Based on the textbook by Verbeek: A Guide to Modern Econometrics Robert M. Kunst robert.kunst@univie.ac.at University of Vienna and Institute for Advanced Studies Vienna May 16, 2013
Outline Univariate time series Multivariate time series Panels Basics The dynamic linear panel model
Basics Definition of a panel A panel data set contains repeated observations for the same units, collected over a number of periods. There is a cross-section dimension (i = 1,...,N) and a time dimension (t = 1,...,T). If there is at least some constancy across time and individuals, panels are certainly more informative than pure time series or pure cross sections. Micro panels have N T, macro panels have N in the same magnitude as T.
Basics Linear regression in a panel Consider the model y it = β 0 +x itβ +ε it, with an intercept β 0 and a K dimensional vector of regressors (or covariates). Interest focuses on the K dimensional coefficient vector β = (β 1,...,β K ). There is an individual index i and a time index t.
Basics Random effects In the random effects model, errors ε it are assumed to be systematically correlated over time. We write ε it = α i +u it, with α i representing individual though unobserved characteristics ( effects ) and u it the also unobserved (white-noise) idiosyncratic errors. The random-effects model assumes that observed covariates x and unobserved characteristics α are uncorrelated.
Basics Fixed effects In the fixed effects model, individual unspecific characteristics are represented by individual dummies y it = α i +x itβ +u it, with α i treated as coefficients to be estimated, just like β, and u it uncorrelated across time and individuals.
The fixed-effects estimator The linear efficient estimator for the fixed-effects regression is the fixed-effects (FE) estimator or least-squares dummy variables estimator (LSDV). There are two ways to implement it. The straightforward way to calculate FE is to run OLS on the regression N y it = α j δ j i +x itβ +u it, j=1 with N individual dummy variables (δ j i = 1 if i = j and 0 otherwise). This is inconvenient if N is large.
FE estimates by mean correction Denote time averages for individuals by ȳ i, x i,ū i. Then, and also ȳ i = α i + x iβ +ū i, y it ȳ i = (x it x i ) β +u it ū i, a regression model that satisfies the Gauss-Markov conditions and is estimated by OLS. This estimate is numerically identical to the first FE version. The second way is more convenient, but it does not directly yield effect estimates ˆα i.
Properties of the FE estimator With strictly exogenous covariates (x uncorrelated with u) and white-noise errors, the FE estimator is linear efficient (BLUE) and consistent (for T or N ). Estimates of α i can only be consistent for T. If the covariates are not strictly exogenous for example, lagged y or variables depending on lagged y all static-panels estimators can be severely biased and are inconsistent for N and T fixed.
Covariance matrix of the FE estimator Under the mentioned conditions, the FE estimator has the variance matrix { N } 1 T varˆβ FE = σu 2 (x it x i )(x it x i ). i=1 t=1 An estimate for σ 2 u is provided by ˆσ 2 u = 1 (T 1)N N i=1 t=1 T ûit 2, for the residuals û it.
The first-difference estimator A simple alternative way to remove the effects is to take first differences y it = y it y i,t 1 : y it = x it β + u it. If u are white noise, u are autocorrelated MA(1), and GLS can be applied to this equation. The first-differences (FD) estimator is slightly less efficient than FE, though it works for some deviations from exogeneity. Thus, strong discrepancies between FE and FD indicate a violation of exogeneity.
Differences in differences If the covariate is a treatment indicator ( 0 for non-treated and 1 for treated), the FD estimate ˆβ FD represents the response of those individuals whose classification to treatment and control vary over time. An extreme case is T = 2. This is why ˆβ FD is often called the difference-in-difference (DID) estimator: differences in treatment assignment and differences over time.
The random-effects estimator The model y it = β 0 +x itβ +α i +u it, α i IID(0,σ 2 α),u IID(0,σ 2 u), is essentially a GLS model. Defining u i = (u i1,...,u it ) and ι T as a T vector of ones, one has E(α i ι T +u i )(α i ι T +u i ) = Ω = σ 2 αι T ι T +σ2 ui T, which has a relatively simple inverse for the GLS calculation. Note that ι T ι T is a T T matrix of ones.
Implementing the RE estimator The inverse Ω 1 has the representation with Ω 1 = σ 2 u {(I T ι T ι T )+ψι Tι T }, 1 ψ = 1+Tσα 2. /σ2 u Estimates for σ 2 u and σ 2 α are obtained from the residuals in a preliminary FE estimation. The so defined feasible GLS estimator is called the random-effects (RE) estimator. Note that ψ = 1 defines OLS and that ψ = 0 yields FE.
Properties of the RE estimator If covariates are exogenous and uncorrelated with the effects, the RE estimator is linear efficient (BLUE). If covariates are not exogenous, the RE estimator will be biased and inconsistent. If covariates are correlated with effects, the RE estimator will also be biased and inconsistent. The RE estimator allowing for this correlation will be identical to the FE estimator.
Covariance matrix of the RE estimator Under the mentioned conditions, the RE estimator ˆβ RE = { N } 1 T (x it x i )(x it x i ) +ψt( x i x)( x i x) i=1 t=1 { N i=1 t=1 } T (x it x i )(y it ȳ i )+ψt( x i x)(ȳ i ȳ) has the variance matrix { N } 1 T varˆβ RE = σu 2 (x it x i )(x it x i ) +ψt( x i x)( x i x), i=1 t=1 which is estimated by plugging in an estimate for σ 2 u.
Fixed effects or random effects? FE works even in the RE model, while RE does not work in the FE model. In doubtful cases, it makes sense to prefer FE; FE is usually preferred if N is small or heterogeneous by definition. RE is often preferred if N is large or if individuals can be seen as a random sample from a homogeneous population; If T is large, RE approaches FE, FE will often be used; If the response to time-constant covariates is of interest, some RE model must be used, as FE does not work; If by definition or by statistics some covariates and effects correlate, FE should be used.
The Hausman test The Hausman test principle applies to situations where There exist two estimators ˆθ and θ; θ is efficient under the null and inconsistent under the alternative; ˆθ is consistent under the null and under the alternative. This describes ˆβ RE and ˆβ FE for the null of the RE model and the alternative of a model with correlation of covariates and effects that is equivalent to the FE model. The Hausman test rejects if the RE model is invalid and FE should be used. Some authors sound a warning that the test is not always reliable.
The Hausman test statistic The Hausman test statistic is calculated as ξ H = (ˆβ FE ˆβ RE ) {var(ˆβ FE ) var(ˆβ RE )} 1 (ˆβ FE ˆβ RE ), which is, under the null, asymptotically χ 2 distributed with K degrees of freedom, where K is the dimension of β.
The dynamic linear panel model Dynamic linear regression with panel data Consider the simplest model of this type, with a first-order autoregressive term y it = x itβ +γy i,t 1 +α i +u it Because y i,t 1 depends on α i by definition, regressor and effects will always be correlated. OLS assumes an error ε it = α i +u it and is clearly biased. FE will also be biased, as the correction term T 1 T t=1 y it = ȳ i is correlated with u it.
The dynamic linear panel model The Nickell bias The Nickell bias is the bias of the FE estimator in a dynamic panel model for finite T. For small T, the bias is substantial. For T, the bias disappears. However, for N, it does not disappear. FE is inconsistent for N. The solution suggested in the literature is using instrumental variables (IV) estimation. The instrument must be correlated with the lag y i,t 1 but uncorrelated with u it. Skilfully designed IV estimators have a small bias but they are consistent under reasonable assumptions.
The dynamic linear panel model The Arellano-Bond estimator: the idea The Arellano-Bond estimator is a particular IV estimator. It considers the model in differences y it = x it β +γ y i,t 1 + u it, t = 3,...,T, thus eliminating the effects α i. y i,1 can be an instrument for y i,3 ; y i,1 and y i,2 for y i,4 ;...; y i,1,...,y i,t 3 for y i,t. The special structure of the residual covariance matrix for u it is exploited in a GLS step. This IV-GLS method is the most popular estimator in dynamic panel regressions.