New York University/Econometrics I

Stern School of Business

Econometrics I

Stern School of Business/B30.3351

Professor W. Greene
Department of Economics
Office:;MEC 7-90, Ph. 998-0876, Fax. 995-4218
e-mail: wgreene@stern.nyu.edu
WWW: http://www.stern.nyu.edu/~wgreene

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Abstract: This is an intermediate level, Ph.D. course in Applied Econometrics. Topics to be studied include specification, estimation, and inference in the context of models that include then extend beyond the standard linear multiple regression framework. After a review of the linear model, we will develop the asymptotic distribution theory necessary for analysis of generalized linear and nonlinear models. We will then turn to instrumental variables, maximum likelihood, generalized method of moments (GMM), and two step estimation methods. Inference techniques used in the linear regression framework such as t and F tests will be extended to include Wald, Lagrange multiplier and likelihood ratio and tests for nonnested hypotheses such as the Hausman specification test and Davidson and MacKinnon’s J test. Specific modelling frameworks will include the linear regression model and extensions to models for panel data, multiple equation models, time series models and models for discrete choice.

Prerequisites: Multivariate calculus, matrix algebra, probability and distribution theory, statistical inference, and an introduction to the multiple linear regression model. Appendices A and B in Greene (2008) are assumed. We will survey the parts of Appendix C and Chapter 2 that would have appeared in prerequisite courses. A significant part of this course will focus on the advanced parts of Appendices C and D and Chapters 4 through 7. We will also make use of a few of the results in Appendix E.

Course Requirements: Grades for the course will be based on:

Midterm examination (30%), The midterm examination will be given in class.
Take home final exam (40%)
Several problem sets and small projects (total 30%).

Course Materials:

Text: The required text for the course is Greene, W., Econometric Analysis, 6^th Edition, Prentice Hall, 2008. (You may use the 5^th edition if you prefer.) Other texts that might be useful are: Davidson, R., and MacKinnon, J., Econometric Theory and Methods, Oxford University Press, 2004, which is more advanced than Greene; Johnston, J. and DiNardo, J., Econometric Methods, 4^th Edition, McGraw-Hill, 1997, which is comparable to Greene; and Kennedy, P., A Guide to Econometrics, 4^th Edition, MIT Press, 1998, Woolridge, J., Introductory Econometrics: A Modern Approach, 3rd Edition (or later), Southwestern, 2006 (or later) or Gujarati, D., Basic Econometrics, 4^rd Edition, McGraw-Hill, 2004, all of which are less advanced. Note: A useful list of errata and comments submitted by readers of the text are listed at the website for the text, http://www.stern.nyu.edu/~wgreene/Text/econometricanalysis.htm where there is a button for the errata/discussion list.

Software: Some of the outside work for this course will involve using a computer. Students may use any computer software that they are familiar with for this purpose. A student version of NLOGIT is available - I will provide a copy to anyone who wishes to use it. This is a restricted version of the full package (www.nlogit.com). But, it is adequate for what we'll be doing in this class. Data sets needed for some of the exercises will be distributed to the class via the course website. The data sets used for the examples in the text are all available at the text website.

Readings: Some relevant articles from the literature will be suggested (not required). The papers listedA few (marked by enclosure in boxes) are useful pedagogical literature, and students intending to do empirical research for their dissertations will probably find them worthwhile reading. The others are a selection from a huge literature that should be both interesting and accessible to students in this course.

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Course Outline: Reading assignments given in red in brackets refer to sections in Greene (2008). For convenience, reading assignments in Greene (2003) are retained in <[Assignment]>

I. The Paradigm of Econometrics: [Chapter 1 (pp. 1-6)] <[Chapter 1 (pp. 1-6)]>

A. Modeling in economics
B. Econometrics: statistics, economics, mathematics
C. Econometric modeling: understanding, prediction, control
D. Modeling frameworks:

1. Bayesian and Classical (frequentist) approaches [14.2, Chapter 18] <[16.2 - 16.2.2, pp. 425-429]>
2. Estimation and inference: Nonparametric, semiparametric, parametric [14.3, 14.4]

E. Estimation and inference in econometrics, methodological issues

II. The Classical Linear Regression Model. Part 1. Specification and Computation

A. The conditional mean function [B.1-B.3, B.7-B.8] <[B.1-B.3, B.7-B.8]>; regression (Waugh)
B. The classical linear regression model and its functional form

1. The linear regression model [2.1] <[2.1]>
2. Linear models and intrinsic linearity [2.1-2.3] <[2.1-2.3]>
3. Logs and levels, estimating elasticities [2.3.1] <[2.3.1]>
4. Functional form and linearity. Transformations and dummy variables [6.1-6.3] <[7.1-7.3]>
5. Linearized regression and Taylor series, linearity in economic modelling [2.3.1, 11.1-11.2] <[2.3.1, pp. 162-163]>

C. Least squares regression [Chapter 3] <[Ch. 3]> (Frisch and Waugh, Longley)

1. Least squares regression [3.1-3.2] <[3.1-3.2]>
2. Partitioned regression and the Frisch-Waugh theorem [3.3, 3.4] <[3.3, 3.4]>
3. Applications of partitioned regression: a fixed effects model [9.4.1] <[13.3.1 up to result (13-6)]>

D. Evaluating the fit of the regression [3.5] <[3.5]>, ANOVA, Adjusted R² [3.5, 7.4] <[3.5, 8.4]>
E. Least squares with restrictions [5.3.2 – 5.3.3] <[6.3.2-6.3.3]>

III. The Classical Linear Regression Model. Part 2. Statistical Inference in Finite Samples

A. Statistical properties of the least squares estimator in finite samples [4.1-4.5] <[4.1-4.5]>

1. Why least squares? [4.2] <[4.2]>
2. Sampling distributions [Example 4.1] <[Example 4.1]>
3. Expectation [4.3] <[4.3]>
4. The effects of omitted and superfluous variables - The Omitted Variable Formula (A VIR) [7.2.1-7.2.3] <[8.2.1, 8.2.3]>
5. Variance of the least squares estimator [4.4, 7.2.2] <[4.4, 8.2.2]>
6. The Gauss-Markov theorem [4.4, 4.5] <[4.4, 4.5]>
7. The Least Absolute Deviations Estimator [14.3.2] <[16.3.2]>

B. Estimating the Variance of the least squares estimator

1. Conventional estimation [4.6] <[4.6]>
2. The effect of multicollinearity [4.8.1] <[4.9.1]>
3. Introduction to bootstrapping;.least absolute deviations [17.6, 14.3.2] <[pp. 924-925, 16.3.2]>

C. The sampling distribution of the least squares coefficient vector [Chapter 4] <[6.6.3]>

1. Generalities about sampling distributions [C.2-C.4] <[C.2 - C.4]>
2. Sampling distributions and properties of estimators [C.5, 4.3 - 4.5] <[C.5, 4.3 - 4.5]>
3. Linear estimation and normality [4.7] <[4.7 and esp. result 4-4 on p. 44, 4.7.1]>
4. Efficient estimation, precision, mean squared error [4.4, 7.2.2] <[4.4, Thm. 4.2, 8.2.2]>
5. Describing the sampling distribution of the estimator - kernel density estimator [4.7, p. 1023, 14.4.1] <[4.7, p. 881, 16.4.1]>

D. Statistical Inference in the linear model [Appendix C, 4.7.1 - 4.7.5, 5.1-5.3, 6.1 - 6.5] <[Appendix C, 4.7.1 - 4.7.5, 6.1-6.3, 7.1 to 7.6]> (Greene and Seaks)

1. Standard results for testing [5.1 - 6.4] <[6.1 to 7.3]>
2. Structural change [6.4] <[7.4, 7.5]>, Model selection [7.3, 7.4] <[8.3, 8.4]>
3. The J test for nonnested models [7.3.3] <[8.3.3]>

E. Prediction using the linear model [5.6] <[6.6]>, the Oaxaca decomposition [5.6] <[4.7.3]>

IV. Asymptotic Theory and Instrumental Variables Estimation

A. Large sample distributions, asymptotic and limiting distributions [Appendix D]
B. Basic large sample results for the classical model [4.9] <[5.1-5.3]> (McCallum)
C. Large sample results for a function of a statistic - the delta method [4.9.4] <[5.2.4]>
D. Instrumental variables estimation and measurement error [12.1, 12.3, 12.5] <[5.4, 5.6]> (Ashenfelter)
E. Test procedures for large samples; t, F, chi-squared, Wald statistic [5.3, 5.4] <[6.4, 6.5]>
F. The Hausman specification test [12.4] <[5.5]>
G. A test for nonnested models, the J and Cox tests: variables [7.3] <[8.3]>, levels vs. logs [none] <[9.4.3]>

V. Nonlinear Regression Models [11.1-11.4] <[9.1-9.4]>

A. The Box-Cox transformation [11.3.2] <[9.3.2]>
B. Nonlinear regression and nonlinear least squares [11.1-11.4] <[9.1-9.4]> (McCullough and Vinod)
C. Two step estimation [11.6] <[9.5]>

MIDTERM
VI. The Generalized Regression Model

A. Nonspherical disturbances [8.1] <[10.1]>

1. General formulation [8.1-8.3] <[10.1]>
2. Heteroscedasticity [8.4] <[11.1]> (Harvey)
3. Autocorrelation [19.1-19.2] <[12.1,12,2]>

B. Implications for least squares [8.4, 19.5] <[10.2 to 10.3, 11.2, 12.5]>

1. Robust covariance matrix estimation [8.3-8.4, 19.5] <[10.3-10.4, 11.3, 12.5-12.6]> (White, Newey/West)
2. Bootstrapping [14.3.2] <[16.3.2]>

C. Testing for nonspherical disturbances [8.5, 19.7] <[11.4, 12.7]>
D. Generalized least squares and weighted least squares [8.3, 8.6-8.8, 19.5-19.6, 19.8-19.9] <[10.5, 11.5-11.7, 12.8-12.9]>
E. Two step feasible GLS estimation, familiar applications [8.3.2, 8.8, 19.8-19.9] <[10.5.2,11.7, 12.9]>

1. Heteroscedasticity [8.8] <[11.7]>
2. Autocorrelation [19.8-19.9] <[13.1 to 13.7]>

F. Applications of two step, feasible GLS estimation

1. A panel data model with random effects [9.5] <[13.4]>
2. Hausman tests for specification [9.5.4] <[13.4.4]>
3. Covariance structures [none] <[13.9]>
4. A random coefficients model [9.8] <[13.8]>
5. Seemingly unrelated regressions (SUR) [10.1-10.4] <[14.1 to 14.3]> (Christensen/Greene)

VII. Methods of Estimation

A. Instrumental Variables Estimation [Chapter 12] <[5.4]>

1. Measurement error [12.5] <[5.4]>
2. Lagged dependent variables and autocorrelation [19.9.3] <[12.9.4]>
3. Two stage least squares [12.3] <(no reading)>

B. Maximum likelihood estimation [Chapter 16] <[Chapter 17, 11.7, 11.8]> (Harvey)

1. Computation [E.2, E.3] <[E.4 and E.5 (both optional)]>
2. Covariance matrix estimation [16.4.6] <[17.4.6]> (BHHH)
3. GARCH models [19.13] <[11.8]>
4. Likelihood ratio, Lagrange multiplier tests [16.6, 8.5.2, 16.9.2, 9.5.3] <[17.5, 11.4.3, 11.6.3, 13.4.3]>

C. Generalized method of moments (GMM) estimation [Chapter 15] <[Chapter 18]> (Hansen, Newey/West)
D. Dynamic panel data models [15.6.5] <[18.5]>: (Hausman and Taylor, Arellano and Bover, Dahlberg and Johansson)
E. Two step estimation [16.7] <[17.7]> (Heckman, Murphy/Topel, Newey)
F. Estimation by simulation; models with unobserved heterogeneity [Chapter 17] <[17.8]> (Revelt and Train)

VIII. Techniques for Analyzing Panel Data

A. Traditional Models: Fixed and Random Effects [9.1-9.6] <[13.1 - 13.4]>
B. Recent Developments: Random Parameters and Latent Class Models [9.8, 16.9.7] <[Greene (two papers), Class notes]>

IX. Brief Survey of Nonregression Models and Techniques (as time permits, and according to interests)

A. Binary choice [23.1-23.5] <[21.1 to 21.5]>
B. Discrete choice - logit models [23.11] <[21.7]>
C. Time series models [20.1, 20.2, 20.5, Chapter 21] <[19.1, 19.2, 19.5, 20.1-20.5]> (not likely we will have time for much of this)
D. Models for counts [25.1-25.5] <[21.9]> (Cameron and Trivedi, Montalvo)
E. Censoring, truncation and selection [24.1-24.5] <[22.1- 22.4]>

X. Non- and Semiparametrics

A. Kernel density estimation [14.4.1, 14.3.2]
B. Nonparametric regression [14.4.2]
C. LAD and Quantile regression [14.3.2]

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Reading List (annotated)

Arellano, M. and O. Bover "Another Look at the Instrumental Variable Estimation of Error-Components Models," Journal of Econometrics, 68, 1995, pp. 29-51.

Ashenfelter, O., and Krueger, A., "Estimates of the Economic Returns to Schooling from a New Sample of Twins," American Economic Review, 84, 5, 1994, pp. 1157-1173. An intriguing study of measurement error and instrumental variables estimation. Intriguing study of data on twins gathered at a convention of twins in Twinsburg, Ohio.

Berndt, E., Hall, B., Hall, R., and Hausman, J., "Estimation and Inference in Nonlinear Structural Models," Annals of Economic and Social Measurement, 3/4, 1974, pp. 653-665. Landmark paper which presents the BHHH, BH³, or outer product of gradients (OPG, lately called the "sandwich") estimator for the asymptotic covariance matrix of the MLE.

Breusch, T., and Pagan, A., "The LM Test and Its Applications to Model Specification in Econometrics," Review of Economic Studies, 47, 1980, pp. 239-254. Began a methodological shift in econometrics toward a reinterpretation of existing tests and development of many new ones. Short lived paradigm shift, as the tests are strongly parametric, and conflict with the current trend toward less stringently parameterized models. An excellent book with similar material, developed at length is Godfrey, L., Misspecification Tests in Econometrics, Cambridge University Press, 1988. (Important contribution to methodology.)

Breusch, T., and Pagan, A., "A Simple Test for Heteroscedasticity and Random Coefficients Variation," Econometrica, 47, 1979, pp. 1287-1294. Application of the LM methodology developed in fuller detail in the 1980 paper (they were done simultaneously) to a common problem. Has become essentially the standard test for heteroscedasticity - soon to be supplanted by the conditional moment test. (See Pagan and Vella.)

Cameron, C., and Trivedi, P., "Econometric Models Based on Count Data: Comparisons and Applications of Some Estimators and Tests," Journal of Applied Econometrics, 1, 1986, pp. 29-53. One of the main references for economists, with Hausman, et al., on the Poisson regression model.

Christensen, L., and Greene, W., "Economies of Scale in U.S. Electric Power Generation," Journal of Political Economy, 84, 1976, pp. 655-676. Application of specification testing in a model, showing how theoretical propositions produce testable restrictions on the empirical model. Fairly straightforward application of the seemingly unrelated regressions model, maximum likelihood estimation, and use of likelihood ratio tests.

Frisch, R., and Waugh, F., "Partial Time Regressions as Compared with Individual Trends," Econometrica, 1, 1933, pp. 387-401. Purely empirical discovery of one of the fundamental pillars of econometrics, the Frisch-Waugh theorem for partitioning a linear projection. Another high water mark in the literature.

Fiorentini, G., Calzolari, G., and Panattoni, L., "Analytic Derivatives and the Computation of GARCH Estimates," Journal of Applied Econometrics, 11,4, 1996, pp. 399-418. How to compute the parameters of a very complicated dynamic econometric model.

Greene, W. "The Behavior of the Fixed Effects Estimator in Nonlinear Models," The Econometrics Journal, 7, 1, 2004, pp. 98-119. Also working Paper 01-01, Stern School of Business, Department of Economics, New York University. You can download this from the web at http://www.stern.nyu.edu/~wgreene/panel.doc

Greene, W., and Seaks, T., "The Restricted Least Squares Estimator: A Pedagogical Note," Review of Economics and Statistics, 73, 1991, pp. 563-567. Some interesting matrix algebra for the linear regression model and restricted least squares. Surprise discovery of an apparently theretofore overlooked (by econometricians, though not statisticians) aspect of linear regression.

Harvey, A., "Estimating Regression Models with Multiplicative Heteroscedasticity," Econometrica, 44, 1976, pp. 461-465. Very general model for heteroscedasticity. A good companion to Breusch and Pagan. Also illustrates an interesting application of Newton's method and the method of scoring for maximum likelihood estimation.

Hansen, L., "Large Sample Properties of Generalized Method of Moments Estimators," Econometrica, 50, 1982, pp. 1029-1054. Soon to be classic, if not already, study of the method of moments. Pioneering paper that has produced a major shift in the direction of econometric methodology. One of the most influential methodological pieces since 1980, in close competition with Dickey-Fuller on unit roots. Shows how estimators of model parameters can be developed without need to make strong distributional assumptions. Innovation in the literature - extremely influential.

Hausman, J., "Specification Tests in Econometrics," Econometrica, 46, 1978, pp. 1251-1271. Develops the "Hausman Test," a now widely used specification test that gets around the need for nested models imposed by the conventional likelihood, Neyman-Pearson based tests.

Hausman, J., Hall, B., and Griliches, Z., "Econometric Models for Count Data with an Application to the Patents-R&D Relationship," Econometrica, 52, 1984, pp. 909-938. The first major reference on count data models for economists. Includes extensions for panel data. Interesting for proposing an entire class of models for a nonlinear regression setting.

Hausman, J., and Taylor, W., "Panel Data and Unobservable Individual Effects," Econometrica, 49, 1981, pp. 1377-1398. Extends the familiar fixed and random effects models to some more involved cases. For example, how to deal with fixed effects in models in which group effects are fixed over time.

Heckman, J., "Sample Selection Bias as a Specification Error," Econometrica, 47, 1979, pp. 153-161. First in a literature on two step estimation of models. A clever application of two step estimation in a model of nonrandom sampling. Began a debate (Heckman’s work began on it as a Ph.D. student in 1970-1972) on sample selection models that continues. Interesting application for the form that methodological progress takes place.

McCallum, B., "A Note Concerning Asymptotic Covariance Expressions," Econometrica, 41, 1973, pp. 581-583. Shows a common (he alleges) error in developing asymptotic results for the linear regression model. A good article for students to use to assess their command of the basic results in asymptotics of the classical model.

Montalvo, J., "GMM Estimation of Count-Panel-Data Models with Fixed Effects and Predetermined Instruments," Journal of Business and Economic Statistics, 15, 1997, pp. 82-89. Current application that shows the use of the GMM estimation method. Straightforward reading, accessible to students in this course.

Murphy, K., and Topel, R., "Estimation and Inference in Two Step Econometric Models," Journal of Business and Economic Statistics, 3, 1985, pp. 370-379. Lays out the computations needed for handling two step maximum likelihood or least squares estimation. A now standard result. Applications becoming increasingly common. Worth reading.

Newey, W., "A Method of Moments Interpretation of Sequential Estimators," Economics Letters, 14, 1984, pp. 201-206. Similar to Murphy and Topel. Develops a similar set of results for GMM estimation - M&T is for ML and least squares (though it can be extended to some GMM estimators).

Newey, W., and West, K., "A Simple, Positive Semi-definite, Heteroscedasticity and Autocorrelation Consistent Covariance Matrix," Econometrica, 55, 1987, pp. 703-708. The canonical presentation of one of the most important tools in the applied econometricians toolkit. Generalizes White's estimator, and makes feasible, many GMM estimators in time series settings.

Pagan, A., and Vella, F., "Diagnostic Tests for Models Based on Individual Data: A Survey," Journal of Applied Econometrics, 4, Supplement, 1989, pp. S29-S59. Develops the theory of conditional moment tests and applies them to several models including the censored regression model.

Revelt, D. and K. Train, "Mixed Logit with Repeated Choices: Households' Choices of Appliance Efficiency Level," Review of Economics and Statistics, 1998, 80, , pp. 1-11.

Waugh, F., "The Place of Least Squares in Econometrics," Econometrica, 29, 1961, pp. 386-396. Historical piece. Argues that OLS, which at that time, was becoming "old fashioned" and ordinary was underappreciated in economics and produced important results. Sounds like he was about 40 years before his time.

White, H., "A Heteroscedasticity-Consistent Covariance Matrix Estimator and Direct Test for Heteroscedasticity," Econometrica, 48, 1980, 817-838. The White estimator for unknown heteroscedasticity. Remarkably simple yet powerful estimator. A major step toward robust estimation in econometrics. Very important paper. (Unfortunately) not simple reading.

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