LIMDEP

Version 10

Econometric Modeling Guide

by

William H. Greene Econometric Software, Inc.

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LIMDEP 10 Econometric Modeling Guide Table of Contents v

Table of Contents Table of Contents.................................................................................................................... v

E1: Econometric Model Estimation .................................................................................... E-1 E1.1 Introduction ........................................................................................................................ E-1 E1.2 Econometric Models .......................................................................................................... E-1 E1.3 Model Commands .............................................................................................................. E-2 E1.4 The Command Builders ..................................................................................................... E-4 E1.5 Model Groups .................................................................................................................... E-7 E1.6 General Model Specifications .......................................................................................... E-10

E1.6.1 Variable Specifications in Model Commands .................................................. E-10 E1.6.2 Controlling Output from Model Commands .................................................... E-10 E1.6.3 Robust Asymptotic Covariance Matrices ......................................................... E-11 E1.6.4 Optimization Controls for Nonlinear Optimization ......................................... E-11 E1.6.5 Predictions and Residuals ................................................................................ E-12 E1.6.6 Hypothesis Tests and Restrictions ................................................................... E-12 E1.6.7 Setup for Panel Data Models ............................................................................ E-12

E2: Descriptive Statistics for Cross Section and Panel Data.......................................... E-14 E2.1 Introduction ...................................................................................................................... E-14 E2.2 Univariate Summary Statistics ......................................................................................... E-14

E2.2.1 Weights ............................................................................................................ E-14 E2.2.2 Missing Observations in Descriptive Statistics ................................................ E-15 E2.2.3 Display of Descriptive Statistics ...................................................................... E-15 E2.2.4 Command Builder Dialog Box ......................................................................... E-16

E2.3 Standard Error of the Mean .............................................................................................. E-17 E2.4 Clustered Data .................................................................................................................. E-18 E2.5 Skewness and Kurtosis ..................................................................................................... E-20 E2.6 Display Format ................................................................................................................. E-21

E2.6.1 Fixed Width Format ......................................................................................... E-21 E2.6.2 Matrix Output ................................................................................................... E-22

E2.7 Stratified Data .................................................................................................................. E-23 E2.8 Tables for Stratified Samples ........................................................................................... E-24

E2.8.1 Groups in the Sample ....................................................................................... E-25 E2.8.2 Weights ............................................................................................................ E-26

E2.9 Sample Quantiles ............................................................................................................. E-27 E2.9.1 Box and Whisker Plots ..................................................................................... E-28 E2.9.2 Related Procedures for Quantiles ..................................................................... E-30

E2.10 Analysis of Variance and Panel Data ............................................................................. E-30 E2.10.1 Analysis of Variance ...................................................................................... E-30 E2.10.2 Matrix Functions for Describing Panel Data.................................................. E-31

E2.11 Discriminant Analysis .................................................................................................... E-33 E2.12 Accuracy and the NIST Benchmarks ............................................................................. E-36

E2.12.1 NIST Benchmarks for Univariate Statistics ................................................... E-37 E2.12.2 Accuracy in ANOVA Computations The NIST Benchmarks ...................... E-38

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E3: Histograms and Kernel Density Estimators .............................................................. E-40 E3.1 Introduction ...................................................................................................................... E-40 E3.2 Normal-Quantile Plots ..................................................................................................... E-40 E3.3 HISTOGRAM Command ................................................................................................ E-42 E3.4 Histograms for Continuous Data ...................................................................................... E-44

E3.4.1 Fixed Number of Bins ...................................................................................... E-45 E3.4.2 Trimming Data for Histograms ........................................................................ E-47 E3.4.3 Fixed Bin Limits .............................................................................................. E-47 E3.4.4 Fixed Number of Bins in a Range .................................................................... E-48 E3.4.5 Fixed Width Bins in a Range ........................................................................... E-49 E3.4.6 Fixed Interval Widths ....................................................................................... E-49 E3.4.7 Fixed Proportion of Observations in Each Bin ................................................ E-50 E3.4.8 Comparison to a Normal Distribution .............................................................. E-50

E3.5 Histograms for Discrete Data ........................................................................................... E-51 E3.5.1 Bin Labels Scaled to Sample Proportions ........................................................ E-52 E3.5.2 Multiple Histograms ........................................................................................ E-53 E3.5.3 Stratification ..................................................................................................... E-54 E3.5.4 Labels for Bins ................................................................................................. E-55

E3.6 Kernel Density Estimation ............................................................................................... E-56 E3.6.1 Options for Kernel Density Estimation ............................................................ E-60 E3.6.2 Multiple Kernel Estimators .............................................................................. E-62 E3.6.3 Sample Strata ................................................................................................... E-63 E3.6.4 Comparison to Normal ..................................................................................... E-64

E3.7 Testing for Normality ....................................................................................................... E-65 E3.7.1 Normality Test Based on Skewness and Kurtosis ............................................ E-65 E3.7.2 Kolmogorov Smirnov Test of Normality ........................................................ E-66

E4: Covariance and Correlation ........................................................................................ E-67 E4.1 Introduction ...................................................................................................................... E-67 E4.2 Covariance and Correlation for Two Variables ............................................................... E-67

E4.2.1 Kendalls Tau ................................................................................................... E-67 E4.2.2 Rank Correlation .............................................................................................. E-67

E4.3 Covariance and Correlation Matrices ............................................................................... E-68 E4.3.1 Matrix Output from DSTAT ............................................................................ E-68 E4.3.2 Correlation and Covariance Matrices with MATRIX ...................................... E-69

E4.4 Correlations for Discrete Variables .................................................................................. E-69 E4.4.1 Tetrachoric Correlation for Binary Variables .................................................. E-69 E4.4.2 Polychoric Correlation for Two Ordered Qualitative Variables ........................ E-71 E4.4.3 Biserial Correlation .......................................................................................... E-73

E4.5 Cross Tabulations ............................................................................................................. E-73 E4.5.1 Output............................................................................................................... E-75 E4.5.2 Testing the Independence Assumption ............................................................ E-76 E4.5.3 Analyzing Frequency Data ............................................................................... E-76 E4.5.4 An Application ................................................................................................. E-77

E5: Descriptive Statistics for Time Series Data ............................................................... E-79 E5.1 Introduction ...................................................................................................................... E-79

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E5.2 Box-Jenkins Time Series Identification ........................................................................... E-79 E5.2.1 Command Builder ............................................................................................ E-79 E5.2.2 Computations ................................................................................................... E-79 E5.2.3 The Burg Variant of the PACF ........................................................................ E-81 E5.2.4 Application ....................................................................................................... E-82

E5.3 Spectral Density Estimation ............................................................................................. E-85 E5.4 Phillips-Perron Test for a Unit Root ................................................................................ E-88 E5.5 Augmented Dickey-Fuller Tests ...................................................................................... E-91

E6: Scatter Diagrams and Plotting ................................................................................... E-92 E6.1 Introduction ...................................................................................................................... E-92 E6.2 Printing and Exporting Figures ........................................................................................ E-92

E6.2.1 Printing ............................................................................................................. E-93 E6.2.2 Saving a Graph as a Graphics File ................................................................... E-94 E6.2.3 Pasting a Graph into a Document or Spreadsheet ............................................ E-94

E6.3 The PLOT Command ....................................................................................................... E-95 E6.3.1 Scatter Plot of One Variable Against Another ................................................. E-96 E6.3.2 Plotting a Simple Linear Regression ................................................................ E-97 E6.3.3 Time Series Plots ............................................................................................. E-98 E6.3.4 Plotting Several Variables Against One Variable ............................................ E-99 E6.3.5 Combining Plots ............................................................................................. E-101 E6.3.6 Options for Scaling and Labeling the Figure ................................................. E-101 E6.3.7 Fenceposts Plot .............................................................................................. E-106 E6.3.8 Centipede Plot ................................................................................................ E-107 E6.3.9 A Program for Plotting Confidence Regions ................................................. E-108 E6.3.10 Sorting the Data Before Plotting .................................................................. E-111 E6.3.11 Plotting a Function ....................................................................................... E-113 E6.3.12 Stratified Scatter Plots .................................................................................. E-116

E6.4 Multiple Scatter Plots The SPLOT Command ............................................................ E-117 E6.5 Plotting Matrices The MPLOT Command .................................................................. E-117

E6.5.1 Plotting Autocorrelation and Partial Autocorrelation Functions ...................... E-118 E6.5.2 Examining an Estimation Criterion (Log Likelihood) Function ...................... E-119

E6.6 Plotting Functions The FPLOT Command ................................................................. E-121 E6.7 Contour Plots The CPLOT Command ........................................................................ E-123

E7: Linear Regression Estimation ............................................................................... E-126 E7.1 Introduction .................................................................................................................... E-126 E7.2 Least Squares Regression Command ............................................................................. E-126 E7.3 Computing the Least Squares Coefficients .................................................................... E-127

E7.3.1 Results Produced by REGRESS .................................................................... E-128 E7.3.2 Retrievable Results ........................................................................................ E-131 E7.3.3 Results that Can Be Computed with MATRIX and CALC ........................... E-132 E7.3.4 Beta Coefficients ............................................................................................ E-133

E7.4 Stepwise Regression....................................................................................................... E-134 E7.5 Interactions and Partial Effects ...................................................................................... E-137 E7.6 Predictions and Residuals .............................................................................................. E-139

E7.6.1 Plotting Residuals .......................................................................................... E-142

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E7.6.2 Standardized Residuals and Regression Diagnostics ..................................... E-144 E7.7 Multicollinearity ............................................................................................................. E-146 E7.8 Variance Inflation Factors .............................................................................................. E-147 E7.9 Specification Analysis .................................................................................................... E-148

E7.9.1 Breusch and Pagan Test for Heteroscedasticity ............................................. E-148 E7.9.2 RESET Specification Test.............................................................................. E-149 E7.9.3 Omitted Variables .......................................................................................... E-150 E7.9.4 The CUSUM Test of Model Stability ............................................................ E-151

E7.10 Robust Covariance Matrix Estimation ......................................................................... E-154 E7.10.1 Heteroscedasticity The White Estimator ................................................... E-154 E7.10.2 Autocorrelation The Newey-West Estimator ............................................ E-155 E7.10.3 Clustering ..................................................................................................... E-157

E7.11 Accuracy in Linear Regression NIST Benchmarks .................................................... E-158

E8: Linear Regression Hypothesis Tests and Restrictions ....................................... E-163 E8.1 Introduction .................................................................................................................... E-163 E8.2 Hypothesis Tests in the Linear Regression Model ......................................................... E-163

E8.2.1 Testing Significance of Individual Coefficients ............................................. E-164 E8.2.2 Linear Function of Coefficients ..................................................................... E-165 E8.2.3 Linear Function with Interaction Terms and Nonlinearities .......................... E-166 E8.2.4 More Than One Linear Restriction ................................................................ E-167 E8.2.5 Testing Nonlinear Restrictions ....................................................................... E-168 E8.2.6 Tests of Structural Change ............................................................................. E-170 E8.2.7 Homogeneity Test .......................................................................................... E-173 E8.2.8 J Tests for Nonnested Hypotheses ................................................................. E-173

E8.3 Restricted Least Squares ................................................................................................ E-174 E8.3.1 Equality Restrictions ...................................................................................... E-174 E8.3.2 Equality Restrictions and Singularity ............................................................. E-176 E8.3.3 Inequality Restricted Least Squares ............................................................... E-179

E9: Non- and Semiparametric Regression Models ........................................................ E-181 E9.1 Introduction .................................................................................................................... E-181 E9.2 Nonparametric (Kernel Density) Regression Estimation ................................................. E-182

E9.2.1 Nonparametric Regression on a Single Variable ........................................... E-182 E9.2.2 Estimating a Nonparametric Single Index Regression Function ...................... E-183 E9.2.3 Options for NPREG ....................................................................................... E-185 E9.2.4 Output from NPREG ...................................................................................... E-188

E9.3 The Least Absolute Deviations Estimator ...................................................................... E-190 E9.4 Quantile Regressions ...................................................................................................... E-194 E9.5 LOWESS ........................................................................................................................ E-196

E9.5.1 Graphical Smoothing with LOWESS ............................................................ E-198 E9.5.2 Local Multiple Regression ............................................................................. E-200 E9.5.3 Technical Details for LOWESS Computations .............................................. E-201

E10: Heteroscedasticity and GARCH Models ................................................................ E-202 E10.1 Introduction .................................................................................................................. E-202 E10.2 Correcting the OLS Covariance Matrix ....................................................................... E-202

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E10.3 Estimating Models with Heteroscedasticity ................................................................. E-206 E10.3.1 Weighted Least Squares ............................................................................... E-206 E10.3.2 Variance Proportional to the Square of the Mean ........................................ E-207 E10.3.3 Testing for Heteroscedasticity ...................................................................... E-208

E10.4 Multiplicative Heteroscedasticity ................................................................................. E-211 E10.4.1 Results .......................................................................................................... E-213 E10.4.2 Application 1 Heteroscedastic Regression ................................................ E-213 E10.4.3 Application 2 Groupwise Heteroscedasticity ............................................ E-214 E10.4.4 Restrictions ................................................................................................... E-216 E10.4.5 Technical Details on Computation of the HREG Model ............................. E-217

E10.5 ARCH(m) and GARCH(m) Models ............................................................................ E-219 E10.5.1 Example: ARCH(0,1) Model for Expected Inflation ................................... E-222 E10.5.2 A Benchmark GARCH(1,1) Model for Exchange Rates ............................. E-223 E10.5.3 The GARCH in Mean Model ....................................................................... E-225 E10.5.4 Technical Details on Estimation of the GARCH(m) Model ........................ E-227

E11: Autocorrelation in the Linear Model ...................................................................... E-229 E11.1 Introduction .................................................................................................................. E-229 E11.2 Correcting the OLS Covariance Matrix ....................................................................... E-229 E11.3 Correcting for First Order Autocorrelation .................................................................. E-232 E11.4 Autocorrelation with a Lagged Dependent Variable .................................................... E-235 E11.5 Differencing and Higher Order Autocorrelation .......................................................... E-237 E11.6 Testing for Autocorrelation .......................................................................................... E-239

E12: ARIMA, ARMAX and Distributed Lag Models ........................................................ E-240 E12.1 Introduction .................................................................................................................. E-240 E12.2 Box-Jenkins ARIMA and ARMAX Models ................................................................ E-240

E12.2.1 Model Command .......................................................................................... E-240 E12.2.2 Model Output ............................................................................................... E-242 E12.2.3 Examples ...................................................................................................... E-244 E12.2.4 Technical Details .......................................................................................... E-247

E12.3 The Geometric Lag Model ........................................................................................... E-249 E12.4 Roots of Dynamic Equations ....................................................................................... E-254

E13: The Box-Cox Regression Model ............................................................................. E-257 E13.1 Introduction .................................................................................................................. E-257 E13.2 Model Commands ........................................................................................................ E-257

E13.2.1 Specification of the Model ........................................................................... E-258 E13.2.2 Specification of the Estimation Method ....................................................... E-258 E13.2.3 Starting Values ............................................................................................. E-259 E13.2.4 The Asymptotic Covariance Matrix ............................................................. E-259 E13.2.5 Model Specifications .................................................................................... E-260

E13.3 Model Components ...................................................................................................... E-261 E13.3.1 Heteroscedasticity ........................................................................................ E-261 E13.3.2 Restrictions on Parameters ........................................................................... E-261

E13.4 Output and Saved Results ............................................................................................ E-261 E13.5 Application ................................................................................................................... E-263

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E13.6 Technical Details .......................................................................................................... E-267

E14: Nonlinear Least Squares......................................................................................... E-271 E14.1 The Nonlinear Regression Model ................................................................................ E-271 E14.2 Command for Nonlinear Regression ............................................................................ E-272 E14.3 Specification of the Regression Function ..................................................................... E-275

E14.3.1 Parameterization and Reparameterization .................................................... E-276 E14.3.2 Functions that May Appear in NLSQ Commands ....................................... E-277 E14.3.3 Linear Functions and Dot Products .............................................................. E-278 E14.3.4 Bilinear and Quadratic Forms ...................................................................... E-280 E14.3.5 Automatically Generating a List of Labels .................................................. E-280 E14.3.6 Lists of Labels .............................................................................................. E-280

E14.4 Quadrature and Simulation .......................................................................................... E-281 E14.5 Recursive Functions ..................................................................................................... E-284 E14.6 Providing Analytic Derivatives .................................................................................... E-285 E14.7 Options for Nonlinear Least Squares ........................................................................... E-290

E14.7.1 Fixing Some of the Parameters .................................................................... E-290 E14.7.2 Setting the Algorithm ................................................................................... E-291 E14.7.3 Heteroscedasticity Robust Covariance Matrix ............................................. E-292 E14.7.4 Degrees of Freedom Correction ................................................................... E-292

E14.8 Model Output and Retrievable Results ........................................................................ E-293 E14.9 Partial Effects for Nonlinear Regressions .................................................................... E-295 E14.10 Imposing Restrictions and Testing Hypotheses ......................................................... E-298 E14.11 An Application ........................................................................................................... E-299 E14.12 Technical Details ........................................................................................................ E-301 E14.13 The NIST Accuracy Benchmarks .............................................................................. E-303

E14.13.1 Setting up the NIST Benchmarks ............................................................... E-304 E14.13.2 Application Dan Wood ........................................................................... E-309

E15: Linear Models for Time Series/Cross Section Data .............................................. E-312 E15.1 Introduction .................................................................................................................. E-312 E15.2 Panel Data Arrangement and Setup ............................................................................. E-313 E15.3 Groupwise Heteroscedasticity, Correlation and Autocorrelation ................................. E-314

E15.3.1 Command and Options ................................................................................. E-315 E15.3.2 Results .......................................................................................................... E-316 E15.3.3 Application ................................................................................................... E-319 E15.3.4 Technical Details .......................................................................................... E-320

E15.4 Hildreth, Houck, and Swamys Random Coefficients Model ...................................... E-323 E15.4.1 Command ..................................................................................................... E-323 E15.4.2 Application ................................................................................................... E-325 E15.4.3 Technical Details for the Random Coefficients Estimator ........................... E-329

E16: Linear Regression Models for Panel Data ............................................................. E-331 E16.1 Introduction .................................................................................................................. E-331 E16.2 Commands for Panel Data Regressions ....................................................................... E-331 E16.3 One Way Analysis of Variance .................................................................................... E-333

E16.3.1 Computations and Saved Results ................................................................. E-333

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E16.3.2 Applications ................................................................................................. E-334 E16.4 The Group Means Estimator ........................................................................................ E-336 E16.5 The Pooled Regression ................................................................................................. E-337 E16.6 Specification Test for the One Factor Panel Models .................................................... E-338 E16.7 One Way Fixed and Random Effects Models .............................................................. E-339

E17: Fixed Effects Linear Regression ............................................................................ E-340 E17.1 Introduction .................................................................................................................. E-340 E17.2 One Way Fixed Effects Model ..................................................................................... E-340

E17.2.1 Command for One Factor Models ................................................................ E-340 E17.2.2 Program Output for One Way Fixed Effects Models ................................... E-342 E17.2.3 Saved Results ............................................................................................... E-343 E17.2.4 Application ................................................................................................... E-344 E17.2.5 Robust Estimation of the Fixed Effects Covariance Matrix......................... E-348 E17.2.6 Fixed Effects Models with Time Invariant Variables .................................. E-349 E17.2.7 Restricted Least Squares .............................................................................. E-351 E17.2.8 Technical Details on Estimation of One Way Fixed Effects Models ........... E-352

E17.3 Two Way Fixed and Random Effects Models ............................................................. E-354 E17.3.1 Program Output for Two Factor Models ...................................................... E-355 E17.3.2 Application ................................................................................................... E-356

E17.4 Autocorrelation ............................................................................................................ E-357 E17.5 Heteroscedasticity and Autocorrelation Robust Covariance Matrix ............................ E-360

E17.5.1 Heteroscedasticity ........................................................................................ E-360 E17.5.2 Autocorrelation ............................................................................................ E-362

E18: Random Effects Linear Models for Panel Data ...................................................... E-363 E18.1 Introduction .................................................................................................................. E-363 E18.2 One Way Random Effects Model ................................................................................ E-363

E18.2.1 Command ..................................................................................................... E-364 E18.2.2 Output........................................................................................................... E-364 E18.2.3 Specification Tests for Random vs. Fixed Effects ....................................... E-366 E18.2.4 Saved Results ............................................................................................... E-369 E18.2.5 Technical Details .......................................................................................... E-369 E18.2.6 Robust Covariance Matrix ........................................................................... E-374

E18.3 ML Estimation of One Way Random Effects Models ................................................. E-375 E18.3.1 Application ................................................................................................... E-375 E18.3.2 Technical Notes on ML Estimation of the Random Effects Model ............. E-376

E18.4 Groupwise Heteroscedasticity in Random Effects ....................................................... E-378 E18.4.1 A Model with Stratification and Grouping .................................................. E-380 E18.4.2 Exponential Heteroscedasticity in Random Effects ..................................... E-381

E18.5 Autocorrelation ............................................................................................................ E-383 E18.6 Two Way Random Effects Model ................................................................................ E-384

E18.6.1 Program Output for Two Factor RE Models ................................................ E-384 E18.6.2 Application ................................................................................................... E-385 E18.6.3 Technical Details .......................................................................................... E-386

E18.7 Two and Three Way Nested Random Effects .............................................................. E-388 E18.7.1 Command ..................................................................................................... E-389

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E18.7.2 Results .......................................................................................................... E-389 E18.7.3 Application ................................................................................................... E-390 E18.7.4 Technical Details .......................................................................................... E-391

E18.8 Multilevel and Multiple Effects in the RP Model ........................................................ E-393 E18.8.1 Command ..................................................................................................... E-394 E18.8.2 Application ................................................................................................... E-394 E18.8.3 Technical Details .......................................................................................... E-397

E19: Random Parameters Linear Models ....................................................................... E-398 E19.1 Introduction .................................................................................................................. E-398 E19.2 Random Parameters Linear Models ............................................................................. E-399 E19.3 Command for the Random Parameters Models ........................................................... E-400

E19.3.1 Specifying Random Parameters ................................................................... E-400 E19.3.2 Constraining the Sign of a Parameter Lognormal and Triangular ............ E-401 E19.3.3 Correlated Random Parameters .................................................................... E-402 E19.3.4 Autocorrelation ............................................................................................ E-402

E19.4 Hierarchical Model Heterogeneity in the Means ...................................................... E-403 E19.5 Saved Results ............................................................................................................... E-404 E19.6 Controlling the Simulation ........................................................................................... E-404 E19.7 Other Options ............................................................................................................... E-405 E19.8 Individual Specific Estimates ....................................................................................... E-406 E19.9 Applications ................................................................................................................. E-406

E19.9.1 Random Parameters Linear Regression Model ............................................ E-406 E19.9.2 Conditional Estimates of Means of Random Parameters ............................. E-408

E19.10 The Parameter Vector and Starting Values ................................................................ E-413 E19.11 Technical Details on the RP Model............................................................................ E-414

E20: Latent Class Linear Models .................................................................................... E-418 E20.1 Introduction .................................................................................................................. E-418 E20.2 Latent Class Linear Regression Model ........................................................................ E-418 E20.3 Command for Latent Class Regression ........................................................................ E-419 E20.4 Restricted Models......................................................................................................... E-420 E20.5 Modeling Class Probabilities ....................................................................................... E-422 E20.6 Posterior Class Probabilities and Predicting Class Membership .................................. E-422 E20.7 Applications ................................................................................................................. E-425

E20.7.1 Finite Mixture of Normals ........................................................................... E-425 E20.7.2 Latent Class Linear Model ........................................................................... E-426

E20.8 Technical Details and the EM Algorithm .................................................................... E-429

E21: Single Equation Instrumental Variables Estimation ............................................. E-430 E21.1 Introduction .................................................................................................................. E-430 E21.2 Two Stage Least Squares ............................................................................................. E-430

E21.2.1 Command ..................................................................................................... E-431 E21.2.2 Model Output for the 2SLS Command ........................................................ E-432 E21.2.3 Robust Estimation of the 2SLS Covariance Matrix ..................................... E-432 E21.2.4 Application ................................................................................................... E-432 E21.2.5 Specification Tests: Hausman and Wu........................................................ E-435

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E21.3 Autocorrelation with a Lagged Dependent Variable .................................................... E-437 E21.4 Alternatives to 2SLS .................................................................................................... E-439

E21.4.1 LIML ............................................................................................................ E-439 E21.4.2 JIVE Estimator ............................................................................................. E-441

E21.5 Nonlinear IV Estimation .............................................................................................. E-443 E21.6 NLSQ/GMM Estimation .............................................................................................. E-446

E21.6.1 GMM Estimation of Single Equation Nonlinear Models ............................. E-447 E21.6.2 Technical Note on Optimization .................................................................. E-449

E21.7 General Specifications of the GMM Estimator ............................................................ E-451 E21.7.1 GMM Estimation ......................................................................................... E-451 E21.7.2 The Weighting Matrix .................................................................................. E-453 E21.7.3 The Optimal Weighting Matrix .................................................................... E-454 E21.7.4 Other Options ............................................................................................... E-455 E21.7.5 Application ................................................................................................... E-455 E21.7.6 Technical Details for the GMM Estimator ................................................... E-458

E22: 2SLS for Panel Data ................................................................................................ E-460 E22.1 Introduction .................................................................................................................. E-460 E22.2 Application ................................................................................................................... E-460 E22.3 2SLS Estimation with Fixed Effects ............................................................................ E-461 E22.4 IV Estimators for Panel Data ....................................................................................... E-463

E23: Hausman-Taylor and Arellano-Bond Estimators .................................................. E-468 E23.1 Introduction .................................................................................................................. E-468 E23.2 The Hausman and Taylor Estimator for Random Effects ............................................ E-468 E23.3 Arellano, Bond, and Bovers Estimator for Dynamic Panel Data Models ................... E-475

E23.3.1 Technical Background ................................................................................. E-476 E23.3.2 Command ..................................................................................................... E-478 E23.3.3 A Test Statistic for the Specification............................................................ E-481 E23.3.4 Technical Notes ............................................................................................ E-481 E23.3.5 An Application ............................................................................................. E-483

E24: Linear Systems of Regression Equations SURE and 3SLS............................... E-486 E24.1 Introduction .................................................................................................................. E-486 E24.2 Linear SURE Models Estimated by GLS ..................................................................... E-486

E24.2.1 Command for SURE Estimation .................................................................. E-487 E24.2.2 Options for the Generalized Least Squares Procedure ................................. E-487 E24.2.3 Model Output for the GLS Estimator ........................................................... E-489 E24.2.4 The Translog System.................................................................................... E-491 E24.2.5 Generalized Least Squares ........................................................................... E-492 E24.2.6 Technical Details for Generalized Least Squares ........................................ E-498

E24.3 Maximum Likelihood Estimation of Constrained Linear Systems .............................. E-499 E24.3.1 Command for ML Estimation of Constrained SURE Systems ...................... E-500 E24.3.2 Model Output for the Maximum Likelihood Estimator ............................... E-501 E24.3.3 Application ................................................................................................... E-502 E24.3.4 Technical Details .......................................................................................... E-506

E24.4 Instrumental Variables (3SLS) Estimation of a Set of Linear Equations ..................... E-508

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E25: Nonlinear Systems of Regression Equations ........................................................ E-511 E25.1 Introduction .................................................................................................................. E-511 E25.2 Nonlinear Systems The NLSUR Command ............................................................. E-512

E25.2.1 OLS Estimation, Equation by Equation (NLOLS) ....................................... E-513 E25.2.2 Weighted Least Squares, Equation by Equation (NLWLS) ......................... E-513 E25.2.3 IV Estimation, Equation by Equation (NL2SLS) ......................................... E-514 E25.2.4 Weighted IV Estimation, Equation by Equation (WNL2SLS) ...................... E-514 E25.2.5 Nonlinear GLS Estimation (NLSURE) ........................................................ E-515 E25.2.6 Nonlinear IV Systems Estimation (NL3SLS) .............................................. E-515 E25.2.7 GMM Estimation (GMM) ............................................................................ E-516 E25.2.8 Weighting Observations in Equation Systems ............................................. E-516 E25.2.9 Model Specifications for the NLSUR Procedure ......................................... E-517

E25.3 Output and Saved Results from NLSUR...................................................................... E-518 E25.4 Application ................................................................................................................... E-519 E25.5 Technical Details .......................................................................................................... E-522

E26: Models for Binary Choice ....................................................................................... E-523 E26.1 Introduction .................................................................................................................. E-523 E26.2 Modeling Binary Choice .............................................................................................. E-523

E26.2.1 Underlying Processes ................................................................................... E-523 E26.2.2 Modeling Approaches .................................................................................. E-525 E26.2.3 The Linear Probability Model ...................................................................... E-526

E26.3 Grouped and Individual Data for Binary Choice Models .............................................. E-526 E26.4 Variance Normalization ............................................................................................... E-526 E26.5 The Constant Term in Index Function Models ............................................................ E-527

E27: Probit and Logit Models: Estimation ...................................................................... E-528 E27.1 Introduction .................................................................................................................. E-528 E27.2 Parametric Models for Binary Choice .......................................................................... E-528

E27.2.1 Functional Forms for Parametric Models..................................................... E-528 E27.2.2 Data Used in Estimation of Parametric Models ........................................... E-531

E27.3 Model Commands ........................................................................................................ E-537 E27.4 Output ........................................................................................................................... E-540

E27.4.1 Reported Estimates....................................................................................... E-540 E27.4.2 Fit Measures ................................................................................................. E-542 E27.4.3 Covariance Matrix ........................................................................................ E-544 E27.4.4 Retained Results and Generalized Residuals ............................................... E-545

E27.5 Robust Covariance Matrix Estimation ......................................................................... E-546 E27.5.1 The Sandwich Estimator .............................................................................. E-546 E27.5.2 Clustering ..................................................................................................... E-546 E27.5.3 Stratification and Clustering ........................................................................ E-549

E27.6 Analysis of Partial Effects ............................................................................................ E-550 E27.6.1 The Krinsky and Robb Method .................................................................... E-551

E27.7 Simulation and Analysis of a Binary Choice Model .................................................... E-556 E27.8 Measuring Fit in Binary Choice Models ...................................................................... E-558 E27.9 Saving Predictions and Residuals ................................................................................ E-562 E27.10 Using Weights and Choice Based Sampling .............................................................. E-563

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E27.11 Heteroscedasticity in Probit and Logit Models .......................................................... E-565 E27.12 Estimation Methods and Technical Details ................................................................ E-571

E27.12.1 Maximum Likelihood Estimation .............................................................. E-571 E27.12.2 Minimum Chi Squared Estimation with Grouped Data ............................. E-574 E27.12.3 Binary Choice Models with Heteroscedasticity ......................................... E-576

E28: Tests and Restrictions in Models for Binary Choice ............................................ E-578 E28.1 Introduction .................................................................................................................. E-578 E28.2 Testing Hypotheses ...................................................................................................... E-578

E28.2.1 Wald Tests .................................................................................................... E-578 E28.2.2 Likelihood Ratio Tests ................................................................................. E-580 E28.2.3 Lagrange Multiplier Tests ............................................................................ E-582

E28.3 Two Specification Tests ............................................................................................... E-584 28.3.1 A Test for Nonnested Probit Models ........................................................... E-584 E28.3.2 A Test for Normality in the Probit Model .................................................... E-585

E28.4 The WALD Command ................................................................................................. E-586 E28.5 Imposing Linear Restrictions ....................................................................................... E-588

E29: Extended Binary Choice Models ............................................................................ E-589 E29.1 Introduction .................................................................................................................. E-589 E29.2 Sample Selection in Probit and Logit Models .............................................................. E-589 E29.3 Endogenous Variable in a Probit Model ...................................................................... E-590 E29.4 Using MAXIMIZE to Estimate Other Parametric Models ............................................ E-594 E29.5 Two Step Estimation Using Binary Choice Models .................................................... E-595 E29.6 Other Models that Build on the Binary Choice Models ............................................... E-601

E30: Fixed and Random Effects Models for Binary Choice .......................................... E-603 E30.1 Introduction .................................................................................................................. E-603 E30.2 Commands ................................................................................................................... E-605 E30.3 Clustering, Stratification and Robust Covariance Matrices ......................................... E-607 E30.4 One and Two Way Fixed Effects Models .................................................................... E-609

E30.4.1 Application ................................................................................................... E-611 E30.4.2 Technical Details .......................................................................................... E-616

E30.5 Conditional MLE of the Fixed Effects Logit Model .................................................... E-619 E30.5.1 Command ..................................................................................................... E-620 E30.5.2 Application ................................................................................................... E-621 E30.5.3 Estimating the Individual Constant Terms ................................................... E-623 E30.5.4 A Hausman Test for Fixed Effects in the Logit Model ................................ E-624

E30.6 Random Effects Models for Binary Choice ................................................................. E-625 E30.6.1 Application ................................................................................................... E-628 E30.6.2 Technical Details for the Random Effects Models ...................................... E-632

E31: Random Parameter Models for Binary Choice ...................................................... E-635 E31.1 Introduction .................................................................................................................. E-635 E31.2 Binary Choice Models with Random Parameters ........................................................ E-636

E31.2.1 Command for the Random Parameters Models ........................................... E-637 E31.2.2 Results from the Estimator and Applications ............................................... E-639

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E31.2.3 Controlling the Simulation ........................................................................... E-646 E31.2.4 Other Options ............................................................................................... E-647 E31.2.5 The Parameter Vector and Starting Values .................................................. E-648 E31.2.6 A Dynamic Probit Model ............................................................................. E-649

E31.3 Latent Class Models for Binary Choice ....................................................................... E-650 E31.3.1 Application ................................................................................................... E-653

E32: Semiparametric and Nonparametric Models for Binary Choice ........................... E-660 E32.1 Introduction .................................................................................................................. E-660 E32.2 Maximum Score Estimation - MSCORE ..................................................................... E-661

E32.2.1 Command for MSCORE .............................................................................. E-662 E32.2.2 Options Specific to the Maximum Score Estimator ..................................... E-662 E32.2.3 General Options for MSCORE .................................................................... E-664 E32.2.4 Output from MSCORE ................................................................................ E-665 E32.2.5 Technical Details .......................................................................................... E-666 E32.2.6 Extensions .................................................................................................... E-667

E32.3 Klein and Spadys Semiparametric Binary Choice Model............................................. E-668 E32.3.1 Command ..................................................................................................... E-668 E32.3.2 Output........................................................................................................... E-669 E32.3.3 Application ................................................................................................... E-670 E32.3.4 Technical Details .......................................................................................... E-672

E32.4 Nonparametric Binary Choice Model .......................................................................... E-673 E32.4.1 Output from NPREG .................................................................................... E-674 E32.4.2 Application ................................................................................................... E-675

E33: Bivariate and Multivariate Probit and Partial Observability Models .................... E-677 E33.1 Introduction .................................................................................................................. E-677 E33.2 Estimating the Bivariate Probit Model ......................................................................... E-677

E33.2.1 Options for the Bivariate Probit Model ........................................................ E-679 E33.2.2 Starting values .............................................................................................. E-682 E33.2.3 Proportions Data .......................................................................................... E-683 E33.2.4 Heteroscedasticity ........................................................................................ E-683 E33.2.5 Specification Tests ....................................................................................... E-684 E33.2.6 Model Results for the Bivariate Probit Model ............................................. E-685 E33.2.7 Partial Effects ............................................................................................... E-687 E33.2.8 Application ................................................................................................... E-693 E33.2.9 Technical Details .......................................................................................... E-700

E33.3 Tetrachoric Correlation ................................................................................................ E-704 E33.4 Bivariate Probit Model with Sample Selection ............................................................ E-706

E33.4.1 Technical Details .......................................................................................... E-706 E33.5 Simultaneity in the Binary Variables ........................................................................... E-708 E33.6 Recursive Bivariate Probit Model ................................................................................ E-711 E33.7 Bivariate Probit Models with Partial Observability ..................................................... E-714

E33.7.1 Example ....................................................................................................... E-715 E33.7.2 Technical Details .......................................................................................... E-718

E33.8 Panel Data Bivariate Probit Models ............................................................................. E-719 E33.8.1 Application ................................................................................................... E-720

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E33.8.2 Simulation and Partial Effects ...................................................................... E-725 E33.9 Simultaneous Equations Models .................................................................................. E-727

E33.9.1 Maddalas Models ........................................................................................ E-728 E33.9.2 Model 3: y1* Observed Directly, y2* Observed as Binary y2 ....................... E-728 E33.9.3 Model 6: Both y1* and y2* Observed as Binary y1 and y2 ........................... E-731

E33.10 Multivariate Probit Model .......................................................................................... E-732 E33.10.1 Other Options ............................................................................................. E-733 E33.10.2 Retrievable Results..................................................................................... E-734 E33.10.3 Marginal Effects ......................................................................................... E-734 E33.10.4 Technical Details ........................................................................................ E-735 E33.10.5 Example ..................................................................................................... E-736 E33.10.6 Sample Selection Model ............................................................................ E-738 E33.10.7 Sequential Selection or Attrition ................................................................ E-739

E34: Ordered Choice Models .......................................................................................... E-740 E34.1 Introduction .................................................................................................................. E-740 E34.2 Command for Ordered Probability Models .................................................................. E-741

E34.2.1 Data Problems .............................................................................................. E-742 E34.2.2 Other Standard Options ................................................................................ E-742

E34.3 Output from the Ordered Probability Estimators ......................................................... E-743 E34.3.1 Robust Covariance Matrix Estimation ......................................................... E-746 E34.3.2 Saved Results ............................................................................................... E-747

E34.4 Model Structure and Data ............................................................................................ E-748 E34.4.1 Constant Term .............................................................................................. E-748 E34.4.2 Censored Data .............................................................................................. E-748

E34.5 Partial Effects and Simulations .................................................................................... E-750 E34.6 Technical Details for Ordered Choice Models ............................................................. E-754

E35: Extended Ordered Choice Models ......................................................................... E-757 E35.1 Introduction .................................................................................................................. E-757 E35.2 Weighting and Heteroscedasticity ................................................................................ E-757 E35.3 Multiplicative Heteroscedasticity ................................................................................. E-758

E35.3.1 Testing for Heteroscedasticity ...................................................................... E-759 E35.3.2 Partial Effects in the Heteroscedasticity Model ........................................... E-763

E35.4 Sample Selection and Treatment Effects ..................................................................... E-765 E35.4.1 Command ..................................................................................................... E-766 E35.4.2 Saved Results ............................................................................................... E-766 E35.4.3 Applications ................................................................................................. E-767 E35.4.4 Technical Details for the Selection Model ................................................... E-771

E35.5 Generalized Ordered Choice and Parallel Regressions .................................................. E-772 E35.5.1 The Proportional Odds Assumption ............................................................. E-772 E35.5.2 Brant Test of the Parallel Regressions Assumption ..................................... E-773

E35.6 Generalized Ordered Choice Models ........................................................................... E-776 E35.7 Hierarchical Ordered Probit Models ............................................................................ E-777 E35.8 Zero Inflated Ordered Probit (ZIOP, ZIHOP) Models ................................................. E-780 E35.9 Bivariate Ordered Probit and Polychoric Correlation .................................................. E-782

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E36: Panel Data Models for Ordered Choice .................................................................. E-787 E36.1 Introduction .................................................................................................................. E-787 E36.2 Fixed Effects Ordered Choice Models ......................................................................... E-788

E36.2.1 Standard Model Specifications for Panel Data Ordered Choice Models ..... E-790 E36.2.2 Application ................................................................................................... E-791

E36.3 Random Effects Ordered Choice Models ..................................................................... E-793 E36.3.1 Commands ................................................................................................... E-793 E36.3.2 Output and Results ....................................................................................... E-794 E36.3.3 Application ................................................................................................... E-795 E36.3.4 Technical Details for the Random Effects Models ...................................... E-798

E36.4 Random Parameters and Random Thresholds Ordered Choice Models ...................... E-801 E36.4.1 Model Commands ........................................................................................ E-802 E36.4.2 Results .......................................................................................................... E-806 E36.4.3 Application ................................................................................................... E-806 E36.4.4 Random Parameters HOPIT Model ............................................................. E-811

E36.5 Latent Class Ordered Choice Models........................................................................... E-817 E36.5.1 Command ..................................................................................................... E-818 E36.5.2 Results .......................................................................................................... E-819

E36.6 Stratification by Thresholds ......................................................................................... E-828

E37: Multinomial Logit Models ........................................................................................ E-832 E37.1 Introduction .................................................................................................................. E-832 E37.2 The Multinomial Logit Model MLOGIT .................................................................. E-833 E37.3 Model Command for the Multinomial Logit Model .................................................... E-834 E37.4 Choice Based Sampling and Robust Covariance Matrices .......................................... E-837 E37.5 Output for the Logit Models ........................................................................................ E-840 E37.6 Partial Effects ............................................................................................................... E-843

E37.6.1 Internal Computation of Partial Effects ....................................................... E-844 E37.6.2 Partial Effects Using PARTIALS ................................................................ E-847

E37.7 Predicted Probabilities ................................................................................................. E-848 E37.8 Generalized Maximum Entropy (GME) Estimation .................................................... E-849 E37.9 Technical Details on Optimization ............................................................................... E-851 E37.10 Panel Data Multinomial Logit Models ....................................................................... E-852

E37.10.1 Random Effects and Common (True) Random Effects ............................. E-852 E37.10.2 A Dynamic Multinomial Logit Model ....................................................... E-858

E38: Conditional Logit Models ........................................................................................ E-860 E38.1 Introduction .................................................................................................................. E-860 E38.2 The Conditional Logit Model CLOGIT .................................................................... E-862 E38.3 Clogit Data for the Applications .................................................................................. E-863

E38.3.1 Setting Up the Data ...................................................................................... E-865 E38.3.2 Checking Data Validity ................................................................................ E-866 E38.3.3 Types of Data on the Choice Variable ......................................................... E-867 E38.3.4 Simulated Choice Data ................................................................................. E-868 E38.3.5 Entering Data on a Single Line .................................................................... E-868 E38.3.6 Converting Wide Data Sets to the Long Format .......................................... E-870

E38.4 Command for the Discrete Choice Model.................................................................... E-873

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E38.5 Results for the Conditional Logit Model...................................................................... E-877 E38.5.1 Robust Standard Errors ................................................................................ E-880 E38.5.2 Descriptive Statistics .................................................................................... E-881

E38.6 Estimating and Fixing Coefficients .............................................................................. E-883 E38.7 MLOGIT and CLOGIT ................................................................................................ E-885

E39: Specifications of the Conditional Logit Models .................................................... E-887 E39.1 Introduction .................................................................................................................. E-887 E39.2 Choice Sets ................................................................................................................... E-887

E39.2.1 Fixed and Variable Numbers of Choices ..................................................... E-887 E39.2.2 Restricting the Choice Set ............................................................................ E-889 E39.2.3 Very Large Choice Sets ............................................................................... E-891

E39.3 Weighting ..................................................................................................................... E-893 E39.4 Choice Based Sampling ............................................................................................... E-893 E39.5 Building the Utility Functions ...................................................................................... E-895

E39.5.1 Alternative Specific Constants and Choice Invariant Variables .................. E-896 E39.5.2 Building the Utility Functions ...................................................................... E-899 E39.5.3 Shorthand Notations for Sets of Utility Functions ....................................... E-901 E39.5.4 Alternative Specific Constants and Interactions .......................................... E-901 E39.5.5 Equality Constraints ..................................................................................... E-902

E39.6 Starting and Fixed Values for Parameters .................................................................... E-903 E39.6.1 Fixed Values ................................................................................................ E-903 E39.6.2 Starting Values and Fixed Values from a Previous Model .......................... E-904

E39.7 Modeling Choice Strategy ............................................................................................ E-904 E39.8 Generalized Maximum Entropy Estimator .................................................................. E-905

E40: Post Estimation Results for Conditional Logit Models ......................................... E-907 E40.1 Introduction .................................................................................................................. E-907 E40.2 Partial Effects and Elasticities ...................................................................................... E-907

E40.2.1 Elasticities .................................................................................................... E-909 E40.2.2 Saving Elasticities in the Data Set ............................................................... E-912 E40.2.3 Exporting Results in a Spreadsheet .............................................................. E-913

E40.3 Predicted Probabilities and Logsums (Inclusive Values) ............................................... E-916 E40.3.1 Fitted Probabilities ....................................................................................... E-916 E40.3.2 Computing and Listing Model Probabilities ................................................ E-917 E40.3.3 Utilities and Inclusive Values ...................................................................... E-918 E40.3.4 Fitted Values of the Choice Variable ........................................................... E-919

E40.4 Hypothesis and Specification Tests of IIA ................................................................... E-920 E40.4.1 Testing the IIA Assumption ......................................................................... E-920 E40.4.2 Lagrange Multiplier, Wald, and Likelihood Ratio Tests.............................. E-923

E40.5 Examining Scenarios and Model Simulations ............................................................. E-924

E41: Models for Count Data ............................................................................................ E-931 E41.1 Introduction .................................................................................................................. E-931 E41.2 The Poisson Regression Model .................................................................................... E-933

E41.2.1 Results for the Poisson Model ...................................................................... E-936 E41.2.2 Application of the Poisson Model ................................................................ E-937

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E41.2.3 Testing for Overdispersion ........................................................................... E-940 E41.2.4 Robust Covariance Matrices ........................................................................ E-941 E41.2.5 Scaling the Asymptotic Covariance Matrix MLE ........................................ E-943 E41.2.6 Technical Details for the Poisson Model ..................................................... E-944

E41.3 Quantile Regression for Count Data ............................................................................ E-945 E41.4 Overdispersion: The Negative Binomial Model .......................................................... E-950

E41.4.1 The Negative Binomial Model ..................................................................... E-950 E41.4.2 Application ................................................................................................... E-952 E41.4.3 Heterogeneous Negative Binomial Model ................................................... E-953 E41.4.4 Negbin 1, Negbin 2 and Negbin P ............................................................... E-955 E41.4.5 Technical Details .......................................................................................... E-959

E41.5 Other Models for Count Data ....................................................................................... E-963 E41.5.1 Gamma Model with Under- or Overdispersion............................................ E-963 E41.5.2 Generalized Poisson Models GP1, GP2, GPP ........................................... E-965 E41.5.3 Polya-Aeppli Model ....................