Consider the panel GMM estimator of Section 22.2.1.

(a) Show that minimization with respect to \(\beta\) of the quadratic function \(Q_{N}(\beta)\) given after (22.3) yields the panel GMM esimator given after \(Q_{N}(\beta)\) that is expressed using summation notation.

(b) Show that this estimator is equivalent to the estimator defined in (22.4).

(c) For simplicity suppose that the matrices \(\mathbf{Z}\) and \(\mathbf{X}\) in (22.4) are nonstochastic and that \(\mathbf{y}=\mathbf{X} \boldsymbol{\beta}+\mathbf{u}\) where \(\mathbf{u}\) has mean 0 and variance \(\Omega\). Obtain the finite sample variance matrix of the estimator in (22.4) and compare this to the asymptotic results in (22.5).

(d) Simplify the panel GMM estimator in the case that \(r=K\).

Transcribed Image Text:

Consider the linear panel model 22.2.1. Panel GMM Yit = x+uit, (22.1) where the regressors Xit may have both time-varying and time-invariant components and may include an intercept. Here there is no individual-specific effect i, an as- sumption relaxed from Section 22.3 on, and xit is assumed to include only current- period variables, an assumption relaxed in Section 22.5. Observations are assumed to be independent over i and a short panel with T fixed and N is assumed. Begin by stacking all T observations for the ith individual, yi = X;+u;, where y; and u; are T x 1 vectors and X; is a T x K matrix with tth row x, so -0-0-0 (22.2)