Consistent Estimation with Weak Instruments in Panel Data

This note analyzes the asymptotic distribution for instrumental variables regression for panel data when the available instruments are weak. We show that consistency can be established in panel data.


Motivation and Results
In recent year, economists have been concerned with the problem of weak instruments or partial identi…cation, see Stock, Wright and Yogo (2002) for an excellent survey. Economists found that the …rst stage F statistic in the two stage least squares (2SLS) regression is often low, say, less than 10. In this case, the usual asymptotic normal approximations can be quite poor, even if the number of observations is large. To provide better asymptotic approximations in this case, Staiger and Stock (1997) derive the weak-instrument asymptotics for instrumental variables estimators. Staiger and Stock show that the 2SLS is inconsistent (i.e., converges to a random variable) and has a nonstandard limiting distribution. In this note we study the asymptotics of 2SLS with weak instruments in panel models. We show that the consistency of 2SLS can be established in panel data. We use (n; T ) seq ! 1 to denote the sequential limit, i.e., n ! 1 followed by T ! 1.
Consider the following panel linear IV regression model with a single endogenous regressor and ; T; where y t and Y t are n 1 vectors of observations on endogenous variables, Z t is a n k matrix of instruments, is a k 1 coe¢ cient vector, and u t and v t are n 1 vectors of disturbance terms.  where the elements of are 2 u ; uv and 2 v , and let = uv = ( u v ). In this note, the errors are assumed be iid for simplicity. This assumption can be relaxed to weak dependence across time series and cross-section at the expense of complicated notations and will be studied in a di¤erent paper. Equation (1) is the structural equation and is the scalar parameter of interest. The reduced-form equation (2) relates the endogenous regressor to the instruments. As proposed by Staiger and Stock (1997), the following assumption is used to describe the nature of weak instruments.
The strength of instrument can be measured by the concentration parameter 2 t , which is de…ned as Under Assumption 1 we obtain as n ! 1 where we assume for a give t, Then the panel 2SLS estimator iŝ Theorem 1 Under assumption 1, as (n; T ) seq ! 1.

Proof and Discussion
We …rst present the following lemma: Proof. Consider (1) and (2). Following Rothenberg (1984), we know As stated in Rothenberg (1984), the (z tu ; z tv ) 0 is bivariate normal with zero means, unit variances, and correlation coe¢ cient . The random variable s tuv has mean k and variance k 1 + 2 and s tvv has mean k and variance 2k.
It is clear that as n ! 1 where 1t = v u ( z tu + s tuv ) and 2t = 2 v 2 + 2 z tv + s tvv : by a law of large numbers (LLN) and E (z tu ) = 0 and E (s tuv ) = k. Similarly,

Consider (2). First we write
where t and u t are independent. Then e.g., Corollary 10.9.2 in Graybill (1983).
By a central limit theorem we have Similarly, Since u ; v ; 2 and k are all positive, the sign of the bias term of panel 2SLS,^ 2SLS ; is determined by which is the sample correlation of disturbance terms u t and v t . When = 0, i.e., Y t is uncorrelated with u t , 2SLS becomes consistent; when has the same sign as ,^ 2SLS is overestimated; when has a di¤erent sign from ,^ 2SLS is underestimated. Once u ; v ; and 2 are consistently estimated, the bias can be corrected by using a bias-corrected estimator. For example, the bias-corrected estimator can be constructed as where^ u ;^ v and^ can be estimated from the residualsû it of 2SLS regression andv it of the …rst stage regression. A consistent estimator of can be constructed by following Moon and Phillips (2000).
In the cross-sectional case, when the concentration parameter stays constant as the sample size grows, the signal of the model is too weak comparing to the noise. Hence the model is weakly identi…ed, i.e., the two stage least square estimator is inconsistent, and more importantly, 2SLS converge to a random variable.
However, in the panel set-up, if time series dimension is large, the weak signal can be strengthened by repeating regression across the time series dimension. It is, in spirit, similar to the argument of establishing the consistency in the panel spurious regression, e.g., Phillips and Moon (1999) and Kao (1999).