------------------------------------------------------------------------------------------------------ log: c:\Imbook\bwebpage\Section2\mma05p1mle.txt log type: text opened on: 17 May 2005, 13:48:11 . . ********** OVERVIEW OF MMA05P1MLE.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 5.9 pp.159-63 . * Maximum likelihood analysis. . . * Provides first two columns of Table 5.7 . * (1) OLS using Stata command regress . * (2) MLE using Stata command exp for exponential MLE . * (3) MLE using Stata command ml for user-provided log-likelihood . * using generated data (see below) . . * Related programs: . * mma05p2nls.do NLS, WNLS, FGNLS for same data using nl command . * mma05p3nlsbyml.do NLS, WNLS, FGNLS for same data using ml command . * mma05p4margeffects.do Calculates marginal effects . . ********** SETUP ********** . . set more off . version 8 . . ********** GENERATE DATA and SUMMARIZE ********** . . * Model is y ~ exponential(exp(a + bx)) . * x ~ N[mux, sigx^2] . * f(y) = exp(a + bx)*exp(-y*exp(a + bx)) . * lnf(y) = (a + bx) - y*exp(a + bx) . * E[y] = exp(-(a + bx)) note sign reversal for the mean . * V[y] = exp(-(a + bx)) = E[y]^2 . . * The dgp sets particular values of a, b, mux and sigx . * Here a = 2, b = -1 and x ~ N[1, 1] . scalar a = 2 . scalar b = -1 . scalar mux = 1 . scalar sigx = 1 . . * Set the sample size. Table 5.7 uses N=10,000 . set obs 10000 obs was 0, now 10000 . . * Generate x and y . set seed 2003 . gen x = mux + sigx*invnorm(uniform()) . gen lamda = exp(a + b*x) . gen Ey = 1/lamda . * To generate 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 y = Ey*invgammap(1,uniform()) . gen lny = ln(y) . gen lnfy = ln(lamda) - y*lamda . * twoway scatter Ey x . . * Descriptive Statisitcs . describe Contains data obs: 10,000 vars: 6 size: 280,000 (97.3% of memory free) ------------------------------------------------------------------------------- storage display value variable name type format label variable label ------------------------------------------------------------------------------- x float %9.0g lamda float %9.0g Ey float %9.0g y float %9.0g lny float %9.0g lnfy float %9.0g ------------------------------------------------------------------------------- Sorted by: Note: dataset has changed since last saved . summarize Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- x | 10000 1.014313 1.004905 -2.895741 4.994059 lamda | 10000 4.457478 5.939084 .0500838 133.7191 Ey | 10000 .6185677 .8294007 .0074784 19.96655 y | 10000 .6194352 1.291416 .0000445 30.60636 lny | 10000 -1.554348 1.62358 -10.02114 3.421208 -------------+-------------------------------------------------------- lnfy | 10000 -.0209485 1.419595 -7.52596 4.402257 . . ********** WRITE DATA TO A TEXT FILE ********** . . * Write data to a text (ascii) file . * used for programs mma05p2nlsbyml.do, mma05p3nlsbynl.do . * and mma05p4margeffects.do . * and can also use with programs other than Stata . outfile y x using mma05data.asc, replace . . ********** DO THE ANALYSIS: OLS and MLE ********** . . ** (1) OLS ESTIMATION . . * OLS is inconsistent in this example . regress y x Source | SS df MS Number of obs = 10000 -------------+------------------------------ F( 1, 9998) = 3030.74 Model | 3879.13606 1 3879.13606 Prob > F = 0.0000 Residual | 12796.7438 9998 1.27993037 R-squared = 0.2326 -------------+------------------------------ Adj R-squared = 0.2325 Total | 16675.8799 9999 1.66775476 Root MSE = 1.1313 ------------------------------------------------------------------------------ y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- x | .6198182 .0112587 55.05 0.000 .5977488 .6418876 _cons | -.0092545 .016075 -0.58 0.565 -.0407648 .0222558 ------------------------------------------------------------------------------ . estimates store rols . regress y x, robust Regression with robust standard errors Number of obs = 10000 F( 1, 9998) = 596.30 Prob > F = 0.0000 R-squared = 0.2326 Root MSE = 1.1313 ------------------------------------------------------------------------------ | Robust y | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+---------------------------------------------------------------- x | .6198182 .0253823 24.42 0.000 .5700638 .6695725 _cons | -.0092545 .0171978 -0.54 0.591 -.0429655 .0244566 ------------------------------------------------------------------------------ . estimates store rolsrobust . . ** (2) ML ESTIMATION USING STATA COMMAND FOR EXPONENTIAL MLE . . * The following uses Stata duration model commands. . * First need to define the duration variable (here y) . stset y failure event: (assumed to fail at time=y) obs. time interval: (0, y] exit on or before: failure ------------------------------------------------------------------------------ 10000 total obs. 0 exclusions ------------------------------------------------------------------------------ 10000 obs. remaining, representing 10000 failures in single record/single failure data 6194.352 total analysis time at risk, at risk from t = 0 earliest observed entry t = 0 last observed exit t = 30.60636 . streg x, dist(exp) nohr failure _d: 1 (meaning all fail) analysis time _t: y Iteration 0: log likelihood = -20754.005 Iteration 1: log likelihood = -17232.884 Iteration 2: log likelihood = -15760.556 Iteration 3: log likelihood = -15752.193 Iteration 4: log likelihood = -15752.19 Iteration 5: log likelihood = -15752.19 Exponential regression -- log relative-hazard form No. of subjects = 10000 Number of obs = 10000 No. of failures = 10000 Time at risk = 6194.352495 LR chi2(1) = 10003.63 Log likelihood = -15752.19 Prob > chi2 = 0.0000 ------------------------------------------------------------------------------ _t | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- x | -.9896276 .0098692 -100.27 0.000 -1.008971 -.9702842 _cons | 1.982921 .0141496 140.14 0.000 1.955188 2.010654 ------------------------------------------------------------------------------ . estimates store rexp . streg x, dist(exp) nohr robust failure _d: 1 (meaning all fail) analysis time _t: y Iteration 0: log pseudo-likelihood = -20754.005 Iteration 1: log pseudo-likelihood = -17232.884 Iteration 2: log pseudo-likelihood = -15760.556 Iteration 3: log pseudo-likelihood = -15752.193 Iteration 4: log pseudo-likelihood = -15752.19 Iteration 5: log pseudo-likelihood = -15752.19 Exponential regression -- log relative-hazard form No. of subjects = 10000 Number of obs = 10000 No. of failures = 10000 Time at risk = 6194.352495 Wald chi2(1) = 9914.62 Log pseudo-likelihood = -15752.19 Prob > chi2 = 0.0000 ------------------------------------------------------------------------------ | Robust _t | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- x | -.9896276 .0099388 -99.57 0.000 -1.009107 -.9701479 _cons | 1.982921 .0144307 137.41 0.000 1.954637 2.011205 ------------------------------------------------------------------------------ . estimates store rexprobust . . ** (3) ML ESTIMATION USING STATA ML COMMAND . . * For MLE computation can use the following Stata commands . * ml model lf provide the log-density . * ml model D0 provide the log-likelihood . * ml model D1 provide the log-likelihood and gradient . * ml model D2 provide the log-likelihood, gradient and hessian . . * At a minimum need to provide . * (A) program define fcn where fcn is the function name . * defines the log-density (independent observations assumed) . * (B) ml model lf fcn + some extras . * the extras give the dependent variable and regressors . * (C) ml maximize . * obtains the mle . * (D) ml model lf fcn + some extras, robust . * provides robust sandwich standard errors . . * Here we provide the log-density (ml model lf) as this is simplest, . * and the Stata manual says that numerically only D2 is better. . . * (A) Define the log-density . * lnf(y) = (a+bx) - y*exp(a+bx) = theta - y*exp(theta) where theta = x'b . program define mleexp0 1. version 8.0 2. args lnf theta /* Must use lnf while could use name other than theta */ 3. quietly replace `lnf' = `theta' - \$ML_y1*exp(`theta') 4. end . . * (B) Say that dependent variable is y and regressors are x plus a constant . ml model lf mleexp0 (y = x) . . * (C) Obtain the MLE . ml search /* Optional - can provide better starting values */ initial: log likelihood = -6194.3525 improve: log likelihood = -6194.3525 alternative: log likelihood = -5212.7607 rescale: log likelihood = -5212.7607 . ml maximize initial: log likelihood = -5212.7607 rescale: log likelihood = -5212.7607 Iteration 0: log likelihood = -5212.7607 Iteration 1: log likelihood = -1563.9176 Iteration 2: log likelihood = -217.6055 Iteration 3: log likelihood = -208.73633 Iteration 4: log likelihood = -208.71383 Iteration 5: log likelihood = -208.71383 Number of obs = 10000 Wald chi2(1) = 10054.85 Log likelihood = -208.71383 Prob > chi2 = 0.0000 ------------------------------------------------------------------------------ y | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- x | -.9896276 .0098692 -100.27 0.000 -1.008971 -.9702842 _cons | 1.982921 .0141496 140.14 0.000 1.955188 2.010654 ------------------------------------------------------------------------------ . estimates store rmle . . * (D) Obtain robust standard errors . ml model lf mleexp0 (y = x), robust . ml search initial: log pseudo-likelihood = -6194.3525 improve: log pseudo-likelihood = -6194.3525 alternative: log pseudo-likelihood = -5212.7607 rescale: log pseudo-likelihood = -5212.7607 . ml maximize initial: log pseudo-likelihood = -5212.7607 rescale: log pseudo-likelihood = -5212.7607 Iteration 0: log pseudo-likelihood = -5212.7607 Iteration 1: log pseudo-likelihood = -1563.9176 Iteration 2: log pseudo-likelihood = -217.6055 Iteration 3: log pseudo-likelihood = -208.73633 Iteration 4: log pseudo-likelihood = -208.71383 Iteration 5: log pseudo-likelihood = -208.71383 Number of obs = 10000 Wald chi2(1) = 9914.62 Log pseudo-likelihood = -208.71383 Prob > chi2 = 0.0000 ------------------------------------------------------------------------------ | Robust y | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- x | -.9896276 .0099388 -99.57 0.000 -1.009107 -.9701479 _cons | 1.982921 .0144307 137.41 0.000 1.954637 2.011205 ------------------------------------------------------------------------------ . estimates store rmlerobust . . * (E) Calculate R-squared and log-likelihood at the ML estimates . * lnL sums lnf(y) = ln(lamda) - y*lamda . gen lamdaml = exp(_b[_cons] + _b[x]*x) . gen lnfml = ln(lamdaml) - y*lamdaml . quietly means lnfml . scalar LLml = r(mean)*r(N) . * R-squared = 1 - Sum_i(y_i - yhat_i)^2 / Sum_i(y_i - ybar)^2 . gen yhatml = 1/lamdaml . egen ybar = mean(y) . * quietly means y . * scalar ybar = r(mean) . gen y_yhatsqml = (y - yhatml)^2 . gen y_ybarsq = (y - ybar)^2 . quietly means y_yhatsqml . scalar SSresidml = r(mean) . quietly means y_ybarsq . scalar SStotal = r(mean) . scalar Rsqml = 1 - SSresidml/SStotal . di LLml " " Rsqml -208.71383 .39062307 . . ********** DISPLAY RESULTS: First two columns of Table 5.7 p.161 . . * (1) OLS - nonrobust and robust standard errors . * Here OLS is inconsistent. . * And expect sign reversal for slope as in true model mean E[y] = exp(-x'b) . estimates table rols rolsrobust, b(%10.4f) se(%10.4f) t stats(N ll r2) keep(_cons x) ---------------------------------------- Variable | rols rolsrobust -------------+-------------------------- _cons | -0.0093 -0.0093 | 0.0161 0.0172 | -0.58 -0.54 x | 0.6198 0.6198 | 0.0113 0.0254 | 55.05 24.42 -------------+-------------------------- N | 10000.0000 10000.0000 ll | -1.542e+04 -1.542e+04 r2 | 0.2326 0.2326 ---------------------------------------- legend: b/se/t . . * (2) MLE by command ereg - nonrobust and robust standard errors . estimates table rexp rexprobust, b(%10.4f) se(%10.4f) t stats(N ll) keep(_cons x) ---------------------------------------- Variable | rexp rexprobust -------------+-------------------------- _cons | 1.9829 1.9829 | 0.0141 0.0144 | 140.14 137.41 x | -0.9896 -0.9896 | 0.0099 0.0099 | -100.27 -99.57 -------------+-------------------------- N | 10000.0000 10000.0000 ll | -1.575e+04 -1.575e+04 ---------------------------------------- legend: b/se/t . . * (3) MLE by command ml - nonrobust and robust standard errors . estimates table rmle rmlerobust, b(%10.4f) se(%10.4f) t stats(N ll) keep(_cons x) ---------------------------------------- Variable | rmle rmlerobust -------------+-------------------------- _cons | 1.9829 1.9829 | 0.0141 0.0144 | 140.14 137.41 x | -0.9896 -0.9896 | 0.0099 0.0099 | -100.27 -99.57 -------------+-------------------------- N | 10000.0000 10000.0000 ll | -208.7138 -208.7138 ---------------------------------------- legend: b/se/t . * And ML log-likelihood (check) and R-squared (needed to be computed) . di "Log likeihood for ML: " LLml Log likeihood for ML: -208.71383 . di "R-squared for MLE: " Rsqml R-squared for MLE: .39062307 . . ********** CLOSE OUTPUT ********** . log close log: c:\Imbook\bwebpage\Section2\mma05p1mle.txt log type: text closed on: 17 May 2005, 13:48:18 ----------------------------------------------------------------------------------------------------