Tài liệu Advanced econometrics panel data econometrics and gmm estimation

DEEQA,Ecole Doctorale MPSE Academic year 2003-2004 Advanced Econometrics Panel data econometrics and GMM estimation Alban Thomas MF 102, [email protected] 2 Purpose of the course  Present recent developments in econometrics, that allow for a consistent treatment of the impact of unobserved heterogeneity on model predictions: Panel data analysis.  Present a convenient econometric framework for dealing with restrictions imposed by theory:  Method of Moments estimation. Deal with discrete-choice models with unobserved hetero- geneity. Two keywords: unobserved heterogeneity and endogeneity. Methods: - Fixed Eects Least Squares - Generalized Least Squares - Instrumental Variables - Maximum Likelihood estimation for Panel Data models - Generalized Method of Moments for Times Series - Generalized Method of Moments for Panel Data - Heteroskedasticity-consistent estimation - Dynamic Panel Data models - Logit and Probit models for Panel Data - Simulation-based inference - Nonparametric and Semiparametric estimation Statistical software: SAS, GAUSS, STATA (?) 3 4 Contents I Panel Data Models 7 1 Introduction 9 1.1 Gains in pooling cross section and time series . . . 9 1.1.1 Discrimination between alternative models . 9 1.1.2 Examples . . . . . . . . . . . . . . . . . . . 10 1.1.3 Less colinearity between explanatory variables 11 1.1.4 May reduce bias due to missing or unobserved variables 2 . . . . . . . . . . . . . . . 11 1.2 Analysis of variance . . . . . . . . . . . . . . . . . 12 1.3 Some denitions . . . . . . . . . . . . . . . . . . . 15 The linear model 17 2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . 17 2.1.1 Model notation . . . . . . . . . . . . . . . 18 2.1.2 Standard matrices and operators . . . . . . 19 2.1.3 Important properties of operators . . . . . 20 The One-Way Fixed Eects model . . . . . . . . . 21 2.2 2.2.1 The estimator in terms of the Frisch-WaughLovell theorem . . . . . . . . . . . . . . . . 21 2.2.2 Interpretation as a covariance estimator . . 23 2.2.3 Comments . . . . . . . . . . . . . . . . . . 24 2.2.4 Testing for poolability and individual eects 25 5 6 CONTENTS 2.3 The Random Eects model . . . . . . . . . . . . . 26 2.3.1 Notation and assumptions . . . . . . . . . 26 2.3.2 GLS estimation of the Random-eect model 27 2.3.3 Comparison between GLS, OLS and Within 29 2.3.4 Fixed individual eects or error components? 29 2.3.5 Example: Wage equation, Hausman (1978) 2.3.6 Best Quadratic Unbiased Estimators (BQU) of variances 3 31 Extensions 33 3.1 The Two-way panel data model . . . . . . . . . . . 33 3.1.1 The Two-way xed-eect model 33 3.1.2 Example: Production function (Hoch 1962) 3.2 3.3 4 . . . . . . . . . . . . . . . . . 30 More on non-spherical disturbances . . . . . . . . . . . . . . 36 37 3.2.1 Heteroskedasticity in individual eect . . . 37 3.2.2 `Typical heteroskedasticity . . . . . . . . . 38 Unbalanced panel data models . . . . . . . . . . . 39 3.3.1 Introduction . . . . . . . . . . . . . . . . . 39 3.3.2 Fixed eect models for unbalanced panels . 40 Augmented panel data models 47 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . 47 4.2 Choice between Within and GLS . . . . . . . . . . 48 4.3 An important test for endogeneity 49 4.4 Instrumental Variable estimation: Hausman-Taylor . . . . . . . . . GLS estimator . . . . . . . . . . . . . . . . . . . . 51 4.4.1 Instrumental Variable estimation . . . . . . 51 4.4.2 IV in a panel-data context 51 4.4.3 Exogeneity assumptions and a rst instru- . . . . . . . . . ment matrix . . . . . . . . . . . . . . . . . 52 7 CONTENTS 4.4.4 More ecient procedures: Amemiya-MaCurdy and Breusch-Mizon-Schmidt 4.5 4.5.1 . . . . . . . . . . . . . . . . . . . . . . Full IV-GLS estimation procedure Example: Wage equation 4.6.1 4.7 55 . . . . . 56 . . . . . . . . . . . . . . 56 . . . . . . . . . . . . . 56 Model specication Application: returns to education . . . . . . . . . 4.7.1 Variables related to job status 4.7.2 Variables related to characteristics of households heads 5 53 Computation of variance-covariance matrix for IV estimators 4.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 58 58 Dynamic panel data models 63 5.1 63 Motivation . . . . . . . . . . . . . . . . . . . . . . 5.1.1 5.2 5.3 Dynamic formulations from dynamic programming problems . . . . . . . . . . . . . 63 5.1.2 Euler equations and consumption . . . . . . 65 5.1.3 Long-run relationships in economics . . . . 67 The dynamic xed-eect model . . . . . . . . . . . 69 5.2.1 Bias in the Fixed-Eects estimator . . . . . 70 5.2.2 Instrumental-variable estimation . . . . . . 73 The Random-eects model . . . . . . . . . . . . . 75 5.3.1 Bias in the ML estimator . . . . . . . . . . 75 5.3.2 An equivalent representation . . . . . . . . 76 5.3.3 The role of initial conditions . . . . . . . . 77 5.3.4 Possible inconsistency of GLS . . . . . . . . 78 5.3.5 Example: The Balestra-Nerlove study 78 . . . 8 II 6 CONTENTS Generalized Method of Moments estimation The GMM estimator 6.1 6.2 6.3 85 Moment conditions and the method of moments . 85 . . . . . . . . . . . . . 85 6.1.1 Moment conditions 6.1.2 Example: Linear regression model 6.1.3 Example: Gamma distribution . . . . . 86 . . . . . . . 87 6.1.4 Method of moments estimation . . . . . . . 87 6.1.5 Example: Poisson counting model . . . . . 88 6.1.6 Comments . . . . . . . . . . . . . . . . . . 89 The Generalized Method of Moments (GMM) . . . 91 6.2.1 Introduction . . . . . . . . . . . . . . . . . 91 6.2.2 Example: Just-identied IV model . . . . . 91 6.2.3 A denition 92 6.2.4 Example: The IV estimator again . . . . . . . . . . . . . . . . . . . . . . 92 Asymptotic properties of the GMM estimator . . . 93 6.3.1 Consistency . . . . . . . . . . . . . . . . . 94 6.3.2 Asymptotic normality . . . . . . . . . . . . 95 6.4 Optimal and two-step GMM . . . . . . . . . . . . 97 6.5 Inference with GMM . . . . . . . . . . . . . . . . 99 6.6 Extension: optimal instruments for GMM . . . . . 102 6.6.1 Conditional moment restrictions . . . . . . 102 6.6.2 A rst feasible estimator . . . . . . . . . . 104 6.6.3 Nearest-neighbor estimation of optimal instruments 6.6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . GMM estimators for time series models 7.1 GMM and Euler equation models 7.1.1 106 Generalizing the approach: other nonparametric estimators 7 83 109 115 . . . . . . . . . 115 Hansen and Singleton framework . . . . . . 115 9 CONTENTS 7.1.2 7.2 7.3 7.4 8 GMM estimation . . . . . . . . . . . . . . 117 GMM Estimation of MA models . . . . . . . . . . 118 7.2.1 A simple estimator . . . . . . . . . . . . . 118 7.2.2 A more ecient estimator . . . . . . . . . . 120 7.2.3 Example: The Durbin estimator . . . . . . 121 . . . . . . . . 122 . . . . . . . . . . . 122 . . . . . . . . . . . . . . . . 123 Covariance matrix estimation . . . . . . . . . . . . 125 7.4.1 Example 1: Conditional homoskedasticity . 126 7.4.2 Example 2: Conditional heteroskedasticity . 126 7.4.3 Example 3: Covariance stationary process . 127 7.4.4 The Newey-West estimator . . . . . . . . . 128 7.4.5 Weighted autocovariance estimators . . . . 130 7.4.6 Weighted periodogram estimators . . . . . 133 GMM Estimation of ARMA models 7.3.1 The ARMA(1,1) model 7.3.2 IV estimation GMM estimators for dynamic panel data 135 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . 135 8.2 The Arellano-Bond estimator . . . . . . . . . . . . 136 8.2.1 Model assumptions 136 8.2.2 Implementation of the GMM estimator . . . . . . . . . . . . . . . 137 More ecient procedures (Ahn-Schmidt) . . . . . . 139 8.3.1 Additional assumptions . . . . . . . . . . . 139 8.4 The Blundell-Bond estimator . . . . . . . . . . . . 140 8.5 Dynamic models with Multiplicative eects . . . . 141 8.5.1 Multiplicative individual eects . . . . . . . 141 8.5.2 Mixed structure 143 8.3 8.6 . . . . . . . . . . . . . . . Example: Wage equation . . . . . . . . . . . . . . 145 10 III 9 CONTENTS Discrete choice models 149 Nonlinear panel data models 9.1 9.2 151 Brief review of binary discrete-choice models . . . 151 . . . . . . . . . . 151 9.1.1 Linear Probability model 9.1.2 Logit model . . . . . . . . . . . . . . . . . 152 9.1.3 Probit model . . . . . . . . . . . . . . . . . 152 Logit models for panel data . . . . . . . . . . . . . 153 9.2.1 Sucient statistics . . . . . . . . . . . . . . 153 9.2.2 Conditional probabilities . . . . . . . . . . 155 9.2.3 Example: . . . . . . . . . . . . . . . 156 . . . . . . . . . . . . . . . . . . . . 157 T =2 9.3 Probit models 9.4 Semiparametric estimation of discrete-choice models 158 9.5 9.4.1 The binary choice model . . . . . . . . . . 159 9.4.2 The IV estimator . . . . . . . . . . . . . . 162 SML estimation of selection models . . . . . . . . 164 9.5.1 The GHK simulator . . . . . . . . . . . . . 164 9.5.2 Example 168 . . . . . . . . . . . . . . . . . . . Appendix 1. Maximum-Likelihood estimation of the Random-eect model Appendix 2. The two-way random eects model 171 173 Appendix 3. The one-way unbalanced random eects model 179 Appendix 4. ML estimation of dynamic panel models181 Appendix 5. GMM estimation of static panel models185 11 CONTENTS Appendix 6. A framework for simulation-based inference 194 c Software c Appendix 8. A crash course in Gauss c Appendix 9. Example: The Gauss software Appendix 7. Example: the SAS 203 211 219 c 224 Appendix 10. IV and GMM estimation with Gauss c Appendix 11. DPD estimation with Gauss 232 References 238 12 CONTENTS Part I Panel Data Models 13 Chapter 1 Introduction Panel data: Sequential observations on a number of units (individuals, rms). cross-sections over time, longitudinal data cross-section time-series data. Also called or pooled 1.1 Gains in pooling cross section and time series 1.1.1 Discrimination between alternative models Many economic models in the form: F (Y; X; Z; ) = 0; where Y: individual control variables (workers, rms); policy or principal's) variables; : Z: (public (xed) individual attributes; parameters. Linear model: Y = 0 + xX + z Z + u: 15 X: 16 CHAPTER 1. INTRODUCTION Alternative views concerning this model:  Policy variables have a signicant impact whatever individual characteristics, or  Dierences across individuals are due to idiosyncratic individual features, not included in Z . In practice, observed dierences across individuals may be due to both inter-individual dierences and the impact of policy vari- ables. 1.1.2 Examples a) W AGE = 0 + 1EDUCAT ION + 2Z .  People with higher education level have higher wages because rms value those people more;  People have higher education because they have higher ability (expected productivity) anyway, and rms value worker ability more. b) SALES = 0 + 1ADV ERT ISEMENT + 2Z .  Advertisement expenditures boost sales;  More ecient rms enjoy more sales, and thus have more money for advertisement expenditures. c) OUT P UT = 0 + 1REGULAT ION + 2Z .  Regulatory control aects rm output;  Firms with higher output are more regulated on average. d) W AGE = 0 + 11I(UNION ) + 2Z .  Belonging to a union signicantly raises wages; 1.1. GAINS IN POOLING CROSS SECTION AND TIME SERIES 17  Firms react to higher wages imposed by unions by hiring higherquality workers, and 1.1.3 1I(UNION ) is a proxy for worker quality. Less colinearity between explanatory variables In consumer or production economics, input, output or consumer prices are dicult to use, because:  Time-series: Aggregated macro price indexes are highly cor- related;  Cross-sections: Not enough price variation across individuals or rms. With panel data, variations across individuals and across time periods are accounted for.  Time-series: no information on the impact of individual characteristics (socioeconomic variables,...);  Cross-sections: no information on adjustment dynamics. Estimates may reect inter-individual dierences inherent in comparisons of 1.1.4 dierent people or rms. May reduce bias due to missing or unobserved variables With panel data, easy to control for unobserved heterogeneity across individuals. This is critical in practice, explains why panel data models are now so popular in micro- and macro-econometrics. Point related to endogeneity and omitted variables issues. 18 CHAPTER 1. INTRODUCTION Example: Output supply function under perfect competition max  = pQ C (; Q) where C (; Q) = c(Q) (Q) , p =  @c@Q = AQ 1 (Cobb-Douglas) = (0 + 1Q) (Quadratic). 1 Cobb-Douglas case: log Q =  1 (log p log  A  ). From equilibrium condition to estimable equation: Observations (Qit ; pit ), unobserved heterogeneity i , rm i, period t. 1 (log pit log i A  ) log Qit =  1 Identication issue: estimable equation is Q~ it = a0 + a1p~it + uit; i = 1; 2; : : : ; N; t = 1; 2; : : : ; T; ~ it = log Qit, p~it = log pit, a1 = 1=( 1), where Q a0 = ( A  E log i) =( 1), Euit = 0. Model identied if E log i = 0, i.e., Ei = 1, otherwise A is biased if i is overlooked and E log i 6= 0. Empirical issue: possible correlation between output price and eciency term i.  pit 1.2 Analysis of variance Consider the model yit = i + xiti + "it; where xit is scalar, i and i = 1; 2; : : : ; N; t = 1; 2; : : : ; Ti; i are parameters, and time periods available for individual i. Ti: number of 1.2. 19 ANALYSIS OF VARIANCE Useful rst-order empirical moments are Ti 1X y ; yi = T t=1 it Sxxi = Ti X t=1 x )2; (xit and Syyi = i Ti X t=1 (yit Ti 1X x ; xi = T t=1 it Sxyi = yi)2; Ti X t=1 (xit xi)(yit yi); i = 1; 2; : : : ; N: Least-square parameter estimates are computed as ^ i = Sxyi=Sxxi and xi^ ^ i = y i and the Residual Sum of Squares (RSS) for individual 2 =S ; Sxyi xxi RSSi = Syyi with (Ti i is 2) degrees of freedom: Consider now a restricted model with constant slopes and constant intercepts: yit =  + xit + "it; which obtains by imposing the following restrictions  1 = 2 =


= N (= ) 1 = 2 =


= N (=  ): Under these restrictions, least-squares parameter estimates would be ^ = PN PTi )(yit i=1 t=1(xit x PN PTi )2 i=1 t=1 (xit x y) 20 CHAPTER 1. INTRODUCTION and ^ = y x^ , where y = N Ti N X X 1 P i Ti i=1 t=1 yit; x = N 1 P Ti N X X i Ti i=1 t=1 xit: The Residual Sum of Squares is RSS = hP Ti N X X i=1 t=1 (yit y)2 with as number of degrees of N PTi i=1 t=1(yit y)(xit PN PTi )2 i=1 t=1(xit x PN freedom: i=1 Ti 2. i2 x) ; For a majority of applications, the rst model is too general and estimation would require a great number of time observations. If unobserved heterogeneity is additive in the model, we might consider the following specication with constant slope and dierent intercepts: Minimizing P P i t (yit yit = i + xit + "it: i xit )2 with respect to i and  , we have XX t i (yit i xit ) = 0; XX i t xit(yit i xit ) = 0; so that P P x (y y ) ^ i = yi xi and ^ = P i P t it it i : i ) i t xit (xit x P Residual Sum of Squares has now i Ti (N + 1) degrees of N + 1 parameters are estimated). free- dom ( This is the most popular model encountered in empirical applications.