Abstract
This paper studies the generalized spatial two stage least squares (GS2SLS) estimation of spatial autoregressive models with autoregressive disturbances when there are endogenous regressors with many valid instruments. Using many instruments may improve the efficiency of estimators asymptotically, but the bias might be large in finite samples, making the inference inaccurate. We consider the case that the number of instruments K increases with, but at a rate slower than, the sample size, and derive the approximate mean square errors (MSE) that account for the trade-offs between the bias and variance, for both the GS2SLS estimator and a bias-corrected GS2SLS estimator. A criterion function for the optimal K selection can be based on the approximate MSEs. Monte Carlo experiments are provided to show the performance of our procedure of choosing K.
Highlights
This paper considers the instrumental variable (IV) estimation of the spatial autoregressive (SAR)model with SAR disturbances (SARAR model) in the presence of endogenous regressors and many instruments
We study the case where the number of instruments increases with the sample size and derive asymptotic distributions of the generalized spatial two stage least squares (GS2SLS) estimator and a bias-corrected GS2SLS (CGS2SLS) estimator based on the leading-order many-instrument bias
For the CGS2SLS estimator, the expression for the approximate mean square errors (MSE) is more complicated than that in [3], because the term resulting from the estimation error of the leading order bias involves the asymptotic distributions of the first two stage estimators and an additional term appears due to the estimation of the spatial autoregressive parameter in the error process
Summary
This paper considers the instrumental variable (IV) estimation of the spatial autoregressive (SAR). For the CGS2SLS estimator, the expression for the approximate MSE is more complicated than that in [3], because the term resulting from the estimation error of the leading order bias involves the asymptotic distributions of the first two stage estimators and an additional term appears due to the estimation of the spatial autoregressive parameter in the error process. If Wn = Mn , Mn FK,n generates some identical IVs as those in FK,n In this case, we can take QK,n = [FK,n , Wnp+1 ψq,n ].) The asymptotic variance of the 2SLS estimator decreases when a linear combination of IVs approximates the conditional mean of the endogenous variables more closely. A list of notations, lemmas and proofs are collected in the appendices
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