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Econometrics of Panel Data Jakub Mućk Meeting # 6 Jakub Mućk Econometrics of Panel Data Meeting # 6 1 / 36

Outline 1 The First-Difference (FD) estimator 2 Dynamic panel data models 3 The Anderson and Hsiao estimator 4 Generalized Method of Moments (GMM) 5 The Arellano Bond estimator 6 A system GMM estimator Jakub Mućk Econometrics of Panel Data The First-Difference (FD) estimator Meeting # 6 2 / 36

The First-Difference (FD) estimator I The First-Difference (FD) estimator is an alternative estimation technique that eliminates the fixed effect as well as time invariant regressors. Note that y it = α i + β 1 x 1it +... + β k x kit + u it for t = 1,... T, (1) y it 1 = α i + β 1 x 1it 1 +... + β k x kit 1 + u it 1 for t = 2,... T, (2) and differencing both equations yields: y it = β 1 x 1it +... + β k x kit + u it, (3) where is the well-known (from time series analysis) first-difference operator, i.e. z t = z t z t 1. The parameters in (3) can be estimated with the least squares. In the matrix form: β FD = ( X X) 1 X y. (4) Jakub Mućk Econometrics of Panel Data The First-Difference (FD) estimator Meeting # 6 3 / 36

The First-Difference (FD) estimator II The estimates of fixed effects can be also recovered: ˆα FD i = ȳ i x i ˆβFD. (5) If the error term in (3) is not correlated with independent variable (weak exogeneity) then the least squares estimator is unbiased and consistent. The above assumption is less restrictive than in standard FE model: E ( u it x it ) = E (u it u it 1 x it x it 1 ) = 0. (6) The FE estimator is more efficient when the disturbances are not serially correlated and homoskedastic. But If uit is driven by random walk (autocorrelation with ρ = 1) then the FD estimator is more efficient. Jakub Mućk Econometrics of Panel Data The First-Difference (FD) estimator Meeting # 6 4 / 36

Dynamic panel data models Dynamic linear panel data model: y it = γy it 1 + x itβ + u it, (7) where uit = µ i + ε it and ε it N (0, σε), 2 γ is the autoregressive parameter, yit 1 is the lagged dependent variable, xit is the vector of independent variables. Jakub Mućk Econometrics of Panel Data Dynamic panel data models Meeting # 6 6 / 36

Dynamic panel data models Dynamic linear panel data model: where y it = γy it 1 + x itβ + u it, (7) uit = µ i + ε it and ε it N (0, σε), 2 γ is the autoregressive parameter, yit 1 is the lagged dependent variable, xit is the vector of independent variables. Remarks: We assume that yit is the stable (conditional on x it) process = γ < 1. In other words, the effect of idiosyncratic shock (ε it) dies out. The independent variables (xit) are assumed to be strictly exogenous. µi is the individual-specific (random or fixed) effect. Each observation can be written as: t y it = γ t y i0 + γ j β x it j + 1 t 1 γt 1 γ µi + γ j u it j, (8) j=0 where y i0 is the (non-stochastic) initial value. Jakub Mućk Econometrics of Panel Data Dynamic panel data models Meeting # 6 6 / 36 j=0

Dynamic panel data models bias of the FE estimators The demeaning transformation used to get the within estimator new creates new independent variables that are correlated with the error term. As a result, the standard OLS estimator is inconsistent. General intuition: The within estimator for the panel AR(1) model: y it ȳ i = (µ i µ i) + γ (y it 1 ȳ i 1) + (ε it ε i), (9) where ȳ i 1 = 1/(T 1) T yit 1. t=2 The mean of the lagged dependent variable (ȳi 1) is correlated with ε i even if the error term is not autocorrelated. The average ε i contains the lagged error term ε it 1 and, therefore, it is correlated with y it 1. Taking the probability limit (plim) of the FE estimator (as N ): plimˆγ FE = γ + 1 NT (y it 1 ȳ i 1 ) (ε it ε i ) 1 NT (y it 1 ȳ i 1 ) 2 (10) it can be observed that the correlations between the lagged dependent variable (ȳ i 1 ) and error term will lead to inconsistency of the OLS estimator. Jakub Mućk Econometrics of Panel Data Dynamic panel data models Meeting # 6 7 / 36

Dynamic panel data models bias of the FE estimators Nickell s (1981) bias. The small T bias of the FE estimator as N : plim (ˆγ ) ( ) [ ( )] FE (1 + γ) γ = 1 1 1 γ T 1 1 T T 1 γ T 2γ 1 1 1 1 γ T (1 γ) T T 1 γ (11) The bias of the FE estimator depends on T as well as γ. For reasonably large T it can be approximated: plim (ˆγ FE γ ) (1 + γ) T 1 (12) but when T = 2 then plim (ˆγ FE γ ) (1 + γ) 2 (13) The bias in the dynamic fixed effect model is caused by elimination of the individual-specific effect from each observation. It creates a correlation of order 1/T between explanatory variables and error term. Jakub Mućk Econometrics of Panel Data Dynamic panel data models Meeting # 6 8 / 36

Dynamic panel data models bias of the RE estimators Consider the RE AR(1) model: where ε it N (0, σ 2 ε) and µ it N (0, σ 2 µ). y it = γy it 1 + ε it + µ i, (14) In the RE model, the quasi-demeaning also leads to correlation between the transformed lagged dependent variable (ỹ it 1 = y it 1 θȳ i 1 ) and the transformed error term ( ε it = ε it θ ε i ). Therefore, the RE estimates will be biased. For t 1 the dependent variable: y it 1 = γy it 2 + ε it 1 + µ i, (15) also depends on the random individual-specific effect. If so, then the assumption that the individual effects are independent of the explanatory variable (in our case also y it 1 ) is not satisfied and E (µ i y it 1 ) 0. (16) Jakub Mućk Econometrics of Panel Data Dynamic panel data models Meeting # 6 9 / 36

Dynamic panel data models bias of the FD estimators The FD AR(1) estimator: is also biased. y it y it 1 = (µ i µ i ) + γ (y it 1 y it 2 ) + ε it ε it 1 (17) To illustrate the bias of the FD estimator it s useful to recall y i,t 1. y it 1 = γy it 2 + ε it 1 + µ i + ε it 1. (18) In the (18) y it 1 depends on the error term ε it 1. At the same time, in the (17) y it 1 is the explanatory variable and the error term, given by ε it ε it 1, contains the lagged error term from the non-transformed model. Therefore, the lagged dependent variable is correlated with the error term also in the FD model. Jakub Mućk Econometrics of Panel Data Dynamic panel data models Meeting # 6 10 / 36

The MC exercise To illustrate the magnitude of the Nickell s bias we run the MC simulations. Let s assume that the true DGP (data generating process) is a simple panel AR(1) process: y it = γy it 1 + u it, (19) where the error component is quite standard: where µ i N (0, σ 2 µ) and ε it N (0, σ 2 ε). We will consider the FE estimator. The MC settings: γ = 0.9 ( in the second exercise also 0.5 and 0.95) T {3, 5, 10, 30}. σµ = 0.5 and σ ε = 0.25. N = 100 (the cross-sectional dimension). 1000 replications. u it = µ i + ε it (20) Jakub Mućk Econometrics of Panel Data Dynamic panel data models Meeting # 6 11 / 36

0.5 0.6 0.7 0.8 0.9 T=3 T=5 0 10 20 30 0 1 2 3 4 5 6 0 5 10 15 T=10 0.5 0.6 0.7 0.8 0.9 T=30 0 20 40 60 80 0.5 0.6 0.7 0.8 0.9 0.5 0.6 0.7 0.8 0.9 Jakub Mućk Econometrics of Panel Data Dynamic panel data models Meeting # 6 12 / 36

0.5 0.6 0.7 0.8 0.9 rho=.5 0 20 40 60 80 rho=.9 T=3; T=5; T=10; T=30. 0 5 10 15 20 25 0 50 100 150 0.2 0.1 0.0 0.1 0.2 0.3 0.4 0.5 rho=.95 0.6 0.7 0.8 0.9 Jakub Mućk Econometrics of Panel Data Dynamic panel data models Meeting # 6 13 / 36

Dynamic panel data models The standard estimators (FE, RE, FD) fail to account for dynamics in the dynamic panel data models. This is due to the fact that the lagged dependent variable becomes endogenous (correlated with error term). The dynamic panel data (DPD) models are designed to account for this endogeneity. It is important when T is relatively small = micro data. When T is large the Nickell s bias is relatively small. Which T is sufficiently large to ignore the Nickell s bias? Jakub Mućk Econometrics of Panel Data Dynamic panel data models Meeting # 6 14 / 36

The Anderson and Hsiao estimator I Anderson and Hsiao (1981) propose estimator that simply uses the IV. Starting point: the FD estimator: y ti = γ y ti 1 + β 1 x 1ti +... + β k x kti + ε it. (21) Problem: y ti 1 is correlated with the error term ε it = ε it ε it 1. Use twice lagged level of dependent variable y it 2 as an instrument for y ti 1. By construction, y it 2 is not correlated with the error term ε it but is correlated with endogenous variable, i.e. y ti 1. In general, one might use the twice lagged differences y ti 2 = y ti 2 y ti 3 as a valid instrument for endogenous variable y ti 1. But: Using yti 2 as the instrumental variable = more data. Using yti 2 as the instrumental variable = larger asymptotic variance of estimator. The AH estimator delivers consistent but not efficient estimates of the parameters in the model. This is due to the fact that the IV doesn t exploit all the available moments conditions. Jakub Mućk Econometrics of Panel Data The Anderson and Hsiao estimator Meeting # 6 16 / 36

The Anderson and Hsiao estimator II The IV estimator also ignores the structure of the error component in the transformed model. The autocorrelation in the first differences errors leads to inconsistency of the IV estimates. The IV estimates would be inconsistent when other regressors are correlated with the error term. Jakub Mućk Econometrics of Panel Data The Anderson and Hsiao estimator Meeting # 6 17 / 36

Empirical example Arellano Bond dataset which contains unbalanced panel for 140 UK companies. Model: 1 2 2 5 n it = γn it 1 +β 1 n it 2 + β 2+j w it j + β 4+j k it j + β 7+j ys it j + β 10+j D ji j=0 j=0 j=0 j=0 (22) nit - the employment, wit - the wage level, kit - the capital stock, ysit - the output in the firm s sector, Djit - the period-specific dummy variables. Jakub Mućk Econometrics of Panel Data The Anderson and Hsiao estimator Meeting # 6 18 / 36

Empirical example OLS FE n it 1 1.045 0.733 (0.052) (0.060) n it 2 0.077 0.139 (0.049) (0.078) w it 0.524 0.560 (0.174) (0.160) w it 1 0.477 0.315 (0.172) (0.143) k it 0.343 0.388 (0.049) (0.057) k it 1 0.202 0.081 (0.065) (0.054) k it 2 0.116 0.028 (0.036) (0.043) ys it 0.433 0.469 (0.179) (0.171) ys it 1 0.768 0.629 (0.251) (0.207) ys it 2 0.312 0.058 (0.132) (0.133) Note: the superscripts, and denote the rejection of null about parameters insignificance at 1%, 5% and 10% significance level, respectively. Jakub Mućk Econometrics of Panel Data The Anderson and Hsiao estimator Meeting # 6 19 / 36

Empirical example OLS FE AH n it 1 1.045 0.733 2.308 (0.052) (0.060) (1.973) n it 2 0.077 0.139 0.224 (0.049) (0.078) (0.179) w it 0.524 0.560 0.810 (0.174) (0.160) (0.262) w it 1 0.477 0.315 1.422 (0.172) (0.143) (1.179) k it 0.343 0.388 0.253 (0.049) (0.057) (0.145) k it 1 0.202 0.081 0.552 (0.065) (0.054) (0.615) k it 2 0.116 0.028 0.213 (0.036) (0.043) (0.240) ys it 0.433 0.469 0.991 (0.179) (0.171) (0.463) ys it 1 0.768 0.629 1.938 (0.251) (0.207) (1.438) ys it 2 0.312 0.058 0.487 (0.132) (0.133) (0.510) Note: the superscripts, and denote the rejection of null about parameters insignificance at 1%, 5% and 10% significance level, respectively. Jakub Mućk Econometrics of Panel Data The Anderson and Hsiao estimator Meeting # 6 20 / 36

Outline 1 The First-Difference (FD) estimator 2 Dynamic panel data models 3 The Anderson and Hsiao estimator 4 Generalized Method of Moments (GMM) 5 The Arellano Bond estimator 6 A system GMM estimator Jakub Mućk Econometrics of Panel Data Generalized Method of Moments (GMM) Meeting # 6 21 / 36

Generalized Method of Moments (GMM) The standard classical methods, e.g., the Maximum Likelihood (ML) method, requires a complete specification of the model that is considered to be estimated. This includes also the probability of distribution of the variable of interest. Contrary to the ML method, the Generalized Method of Moments (GMM) requires only a set moment conditions that are implied by assumption of the underlying econometric model. The GMM method is attractive when: there is a variety of moment or orthogonality conditions that are deduced from the assumption of the theoretical model; the economic model is complex, i.e., it s difficult to write down a tractable and applicable likelihood function, to overcome the computational complexities associated with the ML estimator. Jakub Mućk Econometrics of Panel Data Generalized Method of Moments (GMM) Meeting # 6 22 / 36

Generalized Method of Moments (GMM) Let s assume that a sample of T observations is drawn from the joint probability distribution: f (w 1, w 2,..., w T, θ 0) (23) where θ 0 is the (q 1) vector of true parameters and w t contains one or more endogenous and/or exogenous variables. Population moments condition: E [m (w T, θ 0)] = 0, for all t. (24) where m( ) is the r-dimensional vector of functions. Three cases: 1 q > r = the parameters in θ are not identified; 2 q = r = the parameters in θ are exactly identified; 3 q < r = the parameters in θ are overidentified and the moments conditions have to be restricted in order to deliver a unique θ in estimation. This can be done by the means of a weighting matrix (A T). Estimation bases on the empirical counterpart of E [m (w T, θ 0)]: M T(θ) = 1 T m (w T, θ 0), (25) T where M T(θ) is the r-dimensional vector of sample moments. t=1 Jakub Mućk Econometrics of Panel Data Generalized Method of Moments (GMM) Meeting # 6 23 / 36

GMM examples Linear regression: Consider the standard linear regression: y t = x tβ + ε t (26) Under the standard (classical) assumption, the population conditions is following: E (x t, ε t) = E [ x t, ( y t x tβ )] = 0 for t 1,..., T. (27) Linear regression with endogenous variables: Consider the standard linear regression with endogenous variables: y t = x tβ + ε t (28) where E(x t, ε t) 0. The population conditions: E (z t, ε t) = E [ z t, ( y t x tβ )] = 0 for t 1,..., T (29) where z T is the set of the instrumental variables that satisfies the above orthogonality conditions. Jakub Mućk Econometrics of Panel Data Generalized Method of Moments (GMM) Meeting # 6 24 / 36

The GMM and GIVE estimators I The GMM estimator of θ bases on: ˆθ T = argmin θ Θ {M T(θ)A T M T (θ)}, (30) where A T is a r r positive semi-define, possibly random weighting matrix. We wish to choose the weighting matrix that minimizes the covariance matrix of ˆθ. This provides the efficient estimator. Other weighting matrices would lead to less efficient estimators of θ. The general instrumental variable estimator (GIVE) combines all available instruments to estimates the unknown parameters. In this case, the number of instruments can be larger than number of parameters to estimate (r > k). Jakub Mućk Econometrics of Panel Data Generalized Method of Moments (GMM) Meeting # 6 25 / 36

The GMM and GIVE estimators II The starting point: the r population conditions: E (z t, ε t ) = E [z t, (y t x tβ)] = 0 for t 1,..., T (31) where z t is the set of (r) instruments, x t is the k-dimensional vector of regressors. The regressors are endogenous, i.e., E (x t, ε t ) 0 while the error term is idiosyncratic, ε t N (0, σ 2 ε). The instruments are correlated with x t but not correlated with the error term. This implies the following sample moments: M T (θ) = 1 T T z t (y t β x t ). (32) t=1 It can be shown that the GIVE estimator is given as: ˆβ GIVE = (X P z X) 1 X P z Y, (33) where P z = Z (Z Z) 1 Z. The matrix Z collects all instruments, the matrix X stands for the regressors while Y denotes the observations of the dependent variable. Jakub Mućk Econometrics of Panel Data Generalized Method of Moments (GMM) Meeting # 6 26 / 36

The GMM and GIVE estimators III The estimator of the variance matrix of ˆβ GIVE is as follows: Var ( ˆβGIVE ) = ˆσ 2 GIVE (X P z X) 1. (34) where the estimated variance of the error term bases on the variance of the residuals from the considered regression: ˆσ 2 GIVE = 1 T K ˆε GIVEˆε GIVE. (35) In analogous fashion to the basic linear models, the robust standard error (e.g. hetereoskedasticity-consistent) can be computed. Jakub Mućk Econometrics of Panel Data Generalized Method of Moments (GMM) Meeting # 6 27 / 36

Sargan s general test for misspecification In the GIVE estimation we use r instruments. Are these instruments valid? Consider following test statistics: χ 2 SM = Q( ˆβ GIVE ) ˆσ GIVE 2, (36) where Q( ˆβ GIVE ) = (y X ˆβ GIVE) Pz ( y X ˆβ GIVE) (37) Under the null the regression is correctly specified and the r instruments Z are valid instruments. Sargan s misspecification statistics is χ 2 distributed with r k degrees of freedom. Jakub Mućk Econometrics of Panel Data Generalized Method of Moments (GMM) Meeting # 6 28 / 36

The Arellano Bond estimator Arellano and Bond (1991) suggest using a GMM approach based on all available conditions. Starting point: the FD estimator: Valid instruments: [t=2 or t=1]: no instruments, y it = γ y it 1 + β x it + ε it (38) [t=3]: the valid instrument for yi2 = (y i2 y i1) is y i1, [t=4]: the valid instruments for yi3 = (y i3 y i2) is y i2 as well as y i1, [t=5]: the valid instruments for yi4 = (y i4 y i3) is y i3 as well as y i2 and y i1, [t=6]: the valid instruments for yi5 = (y i5 y i4) is y i4 as well as y i3, y i2 and y i1, [t=t]: the valid instruments for yit 1 = (y it 1 y it 2) is y it 2 as well as y it 3,..., y i1. Hence, there is a total of (T 1)(T 2)/2 available instruments or moment conditions for y it 1. In general, it can be written as: E [y is ( y it γ y it 1 β x it )] = 0 for s = 0,..., t 2 and t = 2,..., T (39) Jakub Mućk Econometrics of Panel Data The Arellano Bond estimator Meeting # 6 30 / 36

The Arellano Bond estimator Consider the following specification: where y i. = y i2 y i3. y it, yi. 1 = y i. = γ y i. 1 + X i. β + ε i., (40) y i1 y i2. y it 1, Xi. = x i2 x i3. x it, εi. = The corresponding matrix of instruments for the lagged difference: y i1 0... 0 0 y i1, y i2... 0 W i =......, 0 0... y i1, y i2,..., y it 2 Then the moment conditions can be described as: ε i2 ε i3. ε it. E [W i ε i. ] = 0 (41) Jakub Mućk Econometrics of Panel Data The Arellano Bond estimator Meeting # 6 31 / 36

The Arellano Bond estimator Finally, the GMM estimator that takes into account the formulated moment conditions can be applied: where ˆλ GMM = (G ZS N Z G) 1 G ZS N Z y (42) ˆλGMM = [ˆγ ] GMM ˆβGMM, G = ( y 1, X), Z = (W, X). SN is the optimal weighting matrix. The matrix S N is usually calculated from initial estimates, e.g., IV estimates. ( N ) 1 S N = Z i.ê i. ê i.z i., (43) i=1 where ê i. stands for the residuals from the initial estimates. The above procedure refers to two-step GMM estimator. Alternatively, onestep estimator can be applied. One-step estimator takes into account the dynamic structure of the error term. Jakub Mućk Econometrics of Panel Data The Arellano Bond estimator Meeting # 6 32 / 36

The Arellano Bond estimator general remarks The Arellano-Bond (AB) estimator is usually called difference GMM. The AB estimator deteriorates when: yit exhibits a substantial persistence, i.e., γ is close to unity. the variance of unit-specific error component (σµ) increases relatively to the variance of the idiosyncratic error term (σ ε). Note that for long panel (large T) the number of instruments increases dramatically, i.e., r = T/(T 1)/2. Consistency of the GMM estimator bases on the assumption that the transformed error term is not serially correlated, i.e., E ( ε, i,t, ε, i,t 2 ) = 0. It s crucially to test whether the second-order autocorrelation is zero for all periods in the sample. Conventionally, test bases on residuals from the first difference equation. Jakub Mućk Econometrics of Panel Data The Arellano Bond estimator Meeting # 6 33 / 36

A system GMM estimator I Blundell and Bond (1998) propose to include additional moment restrictions. These additional moment restrictions are imposed on the distribution of initial values, i.e., y i0. This set of restrictions is important when γ is close to unity and/or when σ µ/σ ε becomes large. Consider simply panel AR(1) without regressors. Then, y i0 = under the following assumption: µ i 1 γ + ε i0 for i = 1,..., N. (44) E ( y i1 µ i ) = 0 (45) It can be show that if the above condition is satisfied then the following T 1 moment conditions can be used: E [(y it γy it 1 ) y it 1 ] = 0. (46) Jakub Mućk Econometrics of Panel Data A system GMM estimator Meeting # 6 35 / 36

A system GMM estimator II Note that the system estimator combines the standard AB estimator and equation for levels (with the corresponding T 1 moment conditions). The instrument matrix: Z = Z AB 0 0... 0 0 y i2 0... 0 0 0 y i3... 0....... 0 0 0... y it 1 where Z AB is the instrument matrix from the Arellano-Bond estimator. (47) Jakub Mućk Econometrics of Panel Data A system GMM estimator Meeting # 6 36 / 36