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LIMDEP
Version 10
Econometric Modeling Guide
by
William H. Greene Econometric Software, Inc.
1986 - 2012 Econometric Software, Inc. All rights reserved. This software product, including both the program code and the accompanying documentation, is copyrighted by, and all rights are reserved by Econometric Software, Inc. No part of this product, either the software or the documentation, may be reproduced, stored in a retrieval system, or transmitted in any form or by any means without prior written permission of Econometric Software, Inc. LIMDEP and NLOGIT are registered trademarks of Econometric Software, Inc. All other brand and product names are trademarks or registered trademarks of their respective companies. Econometric Software, Inc. 15 Gloria Place Plainview, NY 11803 USA Tel: +1 516-938-5254 Fax: +1 516-938-2441 Email: [email protected] Websites: www.limdep.com and www.nlogit.com. Econometric Software, Australia 215 Excelsior Avenue Castle Hill, NSW 2154 Australia Tel: +61 (0)4-1843-3057 Fax: +61 (0)2-9899-6674 Email: [email protected]
mailto:[email protected]://www.limdep.com/http://www.nlogit.com/mailto:[email protected]End-User License Agreement
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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
LIMDEP 10 Econometric Modeling Guide Table of Contents vi
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
LIMDEP 10 Econometric Modeling Guide Table of Contents vii
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
LIMDEP 10 Econometric Modeling Guide Table of Contents viii
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
LIMDEP 10 Econometric Modeling Guide Table of Contents ix
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
LIMDEP 10 Econometric Modeling Guide Table of Contents x
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
LIMDEP 10 Econometric Modeling Guide Table of Contents xi
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
LIMDEP 10 Econometric Modeling Guide Table of Contents xii
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
LIMDEP 10 Econometric Modeling Guide Table of Contents xvi
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
LIMDEP 10 Econometric Modeling Guide Table of Contents xvii
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 ....................