------------------------------------------------------------------------------------------------------ log: c:\Imbook\bwebpage\Section2\mma07p3montecarlo.txt log type: text opened on: 18 May 2005, 11:28:58 . . ********** OVERVIEW OF MMA07P3MONTECARLO.DO ********** . . * STATA Program . * copyright C 2005 by A. Colin Cameron and Pravin K. Trivedi . * used for "Microeconometrics: Methods and Applications" . * by A. Colin Cameron and Pravin K. Trivedi (2005) . * Cambridge University Press . . * Chapter 7.7.1-7.7.5 pp. 250-4 . * Size and power of the Wald test . . * (1) Figure 7.2 Density of Wald test statistic . * (2) Table 7.2 Actual size of Wald test at various nominal sizes . * (3) Table 7.2 Actual power of Wald test at various nominal sizes . * (4) Table 7.2 Nominal power of Wald test at various nominal sizes . * (5) Alternative way to simulate using postfile rather than simulate . . * on the slope coefficient for a Probit model with simulated data (see below). . . * NOTE: Because this is a simulation using many samples (here 10,000) . * the generated data are not saved in a text file. . . * Problem can arise if in one of the simulations all of sample is y=0 or y=1 . * Then the probit model is not estimable. . * Then need increase sample size, change dgp or reduce number of simulations. . * Here used N=40 with S=10000 for size and for power . * Another possible change is to have same regressors x across simulations . . ********** SETUP ********** . . set more off . version 8.0 . set scheme s1mono /* Graphics scheme */ . . ********** MONTE CARLO OVERVIEW ********** . . * The data generating process is . * - Probit with Pr[y=1] = Phi(b1 + b2*x2) . * - where b1 = 0 and b2 = 1 . * - and regressor x ~ N[0,1] is fixed throughout the simulations . . * The sample size N set below in the global numobs . * The number of simulations S is set below in the global numsims . * A third option is to switch to same x in each sample. This needs to be done manually. . . * The simulation is done using stata command simulate . * At the end of the program, an alternative using postfile is given . . * The program investigates both size and power . * of the Wald test that b2 = 1. . * For power the dgp instead uses b2 = 2. . . ********** INITIAL SIMULATION SET UP ********** . . set seed 10101 . * Change the following for different sample size N . global numobs "40" . * Change the following for different number of simulations S . global numsims "10000" . . ****** ANALYSIS: SIMULATION OF PROBIT MODEL SLOPE ESTIMATES AND WALD TEST . . * The program is rclass. . * This means the results returned by the program are put into r( ) . * Here we return meany, vary, betahat, sebetahat, ztestforbetaeq1 . . * The probit model is Pr[y=1] = Phi(b1 + b2*x2) where b1=0 and b2=1 . * For size calculations: b2 = 1 . * For power calculations: b2 = 1.5 (as an example) . * So pass the argument trueb2 as an argument. . . * The following three lines are only needed . * if the regressors are constant across simulations, . * as then need to generate once and put in a data file to be reused. . * They are commented out here as here (x,y) both resampled. . * Also simprobit and simprobit2 need one line changed if x is fixed. . /* > set obs numobs > gen x = invnorm(uniform()) > save xforsim, replace > */ . * This version of the program instead redraws both x and y in each simulation . . * The program has one argument . * - trueb2 = value of b2 in the dgp . . program simprobit, rclass 1. version 8.0 2. /* define arguments. Here trueb2 = b2 in Phi(b1 + b2*x2) */ . args trueb2 3. /* Generate the data: here x and y */ . drop _all 4. set obs \$numobs 5. gen x = invnorm(uniform()) 6. /* If instead want same x in each simulation, > replace above line with: use xforsim */ . gen y = 0 7. replace y = 1 if 0 + `trueb2'*x + invnorm(uniform()) > 0 8. /* Summarize the generated data as a check */ . summarize y 9. return scalar ymean=r(mean) 10. return scalar yvar=r(Var) 11. /* Do probit and store key results */ . probit y x 12. return scalar b2hat=_b[x] 13. return scalar seb2hat = _se[x] 14. return scalar ztestforb2eq1 = (_b[x]-1)/_se[x] 15. end . . ****** (1) DISTRIBUTION OF WALD TEST STATISTIC (Figure 7.2 p.253) . . * Now call the program simprobit where . * - include values for each argument within the quotes " " . * (here the argument is b2true and is set to 1 for size and 1.5 for power) . * - make sure that ask for each of the returned results . . * For size calculations set trueb2 = 1 . simulate "simprobit 1" ymean=r(ymean) yvar=r(yvar) b2hat=r(b2hat) /* > */ seb2hat=r(seb2hat) ztestforb2eq1=r(ztestforb2eq1), reps(\$numsims) command: simprobit 1 statistics: ymean = r(ymean) yvar = r(yvar) b2hat = r(b2hat) seb2hat = r(seb2hat) ztestfor~1 = r(ztestforb2eq1) . . * Summary of the results returned by simulate . * For Wald test key output is ztestforb2eq1 . describe Contains data obs: 10,000 simulate: simprobit 1 vars: 5 18 May 2005 11:29 size: 240,000 (97.7% of memory free) ------------------------------------------------------------------------------- storage display value variable name type format label variable label ------------------------------------------------------------------------------- ymean float %9.0g r(ymean) yvar float %9.0g r(yvar) b2hat float %9.0g r(b2hat) seb2hat float %9.0g r(seb2hat) ztestforb2eq1 float %9.0g r(ztestforb2eq1) ------------------------------------------------------------------------------- Sorted by: . summarize Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- ymean | 10000 .49946 .0794447 .225 .775 yvar | 10000 .2499373 .0089917 .1788462 .2564103 b2hat | 10000 1.133952 .4516738 -.0306482 9.389184 seb2hat | 10000 .3589645 .1561059 .1902922 4.583915 ztestforb2~1 | 10000 .1141294 .9558451 -4.087344 2.278257 . . * For b2hat there are two ways to estimate the standard deviation. . * One is the average of seb2hat, the standard error of b2hat . * The other is the standard deviation of b2hat. . * These are equal asymptotically, but perhaps not in small samples due to bias. . * Also aveseb2hat is used later in calculating asymptotic power. . sum seb2hat Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- seb2hat | 10000 .3589645 .1561059 .1902922 4.583915 . scalar aveseb2hat = r(mean) . sum b2hat Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- b2hat | 10000 1.133952 .4516738 -.0306482 9.389184 . scalar stdevb2hat = r(sd) . di "Average standard error of b2hat: " aveseb2hat Average standard error of b2hat: .3589645 . di "Standard deviation of b2hat: " stdevb2hat Standard deviation of b2hat: .45167383 . . * The Wald test statistic will be called Wald . gen Wald = ztestforb2eq1 . label var Wald "Wald test statistic" . . * The mean and st.dev. should be 0 and 1 if Wald ~ N[0,1] . sum Wald Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- Wald | 10000 .1141294 .9558451 -4.087344 2.278257 . . * The 2.5 and 97.5 percentiles should be -1.96 and 1.96 if Wald ~ N[0,1] . * They can be used to get size-adjusted Wald test at 5 percent. . _pctile Wald, p(2.5,99.5) . display "Wald: Lower 2.5 percentile = " r(r1) " Upper 2.5 percentile = " r(r2) Wald: Lower 2.5 percentile = -1.904708 Upper 2.5 percentile = 2.0034728 . . * The density of the simulated values of the Wald test should be . * a standard normal density if Wald ~ N[0,1] . * The following plots kernel estimate of density of Wald and a N[0,1] density . * Could also do Student[N-k] but this looks same as N[0,1] if N>=30. . gen N01density = normden(Wald) . sum Wald Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- Wald | 10000 .1141294 .9558451 -4.087344 2.278257 . . graph twoway (kdensity Wald, range(-3 3) clstyle(p1)) /* > */ (connect N01density Wald if Wald>-3 & Wald<3, clstyle(p2) sort(Wald) s(i)), /* > */ scale (1.2) plotregion(style(none)) /* > */ title("Monte Carlo Simulations of Wald Test") /* > */ xtitle("Wald Test Statistic", size(medlarge)) xscale(titlegap(*5)) /* > */ ytitle("Density", size(medlarge)) yscale(titlegap(*5)) /* > */ legend(pos(11) ring(0) col(1)) legend(size(small)) /* > */ legend( label(1 "Monte Carlo") label(2 "Standard Normal") /* > */ label(3 "Test size = 0.01")) . graph export ch7montecarlo.wmf, replace (file c:\Imbook\bwebpage\Section2\ch7montecarlo.wmf written in Windows Metafile format) . . ****** (2) ACTUAL SIZE OF THE WALD TEST STATISTIC (Table 7.2, p.253) . . * Obtain the size properties of a two-sided Wald test . * That rejects if |Wald| > z_alpha/2 where alpha = .01, .05, .1, .2 . . * Convert to two-sided test by taking absolute value . gen absWald = abs(Wald) . . * Give key percentiles of |Wald| . * Percentiles must be in ascending order for Stata . _pctile absWald, p(0.80,0.90,0.95,0.99) . display "I[Upper percentiles of |Wald|: " " 1 " r(r4) " 5 " r(r3) " 10 " r(r2) " 20 " r(r1) I[Upper percentiles of |Wald|: 1 .0115847 5 .01074749 10 .00998338 20 .00923005 . . * Program to calculate actual size given nominal size . * Temporary variables and scalars are in quotes ` ' . program size, rclass 1. version 8.0 2. args nominalsize 3. tempvar reject 4. tempname normalcriticalvalue 5. quietly { 6. scalar `normalcriticalvalue' = invnorm(1-(`nominalsize'/2)) 7. gen `reject' = 0 8. replace `reject' = 1 if absWald > `normalcriticalvalue' 9. summarize `reject' 10. return scalar actualsize = r(mean) 11. } 12. end . . * Calculate actual size for nominal sizes 0.01, 0.05, 0.10 and 0.20 . size 0.01 . scalar actualsize01 = r(actualsize) . size 0.05 . scalar actualsize05 = r(actualsize) . size 0.10 . scalar actualsize10 = r(actualsize) . size 0.20 . scalar actualsize20 = r(actualsize) . . * Following gives Actual Size column of Table 7.2 (p.253) . * Nominal Sizes and Actual Sizes of Two-sided Wald Test . di "0.01: " actualsize01 _new "0.05: " actualsize05 _new /* > */ "0.10: " actualsize10 _new "0.20: " actualsize20 0.01: .0053 0.05: .0294 0.10: .0805 0.20: .1922 . . ****** (3) ACTUAL POWER OF THE WALD TEST STATISTIC (Table 7.2, p.253) . . * Consider power when b2 = 2 rather than 1 . . * Obtain the actual power by simulation . * Use the same program simprobit as for size, . * except the argument b2true is 2.0 rather than 1.0 . . drop _all . . * For size calculations set trueb2 = 2 . simulate "simprobit 2" ymean=r(ymean) yvar=r(yvar) b2hat=r(b2hat) /* > */ seb2hat=r(seb2hat) ztestforb2eq1=r(ztestforb2eq1), reps(10000) command: simprobit 2 statistics: ymean = r(ymean) yvar = r(yvar) b2hat = r(b2hat) seb2hat = r(seb2hat) ztestfor~1 = r(ztestforb2eq1) . . * Calculate |Wald| . gen Wald = ztestforb2eq1 (71 missing values generated) . gen absWald = abs(Wald) (71 missing values generated) . . summarize Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- ymean | 9929 .4998389 .0791531 .225 .825 yvar | 9929 .249985 .0090933 .1480769 .2564103 b2hat | 9929 2.581075 2.73046 .8547966 209.9805 seb2hat | 9929 1.002628 5.799384 .2816004 540.1536 ztestforb2~1 | 9929 1.667773 .3853416 -.4042006 2.59991 -------------+-------------------------------------------------------- Wald | 9929 1.667773 .3853416 -.4042006 2.59991 absWald | 9929 1.668285 .383118 .0033462 2.59991 . . * Calculate actual power for nominal sizes 0.01, 0.05, 0.10 and 0.20 . * This can use the earlier program size . size 0.01 . scalar actualpower01 = r(actualsize) . size 0.05 . scalar actualpower05 = r(actualsize) . size 0.10 . scalar actualpower10 = r(actualsize) . size 0.20 . scalar actualpower20 = r(actualsize) . . * Following gives Actual Power column of Table 7.2 (p.253) . * Nominal Sizes and Actual Power of Two-sided Wald Test . di "0.01: " actualpower01 _new "0.05: " actualpower05 _new /* > */ "0.10: " actualpower10 _new "0.20: " actualpower20 0.01: .0073 0.05: .2257 0.10: .6077 0.20: .8583 . . ****** (4) ASYMPTOTIC POWER OF THE WALD TEST STATISTIC (Table 7.2, p.253) . . * Consider power when b2 = 2 rather than 1 . . * Calculate asymptotic theoretical power using noncentral chisquare . * Asymptotic power = Pr[W > chi-square(alpha) | W ~ noncentral chi-square(alpha,ncp) . * The noncentrality parameter is 0.5*(delta^2)/(se[b2]^2) . * Here size has b2 = 1 and power has b2 = 1+delta . * So delta = b2true - 1. . * Need to find the standard error of b2. . * Use the average from earlier simulations. . . * Program to calculate asymptotic power given nominal size . * Temporary variables and scalars and arguments are in quotes ` ' . * invchi2tail gives cv such that Pr(Chi2 > cv) = nominalsize . * Power is 1 minus cdf of noncentral chisquare . * nchi2 gives the cdf of noncentral chisquare . . drop _all . . * Arguments are alpha (size), lamda and df (degrees of freedom) . program power, rclass 1. version 8.0 2. args alpha lamda df 3. tempname criticalvalue powervianoncentralchi 4. quietly { 5. scalar `criticalvalue' = invchi2tail(`df',`alpha') 6. scalar `powervianoncentralchi' = 1-nchi2(`df',`lamda',`criticalvalue') 7. return scalar asymppower = `powervianoncentralchi' 8. } 9. end . . * scalar criticalvalue = invchi2tail(df,alpha) . * replace power = 1-nchi2(df,lamda,criticalvalue) . . * Calculate df and lamda. . * This uses an estimate of se[beta] obtained earlier . scalar delta = 1 /* Here 2 - 1. Changes for different alternatives */ . scalar lamda = 0.5*(delta*delta)/(aveseb2hat*aveseb2hat) . scalar df = 1 . di "delta: " delta " aveseb2hat: " aveseb2hat " lamda: " lamda " df: " df delta: 1 aveseb2hat: .3589645 lamda: 3.8803151 df: 1 . . * Calculate asymptotic power for nominal sizes 0.01, 0.05, 0.10 and 0.20 . power 0.01 lamda df . scalar asymppower01 = r(asymppower) . power 0.05 lamda df . scalar asymppower05 = r(asymppower) . power 0.10 lamda df . scalar asymppower10 = r(asymppower) . power 0.20 lamda df . scalar asymppower20 = r(asymppower) . . * Following gives Asymptotic Power column of Table 7.2 (p.253) . * Nominal Sizes and Asymptotic Power of Two-sided Wald Test . di "0.01: " asymppower01 _new "0.05: " asymppower05 _new /* > */ "0.10: " asymppower10 _new "0.20: " asymppower20 0.01: .2722675 0.05: .50398701 0.10: .62755902 0.20: .75494224 . . ****** (5) ALTERNATIVE ANALYSIS: SIMULATION METHOD USING POSTFILE . . * This is an alternative, given for completeness. . * This fails if the model is not estimable in any of the simulation samples. . * By contrast, simulate just drops that simulation sample and continues simulating. . . * For each round of the simulation, the variables in `sim' are sent . * as a new line to a stata data set simprobitresults. . * The names of these variables are given in quotes after S_1 . * Need as many names in quotes after S_1 as variables at post . * Then can analyze these using summarize etcetera . . * This program has two arguments . * - numsims = desired number of simulations . * - trueb2 = slope coefficient used to generate the data . . drop _all . . program simprobit2 1. version 8.0 2. args numsims trueb2 3. tempname sim 4. postfile `sim' meany vary beta sterror ztestforbeta using probitsimresults, replace 5. quietly { 6. forvalues i = 1/`numsims' { 7. drop _all 8. set obs \$numobs /* may need to change */ 9. gen x = invnorm(uniform()) 10. /* If instead want same x in each simulation > replace above line with: use xforsim */ . gen y = 0 11. /* Use b2 = 1.0 for size and 1.5 for power */ . replace y = 1 if 0+`trueb2'*x+invnorm(uniform()) > 0 12. summarize y 13. scalar meany=r(mean) 14. scalar vary=r(Var) 15. probit y x 16. scalar beta=_b[x] 17. scalar sterror = _se[x] 18. scalar ztestforbeta = (beta-1)/sterror 19. post `sim' (meany) (vary) (beta) (sterror) (ztestforbeta) 20. } 21. } 22. postclose `sim' 23. end . . simprobit2 \$numsims 1 . use probitsimresults, clear . . * Here we just summarize results for comparison with earlier . * But could do the further analysis as above . sum Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- meany | 10000 .4989575 .0791248 .225 .775 vary | 10000 .2499885 .0090127 .1788462 .2564103 beta | 10000 1.135003 .4315248 .0901358 7.205799 sterror | 10000 .3583266 .133302 .1863547 3.360862 ztestforbeta | 10000 .1218973 .954814 -3.401833 2.299991 . . ********** CLOSE OUTPUT ********** . log close log: c:\Imbook\bwebpage\Section2\mma07p3montecarlo.txt log type: text closed on: 18 May 2005, 11:29:29 ----------------------------------------------------------------------------------------------------