------------------------------------------------------------------------------------------------------ log: c:\Imbook\bwebpage\Section3\mma11p1boot.txt log type: text opened on: 18 May 2005, 15:52:55 . . ********** OVERVIEW OF MMA11P1BOOT.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 11.3 pages 366-368 . * Bootstrap applied to exponential regression model . * Provides . * (1) Bootstrap distribution of beta and t-statistic (Table 11.1) . * (2) Various statistics from bootstrap (pages 366-8) . * (3) Bootstrap density of the t-statistic (Figure 11.1) . * using generated data (see below) . . * Note: To speed up progam reduce breps - the number of bootstrap replications . * But final program should use many repications . . * Note: This program uses ereg which is an old Stata command . * superceded by streg, dist(exp) . . * Note: For bootstrap see also mm07p4boot.do . * which has additional commands / ways to bootstrap . . ********** SETUP ********** . . set more off . version 8 . . ********** GENERATE DATA ********** . . * Model is y ~ exponential(exp(a + bx + cz)) . * where x and z are joint normal (1,1,0.1,0.1,0.5) . * i.e. means 0.1 and 0.1 . * sd's 0.1 and 0.1 and correln 0.5 (so correln^2 = .25) . * variances 0.01 and 0.01 and covariance 0.005 . . * Generate data from joint normal . * Use fact that x is N(mu0.1,0.1) . * and z | x is N(0.1 + .05/.1*(x - .1), .01x.75 = .0075) . * so that st dev = sqrt(0.0075) = 0.0866025 . . set obs 50 obs was 0, now 50 . set seed 10001 . * Generate x and z bivariate normal . scalar mu1=0.1 . scalar mu2=0.1 . scalar sig1=0.1 . scalar sig2=0.1 . scalar rho=0.5 . scalar sig12=rho*sig1*sig2 . gen x = mu1 + sig1*invnorm(uniform()) . gen muzgivx = mu2+(sig12/(sig2*sig2))*(x-mu1) . gen sigzgivx = sqrt(sig2*sig2*(1-rho*rho)) . gen z = muzgivx + sigzgivx*invnorm(uniform()) . * To generate y exponential with mean mu=Ey use . * Integral 0 to a of (1/mu)exp(-x/mu) dx by change of variables . * = Integral 0 to a/mu of exp(-t)dt . * = incomplete gamma function P(0,a/mu) in the terminology of Stata . gen Ey = exp(-2.0+2*x+2*z) . gen y = Ey*invgammap(1,uniform()) . gen logy = log(y) . . * Descriptive Statistics . summarize Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- x | 50 .0935209 .1031485 -.1173506 .2778609 muzgivx | 50 .0967604 .0515742 -.0086753 .1889304 sigzgivx | 50 .0866025 0 .0866025 .0866025 z | 50 .1033014 .0909297 -.0885447 .3137469 Ey | 50 .2114837 .071719 .0945722 .4314067 -------------+-------------------------------------------------------- y | 50 .2024206 .2237202 .0005293 .9601147 logy | 50 -2.282336 1.45494 -7.543878 -.0407026 . ereg y x z Iteration 0: log likelihood = -84.246434 Iteration 1: log likelihood = -80.068104 Iteration 2: log likelihood = -79.871694 Iteration 3: log likelihood = -79.871338 Iteration 4: log likelihood = -79.871338 Exponential regression -- entry time 0 log expected-time form Number of obs = 50 LR chi2(2) = 8.75 Log likelihood = -79.871338 Prob > chi2 = 0.0126 ------------------------------------------------------------------------------ y | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- x | .2670543 1.417339 0.19 0.851 -2.510879 3.044988 z | 4.663384 1.740712 2.68 0.007 1.251652 8.075117 _cons | -2.191619 .2328589 -9.41 0.000 -2.648014 -1.735224 ------------------------------------------------------------------------------ . . save mma11p1boot, replace file mma11p1boot.dta saved . . * Write data to a text (ascii) file so can use with programs other than Stata . outfile y x z using mma11p1boot.asc, replace . . ********** SIMPLE BOOTSTRAP ********** . . * Stata produces four bootstrap 100*(1-alpha) confidence intervals . * (N) and (P) have no asymptotic refinement . * (BC)-(BCA) have asymptotic refinement . * For details see program mma07p4boot.do . . * Change the following for different number of simulations S . * From page 399, for testing better to use 999 than 1000 . global breps = 999 /* The number of bootstrap reps used below */ . . set seed 20001 . . * A simple and adequate bootstrap command for the slope coefficients is . bs "ereg y x z" "_b[x] _b[z]", reps(\$breps) level(95) command: ereg y x z statistics: _bs_1 = _b[x] _bs_2 = _b[z] Bootstrap statistics Number of obs = 50 Replications = 999 ------------------------------------------------------------------------------ Variable | Reps Observed Bias Std. Err. [95% Conf. Interval] -------------+---------------------------------------------------------------- _bs_1 | 999 .2670543 -.1885509 1.420956 -2.52135 3.055458 (N) | -2.9054 2.696445 (P) | -2.590993 2.864327 (BC) _bs_2 | 999 4.663384 .0524786 1.939086 .8582302 8.468539 (N) | .5006047 8.483892 (P) | .231034 8.174835 (BC) ------------------------------------------------------------------------------ Note: N = normal P = percentile BC = bias-corrected . . ********** MORE DETAILED BOOTSTRAP ********** . . * The following bootstrap also gives standard error at each replication . * and saves data from replications for further analysis . . * In partiulcar, want to use the percentile-t method, . * which provides asymtptotic refinement . . * Stata does not give this. For methods see . * e.g. Efron and Tibsharani (1993, pp.160-162) . * e.g. Cameron and Trivedi (2005) Chapter 11.2.6-11.2.7 . * For sample s compute t-test(s) = (bhat(s)-bhat) / se(s) . * where bhat is initial estimate . * and bhat(s) and se(s) are for sth round. . * Order the t-test(s) statistics and choose the alpha/2 percentiles . * which give the critical values for the t-test . . * Implementation requires saving the results from each bootstrap replication . * in order to obtain ccritical values from percentiles of bootstrap distribution . . use mma11p1boot.dta, clear . . * Get and store coefficients (b) . * for regressors in the original model and data before bootstrap . quietly ereg y x z . global bx=_b[x] . global sex=_se[x] . global bz=_b[z] . global sez=_se[z] . di " Coefficients bx: " \$bx " and bz: " \$bz Coefficients bx: .26705432 and bz: 4.6633845 . di " Standard error sex: " \$sex " and sez: " \$sez Standard error sex: 1.4173391 and sez: 1.7407119 . . * Bootstrap and save coeff estimates and se's from each replication . set seed 20001 . bs "ereg y x z" "_b[x] _b[z] _se[x] _se[z]", reps(\$breps) level(95) saving(mma11p1bootreps) repl > ace command: ereg y x z statistics: _bs_1 = _b[x] _bs_2 = _b[z] _bs_3 = _se[x] _bs_4 = _se[z] Bootstrap statistics Number of obs = 50 Replications = 999 ------------------------------------------------------------------------------ Variable | Reps Observed Bias Std. Err. [95% Conf. Interval] -------------+---------------------------------------------------------------- _bs_1 | 999 .2670543 -.1885509 1.420956 -2.52135 3.055458 (N) | -2.9054 2.696445 (P) | -2.590993 2.864327 (BC) _bs_2 | 999 4.663384 .0524786 1.939086 .8582302 8.468539 (N) | .5006047 8.483892 (P) | .231034 8.174835 (BC) _bs_3 | 999 1.417339 .0644196 .1718393 1.080131 1.754547 (N) | 1.234399 1.902349 (P) | 1.196068 1.742845 (BC) _bs_4 | 999 1.740712 .0910103 .186631 1.374478 2.106946 (N) | 1.542322 2.257937 (P) | 1.453673 2.058318 (BC) ------------------------------------------------------------------------------ Note: N = normal P = percentile BC = bias-corrected . . * Now use the bootstrap estimates . use mma11p1bootreps, clear (bootstrap: ereg y x z) . sum Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- _bs_1 | 999 .0785034 1.420956 -9.431229 4.278278 _bs_2 | 999 4.715863 1.939086 -1.747643 12.09208 _bs_3 | 999 1.481759 .1718393 1.145421 2.761842 _bs_4 | 999 1.831722 .186631 1.387625 2.910449 . * Order comes from "_b[x] _b[z] _se[x] _se[z]" in earlier bs . gen bxs = _bs_1 . gen bzs = _bs_2 . gen sexs = _bs_3 . gen sezs = _bs_4 . gen ttestxs = (bxs - \$bx)/sexs . gen ttestzs = (bzs - \$bz)/sezs . . ********** (1) TABLE 11.1 (page 367) . . summarize bzs ttestzs, d bzs ------------------------------------------------------------- Percentiles Smallest 1% -.3361366 -1.747643 5% 1.544816 -1.716207 10% 2.270323 -1.366866 Obs 999 25% 3.570291 -1.205571 Sum of Wgt. 999 50% 4.77197 Mean 4.715863 Largest Std. Dev. 1.939086 75% 5.970802 10.10243 90% 7.100958 10.42623 Variance 3.760056 95% 7.810663 10.76733 Skewness -.1344324 99% 9.426978 12.09208 Kurtosis 3.545415 ttestzs ------------------------------------------------------------- Percentiles Smallest 1% -2.66391 -3.921595 5% -1.727528 -3.483456 10% -1.32364 -3.201425 Obs 999 25% -.6209012 -2.975815 Sum of Wgt. 999 50% .0618649 Mean .0261125 Largest Std. Dev. 1.046855 75% .7034938 2.693856 90% 1.323415 3.087892 Variance 1.095904 95% 1.70558 3.11692 Skewness -.1596043 99% 2.529097 3.738328 Kurtosis 3.337749 . . * Additionally need the 2.5 and 97.5 percentiles not given in summarize, d . . * Coefficient of z . _pctile bzs, p(2.5,97.5) . di " Lower 2.5 and upper 2.5 percentile of coeff b for z: " r(r1) " and " r(r2) Lower 2.5 and upper 2.5 percentile of coeff b for z: .50060469 and 8.4838924 . . * t-statistic for z . _pctile ttestzs, p(2.5,97.5) . di " Lower 2.5 and upper 2.5 percentile of ttest on z: " r(r1) " and " r(r2) Lower 2.5 and upper 2.5 percentile of ttest on z: -2.1827998 and 2.0659592 . . ********** (2) RESULTS IN TEXT PAGES 366-7 ********** . . * (2A) Bootstrap standard error estimate (no refinement) . * These are given earlier in bootstrap table output . * Equivalently get the standard deviation of bzs . . quietly sum bzs . scalar bzbootse = r(sd) . di "Bootstrap estimate of standard error: " bzbootse Bootstrap estimate of standard error: 1.9390864 . . * (2B) Test b3 = 0 using percentile-t method (asymptotic refinement) . * Use the 2.5% and 97.5% bootstrap critical values for t-statistic for z . . _pctile ttestzs, p(2.5,97.5) . di " Lower 2.5 and upper 2.5 percentile of ttest on z: " r(r1) " and " r(r2) Lower 2.5 and upper 2.5 percentile of ttest on z: -2.1827998 and 2.0659592 . . * (2D) 95% confidence interval with asymptotic refinement . * Use the preceding critical values . . scalar lbz = \$bz + r(r1)*\$sez /* Note the plus sign here */ . scalar ubz = \$bz + r(r2)*\$sez . di " Percentile-t interval lower and upper bounds: (" lbz "," ubz ")" Percentile-t interval lower and upper bounds: (.86375888,8.2596243) . . * (2B-Var) Variation for symmetric two-sided test on z . . gen absttestzs = abs(ttestzs) . _pctile absttestzs, p(95) . di " Upper 5 percentile of symmetric two-sided test on z: " r(r1) " Upper 5 percentile of symmetric two-sided test on z: 2.0775187 . . * (2C) Test b3 = 0 without asymptotic refinement . * Usual Wald test except use bootstrap estimate of standard error . . scalar Wald = (\$bz - 0) / bzbootse . di "Wald statistic using bootstrap standard error: " Wald Wald statistic using bootstrap standard error: 2.404939 . . * (2E) Bootstrap estimate of bias . * This is given in the earlier bootstrap results table . * and is explained in the text . . ********** (3) FIGURE 11.1 (p.368) PLOTS ESTIMATED DENSITY OF T-STATISTIC FOR Z . . set scheme s1mono . label var ttestzs "Bootstrap t-statistic" . kdensity ttestzs, normal /* > */ scale (1.2) plotregion(style(none)) /* > */ title("Bootstrap Density of 't-Statistic'") /* > */ xtitle("t-statistic from each bootstrap replication", 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 "Bootstrap Estimate") label(2 "Standard Normal")) . graph save ch11boot, replace (file ch11boot.gph saved) . . ********** CLOSE OUTPUT ********** . log close log: c:\Imbook\bwebpage\Section3\mma11p1boot.txt log type: text closed on: 18 May 2005, 15:53:47 ----------------------------------------------------------------------------------------------------