ADVANCED ECONOMETRICS I. Course Description. Contents - Theory 18/10/2017. Theory (1/3)

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1 ADVANCED ECONOMETRICS I Theory (1/3) Instructor: Joaquim J. S. Ramalho E.mail: jjsro@iscte-iul.pt Personal Website: Office: D5.10 Course Website: Fénix: semestre Course Description This course provides an introduction to the modern econometric techniques used in the analysis of cross-sectional and panel data in the area of microeconometrics: The interaction between theory and empirical econometric analysis is emphasized Students will be trained in formulating and testing economic models using real data Pre-requisites (recommended): Introductory Econometrics Grading: Two individual home works (50%) + Final (open book) exam (50%) Minimum grade at the exam of 7/20 No re-sit examinations 2 Contents - Theory i. Introduction 1. Linear Regression Analysis with Cross-Sectional Data 2. Econometric Models with Endogenous Explanatory Variables 3. Static and Dynamic Panel Data Models 4. Limited Dependent Variable Models 3 1

2 Textbooks Recommended: Cameron, A. and P.K. Trivedi (2005), Microeconometrics: Methods and Applications, Cambridge University Press Others: Baltagi, B. (2013), Econometric Analysis of Panel Data, John Wiley and Sons (5th Edition) Davidson, R. and J.G. MacKinnon (2003), Econometric Theory and Methods, Oxford University Press Greene, W. (2011), Econometric Analysis, Pearson (7th Edition) Verbeek, M. (2012), A Guide to Modern Econometrics, Wiley (4th Edition) Wooldridge, J.M. (2010), Econometric Analysis of Cross Section and Panel Data, MIT Press (2nd Edition) Wooldridge, J.M. (2012), Introductory Econometrics: A Modern Approach, South Western (5th Edition). 4 Contents - Illustrations 1. Determinants of Firm Debt 2. Estimating the Returns to Schooling 3. Explaining Individual Wages 4. Explaining Capital Structure 5. Modelling the Choice Between Two Brands 6. Health Care Expenses and Consultations 7. Explaining Firm s Credit Ratings 8. Travel Mode Choice 9. Health Care Expenses and Consultations (revisited) 10. Determinants of Firm Debt (revisited) 5 Software Recommended: : R: Others: Gauss: Matlab: 6 2

3 i. Introduction i.1. Econometric Methodology i.2. The Structure of Economic and Financial Data i.3. Dependent Variables and Econometric Models i.4. Types of Explanatory Variables 7 i. Introduction i.1. Econometric Methodology Econometrics: Definition: Application of statistical techniques to the analysis of economic, financial, social... data aiming at estimating relationships between a dependent variable and a set of explanatory variables Ultimate purpose: Theory validation Prediction / Forecasting Policy recommendation 8 i. Introduction i.1. Econometric Methodology Model specification Data collection Model estimation Model evaluation Unsuitable model Suitable model Result interpretation Theory validation Policy recommendation Prediction / Forecasting 9 3

4 i. Introduction i.2. The Structure of Economic and Financial Data Cross-sectional data: N cross-sectional units (individuals / firms / cities /...) 1 time observation per unit Time series: 1 unit T time observations per unit Panel data N units T time observations per unit 10 i. Introduction i.3. Dependent Variables and Econometric Models Main types of econometric models: Regression Model: Aim: explaining E Y X Probabilistic model: Aim: explaining Pr Y X Usually incorporates also a regression model for E Y X Y: dependent variable X: explanatory variables E Y X : expected value for Y given X Pr Y X : probability of Y being equal to a specific value given X Each type of econometric model has many variants 11 i. Introduction i.3. Dependent Variables and Econometric Models The numeric characteristics of the dependent variable restricts the variants that may be applied in each case: Y Type of outcome Main model ], + [ Unbounded data Linear [0, + [ Nonnegative data Exponential [0,1] Fractional data Fractional Logit,... {0,1} Binary choices Logit,... {0,1,2,, J 1} Multinomial choices Multinomial logit, {0,1,2,, J 1} Ordered choices Ordered logit, {0,1,2, } Count data Poisson, 12 4

5 i. Introduction i.3. Dependent Variables and Econometric Models Model Transformations and Adaptations: Bounded continuous outcomes may often be transformed in such a way that they give rise to unbounded outcomes which may be modelled using a linear model Any econometric model may require adaptations: Data structure: cross-section, time series, panel Non-random samples: stratified, censored, truncated Measurement error Endogenous explanatory variables Corner solutions 13 i. Introduction i.4. Types of Explanatory Variables Quantitative variables: Levels (Euro, kilograms, meters, ) Levels and squares Logs Growth rates Per capita values Qualitative variables Binary (dummy) variables: X = 0,1 Interaction variables: X = Dummy var. Quantitative or dummy var Linear Regression Analysis with Cross-Sectional Data 1.1. The Linear Regression Model Specification Estimation Interpretation Inference 1.2. Model Evaluation RESET Test Tests for Heteroskedascity Chow Test 15 5

6 1.1. The Linear Regression Model Specification Model Specification: Y i = β 0 + β 1 X 1i + + β k X ki + u i i = 1,, N Y = Xβ + u E Y X = Xβ Assumptions: 1. Random sampling 2. E u X = 0 3. No perfect colinearity 4. Homoskedasticity: Var u X = σ 2 5. Normality: u~normal 0, σ 2 u: error term β: parameters k: n. explanatory variables p: n. parameters N: n. observations σ 2 : error variance The Linear Regression Model Estimation Estimation method - Ordinary Least Squares (OLS): N min u 2 i, u i = Y i Y i, Y i = መβ 0 + መβ 1 X መβ k X k i=1 መβ = X X 1 X y Estimator properties: Finite samples: Assumptions 1-3: Unbiasedness Assumptions 1-4: Unbiasedness and efficiency Assumptions 1-5: Unbiasedness, efficiency and normality Asymptotically: Assumptions 1-3: Consistency Assumptions 1-4: Consistency, efficiency and normality u: residual Y: fitted value of Y መβ: estimate for β regress Y X 1 X k The Linear Regression Model Interpretation Effects from unitary changes in a given explanatory variable: X j = 1 Y = β j Needs adaptation for: Transformed dependent variables Transformed quantitative explanatory variables Qualitative explanatory variables Aims: Most ofthe time: testing whether theeffect is null orsignificantly different from zero it is equivalent to test whether a parameter or a set of a parameters or a linear combination of parameters are significantly different from zero Most of the time: analyze the sign of the effect (null, positive, negative) Sometimes: calculate and analyze the magnitude of the effect 18 6

7 1.1. The Linear Regression Model Inference Variance of the parameter estimators: Needed for performing hypothesis tests Alternative estimators: Standard assumes homoskedasticity Robust valid under both homoskedasticity and heteroskedasticity Cluster-robust specific for panel data Bootstrap simulated variance, always valid provided that bootstrap samples are generated under appropriate assumptions regress Y X 1 X k regress Y X 1 X k, vce(robust) regress Y X 1 X k, vce(cluster clustvar) regress Y X 1 X k, vce(bootstrap) The Linear Regression Model Inference Test for the individual significance of a parameter t test: H 0 : β j = 0 H 1 : β j 0 or H 1 : β j > 0 or H 1 : β j < 0 t = መβ j σ βj ~t N p regress Y X 1 X k regress Y X 1 X k, vce(robust) regress Y X 1 X k, vce(cluster clustvar) regress Y X 1 X k, vce(bootstrap) The Linear Regression Model Inference Test for the joint significance of a set of parameters F / Wald tests: Model: Y = β 0 + β 1 X 1 + +β g X g +β g+1 X g β k X k + v Hypotheses: H 0 : β g+1 = = β k = 0 H 1 : No H 0 Test: F = R2 R 2 1 R 2 N p ~F q q N p valid only under homoskedasticity W = N መβ Var መβ 1 መβ ~χ q 2 general formula R 2 : coefficient of determination of the model under H 1 R 2 : coefficient of determination of the model under H 1 q: n. of parameters being tested መβ : vector of parameters to be tested regress Y X 1 X g X g+1 X k, test X g+1 X k 21 7

8 1.2. Model Evaluation Specification tests: Model functional form: RESET test Heteroskedasticity: Breusch-Pagan (BP) test, White test, Structural breaks: Chow tests Model Evaluation RESET Test RESET test: Intuition: Any model of the type E Y X = S Xβ may be approximated by E Y X = Xβ + σ j=1 γ j X መβ j+1 Implementation: Estimate the original model Generate the variables (only test version based on three fitted powers) regress Y X 1 X k estat ovtest X መβ 2, X መβ 3, X መβ 4, Add the generated variables to the original model and estimate the following auxiliary model: Y = β 0 + β 1 X β k X k + γ 1 X መβ 2 + γ 2 X መβ 3 + γ 3 X መβ v Apply an F / Wald test for the significance of the added variables: H 0 : γ 1 = γ 2 = γ 3 = = 0 (suitable model functional form) H 1 : No H 0 (unsuitable model functional form) Model Evaluation Tests for Heteroskedascity BP Test for Heteroskedasticity: Intuition: Because the mean of the error term is zero, the variance of the error term is given bythesum of thesquared error terms The squared residuals are an estimate of the squared error terms Implementation: Generate the variable u 2 Replace Y by u 2 in the original model and estimate the following auxiliary model: u 2 = γ 0 + γ 1 X γ k X k + v Apply an F / Wald test for the joint significance of the right-hand side variables of the previous auxiliary model: H 0 : γ 1 = = γ k = 0 (homoskedasticity) H 1 : Não H 0 (heteroskedasticity) regress Y X 1 X k estat hettest, rhs fstat 24 8

9 1.2. Model Evaluation Chow Test Chow Test for Structural Breaks: Context: Two groups of individuals / firms /...: G A, G B It is suspected that the behaviour of the two groups in which regards the dependent variable may have different determinants Implementation: Generate the dummy variable D = ቊ 1 if the individual belongs to G A 0 if the individual belongs to G B Estimate the original model duplicated : Y = θ 0 + θ 1 X θ k X k + γ 0 D + γ 1 DX γ k DX k + v Apply an F / Wald test for thesignificance of the variables where D is present: H 0 : γ 1 = = γ k = 0 (no structural break) H 1 : Não H 0 (with a structural break) regress Y X 1 X k D DX 1 DX k test D DX 1 DX k Econometric Models with Endogenous Explanatory Variables 2.1. Endogeneity Definition and Consequences Motivation: Omitted variables, Measurement Errors, Simultaneous Equations Solutions: Instrumental Variables, Panel Data 2.2. Methods based on Instrumental Variables Two-Stage Least Squares Generalized Method of Moments (GMM) 2.3. Specification Tests Test for the Exogeneity of an Explanatory Variable Test for the Exogeneity of the Instrumental Variables Tests for Correlation between Instrumental Variables and Explanatory Variables Endogeneity Definition and Consequences Definitions: Exogenous explanatory variables: E u X = 0 essential assumption in any regression model Endogenous explanatory variables: E u X 0 Consequences: Standard estimators become inconsistent Motivation: Omitted variables Covariate measurement error Simultaneity 27 9

10 2.1. Endogeneity Motivation: Omitted variables, Measurement Errors, Simultaneous Equations Omitted variables - example: True model: Y = β 0 + β 1 X 1 + β 2 X 2 + v Estimated model: Y = β 0 + β 1 X 1 + u As u = β 2 X 2 + v: If cov X 1, X 2 = 0, then E u X 1 = 0 X 1 is exogenous If cov X 1, X 2 0, then E u X 1 0 X 1 is endogenous Endogeneity Motivation: Omitted variables, Measurement Errors, Simultaneous Equations Covariate measurement error - example: True model: Y = β 0 + β 1 X 1 + v e: measurement error Instead of X 1, it is observed X 1 = X 1 + e Estimated model: Y = β 0 + β 1 X 1 + u As u = v β 1 e: If cov X 1, e = 0, then E u X 1 = 0 X 1 is exogenous If cov X 1, e 0, then E u X 1 0 X 1 is endogenous most common case because the measurement error is on X Endogeneity Motivation: Omitted variables, Measurement Errors, Simultaneous Equations Measurement error on Y has less serious consequences: True model: Y = β 0 + β 1 X 1 + v Instead of Y, it is observed Y = Y + e Estimated model: Y = β 0 + β 1 X 1 + u As u = v + e: In general, cov X 1, e = 0 and E u X 1 = 0, since the measurement error is on Y and not in X 1 Hence, usually there are no endogeneity problems However, estimation is less precise, since the error term has now two components 30 10

11 2.1. Endogeneity Motivation: Omitted variables, Measurement Errors, Simultaneous Equations Simultaneity - example: True model: ቊ Supply: Q = β 0 + β 1 P + u Demand: Q = α 0 + α 1 P + v Estimated model: ቊ Supply: Q = β 0 + β 1 P + u Demand: Q = α 0 + α 1 P + v As: P = α 0 β 0 v u + β 1 α 1 β 1 α 1 Q = then P is function of v and u; hence: E u P 0 in the supply equation P is endogenous E v P 0 in the demand equation P is endogenous Endogeneity Instrumental Variables What to do in case of endogeneity: Universal solution methods based on instrumental variables : Two-Stage Least Squares Generalized Method of Moments (GMM) When data is in panel form and the endogeneity problem is caused by omitted time-constant variables: Methods based on the removal of the fixed effects Endogeneity Instrumental Variables Instrumental variables: Context: Y = β 0 + β 1 X 1 + β 2 X β k X k + u (structural model) E u X 1 0 X 1 is endogenous Definitions of instrumental variable (IV A,, IV M ): E u IV A = = E u IV M = 0 cov IV A, X 1 0,,cov IV M, X 1 0 The number of instrumental variables must be equal or larger than the number of endogenous explanatory variables 33 11

12 2.2. Methods based on Instrumental Variables Two-Stage Least Squares Implementation: 1. Estimate the reduced form of the model by OLS: ดX 1 = π 0 + π 2 X π k X k + π A IV A + + π M IV M + w End. Expl. Var. Ex. Expl. Var. Instrumental Variables and get X 1 = π 0 + π 2 X π k X k + π A IV A + + π M IV M 2. Estimate the structural model, with X 1 replaced by X 1, by OLS: Y = β 0 + β 1 X 1 + β 2 X β k X k + u (by default, variances are estimated in a standard way; to use another estimator, use the option vce(robust) or similar) ivregress 2sls Y (X 1 = IV A IV M ) X 2 X k Methods based on Instrumental Variables Generalized Method of Moments (GMM) Very general estimation method: Applies to a large variety of cases Includes as particular cases OLS, maximum likelihood, etc. Formulation: Moment conditions: E g Y, X, IV; β = 0 s moment conditions p parameters: β 0, β 1,, β k Optimization: s = p just-identified model: g Y, X, IV; መβ = 0 s > p overidentified model: min J = g Y, X, IV; β Wg Y, X, IV; β where W is a weighting matrix; the first-order conditions are given by: g Y, X, IV; መβ Wg Y, X, IV; መβ = 0 β Methods based on Instrumental Variables Generalized Method of Moments (GMM) Choosing W in overidentified models: To get efficient estimators, W has to be defined as the inverse of the covariance matrix of the moment conditions: where: W = Ω 1, Ω = Var g Y, X, IV; β = E g Y, X, IV; β g Y, X, IV; β (by default, variances are estimated in a standard way; to use another estimator, use the option vce(robust) or similar) ivregress gmm Y (X 1 = IV A IV M ) X 2 X k 36 12

13 2.2. Methods based on Instrumental Variables Generalized Method of Moments (GMM) Particular case OLS: Assumption: E u X = 0 E X u = 0 g Y, X, IV; β = X u = X y Xβ s = p X y X መβ = 0 X y X መβ = 0 X y X X መβ = 0 X X መβ = X y መβ = X X 1 X y Methods based on Instrumental Variables Generalized Method of Moments (GMM) Instrumental Variables (s = p): Exogenous variables: Z = X 2 X k IV A IV M Assumption: E u Z = 0 E Z u = 0 g Y, X, IV; β = Z u = Z y Xβ If s = p Z y X መβ = 0 መβ = Z X 1 Z y In this case, it is identical to the TSLS estimator Methods based on Instrumental Variables Generalized Method of Moments (GMM) Instrumental Variables (s > p): Optimization problem: min J = First-order conditions: y Xβ ZΩ 1 Z y Xβ X ZΩ 1 Z y X መβ = 0 X ZΩ 1 Z y = X ZΩ 1 Z X መβ መβ = X ZΩ 1 Z X 1 X ZΩ 1 Z y where Ω is a preliminar estimate of Ω = E Z uu Z estimation in two steps (Two-step GMM) 39 13

14 2.2. Methods based on Instrumental Variables Generalized Method of Moments (GMM) To estimate Ω, one may assume homoskedasticity, heteroskedasticity, cluster-type structures, etc. The choice for Ω affects the analysis not only in terms of parameter inference but also parameter value Instead of a two-step procedure, it is possible to estimate Ω simultaneously with β Continuously-updating GMM: More complex, without an explicit solution for መβ Less common than two-step GMM Specification Tests Tests relevant for TSLS / GMM: Tests for the exogeneity of an explanatory variable If the explanatory variable is exogenous, it is better to use OLS in order to get efficient estimators Methods based oniv s should be used only if really necessary, since there may be a substantial loss in precision Tests for the exogeneity of the instrumental variables To act as an IV, a variable has to be exogenous when based on IV s that actually are endogenous, TSLS and GMM are inconsistent Tests for correlation between instrumental variables and explanatory variables To act as an IV, a variable has to be correlated with the endogenous explanatory variable when based on IV s uncorrelated with the endogenous regressors, TSLS and GMM are inconsistent If the IV s are only weakly correlated with the endogenous regressors, then the model with be poorly identified in such a case, TSLS and GMM may display a huge variability Specification Tests Tests for the Exogeneity of an Explanatory Variable TSLS - Wu-Hausman test: 1. Estimate the reduced model by OLS: X 1 = π 0 + π 2 X π k X k + π A IV A + + π M IV M + w 2. Calculate the residuals w 3. Add w to the strutural model and re-estimate it, by OLS: Y = β 0 + β 1 X 1 + β 2 X β k X k + δ w + u 4. t test: H 0 : δ = 0 (X 1 is exogenous) H 1 : δ 0 (X 1 is endogenous) ivregress 2sls Y (X 1 = IV A IV M ) X 2 X k estat endogenous GMM - Eichenbaum, Hansen e Singleton s (1988) C test Based on the difference of two J statistics (see the next page) ivregress gmm Y (X 1 = IV A IV M ) X 2 X k estat endogenous 42 14

15 2.3. Specification Tests Tests for the Exogeneity of the Instrumental Variables These tests can be applied only when the model is overidentified If the model is just-identified, then it is only possible to justify the exogeneity of the IV s using theoretical arguments Hansen s J test (also called test for overidentifying restrictions; this is an extension of the Sargan test, very common in the TSLS framework under homoskedasticity): 1. Estimate the model by GMM 2. Test H 0 : E u Z = 0 (IV s are exogenous) H 1 : E u Z 0 (IV s are not exogenous) q: number of overidentifying using the J statistic: restrictions መJ = g Y, X, IV; መβ Wg Y, X, IV; መβ ~χ2 q (applies also to the TSLS estimator, with the obvious adaptation) ivregress gmm Y (X 1 = IV A IV M ) X 2 X k estat overid Specification Tests Tests for Correlation between Instrumental Variables and Explanatory Variables Alternatives: F / Wald tests for the significance of the IV s in the reduced form model Criteria / tests for weak instruments F / Wald tests: 1. Estimate the reduced form model: X 1 = π 0 + π 2 X π k X k + π A VI A + + π M VI M + w 2. Test the hypothesis: H 0 : π A = = π M = 0 (IV s and X 1 are not correlated) (applies also to the GMM estimator, with the obvious adaptation) ivregress 2sls Y (X 1 = IV A IV M ) X 2 X k estat firststage Specification Tests Tests for Correlation between Instrumental Variables and Explanatory Variables Tests for weak instruments : Even when the previous tests reveal that IV s and X 1 are correlated, that correlation may be so weak that TSLS / GMM estimators are very little precise Criterium: F < 10 Suggest a high probability of having weak instruments Tests: Cragg anddonald (2005), Kleibergen and Paap (2006) 45 15

16 3. Static and Dynamic Panel Data Models 3.1. Panel Data 3.3. Inference and Model Evaluation 3.4. Dynamic Panel Data Models Panel Data Definitions Panel data: N cross-sectional units: i = 1,, N T time observations per unit: t = 1,, T Econometric analysis more complex: Cross-sectional data: different units independent observations Panel data: same units dependent observations over time Panel Data Definitions Advantages: Make possible the analysis of the dynamics of individual behaviours Generate more efficient estimators, since samples are larger Allow for endogenous explanatory variables in some special cases It is simple to get instrumental variables Limitations: Prediction and calculation of partial effects not possible in some models 48 16

17 3.1. Panel Data Definitions Temporal dimension: Short panels: Sample comprises many individuals (N ), butthere is a reduced number of time observations (small T) Thetemporal correlation of theobservations for each individual is an issue, but across individuals it is assumed independence Long panels: Large number of time observations for each individual (T ) Need to take into account time series issues (stationarity, cointegration, etc.) Panel Data Definitions Cross-sectional composition: Balanced panels: Every individual is observed in all time periods (T i = T, i) Unbalanced panels: Some individuals are not observed in some time periods (T i T) Motivation: some individuals refuse to continue providing information after some time periods attrition problem Most estimators work with unbalanced panels, provided that one may assume that there is no endogenous selection (the data are missing at random) Panel Data Definitions Variability over time and across individuals: With panel data, the variance of Y it can be decomposed as follows: N T Y it തY 2 = Y it തY i + തY i തY 2 i=1 t=1 N T N T i=1 t=1 = Y it തY 2 i + തY i തY 2 i=1 t=1 i=1 Variability within the Variability between the groups groups To obtain the the total, within and between variance, divide by, respectively, NT 1, N T 1 and N 1 N 51 17

18 3.1. Panel Data Models Base Model Model with individual effects: Y it = α i + x it β + u it i = 1,, N; t = 1,, T α i : individual effects, time-constant x it - explanatory variables, including: x it : changes across individuals and over time x i : time-constant d t : temporal dummy variable t: trend (one may also have t 2, t 3, ) d t.x it : interaction term u it : idiosyncratic error term changes randomly across individuals and over time Panel Data Models Other models: Pooled or Population-averaged model: Y it = α + x it β + u it Direct extension of cross-sectional models Popular in the area of Statistics but not in Economics, where individual heterogeneity is always an issue Random Coefficients Model: Y it = α i + x it β i + u it More complex than thebase model, allowing for a different coefficient also for the explanatory variables Computationally-intensive model, hard to estimate Panel Data Fixed Effects versus Random Effects The individual effects model may be re-written as: Y it = x it β + α i + u it The error term has now two componentes, α i and u it The individual effects α i may be correlated, or not, with the explanatory variables Fixed effects: α i and x it are correlated x it is endogenous Direct estimation of the model is not possible Random effects: α i and x it are not correlated x it is exogenous Direct estimation of the model is possible 54 18

19 Most panel data estimators: Are based on transformed versions of the base model Differ of the type of exogeneity required for the explanatory variables: Contemporaneous: E x it u it = 0 Weak (pre-determined variables): E x it u i,t+j = 0, j 0 Strict: E x it u is = 0, s, t 55 List of estimators: Estimators for the random effects model: Pooled OLS Between estimator Random effects estimator Estimators for the fixed effects model: Fixed effects or Within estimator Least squares dummy variables (LSDV) estimator First-diferences estimator Estimators based on instrumental variables: General estimators Hausman-Taylor estimator Estimators for the Random Effects Model Pooled OLS estimator: Model: Y it = α + x it β + α i α + u it v it Assumption: E x it α i + u it = 0 Requires random effects: α i e x it must be uncorrelated Requires contemporaneous exogeneity between u it and x it Estimation: OLS with a cluster-type estimator for the variance regress Y X 1 X k, vce(cluster clustvar) 57 19

20 ҧ 18/10/ Estimators for the Random Effects Model Between estimator: Model: തY i = 1 T σ i T i t=1 Yit, etc. തY i = α + x i β + α i α + തu i Assumption: E xҧ i α i + തu i = 0 Requires random effects: α i e x it must be uncorrelated Requires strict exogeneity between u it and x it തv i Estimation: OLS xtreg Y X 1 X k, be Estimators for the Random Effects Model Random effects estimator: Model: Y it = α + x it β + α i α + u it v it New assumptions: Var α i = σ2 α Var u it = σ2 u Under the new assumptions: cor v it, v is = σ 2 α / σ 2 α + σ2 u Taking into account this result, more efficient estimators may be obtained All the previous estimators did not fully exploit the panel nature of the data in the estimation process Estimators for the Random Effects Model Estimation: Generalized least squares (GLS) It is equivalent to apply OLS to the following equation: Y it መθ i തY i = 1 መθ i α + x it መθ i xҧ i β + vit θ i = 1 σ u 2 / T i σ α 2 + σ u 2 v it = 1 θ i α i + u it θ i തu i Assumption: E xҧ i v it = 0 Requires random effects: α i e x it must be uncorrelated Requires strict exogeneity between u it and x it xtreg Y X 1 X k, re vce(cluster clustvar) 60 20

21 ҧ 18/10/ Estimators for the Fixed Effects Model Fixed effects / within estimator: Model: Y it തY i = x it xҧ i β + u it തu i Corresponds to the subtraction of the model defining the between estimator from the base individual effects model Assumption: E x it xҧ i u it തu i = 0 Requires strict exogeneity between u it and x it Estimation: OLS with a cluster-type estimator for the variance xtreg Y X 1 X k, fe vce(cluster clustvar) Estimators for the Fixed Effects Model Limitations: It is not possible to include in the model: Time-constant explanatory variables If the model includes time dummies, explanatory variables with identical changes over time for all individuals (e.g. age) Prediction not possible Partial effects conditional not only on x it but also on α i ; how to calculate them? Main advantage: Allow for (time-constant) unobserved individual heterogeneity that may be correlated with the explanatory variables, not requiring the use of instrumental variables Estimators for the Fixed Effects Model LSDV estimator: Model: Y it = j=1 where d ij = 1 if i = j and 0 if i j N α j d ij + x it β + u it Assumptions and estimates of β identical to those of the fixed effects estimator Estimates of α j : Given by α i = തY i x i መβ Consistent only in case of a long panel, in which case it is also possible to make prediction and calculate partial effects conditional on both x it and α i regress Y X 1 X k D 1 D N, vce(cluster clustvar) or areg Y X 1 X k, absorb(idname) vce(cluster clustvar) 63 21

22 Estimators for the Fixed Effects Model First-differences estimator: Model: Y it Y i,t 1 = x it x i,t 1 β + uit u i,t 1 ΔY it = Δx it β + Δu it Corresponds to the subtraction of the first-diferences equation from the base individual effects model Assumption: E Δx it Δu it = 0 Requires E x it u it = E x it u i,t 1 = E x it u i,t+1 = 0 Estimation: OLS Comparison with the fixed-effects estimator: Identical when T = 2 Does not require strict exogeneity regress D.Y D.X 1 D.X k, vce(cluster clustvar) nocons Instrumental Variables Estimators Endogenous explanatory variables - E x it u it 0: Possível IV s for x it : External instruments, as in the cross-sectional case Internal instruments (same explanatory variable but relative to other time periods) Example of internal instruments: If x it is weakly exogenous (apart from the current period), then: All past values (lags) of x it may be used as IV s Possible IV s: x i,t 1 or x i,t 1, x i,t 2 or x i,t 1,, x i,t 5, etc. If x it is strictly exogenous (apart from the current period), then: All past (lags) and future (leads) values of x it may be used as IV s Possible IV s : x i,t 1 and/or x i,t+1, etc. (external instruments TSLS) xtivreg Y (X 1 = IV A IV M ) X 2 X k, options (options: re, fe, be, fd) (internal instruments TSLS) xtivreg Y (X 1 = L. X 1 L2. X 1 ) X 2 X k, options (options: re, fe, be, fd) Instrumental Variables Estimators Hausman-Taylor estimator: Useful when interest lies on time-invariant explanatory variables which are correlated with α i, since: Fixed effects estimators: drop those variables from the model Random effects estimators: inconsistent Intermediate case between the fixed and the random effects cases: Explanatory variables correlated with α i are dealt with under the assumption of fixed effects Explanatory variables uncorrelated with α i are dealt with under the assumption of random effects 66 22

23 Instrumental Variables Estimators Model: Y it = x 1it β 1 + x 2it β 2 + w 1i γ 1 + w 2i γ 2 + α i + u it x 1it : time-varying explanatory variables, uncorrelated with α i x 2it : time-varying explanatory variables, correlated with α i w 1i : time-invariant explanatory variables, uncorrelated with α i w 2i : time-invariant explanatory variables, correlated with α i Assumptions: All explanatory variables are strictly exogenous relative to u it x 1 has a larger dimension than w 2 x 1 and w 2 are correlated Instrumental Variables Estimators Estimation: TSLS / GMM based on the following instruments for the explanatory variables correlated with α i : x 2it : x 2it xҧ 2i w 2i : xҧ 1i xthtaylory X 11 X 12 X 21 X 22 W 11 W 12 W 21 W 22, endog(x 21 X 22 W 21 W 22 ) 68 Static Panel Data Models - Summary Estimator Effects Random Fixed Efficiency Prediction Exogeneity Pooled x x Contemporaneous Internal instruments for x it Between x x Strict Lags / Leads Random Effects x x x Strict Lags / Leads Fixed Effects First- Differences x x Only if long panel Strict E x itu it = E x itu i,t 1 = E x itu i,t+1 = 0 Lags / Leads Lags / Leads (except t 1 and t + 1) 69 23

24 3.3. Inference and Model Evaluation Variance estimators best alternatives: Cluster-robust Bootstrap Hausman test: Random or fixed effects? H 0 : E α i x it = 0 (RE and FE consistent, RE also efficient) H 1 : E α i x it 0 (FE consistent, RE inconsistent) H = መβ FE መβ RE V መβ FE V መβ RE 1 መβ FE መβ RE 2 χ k (models must be estimated using standard estimators of the variance) xtreg Y X 1 X k, fe estimates store ModelFE xtreg Y X 1 X k estimates store ModelRE hausman ModelFE ModelRE Dynamic Panel Data Models Inconsistency of Estimators for Static Models Models with lagged dependent variables: Include Y i,t 1, Y i,t 2, as explanatory variables: Y it = γ 1 Y i,t γ p Y i,t p + x it β + α i + u it, t = p + 1, T All estimators for static panel data models are inconsistent Example AR 1 model: Y it = γ 1 Y i,t 1 + α i + u it This equation holds for alltime periods, so: Y i,t 1 = γ 1 Y i,t 2 + α i + u i,t 1 Y i,t 2 = γ 1 Y i,t 3 + α i + u i,t 2 etc Dynamic Panel Data Models Inconsistency of Estimators for Static Models Random effects estimator: Required assumption: E Y i,t 1 α i = 0 However, α i is one of the components of Y i,t 1, so E Y i,t 1 α i 0 All other lags of Y it are also correlated with α i Fixed effects model: Required assumption: E Y i,t 1 തY i u it തu i = 0 However, u i,t 1 (included in തu i ) is one of the components of Y i,t 1, so E Y i,t 1 തY i u it തu i 0 All other lags of Y it, which are included in തY i, are also correlated with the corresponding element of തu i 72 24

25 3.4. Dynamic Panel Data Models Inconsistency of Estimators for Static Models First-diferences method: Required assumption: E ΔY i,t 1 Δu it = 0 However, u i,t 1 (included in Δu it ) is one of the components of Y i,t 1 (included in ΔY i,t 1 ), so E ΔY i,t 1 Δu it 0 Unlike the previous estimators, Y i,t 2, Y i,t 3, are not correlated with Δu it, provided that there is no autocorrelation Solution: using Y i,t 2, Y i,t 3, (ou functions of these variables) as instruments for ΔY i,t Dynamic Panel Data Models Instrumental Variable Estimators Base Dynamic Panel Data Model: Y it = γ 1 Y i,t 1 + x it β + u it, t = 3,, T Assumption: u it has no autocorrelation Δu it has first-order autocorrelation: Cov u it, u i,t 1 = Cov u it u i,t 1, u i,t 1 u i,t 2 = Cov u i,t 1, u i,t 1 0 Main estimators: Anderson-Hsiao (1981) Arellano-Bond (1991) Difference GMM Blundell-Bond (1998) System GMM Dynamic Panel Data Models Instrumental Variable Estimators Anderson-Hsiao (1981): Two alternative instruments: Y i,t 2 ivregress gmm D.Y (DL. Y = L2. Y) D. X 1 D. X k Y i,t 2 (one observation is lost but in general seems to produce more efficient estimators) xtivreg Y (L. Y = L2. Y) X 1 X k, fd 75 25

26 3.4. Dynamic Panel Data Models Instrumental Variable Estimators Arellano-Bond (1991): Proposed using all available lags of Y i,t as instruments: t = 3: Y i,1 t = 4: Y i,2, Y i,1 t = T: Y i,t 2,, Y i,2, Y i,1 Total number of instruments: T 1 T 2 /2 It is possible to use only a subset of the available instruments More efficient than Anderson-Hsiao s (1981) estimators xtabond Y X 1 X k, maxldep(#) twostep vce(robust) Dynamic Panel Data Models Instrumental Variable Estimators Blundell-Bond (1998): Often, lags of Y i,t are not good instruments for Y i,t Proposed adding the following instruments: Y i,2,, Y i,t 1 Total number of instruments: T 1 2 T 2 + T 2 It is possible to use only a subset of the available instruments More efficient than Arellano-Bond s (1991) estimator Requires heavier assumptions than Arellano-Bond s (1991) estimator xtdpdsys Y X 1 X k, maxldep(#) twostep vce(robust) Dynamic Panel Data Models Tests for Instruments Validity Most common tests: Hansen s J test of overidentifying restrictions (after xtabond or xtdpdsys, with variance estimated in a standard way) estat sargan Test for autocorrelation All estimators for dynamic panel data models assume first-order autocorrelation: Cov u it, u i,t l 0 Cov u it, u i,t l = 0, l>1 (after xtabond or xtdpdsys, with variance estimated in a standard way) estat abond, artests(3) 78 26