnf theta1                 // theta1=x'b, lnf=lnf(y)
  3.   tempvar mu
  4.   local y $ML_y1                  // Define y so program more readable
  5.   generate double `mu'  = exp(`theta1')
  6.   quietly replace `lnf' = - (`y'+$invaglm)*ln(1+$aglm*`mu') + `y'*ln(`mu') 
  7. end

. ml model lf glmnb2 (DVISITS = $XLIST), vce(robust)

. ml maximize

initial:       log pseudolikelihood =  -4958.226
alternative:   log pseudolikelihood = -3936.3498
rescale:       log pseudolikelihood = -3538.1397
Iteration 0:   log pseudolikelihood = -3538.1397  
Iteration 1:   log pseudolikelihood = -3190.1113  
Iteration 2:   log pseudolikelihood = -2945.7753  
Iteration 3:   log pseudolikelihood = -2943.9937  
Iteration 4:   log pseudolikelihood = -2943.9929  
Iteration 5:   log pseudolikelihood = -2943.9929  

                                                  Number of obs   =       5190
                                                  Wald chi2(12)   =     982.46
Log pseudolikelihood = -2943.9929                 Prob > chi2     =     0.0000

------------------------------------------------------------------------------
             |               Robust
     DVISITS |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |   .1878597   .0760759     2.47   0.014     .0387536    .3369658
         AGE |   .5112984   1.362334     0.38   0.707    -2.158826    3.181423
       AGESQ |  -.2272433   1.458772    -0.16   0.876    -3.086385    2.631898
      INCOME |  -.1737962   .1258583    -1.38   0.167     -.420474    .0728815
    LEVYPLUS |   .1128012   .0938866     1.20   0.230    -.0712133    .2968156
    FREEPOOR |  -.4605906   .2760558    -1.67   0.095     -1.00165    .0804687
    FREEREPA |   .1003907   .1239757     0.81   0.418    -.1425971    .3433785
     ILLNESS |   .1980457   .0235064     8.43   0.000     .1519741    .2441174
     ACTDAYS |   .1324919   .0078727    16.83   0.000     .1170617    .1479221
      HSCORE |   .0337695   .0138376     2.44   0.015     .0066484    .0608907
     CHCOND1 |   .1038871   .0873308     1.19   0.234    -.0672781    .2750523
     CHCOND2 |   .1590955   .1201493     1.32   0.185    -.0763928    .3945839
       _cons |  -2.202967   .2518036    -8.75   0.000    -2.696493   -1.709441
------------------------------------------------------------------------------

. 
. estimates table NB2Robust NB2QGPMLE NB2QGPH PRobust, b(%9.3f) se

--------------------------------------------------------------
    Variable | NB2Robust   NB2QGPMLE    NB2QGPH     PRobust   
-------------+------------------------------------------------
DVISITS      |
         SEX |     0.217       0.188       0.188       0.157  
             |     0.074       0.076       0.061       0.079  
         AGE |    -0.216       0.511       0.511       1.056  
             |     1.367       1.362       1.101       1.364  
       AGESQ |     0.609      -0.227      -0.227      -0.849  
             |     1.473       1.459       1.197       1.460  
      INCOME |    -0.142      -0.174      -0.174      -0.205  
             |     0.122       0.126       0.095       0.129  
    LEVYPLUS |     0.118       0.113       0.113       0.123  
             |     0.091       0.094       0.076       0.095  
    FREEPOOR |    -0.497      -0.461      -0.461      -0.440  
             |     0.254       0.276       0.188       0.290  
    FREEREPA |     0.145       0.100       0.100       0.080  
             |     0.121       0.124       0.101       0.126  
     ILLNESS |     0.214       0.198       0.198       0.187  
             |     0.024       0.024       0.020       0.024  
     ACTDAYS |     0.144       0.132       0.132       0.127  
             |     0.009       0.008       0.006       0.008  
      HSCORE |     0.038       0.034       0.034       0.030  
             |     0.014       0.014       0.011       0.014  
     CHCOND1 |     0.099       0.104       0.104       0.114  
             |     0.083       0.087       0.071       0.091  
     CHCOND2 |     0.190       0.159       0.159       0.141  
             |     0.117       0.120       0.091       0.123  
       _cons |    -2.190      -2.203      -2.203      -2.224  
             |     0.249       0.252       0.205       0.254  
-------------+------------------------------------------------
lnalpha      |
       _cons |     0.074                                      
             |     0.108                                      
--------------------------------------------------------------
                                                  legend: b/se

. 
. * Aside for the canonical link use instead in program glmnb2 
. *    generate double `mu'  = $invaglm*exp(`theta1')/(1 - exp(`theta1'))
. * but this does not converge 
. 
. * Following implements QGPPMLE using GMM based on the first-order conditions
. gmm ( (DVISITS - exp({b1}*SEX+{b2}*AGE+{b2}*AGESQ+{b3}*INCOME+{b4}*LEVYPLUS ///
>       +{b5}*FREEPOOR+{b6}*FREEREPA+{b7}*ILLNESS+{b8}*ACTDAYS+{b9}*HSCORE    ///
>       +{b10}*CHCOND1+{b11}*CHCOND2+{b0}))                                   ///
>       / (1 + alphanb2*exp({b1}*SEX+{b2}*AGE+{b2}*AGESQ+{b3}*INCOME+{b4}*LEVYPLUS ///
>       +{b5}*FREEPOOR+{b6}*FREEREPA+{b7}*ILLNESS+{b8}*ACTDAYS+{b9}*HSCORE    ///
>       +{b10}*CHCOND1+{b11}*CHCOND2+{b0})) ), instruments($XLIST) onestep

Step 1
Iteration 0:   GMM criterion Q(b) =  .37229303  
Iteration 1:   GMM criterion Q(b) =  .01131616  
Iteration 2:   GMM criterion Q(b) =  .00003869  
Iteration 3:   GMM criterion Q(b) =  5.412e-06  
Iteration 4:   GMM criterion Q(b) =  5.406e-06  

GMM estimation 

Number of parameters =  12
Number of moments    =  13
Initial weight matrix: Unadjusted                     Number of obs  =    5190

------------------------------------------------------------------------------
             |               Robust
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         /b1 |   .1876506   .0760512     2.47   0.014      .038593    .3367081
         /b2 |   .1520978   .1110665     1.37   0.171    -.0655885    .3697841
         /b3 |  -.1674461   .1224328    -1.37   0.171      -.40741    .0725179
         /b4 |   .1163823   .0924398     1.26   0.208    -.0647963    .2975609
         /b5 |  -.4599732   .2753538    -1.67   0.095    -.9996566    .0797103
         /b6 |   .1052424   .1220778     0.86   0.389    -.1340257    .3445106
         /b7 |    .197823    .023411     8.45   0.000     .1519383    .2437077
         /b8 |   .1324823     .00787    16.83   0.000     .1170573    .1479072
         /b9 |   .0340912   .0137577     2.48   0.013     .0071266    .0610558
        /b10 |   .1057899   .0862754     1.23   0.220    -.0633068    .2748865
        /b11 |   .1637277    .116111     1.41   0.159    -.0638457    .3913011
         /b0 |  -2.142931   .1353147   -15.84   0.000    -2.408142   -1.877719
------------------------------------------------------------------------------
Instruments for equation 1: SEX AGE AGESQ INCOME LEVYPLUS FREEPOOR FREEREPA ILLNESS ACTDAYS HSCORE CHCOND1 CHCOND2 _cons

. 
. estimates table NB2Robust NB2QGPMLE NB2QGPH PRobust, b(%9.3f) se

--------------------------------------------------------------
    Variable | NB2Robust   NB2QGPMLE    NB2QGPH     PRobust   
-------------+------------------------------------------------
DVISITS      |
         SEX |     0.217       0.188       0.188       0.157  
             |     0.074       0.076       0.061       0.079  
         AGE |    -0.216       0.511       0.511       1.056  
             |     1.367       1.362       1.101       1.364  
       AGESQ |     0.609      -0.227      -0.227      -0.849  
             |     1.473       1.459       1.197       1.460  
      INCOME |    -0.142      -0.174      -0.174      -0.205  
             |     0.122       0.126       0.095       0.129  
    LEVYPLUS |     0.118       0.113       0.113       0.123  
             |     0.091       0.094       0.076       0.095  
    FREEPOOR |    -0.497      -0.461      -0.461      -0.440  
             |     0.254       0.276       0.188       0.290  
    FREEREPA |     0.145       0.100       0.100       0.080  
             |     0.121       0.124       0.101       0.126  
     ILLNESS |     0.214       0.198       0.198       0.187  
             |     0.024       0.024       0.020       0.024  
     ACTDAYS |     0.144       0.132       0.132       0.127  
             |     0.009       0.008       0.006       0.008  
      HSCORE |     0.038       0.034       0.034       0.030  
             |     0.014       0.014       0.011       0.014  
     CHCOND1 |     0.099       0.104       0.104       0.114  
             |     0.083       0.087       0.071       0.091  
     CHCOND2 |     0.190       0.159       0.159       0.141  
             |     0.117       0.120       0.091       0.123  
       _cons |    -2.190      -2.203      -2.203      -2.224  
             |     0.249       0.252       0.205       0.254  
-------------+------------------------------------------------
lnalpha      |
       _cons |     0.074                                      
             |     0.108                                      
--------------------------------------------------------------
                                                  legend: b/se

. 
. *** TABLE 3.4: NB2 and NB1 MODEL ESTIMATORS AND STANDARD ERRORS
. 
. estimates table NB2Robust NB2QGPMLE NB1Robust PRobust, b(%9.3f) se

--------------------------------------------------------------
    Variable | NB2Robust   NB2QGPMLE   NB1Robust    PRobust   
-------------+------------------------------------------------
DVISITS      |
         SEX |     0.217       0.188       0.164       0.157  
             |     0.074       0.076       0.071       0.079  
         AGE |    -0.216       0.511       0.279       1.056  
             |     1.367       1.362       1.208       1.364  
       AGESQ |     0.609      -0.227       0.021      -0.849  
             |     1.473       1.459       1.315       1.460  
      INCOME |    -0.142      -0.174      -0.135      -0.205  
             |     0.122       0.126       0.110       0.129  
    LEVYPLUS |     0.118       0.113       0.212       0.123  
             |     0.091       0.094       0.084       0.095  
    FREEPOOR |    -0.497      -0.461      -0.538      -0.440  
             |     0.254       0.276       0.254       0.290  
    FREEREPA |     0.145       0.100       0.208       0.080  
             |     0.121       0.124       0.113       0.126  
     ILLNESS |     0.214       0.198       0.196       0.187  
             |     0.024       0.024       0.022       0.024  
     ACTDAYS |     0.144       0.132       0.112       0.127  
             |     0.009       0.008       0.007       0.008  
      HSCORE |     0.038       0.034       0.036       0.030  
             |     0.014       0.014       0.013       0.014  
     CHCOND1 |     0.099       0.104       0.133       0.114  
             |     0.083       0.087       0.080       0.091  
     CHCOND2 |     0.190       0.159       0.174       0.141  
             |     0.117       0.120       0.107       0.123  
       _cons |    -2.190      -2.203      -2.202      -2.224  
             |     0.249       0.252       0.228       0.254  
-------------+------------------------------------------------
lnalpha      |
       _cons |     0.074                                      
             |     0.108                                      
-------------+------------------------------------------------
lndelta      |
       _cons |                            -0.787              
             |                             0.125              
--------------------------------------------------------------
                                                  legend: b/se

. 
. /* Following not run to save time but cited in discussion of Table 3.4
> * Two checks: 
> * (1) correct standard errors if DVISITS_se observed Coef. 
> *     is close to DVISITS Bootstrap Std. Error
> * (2) variablity of the s.e. is DVISITS_se Bootstrap Std. Error
> * NB2 Robust sandwich se's
> bootstrap _b _se, reps(400) seed(10101): nbreg DVISITS $XLIST, dispersion(mean) vce(robust)
> * NB2 default se's
> bootstrap _b _se, reps(400) seed(10101): nbreg DVISITS $XLIST, dispersion(mean)
> */
. 
. *** FIGURE 3.1 
. 
. * The following creates Figure 3.1 manually
. * where the predicted probabilities come from 
. * Average Predicted probabilities for y = 0, 1, ... , 10
. * countfit DVISITS, maxcount(10) prm nograph noestimates nofit
. * countfit DVISITS, maxcount(10) nbreg nograph noestimates nofit 
. * countfit DVISITS $XLIST, maxcount(10) prm nograph noestimates nofit
. * countfit DVISITS $XLIST, maxcount(10) nbreg nograph noestimates nofit
. clear 

. input count sample poissintonly nb2intonly poissreg nb2reg 

         count     sample  poissin~y  nb2into~y   poissreg     nb2reg
  1.  0 .7979 .7395347 .8011167 .7733644 .803997
  2.  1 .1507 .2231428 .1343514 .1788264 .1398705
  3.  2 .0335 .0336649 .0411049 .0323732 .032438
  4.  3 .0058 .0033859 .0144702 .0087688 .0104049  
  5.  4 .0046 .0002554 .0054274 .0036505 .004566  
  6.  5 .0017 .0000154 .0021107 .0017036 .0025236 
  7.  6 .0023 .0000008 .0008403 .0007746 .0016031 
  8.  7 .0023 .0000000 .0003401 .0003316 .0011016 
  9.  8 .0010 .0000000 .0001393 .0001325 .0007925
 10.  9 .0002 .0000000 .0000576 .0000494 .0005869 
 11. end

. label variable sample "Number of doctor visits"

. label variable sample "Sample frequency"

. label variable poissintonly "Poisson no regressors"

. label variable nb2intonly "NB2 no regressors"

. label variable poissreg "Poisson with regressors"

. label variable nb2reg "NB2 with regressors"

. drop if count > 4
(5 observations deleted)

. set scheme s1mono

. graph bar (mean) sample poissintonly poissreg, over(count)                  ///
>   saving(racd03graph1, replace) ytitle("Sample and predicted frequencies") ///
>   legend( ring(0) rows(3) pos(3) label(1 "Sample frequency")               ///
>     label(2 "Poisson no regressors") label(3 "Poisson with regressors") )
(file racd03graph1.gph saved)

. graph bar (mean) sample nb2intonly nb2reg, over(count)                      ///
>   saving(racd03graph2, replace)  ytitle("Sample and predicted frequencies") ///
>   legend( ring(0) rows(3) pos(3) label(1 "Sample frequency")                ///
>     label(2 "NB2 no regressors") label(3 "NB2 with regressors") )
(file racd03graph2.gph saved)

. graph combine racd03graph1.gph racd03graph2.gph, iscale(0.9) ysize(3) xsize(6) 

. graph export racd03fig1.eps, replace
(file racd03fig1.eps written in EPS format)

. graph export racd03fig1.wmf, replace
(file c:\acdbookrevision\stata_final_programs_2013\racd03fig1.wmf written in Windows Metafile format)

. use racd03data.dta, replace

. 
. ********** 3.3.6 SIMULATION
. 
. * From Cameron and Trivedi (1986) NB with mean mu and variance mu + a*mu^j
. * is generated from Poisson(xgamma) where 
. * xgamma is gamma with mean mu and variance alpha*mu^j
. * Since rgamma(a,b) yields gamma with mean ab and variance ab^2
. * we need rgamma(mu^(2-j)/a, a*mu^(j-1))
. 
. * Test that code works with mu=2 and a=2 
. * Should yield NB2 with mean 3 and variance 3 + 2*3^2 = 21
. *          and NB1 with meam 3 and variance 3 + 2*3 = 9
. clear 

. set seed 10101

. set obs 100000
obs was 0, now 100000

. generate mu = 3

. scalar a = 2

. * NB2 has variance mu + a*mu^2 so set j = 2
. generate gammaNB2 = rgamma(1/a, a*mu)

. generate xNB2 = rpoisson(gammaNB2)
(43 missing values generated)

. * NB1 has variance j + a*mu^2 so set j = 1
. generate gammaNB1 = rgamma(mu/a, a)

. generate xNB1 = rpoisson(gammaNB1)

. tabstat xNB2 xNB1, stat(mean var min max) col(stat)

    variable |      mean  variance       min       max
-------------+----------------------------------------
        xNB2 |  2.994208  20.80347         0        56
        xNB1 |   2.99792  8.941825         0        32
------------------------------------------------------

. 
. *** Generate Poisson, NB1 and NB2 with n = 10,000
. clear

. set obs 10000
obs was 0, now 10000

. set seed 10101 

. generate x = runiform()

. scalar a = 2

. generate mu = exp(0 + 2*x)

. generate yP = rpoisson(mu)

. generate gammaNB2 = rgamma(1/a, a*mu)

. generate yNB2 = rpoisson(gammaNB2)
(3 missing values generated)

. generate gammaNB1 = rgamma(mu/a, a)

. generate yNB1 = rpoisson(gammaNB1)

. summarize x mu gammaNB2 gammaNB1 yP yNB2 yNB1

    Variable |       Obs        Mean    Std. Dev.       Min        Max
-------------+--------------------------------------------------------
           x |     10000    .4987486    .2886023   .0001192   .9998879
          mu |     10000    3.185981    1.782515   1.000238     7.3874
    gammaNB2 |     10000    3.169733    5.573155   1.51e-07   87.29626
    gammaNB1 |     10000    3.199752    3.076172   1.33e-06   23.13055
          yP |     10000      3.1821    2.497632          0         16
-------------+--------------------------------------------------------
        yNB2 |      9997    3.204861     5.93189          0        102
        yNB1 |     10000      3.1984    3.520835          0         27

. 
. * POISSON regressions - with P, NB1 and NB2 as dgp
. * All should be consistent
. poisson yP x, vce(robust)

Iteration 0:   log pseudolikelihood = -18640.465  
Iteration 1:   log pseudolikelihood =  -18640.46  
Iteration 2:   log pseudolikelihood =  -18640.46  

Poisson regression                                Number of obs   =      10000
                                                  Wald chi2(1)    =    8606.40
                                                  Prob > chi2     =     0.0000
Log pseudolikelihood =  -18640.46                 Pseudo R2       =     0.2018

------------------------------------------------------------------------------
             |               Robust
          yP |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           x |   1.975465   .0212941    92.77   0.000      1.93373    2.017201
       _cons |    .014833   .0149929     0.99   0.322    -.0145525    .0442186
------------------------------------------------------------------------------

. estimates store P_Prob

. poisson yNB1 x, vce(robust)

Iteration 0:   log pseudolikelihood = -24982.318  
Iteration 1:   log pseudolikelihood = -24982.312  
Iteration 2:   log pseudolikelihood = -24982.312  

Poisson regression                                Number of obs   =      10000
                                                  Wald chi2(1)    =    2930.24
                                                  Prob > chi2     =     0.0000
Log pseudolikelihood = -24982.312                 Pseudo R2       =     0.1595

------------------------------------------------------------------------------
             |               Robust
        yNB1 |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           x |   1.976713   .0365167    54.13   0.000     1.905141    2.048284
       _cons |   .0191272   .0258091     0.74   0.459    -.0314577    .0697121
------------------------------------------------------------------------------

. estimates store NB1_Prob

. poisson yNB2 x, vce(robust)

Iteration 0:   log pseudolikelihood = -37654.621  
Iteration 1:   log pseudolikelihood = -37654.614  
Iteration 2:   log pseudolikelihood = -37654.614  

Poisson regression                                Number of obs   =       9997
                                                  Wald chi2(1)    =    1057.75
                                                  Prob > chi2     =     0.0000
Log pseudolikelihood = -37654.614                 Pseudo R2       =     0.1161

------------------------------------------------------------------------------
             |               Robust
        yNB2 |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           x |   2.021443   .0621542    32.52   0.000     1.899623    2.143262
       _cons |  -.0082794   .0368711    -0.22   0.822    -.0805454    .0639866
------------------------------------------------------------------------------

. estimates store NB2_Prob

. 
. * NB2 regressions - with NB1 and NB2 as dgp
. * Note that do not estimate for Poisson dgp since half the time generated Poisson
. * will be underdispersed and cannot use NB2 then
. * All should be consistent but check standard errors
. nbreg yNB1 x

Fitting Poisson model:

Iteration 0:   log likelihood = -24982.318  
Iteration 1:   log likelihood = -24982.312  
Iteration 2:   log likelihood = -24982.312  

Fitting constant-only model:

Iteration 0:   log likelihood = -23048.372  
Iteration 1:   log likelihood = -23048.257  
Iteration 2:   log likelihood = -23048.257  

Fitting full model:

Iteration 0:   log likelihood = -21963.512  
Iteration 1:   log likelihood = -21693.676  
Iteration 2:   log likelihood =  -21666.46  
Iteration 3:   log likelihood = -21666.353  
Iteration 4:   log likelihood = -21666.353  

Negative binomial regression                      Number of obs   =      10000
                                                  LR chi2(1)      =    2763.81
Dispersion     = mean                             Prob > chi2     =     0.0000
Log likelihood = -21666.353                       Pseudo R2       =     0.0600

------------------------------------------------------------------------------
        yNB1 |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           x |   1.980886    .035352    56.03   0.000     1.911597    2.050174
       _cons |   .0166931   .0221209     0.75   0.450    -.0266631    .0600493
-------------+----------------------------------------------------------------
    /lnalpha |  -.5099326   .0254761                     -.5598648   -.4600004
-------------+----------------------------------------------------------------
       alpha |   .6005361   .0152993                      .5712863    .6312834
------------------------------------------------------------------------------
Likelihood-ratio test of alpha=0:  chibar2(01) = 6631.92 Prob>=chibar2 = 0.000

. estimates store NB1_NB2def

. nbreg yNB1 x, vce(robust)

Fitting Poisson model:

Iteration 0:   log pseudolikelihood = -24982.318  
Iteration 1:   log pseudolikelihood = -24982.312  
Iteration 2:   log pseudolikelihood = -24982.312  

Fitting constant-only model:

Iteration 0:   log pseudolikelihood = -23048.372  
Iteration 1:   log pseudolikelihood = -23048.257  
Iteration 2:   log pseudolikelihood = -23048.257  

Fitting full model:

Iteration 0:   log pseudolikelihood = -21963.512  
Iteration 1:   log pseudolikelihood = -21693.676  
Iteration 2:   log pseudolikelihood =  -21666.46  
Iteration 3:   log pseudolikelihood = -21666.353  
Iteration 4:   log pseudolikelihood = -21666.353  

Negative binomial regression                      Number of obs   =      10000
Dispersion           = mean                       Wald chi2(1)    =    2591.30
Log pseudolikelihood = -21666.353                 Prob > chi2     =     0.0000

------------------------------------------------------------------------------
             |               Robust
        yNB1 |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           x |   1.980886   .0389135    50.90   0.000     1.904617    2.057155
       _cons |   .0166931   .0270345     0.62   0.537    -.0362935    .0696797
-------------+----------------------------------------------------------------
    /lnalpha |  -.5099326   .0270242                      -.562899   -.4569662
-------------+----------------------------------------------------------------
       alpha |   .6005361    .016229                      .5695555    .6332018
------------------------------------------------------------------------------

. estimates store NB1_NB2rob

. nbreg yNB2 x

Fitting Poisson model:

Iteration 0:   log likelihood = -37654.621  
Iteration 1:   log likelihood = -37654.614  
Iteration 2:   log likelihood = -37654.614  

Fitting constant-only model:

Iteration 0:   log likelihood = -23059.016  
Iteration 1:   log likelihood = -21768.085  
Iteration 2:   log likelihood = -21767.104  
Iteration 3:   log likelihood = -21767.104  

Fitting full model:

Iteration 0:   log likelihood = -21241.719  
Iteration 1:   log likelihood = -21153.562  
Iteration 2:   log likelihood = -21148.774  
Iteration 3:   log likelihood = -21148.767  
Iteration 4:   log likelihood = -21148.767  

Negative binomial regression                      Number of obs   =       9997
                                                  LR chi2(1)      =    1236.67
Dispersion     = mean                             Prob > chi2     =     0.0000
Log likelihood = -21148.767                       Pseudo R2       =     0.0284

------------------------------------------------------------------------------
        yNB2 |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           x |   2.022014   .0553378    36.54   0.000     1.913554    2.130475
       _cons |  -.0086253   .0330705    -0.26   0.794    -.0734423    .0561917
-------------+----------------------------------------------------------------
    /lnalpha |    .732058   .0196341                      .6935758    .7705401
-------------+----------------------------------------------------------------
       alpha |   2.079355   .0408263                      2.000857    2.160933
------------------------------------------------------------------------------
Likelihood-ratio test of alpha=0:  chibar2(01) = 3.3e+04 Prob>=chibar2 = 0.000

. estimates store NB2_NB2def

. nbreg yNB2 x, vce(robust)

Fitting Poisson model:

Iteration 0:   log pseudolikelihood = -37654.621  
Iteration 1:   log pseudolikelihood = -37654.614  
Iteration 2:   log pseudolikelihood = -37654.614  

Fitting constant-only model:

Iteration 0:   log pseudolikelihood = -23059.016  
Iteration 1:   log pseudolikelihood = -21768.085  
Iteration 2:   log pseudolikelihood = -21767.104  
Iteration 3:   log pseudolikelihood = -21767.104  

Fitting full model:

Iteration 0:   log pseudolikelihood = -21241.719  
Iteration 1:   log pseudolikelihood = -21153.562  
Iteration 2:   log pseudolikelihood = -21148.774  
Iteration 3:   log pseudolikelihood = -21148.767  
Iteration 4:   log pseudolikelihood = -21148.767  

Negative binomial regression                      Number of obs   =       9997
Dispersion           = mean                       Wald chi2(1)    =    1271.60
Log pseudolikelihood = -21148.767                 Prob > chi2     =     0.0000

------------------------------------------------------------------------------
             |               Robust
        yNB2 |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           x |   2.022014   .0567033    35.66   0.000     1.910878    2.133151
       _cons |  -.0086253   .0339976    -0.25   0.800    -.0752593    .0580087
-------------+----------------------------------------------------------------
    /lnalpha |    .732058   .0197124                      .6934223    .7706936
-------------+----------------------------------------------------------------
       alpha |   2.079355   .0409892                       2.00055    2.161265
------------------------------------------------------------------------------

. estimates store NB2_NB2rob

. 
. * NB1 regressions - with NB1 and NB2 as dgp
. * Note that do not estimate for Poisson dgp since half the time generated Poisson
. * will be underdispersed and cannot use NB2 then
. * All should be consistent but check standard errors
. nbreg yNB1 x, dispersion(constant)

Fitting Poisson model:

Iteration 0:   log likelihood = -24982.318  
Iteration 1:   log likelihood = -24982.312  
Iteration 2:   log likelihood = -24982.312  

Fitting constant-only model:

Iteration 0:   log likelihood = -24338.833  
Iteration 1:   log likelihood = -23048.527  
Iteration 2:   log likelihood = -23048.257  
Iteration 3:   log likelihood = -23048.257  

Fitting full model:

Iteration 0:   log likelihood = -23048.257  
Iteration 1:   log likelihood = -22555.301  
Iteration 2:   log likelihood = -21444.572  
Iteration 3:   log likelihood = -21399.126  
Iteration 4:   log likelihood = -21398.976  
Iteration 5:   log likelihood = -21398.976  

Negative binomial regression                      Number of obs   =      10000
                                                  LR chi2(1)      =    3298.56
Dispersion     = constant                         Prob > chi2     =     0.0000
Log likelihood = -21398.976                       Pseudo R2       =     0.0716

------------------------------------------------------------------------------
        yNB1 |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           x |   1.984823   .0334893    59.27   0.000     1.919185    2.050461
       _cons |   .0138248   .0239108     0.58   0.563    -.0330396    .0606892
-------------+----------------------------------------------------------------
    /lndelta |   .6632732   .0261172                      .6120843     .714462
-------------+----------------------------------------------------------------
       delta |   1.941136   .0506971                      1.844271    2.043087
------------------------------------------------------------------------------
Likelihood-ratio test of delta=0:  chibar2(01) = 7166.67 Prob>=chibar2 = 0.000

. estimates store NB1_NB1def

. nbreg yNB1 x, dispersion(constant) vce(robust)

Fitting Poisson model:

Iteration 0:   log pseudolikelihood = -24982.318  
Iteration 1:   log pseudolikelihood = -24982.312  
Iteration 2:   log pseudolikelihood = -24982.312  

Fitting constant-only model:

Iteration 0:   log pseudolikelihood = -24338.833  
Iteration 1:   log pseudolikelihood = -23048.527  
Iteration 2:   log pseudolikelihood = -23048.257  
Iteration 3:   log pseudolikelihood = -23048.257  

Fitting full model:

Iteration 0:   log pseudolikelihood = -23048.257  
Iteration 1:   log pseudolikelihood = -22555.301  
Iteration 2:   log pseudolikelihood = -21444.572  
Iteration 3:   log pseudolikelihood = -21399.126  
Iteration 4:   log pseudolikelihood = -21398.976  
Iteration 5:   log pseudolikelihood = -21398.976  

Negative binomial regression                      Number of obs   =      10000
Dispersion           = constant                   Wald chi2(1)    =    3506.12
Log pseudolikelihood = -21398.976                 Prob > chi2     =     0.0000

------------------------------------------------------------------------------
             |               Robust
        yNB1 |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           x |   1.984823   .0335204    59.21   0.000     1.919124    2.050522
       _cons |   .0138248   .0239094     0.58   0.563    -.0330367    .0606863
-------------+----------------------------------------------------------------
    /lndelta |   .6632732   .0255088                      .6132768    .7132695
-------------+----------------------------------------------------------------
       delta |   1.941136    .049516                      1.846472    2.040652
------------------------------------------------------------------------------

. estimates store NB1_NB1rob

. nbreg yNB2 x, dispersion(constant)

Fitting Poisson model:

Iteration 0:   log likelihood = -37654.621  
Iteration 1:   log likelihood = -37654.614  
Iteration 2:   log likelihood = -37654.614  

Fitting constant-only model:

Iteration 0:   log likelihood = -27995.674  
Iteration 1:   log likelihood = -22090.287  
Iteration 2:   log likelihood = -21768.866  
Iteration 3:   log likelihood = -21767.104  
Iteration 4:   log likelihood = -21767.104  

Fitting full model:

Iteration 0:   log likelihood = -21767.104  
Iteration 1:   log likelihood = -21399.375  
Iteration 2:   log likelihood = -21381.439  
Iteration 3:   log likelihood = -21381.402  
Iteration 4:   log likelihood = -21381.402  

Negative binomial regression                      Number of obs   =       9997
                                                  LR chi2(1)      =     771.40
Dispersion     = constant                         Prob > chi2     =     0.0000
Log likelihood = -21381.402                       Pseudo R2       =     0.0177

------------------------------------------------------------------------------
        yNB2 |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           x |   1.217482   .0437732    27.81   0.000     1.131688    1.303276
       _cons |   .4964225   .0306802    16.18   0.000     .4362903    .5565546
-------------+----------------------------------------------------------------
    /lndelta |   1.977396   .0251195                      1.928162    2.026629
-------------+----------------------------------------------------------------
       delta |   7.223905   .1814608                      6.876861    7.588462
------------------------------------------------------------------------------
Likelihood-ratio test of delta=0:  chibar2(01) = 3.3e+04 Prob>=chibar2 = 0.000

. estimates store NB2_NB1def

. nbreg yNB2 x, dispersion(constant) vce(robust)

Fitting Poisson model:

Iteration 0:   log pseudolikelihood = -37654.621  
Iteration 1:   log pseudolikelihood = -37654.614  
Iteration 2:   log pseudolikelihood = -37654.614  

Fitting constant-only model:

Iteration 0:   log pseudolikelihood = -27995.674  
Iteration 1:   log pseudolikelihood = -22090.287  
Iteration 2:   log pseudolikelihood = -21768.866  
Iteration 3:   log pseudolikelihood = -21767.104  
Iteration 4:   log pseudolikelihood = -21767.104  

Fitting full model:

Iteration 0:   log pseudolikelihood = -21767.104  
Iteration 1:   log pseudolikelihood = -21399.375  
Iteration 2:   log pseudolikelihood = -21381.439  
Iteration 3:   log pseudolikelihood = -21381.402  
Iteration 4:   log pseudolikelihood = -21381.402  

Negative binomial regression                      Number of obs   =       9997
Dispersion           = constant                   Wald chi2(1)    =     800.22
Log pseudolikelihood = -21381.402                 Prob > chi2     =     0.0000

------------------------------------------------------------------------------
             |               Robust
        yNB2 |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
           x |   1.217482   .0430386    28.29   0.000     1.133128    1.301836
       _cons |   .4964225   .0276706    17.94   0.000      .442189    .5506559
-------------+----------------------------------------------------------------
    /lndelta |   1.977396   .0288683                      1.920815    2.033976
-------------+----------------------------------------------------------------
       delta |   7.223905   .2085418                      6.826519    7.644424
------------------------------------------------------------------------------

. estimates store NB2_NB1rob

. 
. estimates table P_Prob NB1_Prob NB2_Prob, b(%7.4f) se(%7.4f) stats(N ll) stfmt(%9.1f) ///
>    modelwidth(9) equations(1) title("Poisson with dgp Poisson, NB1, NB2")

Poisson with dgp Poisson, NB1, NB2

--------------------------------------------------
    Variable |  P_Prob     NB1_Prob    NB2_Prob   
-------------+------------------------------------
           x |    1.9755      1.9767      2.0214  
             |    0.0213      0.0365      0.0622  
       _cons |    0.0148      0.0191     -0.0083  
             |    0.0150      0.0258      0.0369  
-------------+------------------------------------
           N |     10000       10000        9997  
          ll |  -18640.5    -24982.3    -37654.6  
--------------------------------------------------
                                      legend: b/se

. estimates table NB1_NB2def NB1_NB2rob NB2_NB2def NB2_NB2rob, b(%7.4f) se(%7.4f) ///
>    stats(N ll) stfmt(%9.1f) modelwidth(9) equations(1)                          ///
>    title("NB2 MLE with dgp NB1, NB2; default, robust se's")

NB2 MLE with dgp NB1, NB2; default, robust se's

--------------------------------------------------------------
    Variable | NB1_NB2~f   NB1_NB2~b   NB2_NB2~f   NB2_NB2~b  
-------------+------------------------------------------------
#1           |
           x |    1.9809      1.9809      2.0220      2.0220  
             |    0.0354      0.0389      0.0553      0.0567  
       _cons |    0.0167      0.0167     -0.0086     -0.0086  
             |    0.0221      0.0270      0.0331      0.0340  
-------------+------------------------------------------------
lnalpha      |
       _cons |   -0.5099     -0.5099      0.7321      0.7321  
             |    0.0255      0.0270      0.0196      0.0197  
-------------+------------------------------------------------
Statistics   |                                                
           N |     10000       10000        9997        9997  
          ll |  -21666.4    -21666.4    -21148.8    -21148.8  
--------------------------------------------------------------
                                                  legend: b/se

. estimates table NB1_NB1def NB1_NB1rob NB2_NB1def NB2_NB1rob, b(%7.4f) se(%7.4f) ///
>    stats(N ll) stfmt(%9.1f) modelwidth(9) equations(1)                         ///
>    title("NB2 MLE with dgp NB1, NB2; default, robust se's")   

NB2 MLE with dgp NB1, NB2; default, robust se's

--------------------------------------------------------------
    Variable | NB1_NB1~f   NB1_NB1~b   NB2_NB1~f   NB2_NB1~b  
-------------+------------------------------------------------
#1           |
           x |    1.9848      1.9848      1.2175      1.2175  
             |    0.0335      0.0335      0.0438      0.0430  
       _cons |    0.0138      0.0138      0.4964      0.4964  
             |    0.0239      0.0239      0.0307      0.0277  
-------------+------------------------------------------------
lndelta      |
       _cons |    0.6633      0.6633      1.9774      1.9774  
             |    0.0261      0.0255      0.0251      0.0289  
-------------+------------------------------------------------
Statistics   |                                                
           N |     10000       10000        9997        9997  
          ll |  -21399.0    -21399.0    -21381.4    -21381.4  
--------------------------------------------------------------
                                                  legend: b/se

. 
. *** TABLE 3.5: SIMULATION RESULTS (NB1_NB2rob means NB1 dgp and NB2 MLE with robust se's)
. 
. * For alpha goes to earlier command output as here ln(alpha) is given
. estimates table NB2_Prob NB2_NB1rob NB2_NB2rob NB1_Prob NB1_NB1rob NB1_NB2rob NB1_NB2def, ///
>    b(%7.4f) se(%7.4f) stats(N ll) stfmt(%9.1f) modelwidth(9) equations(1)  ///
>    title("NB2 MLE with dgp NB1, NB2; default, robust se's") 

NB2 MLE with dgp NB1, NB2; default, robust se's

--------------------------------------------------------------------------------------------------
    Variable | NB2_Prob    NB2_NB1~b   NB2_NB2~b   NB1_Prob    NB1_NB1~b   NB1_NB2~b   NB1_NB2~f  
-------------+------------------------------------------------------------------------------------
#1           |
           x |    2.0214      1.2175      2.0220      1.9767      1.9848      1.9809      1.9809  
             |    0.0622      0.0430      0.0567      0.0365      0.0335      0.0389      0.0354  
       _cons |   -0.0083      0.4964     -0.0086      0.0191      0.0138      0.0167      0.0167  
             |    0.0369      0.0277      0.0340      0.0258      0.0239      0.0270      0.0221  
-------------+------------------------------------------------------------------------------------
lndelta      |
       _cons |                1.9774                              0.6633                          
             |                0.0289                              0.0255                          
-------------+------------------------------------------------------------------------------------
lnalpha      |
       _cons |                            0.7321                             -0.5099     -0.5099  
             |                            0.0197                              0.0270      0.0255  
-------------+------------------------------------------------------------------------------------
Statistics   |                                                                                    
           N |      9997        9997        9997       10000       10000       10000       10000  
          ll |  -37654.6    -21381.4    -21148.8    -24982.3    -21399.0    -21666.4    -21666.4  
--------------------------------------------------------------------------------------------------
                                                                                      legend: b/se

. 
. ********** 3.4 OVERDISPERSION TESTS
. 
. use racd03data.dta, clear

. 
. * Raw overdispersion
. quietly summarize DVISITS

. display "Overdispersion: variance/ mean ratio is " r(var) " / " r(mean) " = " r(var)/r(mean)
Overdispersion: variance/ mean ratio is . / .3017341 = .

. 
. * LR test statistic against NB2
. * Stata command lrtest does not work as LR test is for 2 different model commands
. quietly poisson DVISITS $XLIST

. scalar llpoisson = e(ll)

. quietly nbreg DVISITS $XLIST, dispersion(mean)

. scalar llnb2 = e(ll)

. display "LR test against NB2 = 2* (" llnb2 " - " llpoisson ") = " 2*(llnb2 - llpoisson)
LR test against NB2 = 2* (-3198.7438 - -3355.5413) = 313.59502

. 
. * LR test statistic against NB1
. quietly nbreg DVISITS $XLIST, dispersion(constant)

. scalar llnb1 = e(ll)

. display "LR test against NB1 = 2* (" llnb1 " - " llpoisson ") = " 2*(llnb1 - llpoisson)
LR test against NB1 = 2* (-3226.859 - -3355.5413) = 257.36473

. 
. * Wald test against NB2
. * Use output from nbreg DVISITS $XLIST, dispersion(mean)
. 
. * Wald test against NB1
. * Use output from nbreg DVISITS $XLIST, dispersion(constant)
. 
. * LM test against NB2
. capture drop mu

. quietly poisson DVISITS $XLIST 

. predict mu, n

. generate ystar = ((DVISITS - mu)^2 - DVISITS) / mu

. regress ystar mu, noconstant

      Source |       SS       df       MS              Number of obs =    5190
-------------+------------------------------           F(  1,  5189) =   56.32
       Model |  1169.70113     1  1169.70113           Prob > F      =  0.0000
    Residual |  107770.171  5189  20.7689673           R-squared     =  0.0107
-------------+------------------------------           Adj R-squared =  0.0105
       Total |  108939.872  5190  20.9903415           Root MSE      =  4.5573

------------------------------------------------------------------------------
       ystar |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
          mu |   .9574298   .1275783     7.50   0.000     .7073225    1.207537
------------------------------------------------------------------------------

. regress ystar mu, noconstant vce(robust)

Linear regression                                      Number of obs =    5190
                                                       F(  1,  5189) =   66.48
                                                       Prob > F      =  0.0000
                                                       R-squared     =  0.0107
                                                       Root MSE      =  4.5573

------------------------------------------------------------------------------
             |               Robust
       ystar |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
          mu |   .9574298   .1174219     8.15   0.000     .7272335    1.187626
------------------------------------------------------------------------------

. 
. * LM test against NB1
. regress ystar

      Source |       SS       df       MS              Number of obs =    5190
-------------+------------------------------           F(  0,  5189) =    0.00
       Model |           0     0           .           Prob > F      =       .
    Residual |   108048.49  5189  20.8226037           R-squared     =  0.0000
-------------+------------------------------           Adj R-squared =  0.0000
       Total |   108048.49  5189  20.8226037           Root MSE      =  4.5632

------------------------------------------------------------------------------
       ystar |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _cons |   .4144272   .0633408     6.54   0.000     .2902524    .5386019
------------------------------------------------------------------------------

. regress ystar, vce(robust)

Linear regression                                      Number of obs =    5190
                                                       F(  0,  5189) =    0.00
                                                       Prob > F      =       .
                                                       R-squared     =  0.0000
                                                       Root MSE      =  4.5632

------------------------------------------------------------------------------
             |               Robust
       ystar |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       _cons |   .4144272   .0633408     6.54   0.000     .2902524    .5386019
------------------------------------------------------------------------------

. 
. ********** 3.6 POISSON MARGINAL EFFECTS AND PREDICTION
. 
. *** Table 3.6 OLS column
. * OLS coefficients are OLS marginal effects 
. regress DVISITS $XLIST, vce(robust)   // Table 3.5 OLS column

Linear regression                                      Number of obs =    5190
                                                       F( 12,  5177) =   23.04
                                                       Prob > F      =  0.0000
                                                       R-squared     =  0.2018
                                                       Root MSE      =  .71388

------------------------------------------------------------------------------
             |               Robust
     DVISITS |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |    .033811   .0229929     1.47   0.141    -.0112648    .0788868
         AGE |    .203201   .4475822     0.45   0.650    -.6742492    1.080651
       AGESQ |  -.0621028   .5149627    -0.12   0.904    -1.071647    .9474416
      INCOME |  -.0573227   .0348968    -1.64   0.101    -.1257351    .0110897
    LEVYPLUS |   .0351789   .0217826     1.61   0.106    -.0075243     .077882
    FREEPOOR |  -.1033142   .0476909    -2.17   0.030    -.1968086   -.0098198
    FREEREPA |   .0332409    .043345     0.77   0.443    -.0517336    .1182155
     ILLNESS |   .0599457   .0099355     6.03   0.000     .0404678    .0794236
     ACTDAYS |   .1031916   .0097408    10.59   0.000     .0840956    .1222876
      HSCORE |   .0169765   .0071747     2.37   0.018     .0029111    .0310419
     CHCOND1 |   .0043844   .0222637     0.20   0.844     -.039262    .0480307
     CHCOND2 |   .0416174   .0464145     0.90   0.370    -.0493745    .1326094
       _cons |   .0276322   .0733923     0.38   0.707    -.1162477    .1715121
------------------------------------------------------------------------------

. 
. *** Table 3.6 QMLE, AME, MEM and Elast columns
. * Poisson marginal effects done the easy way using calculus method 
. * and ignoring the quadratic in age
. poisson DVISITS $XLIST, vce(robust)

Iteration 0:   log pseudolikelihood = -4923.1976  
Iteration 1:   log pseudolikelihood = -3890.2934  
Iteration 2:   log pseudolikelihood = -3356.8559  
Iteration 3:   log pseudolikelihood = -3355.5431  
Iteration 4:   log pseudolikelihood = -3355.5413  
Iteration 5:   log pseudolikelihood = -3355.5413  

Poisson regression                                Number of obs   =       5190
                                                  Wald chi2(12)   =     964.02
                                                  Prob > chi2     =     0.0000
Log pseudolikelihood = -3355.5413                 Pseudo R2       =     0.1576

------------------------------------------------------------------------------
             |               Robust
     DVISITS |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |    .156882   .0792209     1.98   0.048     .0016118    .3121522
         AGE |   1.056299   1.364474     0.77   0.439    -1.618021     3.73062
       AGESQ |  -.8487041   1.459683    -0.58   0.561    -3.709631    2.012223
      INCOME |  -.2053206   .1292572    -1.59   0.112      -.45866    .0480188
    LEVYPLUS |   .1231854   .0951652     1.29   0.196    -.0633348    .3097057
    FREEPOOR |  -.4400609   .2900225    -1.52   0.129    -1.008494    .1283726
    FREEREPA |   .0797984   .1257953     0.63   0.526    -.1667558    .3263527
     ILLNESS |   .1869484   .0239387     7.81   0.000     .1400295    .2338674
     ACTDAYS |   .1268465   .0077698    16.33   0.000     .1116179     .142075
      HSCORE |    .030081   .0142359     2.11   0.035     .0021791    .0579829
     CHCOND1 |   .1140853   .0908541     1.26   0.209    -.0639854    .2921561
     CHCOND2 |   .1411583   .1227226     1.15   0.250    -.0993737    .3816902
       _cons |  -2.223848   .2544567    -8.74   0.000    -2.722574   -1.725122
------------------------------------------------------------------------------

. margins, dydx(*)           // Table 3.6 AME column

Average marginal effects                          Number of obs   =       5190
Model VCE    : Robust

Expression   : Predicted number of events, predict()
dy/dx w.r.t. : SEX AGE AGESQ INCOME LEVYPLUS FREEPOOR FREEREPA ILLNESS ACTDAYS HSCORE CHCOND1 CHCOND2

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |   .0473366    .024003     1.97   0.049     .0002916    .0943817
         AGE |   .3187216   .4124949     0.77   0.440    -.4897535    1.127197
       AGESQ |   -.256083   .4409922    -0.58   0.561    -1.120412    .6082459
      INCOME |  -.0619522   .0391506    -1.58   0.114     -.138686    .0147816
    LEVYPLUS |   .0371692   .0285961     1.30   0.194     -.018878    .0932165
    FREEPOOR |  -.1327814   .0874291    -1.52   0.129    -.3041394    .0385766
    FREEREPA |   .0240779    .037872     0.64   0.525    -.0501499    .0983057
     ILLNESS |   .0564087    .007417     7.61   0.000     .0418717    .0709457
     ACTDAYS |   .0382739    .003007    12.73   0.000     .0323803    .0441675
      HSCORE |   .0090765   .0042949     2.11   0.035     .0006585    .0174944
     CHCOND1 |   .0344234   .0273604     1.26   0.208    -.0192019    .0880488
     CHCOND2 |   .0425923   .0369898     1.15   0.250    -.0299064    .1150909
------------------------------------------------------------------------------

. margins, dydx(*) atmeans   // Table 3.6 MEM column

Conditional marginal effects                      Number of obs   =       5190
Model VCE    : Robust

Expression   : Predicted number of events, predict()
dy/dx w.r.t. : SEX AGE AGESQ INCOME LEVYPLUS FREEPOOR FREEREPA ILLNESS ACTDAYS HSCORE CHCOND1 CHCOND2
at           : SEX             =    .5206166 (mean)
               AGE             =    .4063854 (mean)
               AGESQ           =    .2070766 (mean)
               INCOME          =    .5831599 (mean)
               LEVYPLUS        =    .4427746 (mean)
               FREEPOOR        =    .0427746 (mean)
               FREEREPA        =    .2102119 (mean)
               ILLNESS         =    1.431985 (mean)
               ACTDAYS         =    .8618497 (mean)
               HSCORE          =    1.217534 (mean)
               CHCOND1         =    .4030829 (mean)
               CHCOND2         =    .1165703 (mean)

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |   .0357194   .0180784     1.98   0.048     .0002864    .0711524
         AGE |   .2405017   .3108328     0.77   0.439    -.3687195    .8497228
       AGESQ |  -.1932357    .332465    -0.58   0.561     -.844855    .4583836
      INCOME |  -.0467481   .0294151    -1.59   0.112    -.1044006    .0109045
    LEVYPLUS |   .0280473   .0215155     1.30   0.192    -.0141224    .0702169
    FREEPOOR |  -.1001945   .0657527    -1.52   0.128    -.2290675    .0286785
    FREEREPA |   .0181688   .0285719     0.64   0.525     -.037831    .0741686
     ILLNESS |    .042565   .0053681     7.93   0.000     .0320437    .0530863
     ACTDAYS |   .0288808   .0020565    14.04   0.000     .0248501    .0329116
      HSCORE |   .0068489   .0032296     2.12   0.034      .000519    .0131789
     CHCOND1 |   .0259753   .0206209     1.26   0.208     -.014441    .0663916
     CHCOND2 |   .0321394   .0278903     1.15   0.249    -.0225246    .0868034
------------------------------------------------------------------------------

. margins, eyex(*)           // Table 3.5 Elast column

Average marginal effects                          Number of obs   =       5190
Model VCE    : Robust

Expression   : Predicted number of events, predict()
ey/ex w.r.t. : SEX AGE AGESQ INCOME LEVYPLUS FREEPOOR FREEREPA ILLNESS ACTDAYS HSCORE CHCOND1 CHCOND2

------------------------------------------------------------------------------
             |            Delta-method
             |      ey/ex   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |   .0816753   .0412437     1.98   0.048     .0008391    .1625116
         AGE |   .4292646   .5545024     0.77   0.439    -.6575401    1.516069
       AGESQ |  -.1757467   .3022662    -0.58   0.561    -.7681777    .4166842
      INCOME |  -.1197347   .0753776    -1.59   0.112    -.2674721    .0280026
    LEVYPLUS |   .0545434   .0421367     1.29   0.196    -.0280431    .1371298
    FREEPOOR |  -.0188234   .0124056    -1.52   0.129    -.0431379    .0054911
    FREEREPA |   .0167746   .0264437     0.63   0.526    -.0350541    .0686032
     ILLNESS |   .2677073   .0342798     7.81   0.000     .2005201    .3348945
     ACTDAYS |   .1093226   .0066964    16.33   0.000     .0961979    .1224473
      HSCORE |   .0366246   .0173327     2.11   0.035     .0026532    .0705961
     CHCOND1 |   .0459858   .0366217     1.26   0.209    -.0257914    .1177631
     CHCOND2 |   .0164549   .0143058     1.15   0.250     -.011584    .0444938
------------------------------------------------------------------------------

. 
. * Following discussed in text
. * Gives Treatment eEffect for binary regressors
. * Gives correct AME and MEM for the quadratic in AGE
. poisson DVISITS i.SEX c.AGE##c.AGE c.INCOME i.LEVYPLUS i.FREEPOOR i.FREEREPA ///
>   c.ILLNESS c.ACTDAYS c.HSCORE i.CHCOND1 i.CHCOND2, vce(robust)

Iteration 0:   log pseudolikelihood = -4923.1976  
Iteration 1:   log pseudolikelihood = -3890.2934  
Iteration 2:   log pseudolikelihood = -3356.8559  
Iteration 3:   log pseudolikelihood = -3355.5431  
Iteration 4:   log pseudolikelihood = -3355.5413  
Iteration 5:   log pseudolikelihood = -3355.5413  

Poisson regression                                Number of obs   =       5190
                                                  Wald chi2(12)   =     964.02
                                                  Prob > chi2     =     0.0000
Log pseudolikelihood = -3355.5413                 Pseudo R2       =     0.1576

------------------------------------------------------------------------------
             |               Robust
     DVISITS |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       1.SEX |    .156882   .0792209     1.98   0.048     .0016118    .3121522
         AGE |   1.056299   1.364474     0.77   0.439    -1.618021    3.730619
             |
 c.AGE#c.AGE |  -.8487036   1.459683    -0.58   0.561     -3.70963    2.012222
             |
      INCOME |  -.2053206   .1292571    -1.59   0.112    -.4586599    .0480188
  1.LEVYPLUS |   .1231854   .0951652     1.29   0.196    -.0633348    .3097057
  1.FREEPOOR |  -.4400609   .2900225    -1.52   0.129    -1.008494    .1283726
  1.FREEREPA |   .0797984   .1257953     0.63   0.526    -.1667558    .3263527
     ILLNESS |   .1869484   .0239387     7.81   0.000     .1400295    .2338674
     ACTDAYS |   .1268465   .0077698    16.33   0.000     .1116179     .142075
      HSCORE |    .030081   .0142359     2.11   0.035     .0021791    .0579829
   1.CHCOND1 |   .1140853   .0908541     1.26   0.209    -.0639854    .2921561
   1.CHCOND2 |   .1411583   .1227226     1.15   0.250    -.0993737    .3816902
       _cons |  -2.223848   .2544567    -8.74   0.000    -2.722574   -1.725122
------------------------------------------------------------------------------

. margins, dydx(*)

Average marginal effects                          Number of obs   =       5190
Model VCE    : Robust

Expression   : Predicted number of events, predict()
dy/dx w.r.t. : 1.SEX AGE INCOME 1.LEVYPLUS 1.FREEPOOR 1.FREEREPA ILLNESS ACTDAYS HSCORE 1.CHCOND1 1.CHCOND2

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       1.SEX |   .0466044   .0232474     2.00   0.045     .0010403    .0921686
         AGE |   .0760413   .0649717     1.17   0.242     -.051301    .2033835
      INCOME |  -.0619522   .0391506    -1.58   0.114     -.138686    .0147816
  1.LEVYPLUS |   .0375736   .0292403     1.28   0.199    -.0197363    .0948836
  1.FREEPOOR |  -.1087454   .0577384    -1.88   0.060    -.2219106    .0044198
  1.FREEREPA |   .0244398   .0390335     0.63   0.531    -.0520644     .100944
     ILLNESS |   .0564087    .007417     7.61   0.000     .0418717    .0709457
     ACTDAYS |   .0382739    .003007    12.73   0.000     .0323803    .0441675
      HSCORE |   .0090765   .0042949     2.11   0.035     .0006585    .0174944
   1.CHCOND1 |   .0346488    .027731     1.25   0.211     -.019703    .0890005
   1.CHCOND2 |   .0443413   .0400876     1.11   0.269    -.0342289    .1229115
------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.

. margins, dydx(*) atmeans

Conditional marginal effects                      Number of obs   =       5190
Model VCE    : Robust

Expression   : Predicted number of events, predict()
dy/dx w.r.t. : 1.SEX AGE INCOME 1.LEVYPLUS 1.FREEPOOR 1.FREEREPA ILLNESS ACTDAYS HSCORE 1.CHCOND1 1.CHCOND2
at           : 0.SEX           =    .4793834 (mean)
               1.SEX           =    .5206166 (mean)
               AGE             =    .4063854 (mean)
               INCOME          =    .5831599 (mean)
               0.LEVYPLUS      =    .5572254 (mean)
               1.LEVYPLUS      =    .4427746 (mean)
               0.FREEPOOR      =    .9572254 (mean)
               1.FREEPOOR      =    .0427746 (mean)
               0.FREEREPA      =    .7897881 (mean)
               1.FREEREPA      =    .2102119 (mean)
               ILLNESS         =    1.431985 (mean)
               ACTDAYS         =    .8618497 (mean)
               HSCORE          =    1.217534 (mean)
               0.CHCOND1       =    .5969171 (mean)
               1.CHCOND1       =    .4030829 (mean)
               0.CHCOND2       =    .8834297 (mean)
               1.CHCOND2       =    .1165703 (mean)

------------------------------------------------------------------------------
             |            Delta-method
             |      dy/dx   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
       1.SEX |   .0369316   .0188214     1.96   0.050     .0000423     .073821
         AGE |   .0864681   .0675555     1.28   0.201    -.0459382    .2188745
      INCOME |  -.0484415   .0313139    -1.55   0.122    -.1098156    .0129326
  1.LEVYPLUS |   .0292874   .0224741     1.30   0.193    -.0147611    .0733358
  1.FREEPOOR |  -.0855881   .0464512    -1.84   0.065    -.1766308    .0054547
  1.FREEREPA |   .0192725   .0308851     0.62   0.533    -.0412613    .0798062
     ILLNESS |   .0441069   .0065024     6.78   0.000     .0313624    .0568515
     ACTDAYS |    .029927   .0029461    10.16   0.000     .0241527    .0357013
      HSCORE |    .007097   .0033326     2.13   0.033     .0005652    .0136289
   1.CHCOND1 |   .0272303   .0216383     1.26   0.208    -.0151799    .0696405
   1.CHCOND2 |    .035185   .0317665     1.11   0.268    -.0270763    .0974463
------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.

. 
. * Following computes standardized coefficients
. * Table 3.6 SSC column
. capture drop one

. generate one = 1

. matrix accum Cov = $XLIST, deviations noconstant
(obs=5190)

. quietly sum one

. matrix Cov = Cov / (r(N)-1)

. matrix Stdev = (vecdiag(cholesky(diag(vecdiag(Cov)))))'

. // Need to add back constant as last entry in Stdev to make conformable with b
. matrix Stdev = Stdev \ 0

. matrix list Stdev

Stdev[13,1]
                 r1
     SEX  .49962291
     AGE  .20478182
   AGESQ  .18563646
  INCOME   .3689067
LEVYPLUS  .49676231
FREEPOOR  .20236797
FREEREPA  .40749832
 ILLNESS  1.3841524
 ACTDAYS  2.8876284
  HSCORE  2.1242665
 CHCOND1   .4905644
 CHCOND2  .32093852
     r13          0

. quietly poisson DVISITS $XLIST, vce(robust)

. matrix b = e(b)'

. matrix bstandardized = hadamard(b,Stdev)

. matrix list bstandardized     // Table 3.6 SSC column

bstandardized[13,1]
                          y1
     DVISITS:SEX   .07838182
     DVISITS:AGE   .21631093
   DVISITS:AGESQ  -.15755043
  DVISITS:INCOME  -.07574414
DVISITS:LEVYPLUS   .06119388
DVISITS:FREEPOOR  -.08905424
DVISITS:FREEREPA   .03251773
 DVISITS:ILLNESS   .25876512
 DVISITS:ACTDAYS    .3662855
  DVISITS:HSCORE   .06390007
 DVISITS:CHCOND1   .05596619
 DVISITS:CHCOND2   .04530313
   DVISITS:_cons           0

. 
. ********** 3.7 OTHER MODELS
. 
. use racd03data.dta, clear

. 
. * Binary Poisson
. generate BINARYVISIT = DVISITS > 0

. tabulate BINARYVISIT

BINARYVISIT |      Freq.     Percent        Cum.
------------+-----------------------------------
          0 |      4,141       79.79       79.79
          1 |      1,049       20.21      100.00
------------+-----------------------------------
      Total |      5,190      100.00

. 
. * Poisson ML program lfpois to be called by command ml method lf
. program lfbinarypois
  1.   version 10.1
  2.   args lnf theta1                  // theta1=x'b, lnf=lnf(y)
  3.   tempvar lnyfact mu p
  4.   local y "$ML_y1"                 // Define y so program more readable
  5.   generate double `lnyfact' = lnfactorial(`y')
  6.   generate double `mu'      = exp(`theta1')
  7.   generate double `p'       = 1 - exp(-`mu')
  8.   quietly replace `lnf'     = `y'*ln(`p') + ln(1-`p') - `y'*ln(1-`p')
  9. end

. ml model lf lfbinarypois (BINARYVISIT = $XLIST), vce(robust) 

. ml check

Test 1:  Calling lfbinarypois to check if it computes log pseudolikelihood and
         does not alter coefficient vector...
         Passed.

Test 2:  Calling lfbinarypois again to check if the same log pseudolikelihood
         value is returned...
         Passed.

Test 3:  Calling lfbinarypois to check if 1st derivatives are computed...
         test not relevant for type lf evaluators.

Test 4:  Calling lfbinarypois again to check if the same 1st derivatives are
         returned...
         test not relevant for type lf evaluators.

Test 5:  Calling lfbinarypois to check if 2nd derivatives are computed...
         test not relevant for type lf evaluators.

Test 6:  Calling lfbinarypois again to check if the same 2nd derivatives are
         returned...
         test not relevant for type lf evaluators.

------------------------------------------------------------------------------
Searching for alternate values for the coefficient vector to verify that
lfbinarypois returns different results when fed a different coefficient
vector:

Searching...
initial:       log pseudolikelihood =     -<inf>  (could not be evaluated)
searching for feasible values +

feasible:      log pseudolikelihood = -14886.656
improving initial values ..........
improve:       log pseudolikelihood = -14886.656

continuing with tests...
------------------------------------------------------------------------------

Test 7:  Calling lfbinarypois to check log pseudolikelihood at the new
         values...
         Passed.

Test 8:  Calling lfbinarypois requesting 1st derivatives at the new values...
         test not relevant for type lf evaluators.

Test 9:  Calling lfbinarypois requesting 2nd derivatives at the new values...
         test not relevant for type lf evaluators.

------------------------------------------------------------------------------
                         lfbinarypois HAS PASSED ALL TESTS
------------------------------------------------------------------------------

Test 10: Does lfbinarypois produce unanticipated output?
         This is a minor issue.  Stata has been running lfbinarypois with all
         output suppressed.  This time Stata will not suppress the output.  If
         you see any unanticipated output, you need to place quietly in front
         of some of the commands in lfbinarypois.

-------------------------------------------------------------- begin execution
---------------------------------------------------------------- end execution

. ml search
initial:       log pseudolikelihood = -14886.656
rescale:       log pseudolikelihood = -4622.1502

. ml maximize 

initial:       log pseudolikelihood = -4622.1502
rescale:       log pseudolikelihood = -4622.1502
Iteration 0:   log pseudolikelihood = -4622.1502  
Iteration 1:   log pseudolikelihood = -2364.0232  
Iteration 2:   log pseudolikelihood = -2304.7414  
Iteration 3:   log pseudolikelihood =  -2303.782  
Iteration 4:   log pseudolikelihood = -2303.7784  
Iteration 5:   log pseudolikelihood = -2303.7784  

                                                  Number of obs   =       5190
                                                  Wald chi2(12)   =     610.57
Log pseudolikelihood = -2303.7784                 Prob > chi2     =     0.0000

------------------------------------------------------------------------------
             |               Robust
 BINARYVISIT |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |   .2061259    .071847     2.87   0.004     .0653084    .3469434
         AGE |  -1.000849   1.283692    -0.78   0.436    -3.516839    1.515141
       AGESQ |   1.473944   1.405104     1.05   0.294    -1.280009    4.227896
      INCOME |   .0018279      .1088     0.02   0.987    -.2114163    .2150721
    LEVYPLUS |   .2218097   .0870455     2.55   0.011     .0512038    .3924157
    FREEPOOR |  -.5717048   .2322152    -2.46   0.014    -1.026838   -.1165713
    FREEREPA |   .3455172   .1223096     2.82   0.005     .1057947    .5852397
     ILLNESS |   .2074296   .0232989     8.90   0.000     .1617646    .2530946
     ACTDAYS |   .0993124   .0082463    12.04   0.000     .0831499    .1154749
      HSCORE |     .04358    .015178     2.87   0.004     .0138317    .0733284
     CHCOND1 |      .1198   .0800389     1.50   0.134    -.0370733    .2766733
     CHCOND2 |   .2267205   .1091817     2.08   0.038     .0127283    .4407128
       _cons |  -2.318976   .2374827    -9.76   0.000    -2.784434   -1.853519
------------------------------------------------------------------------------

. 
. * CLOGLOG model is the same !
. cloglog DVISITS $XLIST, vce(robust)   // Table 3.7 BP column

Iteration 0:   log pseudolikelihood = -2446.2384  
Iteration 1:   log pseudolikelihood = -2304.8611  
Iteration 2:   log pseudolikelihood = -2303.7792  
Iteration 3:   log pseudolikelihood = -2303.7784  
Iteration 4:   log pseudolikelihood = -2303.7784  

Complementary log-log regression                Number of obs     =       5190
                                                Zero outcomes     =       4141
                                                Nonzero outcomes  =       1049

                                                Wald chi2(12)     =     610.57
Log pseudolikelihood = -2303.7784               Prob > chi2       =     0.0000

------------------------------------------------------------------------------
             |               Robust
     DVISITS |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |   .2061259    .071847     2.87   0.004     .0653084    .3469434
         AGE |  -1.000849   1.283692    -0.78   0.436    -3.516839    1.515141
       AGESQ |   1.473944   1.405104     1.05   0.294    -1.280009    4.227896
      INCOME |   .0018279      .1088     0.02   0.987    -.2114163     .215072
    LEVYPLUS |   .2218097   .0870455     2.55   0.011     .0512038    .3924157
    FREEPOOR |  -.5717048   .2322152    -2.46   0.014    -1.026838   -.1165713
    FREEREPA |   .3455172   .1223096     2.82   0.005     .1057947    .5852397
     ILLNESS |   .2074296   .0232989     8.90   0.000     .1617646    .2530946
     ACTDAYS |   .0993124   .0082463    12.04   0.000     .0831499    .1154749
      HSCORE |     .04358    .015178     2.87   0.004     .0138317    .0733284
     CHCOND1 |      .1198   .0800389     1.50   0.134    -.0370732    .2766733
     CHCOND2 |   .2267205   .1091817     2.08   0.038     .0127283    .4407128
       _cons |  -2.318976   .2374827    -9.76   0.000    -2.784434   -1.853519
------------------------------------------------------------------------------

. estimates store CLOGLOG

. 
. * Compare to binary logit and probit
. probit DVISITS $XLIST, vce(robust)

Iteration 0:   log pseudolikelihood = -2612.2652  
Iteration 1:   log pseudolikelihood = -2273.8846  
Iteration 2:   log pseudolikelihood = -2271.6122  
Iteration 3:   log pseudolikelihood = -2271.6115  
Iteration 4:   log pseudolikelihood = -2271.6115  

Probit regression                                 Number of obs   =       5190
                                                  Wald chi2(12)   =     541.46
                                                  Prob > chi2     =     0.0000
Log pseudolikelihood = -2271.6115                 Pseudo R2       =     0.1304

------------------------------------------------------------------------------
             |               Robust
     DVISITS |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |   .1498603   .0462357     3.24   0.001     .0592401    .2404806
         AGE |  -1.180531    .862017    -1.37   0.171    -2.870053    .5089916
       AGESQ |   1.624826    .959646     1.69   0.090    -.2560457    3.505697
      INCOME |   .0032784    .071177     0.05   0.963     -.136226    .1427828
    LEVYPLUS |   .1529976   .0539566     2.84   0.005     .0472446    .2587505
    FREEPOOR |  -.3499512    .134044    -2.61   0.009    -.6126725   -.0872298
    FREEREPA |   .2417192   .0801207     3.02   0.003     .0846854     .398753
     ILLNESS |    .156049   .0166213     9.39   0.000     .1234719    .1886261
     ACTDAYS |   .0935461   .0076002    12.31   0.000     .0786499    .1084423
      HSCORE |   .0356943   .0106316     3.36   0.001     .0148567    .0565318
     CHCOND1 |    .057915   .0502944     1.15   0.250    -.0406603    .1564902
     CHCOND2 |   .1508315   .0733483     2.06   0.040     .0070714    .2945916
       _cons |  -1.353913     .15423    -8.78   0.000    -1.656198   -1.051628
------------------------------------------------------------------------------

. logit DVISITS $XLIST, vce(robust)

Iteration 0:   log pseudolikelihood = -2612.2652  
Iteration 1:   log pseudolikelihood = -2297.7556  
Iteration 2:   log pseudolikelihood = -2278.6779  
Iteration 3:   log pseudolikelihood = -2278.2012  
Iteration 4:   log pseudolikelihood = -2278.2009  
Iteration 5:   log pseudolikelihood = -2278.2009  

Logistic regression                               Number of obs   =       5190
                                                  Wald chi2(12)   =     470.67
                                                  Prob > chi2     =     0.0000
Log pseudolikelihood = -2278.2009                 Pseudo R2       =     0.1279

------------------------------------------------------------------------------
             |               Robust
     DVISITS |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |    .260689   .0836438     3.12   0.002     .0967502    .4246279
         AGE |  -1.976087   1.536669    -1.29   0.198    -4.987903    1.035729
       AGESQ |   2.736668   1.706439     1.60   0.109    -.6078916    6.081227
      INCOME |   .0074574   .1282804     0.06   0.954    -.2439676    .2588824
    LEVYPLUS |   .2670067   .0985694     2.71   0.007     .0738143    .4601992
    FREEPOOR |  -.6803834   .2584529    -2.63   0.008    -1.186942    -.173825
    FREEREPA |   .4162405   .1444865     2.88   0.004     .1330521    .6994288
     ILLNESS |    .263485    .028873     9.13   0.000      .206895     .320075
     ACTDAYS |    .158077   .0138359    11.43   0.000     .1309592    .1851949
      HSCORE |   .0634296   .0186791     3.40   0.001     .0268193    .1000399
     CHCOND1 |    .102007   .0904905     1.13   0.260    -.0753512    .2793652
     CHCOND2 |   .2667977   .1296673     2.06   0.040     .0126545     .520941
       _cons |  -2.289901   .2770794    -8.26   0.000    -2.832967   -1.746836
------------------------------------------------------------------------------

. 
. * Ordered probit
. * Transform to 8 or more (as only one observation 9)
. generate DVISITS8ormore = DVISITS

. replace DVISITS8ormore = 8 if DVISITS > 8
(1 real change made)

. oprobit DVISITS8ormore $XLIST, vce(robust)

Iteration 0:   log pseudolikelihood = -3532.9632  
Iteration 1:   log pseudolikelihood = -3141.8883  
Iteration 2:   log pseudolikelihood = -3138.1006  
Iteration 3:   log pseudolikelihood =  -3138.098  
Iteration 4:   log pseudolikelihood =  -3138.098  

Ordered probit regression                         Number of obs   =       5190
                                                  Wald chi2(12)   =     688.11
                                                  Prob > chi2     =     0.0000
Log pseudolikelihood =  -3138.098                 Pseudo R2       =     0.1118

--------------------------------------------------------------------------------
               |               Robust
DVISITS8ormore |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
---------------+----------------------------------------------------------------
           SEX |   .1318851   .0444632     2.97   0.003     .0447388    .2190313
           AGE |  -.5341869   .8197301    -0.65   0.515    -2.140828    1.072455
         AGESQ |   .8573049   .9059467     0.95   0.344     -.918318    2.632928
        INCOME |   -.062211   .0705469    -0.88   0.378    -.2004804    .0760584
      LEVYPLUS |   .1370307   .0522984     2.62   0.009     .0345278    .2395337
      FREEPOOR |  -.3460449   .1355777    -2.55   0.011    -.6117724   -.0803174
      FREEREPA |   .1783822   .0746215     2.39   0.017     .0321268    .3246376
       ILLNESS |   .1504846    .015439     9.75   0.000     .1202246    .1807445
       ACTDAYS |   .1005754   .0065085    15.45   0.000     .0878189    .1133319
        HSCORE |    .031862   .0094767     3.36   0.001      .013288     .050436
       CHCOND1 |   .0616017   .0487196     1.26   0.206    -.0338869    .1570903
       CHCOND2 |   .1353215    .070455     1.92   0.055    -.0027678    .2734108
---------------+----------------------------------------------------------------
         /cut1 |   1.378705   .1478453                      1.088934    1.668477
         /cut2 |   2.317589    .151183                      2.021276    2.613903
         /cut3 |   2.892994   .1551917                      2.588823    3.197164
         /cut4 |   3.090366   .1584576                      2.779795    3.400937
         /cut5 |   3.331566    .158581                      3.020753    3.642379
         /cut6 |   3.466128   .1610627                      3.150451    3.781805
         /cut7 |   3.712493   .1788635                      3.361927    4.063059
         /cut8 |   4.168501   .2172623                      3.742675    4.594327
--------------------------------------------------------------------------------

. estimates store OPROBIT               

. 
. * Now rescale the Ordered Probit coefficients
. matrix b = e(b)

. quietly regress DVISITS $XLIST

. matrix boprobitrescaled = e(rmse)*b'

. display "Rescale coefficients by multiplying by: " e(rmse)
Rescale coefficients by multiplying by: .71388175

. matrix list boprobitrescaled         // Table 3.7 OrdProb column 

boprobitrescaled[20,1]
                                 y1
     DVISITS8ormore:SEX   .09415035
     DVISITS8ormore:AGE   -.3813463
   DVISITS8ormore:AGESQ   .61201429
  DVISITS8ormore:INCOME  -.04441131
DVISITS8ormore:LEVYPLUS   .09782372
DVISITS8ormore:FREEPOOR  -.24703514
DVISITS8ormore:FREEREPA   .12734379
 DVISITS8ormore:ILLNESS   .10742818
 DVISITS8ormore:ACTDAYS   .07179893
  DVISITS8ormore:HSCORE    .0227457
 DVISITS8ormore:CHCOND1   .04397631
 DVISITS8ormore:CHCOND2   .09660356
             cut1:_cons   .98423264
             cut2:_cons   1.6544848
             cut3:_cons   2.0652554
             cut4:_cons    2.206156
             cut5:_cons    2.378344
             cut6:_cons   2.4744056
             cut7:_cons   2.6502811
             cut8:_cons   2.9758167

. 
. * OLS with dependent variable y
. regress DVISITS $XLIST, vce(robust)

Linear regression                                      Number of obs =    5190
                                                       F( 12,  5177) =   23.04
                                                       Prob > F      =  0.0000
                                                       R-squared     =  0.2018
                                                       Root MSE      =  .71388

------------------------------------------------------------------------------
             |               Robust
     DVISITS |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |    .033811   .0229929     1.47   0.141    -.0112648    .0788868
         AGE |    .203201   .4475822     0.45   0.650    -.6742492    1.080651
       AGESQ |  -.0621028   .5149627    -0.12   0.904    -1.071647    .9474416
      INCOME |  -.0573227   .0348968    -1.64   0.101    -.1257351    .0110897
    LEVYPLUS |   .0351789   .0217826     1.61   0.106    -.0075243     .077882
    FREEPOOR |  -.1033142   .0476909    -2.17   0.030    -.1968086   -.0098198
    FREEREPA |   .0332409    .043345     0.77   0.443    -.0517336    .1182155
     ILLNESS |   .0599457   .0099355     6.03   0.000     .0404678    .0794236
     ACTDAYS |   .1031916   .0097408    10.59   0.000     .0840956    .1222876
      HSCORE |   .0169765   .0071747     2.37   0.018     .0029111    .0310419
     CHCOND1 |   .0043844   .0222637     0.20   0.844     -.039262    .0480307
     CHCOND2 |   .0416174   .0464145     0.90   0.370    -.0493745    .1326094
       _cons |   .0276322   .0733923     0.38   0.707    -.1162477    .1715121
------------------------------------------------------------------------------

. estimates store OLSY                 // Table 3.7 y column

. predict pOLSY, xb

. generate resOLSY = DVISITS - pOLSY

. quietly sum, detail

. display "Skewness: " r(skewness) "  Kurtosis: " r(kurtosis)
Skewness: 3.566847  Kurtosis: 26.445391

. 
. * OLS with dependent variable ln(y+0.1)
. generate LNDVISITS = ln(DVISITS + 0.1)

. regress LNDVISITS $XLIST, vce(robust)  // Table 3.7 lny column

Linear regression                                      Number of obs =    5190
                                                       F( 12,  5177) =   52.19
                                                       Prob > F      =  0.0000
                                                       R-squared     =  0.1721
                                                       Root MSE      =  .97983

------------------------------------------------------------------------------
             |               Robust
   LNDVISITS |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |   .0814928   .0298901     2.73   0.006     .0228955    .1400901
         AGE |  -.5659196   .5828345    -0.97   0.332    -1.708521    .5766821
       AGESQ |   .8771172   .6700677     1.31   0.191    -.4364985    2.190733
      INCOME |  -.0193598   .0451984    -0.43   0.668    -.1079678    .0692483
    LEVYPLUS |   .0796925   .0308905     2.58   0.010      .019134     .140251
    FREEPOOR |  -.1815168   .0572554    -3.17   0.002    -.2937616   -.0692721
    FREEREPA |   .1390223   .0568498     2.45   0.015     .0275726    .2504719
     ILLNESS |   .1096774   .0128764     8.52   0.000     .0844342    .1349205
     ACTDAYS |   .1058193   .0078054    13.56   0.000     .0905174    .1211213
      HSCORE |   .0289177   .0087604     3.30   0.001     .0117435    .0460919
     CHCOND1 |   .0219201   .0314016     0.70   0.485    -.0396403    .0834806
     CHCOND2 |   .1019354   .0565126     1.80   0.071    -.0088532    .2127241
       _cons |  -2.114624   .0987971   -21.40   0.000    -2.308308   -1.920939
------------------------------------------------------------------------------

. estimates store OLSLNY

. predict pOLSLNY, xb

. generate resOLSLNY = LNDVISITS - pOLSLNY

. quietly sum, detail

. display "Skewness: " r(skewness) "  Kurtosis: " r(kurtosis)
Skewness: 1.24342  Kurtosis: 3.9932273

. 
. * OLS with dependent variable sqrt(y)
. generate SQRTDVISITS = sqrt(DVISITS)

. regress SQRTDVISITS $XLIST, vce(robust)  // Table 3.7 sqrty column 

Linear regression                                      Number of obs =    5190
                                                       F( 12,  5177) =   44.72
                                                       Prob > F      =  0.0000
                                                       R-squared     =  0.1886
                                                       Root MSE      =  .44723

------------------------------------------------------------------------------
             |               Robust
 SQRTDVISITS |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |   .0340355   .0137554     2.47   0.013     .0070691    .0610018
         AGE |  -.1605487   .2691814    -0.60   0.551    -.6882579    .3671604
       AGESQ |   .2915928   .3096217     0.94   0.346    -.3153964     .898582
      INCOME |  -.0167669   .0208961    -0.80   0.422     -.057732    .0241983
    LEVYPLUS |   .0337061   .0140193     2.40   0.016     .0062223    .0611898
    FREEPOOR |  -.0810564    .027077    -2.99   0.003    -.1341387    -.027974
    FREEREPA |   .0538427   .0261417     2.06   0.039     .0025939    .1050916
     ILLNESS |   .0483804   .0059633     8.11   0.000     .0366897     .060071
     ACTDAYS |   .0544319   .0041722    13.05   0.000     .0462526    .0626113
      HSCORE |   .0129843   .0041001     3.17   0.002     .0049463    .0210223
     CHCOND1 |   .0086196   .0142406     0.61   0.545    -.0192981    .0365373
     CHCOND2 |   .0428074    .026519     1.61   0.107    -.0091811    .0947958
       _cons |   .0699893   .0453227     1.54   0.123    -.0188623    .1588408
------------------------------------------------------------------------------

. estimates store OLSSQRTY

. predict pOLSSQY, xb

. generate resOLSSQY = SQRTDVISITS - pOLSSQY

. quietly sum, detail

. display "Skewness: " r(skewness) "  Kurtosis: " r(kurtosis)
Skewness: 1.4278829  Kurtosis: 5.4665049

. 
. * Poisson QMLE
. poisson DVISITS $XLIST, vce(robust)

Iteration 0:   log pseudolikelihood = -4923.1976  
Iteration 1:   log pseudolikelihood = -3890.2934  
Iteration 2:   log pseudolikelihood = -3356.8559  
Iteration 3:   log pseudolikelihood = -3355.5431  
Iteration 4:   log pseudolikelihood = -3355.5413  
Iteration 5:   log pseudolikelihood = -3355.5413  

Poisson regression                                Number of obs   =       5190
                                                  Wald chi2(12)   =     964.02
                                                  Prob > chi2     =     0.0000
Log pseudolikelihood = -3355.5413                 Pseudo R2       =     0.1576

------------------------------------------------------------------------------
             |               Robust
     DVISITS |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
         SEX |    .156882   .0792209     1.98   0.048     .0016118    .3121522
         AGE |   1.056299   1.364474     0.77   0.439    -1.618021     3.73062
       AGESQ |  -.8487041   1.459683    -0.58   0.561    -3.709631    2.012223
      INCOME |  -.2053206   .1292572    -1.59   0.112      -.45866    .0480188
    LEVYPLUS |   .1231854   .0951652     1.29   0.196    -.0633348    .3097057
    FREEPOOR |  -.4400609   .2900225    -1.52   0.129    -1.008494    .1283726
    FREEREPA |   .0797984   .1257953     0.63   0.526    -.1667558    .3263527
     ILLNESS |   .1869484   .0239387     7.81   0.000     .1400295    .2338674
     ACTDAYS |   .1268465   .0077698    16.33   0.000     .1116179     .142075
      HSCORE |    .030081   .0142359     2.11   0.035     .0021791    .0579829
     CHCOND1 |   .1140853   .0908541     1.26   0.209    -.0639854    .2921561
     CHCOND2 |   .1411583   .1227226     1.15   0.250    -.0993737    .3816902
       _cons |  -2.223848   .2544567    -8.74   0.000    -2.722574   -1.725122
------------------------------------------------------------------------------

. estimates store POISSON

. predict pPOISS, n

. generate resPOISS = DVISITS - pPOISS

. quietly sum, detail

. display "Skewness: " r(skewness) "  Kurtosis: " r(kurtosis)
Skewness: 3.1246985  Kurtosis: 25.642737

. 
. * Nonlinear least squares with same conditionam mean as Poisson
. generate one = 1

. nl (DVISITS = exp({xb: $XLIST one})), vce(robust) // Table 3.7 NLS 
(obs = 5190)

Iteration 0:  residual SS =  3043.388
Iteration 1:  residual SS =  2729.354
Iteration 2:  residual SS =  2714.721
Iteration 3:  residual SS =  2714.387
Iteration 4:  residual SS =  2714.372
Iteration 5:  residual SS =  2714.371
Iteration 6:  residual SS =  2714.371
Iteration 7:  residual SS =  2714.371
Iteration 8:  residual SS =  2714.371
Iteration 9:  residual SS =  2714.371
Iteration 10:  residual SS =  2714.371
Iteration 11:  residual SS =  2714.371
Iteration 12:  residual SS =  2714.371
Iteration 13:  residual SS =  2714.371
Iteration 14:  residual SS =  2714.371
Iteration 15:  residual SS =  2714.371

Nonlinear regression                                 Number of obs =      5190
                                                     R-squared     =    0.2815
                                                     Adj R-squared =    0.2797
                                                     Root MSE      =   .724095
                                                     Res. dev.     =  11364.56

------------------------------------------------------------------------------
             |               Robust
     DVISITS |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
     /xb_SEX |  -.0570771    .137424    -0.42   0.678    -.3264862     .212332
     /xb_AGE |   3.626041   1.995592     1.82   0.069    -.2861619    7.538243
   /xb_AGESQ |  -3.675901   2.160576    -1.70   0.089    -7.911542     .559741
  /xb_INCOME |  -.3940749   .1953428    -2.02   0.044    -.7770294   -.0111204
/xb_LEVYPLUS |   .2138633   .1451309     1.47   0.141    -.0706545    .4983811
/xb_FREEPOOR |  -.2322004   .4281278    -0.54   0.588    -1.071512    .6071108
/xb_FREEREPA |  -.0029155    .193104    -0.02   0.988    -.3814808    .3756498
 /xb_ILLNESS |   .1396042   .0384706     3.63   0.000     .0641856    .2150227
 /xb_ACTDAYS |   .1209979   .0085272    14.19   0.000      .104281    .1377147
  /xb_HSCORE |   .0229036    .022301     1.03   0.304    -.0208158     .066623
 /xb_CHCOND1 |    .079467   .1457874     0.55   0.586    -.2063378    .3652719
 /xb_CHCOND2 |   -.055081    .177085    -0.31   0.756    -.4022423    .2920803
     /xb_one |  -2.233898   .3645483    -6.13   0.000    -2.948567    -1.51923
------------------------------------------------------------------------------

. estimates store NL

. 
. *** TABLE 3.7: BINARY PROBIT, ORDERED PROBIT, OLS (y, lny, sqrty) POISS NLS (most of table)
. 
. estimates table CLOGLOG OLSY OLSLNY OLSSQRTY POISSON, b(%9.3f) t eq(1) stats(ll)

--------------------------------------------------------------------------
    Variable |  CLOGLOG      OLSY       OLSLNY     OLSSQRTY     POISSON   
-------------+------------------------------------------------------------
         SEX |     0.206       0.034       0.081       0.034       0.157  
             |      2.87        1.47        2.73        2.47        1.98  
         AGE |    -1.001       0.203      -0.566      -0.161       1.056  
             |     -0.78        0.45       -0.97       -0.60        0.77  
       AGESQ |     1.474      -0.062       0.877       0.292      -0.849  
             |      1.05       -0.12        1.31        0.94       -0.58  
      INCOME |     0.002      -0.057      -0.019      -0.017      -0.205  
             |      0.02       -1.64       -0.43       -0.80       -1.59  
    LEVYPLUS |     0.222       0.035       0.080       0.034       0.123  
             |      2.55        1.61        2.58        2.40        1.29  
    FREEPOOR |    -0.572      -0.103      -0.182      -0.081      -0.440  
             |     -2.46       -2.17       -3.17       -2.99       -1.52  
    FREEREPA |     0.346       0.033       0.139       0.054       0.080  
             |      2.82        0.77        2.45        2.06        0.63  
     ILLNESS |     0.207       0.060       0.110       0.048       0.187  
             |      8.90        6.03        8.52        8.11        7.81  
     ACTDAYS |     0.099       0.103       0.106       0.054       0.127  
             |     12.04       10.59       13.56       13.05       16.33  
      HSCORE |     0.044       0.017       0.029       0.013       0.030  
             |      2.87        2.37        3.30        3.17        2.11  
     CHCOND1 |     0.120       0.004       0.022       0.009       0.114  
             |      1.50        0.20        0.70        0.61        1.26  
     CHCOND2 |     0.227       0.042       0.102       0.043       0.141  
             |      2.08        0.90        1.80        1.61        1.15  
       _cons |    -2.319       0.028      -2.115       0.070      -2.224  
             |     -9.76        0.38      -21.40        1.54       -8.74  
-------------+------------------------------------------------------------
          ll | -2303.778   -5608.556   -7252.014   -3181.506   -3355.541  
--------------------------------------------------------------------------
                                                               legend: b/t

. 
. * For TABLE 3.7 OrdProb column use the earlier rescaled coefficients
. matrix list boprobitrescaled 

boprobitrescaled[20,1]
                                 y1
     DVISITS8ormore:SEX   .09415035
     DVISITS8ormore:AGE   -.3813463
   DVISITS8ormore:AGESQ   .61201429
  DVISITS8ormore:INCOME  -.04441131
DVISITS8ormore:LEVYPLUS   .09782372
DVISITS8ormore:FREEPOOR  -.24703514
DVISITS8ormore:FREEREPA   .12734379
 DVISITS8ormore:ILLNESS   .10742818
 DVISITS8ormore:ACTDAYS   .07179893
  DVISITS8ormore:HSCORE    .0227457
 DVISITS8ormore:CHCOND1   .04397631
 DVISITS8ormore:CHCOND2   .09660356
             cut1:_cons   .98423264
             cut2:_cons   1.6544848
             cut3:_cons   2.0652554
             cut4:_cons    2.206156
             cut5:_cons    2.378344
             cut6:_cons   2.4744056
             cut7:_cons   2.6502811
             cut8:_cons   2.9758167

. 
. * For TABLE 3.7 NL column
. estimates table NL, b(%9.3f) t stats(ll)

--------------------------
    Variable |    NL      
-------------+------------
xb_SEX       |
       _cons |    -0.057  
             |     -0.42  
-------------+------------
xb_AGE       |
       _cons |     3.626  
             |      1.82  
-------------+------------
xb_AGESQ     |
       _cons |    -3.676  
             |     -1.70  
-------------+------------
xb_INCOME    |
       _cons |    -0.394  
             |     -2.02  
-------------+------------
xb_LEVYPLUS  |
       _cons |     0.214  
             |      1.47  
-------------+------------
xb_FREEPOOR  |
       _cons |    -0.232  
             |     -0.54  
-------------+------------
xb_FREEREPA  |
       _cons |    -0.003  
             |     -0.02  
-------------+------------
xb_ILLNESS   |
       _cons |     0.140  
             |      3.63  
-------------+------------
xb_ACTDAYS   |
       _cons |     0.121  
             |     14.19  
-------------+------------
xb_HSCORE    |
       _cons |     0.023  
             |      1.03  
-------------+------------
xb_CHCOND1   |
       _cons |     0.079  
             |      0.55  
-------------+------------
xb_CHCOND2   |
       _cons |    -0.055  
             |     -0.31  
-------------+------------
xb_one       |
       _cons |    -2.234  
             |     -6.13  
-------------+------------
Statistics   |            
          ll | -5682.281  
--------------------------
               legend: b/t

.  
. ********** CLOSE OUTPUT
. 
. * log close
. * clear
. * exit
. 
end of do-file

. exit, clear
