Consider the panel data model \(y_{i t}=\alpha+\beta x_{i t}+\gamma w_{i t}+u_{i t}, i=1, \ldots, N, t=\) \(1, \ldots, T\), where for simplicity there is no individual-specific effect. Suppose the scalar regressor \(x_{i t}\) is correlated with \(u_{i s}\) for all \(t\) and \(s\). For each of the following parts state whether consistent IV estimation of \(\beta\) and \(\gamma\) is possible, and if so give all the suitable instruments, based on the discussion in Section 22.2. Assume that three periods of data are available, so \(T=3\), and note that a variable may not be available as an instrument in all years, and that in different years different instruments may be available.

(a) The regressor \(w_{i t}\) satisfies the summation assumption \(\mathrm{E}\left[\sum_{t} w_{i t} u_{i t}\right]=0\).

(b) The regressor \(w_{i t}\) satisfies the contemporaneous exogeneity assumption \(\mathrm{E}\left[w_{i t} u_{i t}\right]=0, t=1, \ldots, 3\).

(c) The regressor \(w_{i t}\) satisfies the weak exogeneity assumption \(\mathrm{E}\left[w_{i s} u_{i t}\right]=\) \(0, s \leq t, t=1, \ldots, 3\).

(d) The regressor \(w_{i t}\) satisfies the strong exogeneity assumption \(E\left[w_{i t} u_{i t}\right]=\) \(0, s, t=1, \ldots, 3\).

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22.2. GMM Estimation of Linear Panel Models The panel regression models in Chapter 21 restricted the scalar dependent variable yit to depend on just the contemporaneous value of regressors Xit, even though potentially all of Xi1,..., XiT could be regressors under the Chapter 21 assumption of strong ex- ogeneity. This introduces the possibility of more efficient estimation using excluded regressors from other periods as instruments in the current period. Furthermore, regressors in other periods may be valid instruments for current- period regressors that are either endogenous or lags of the dependent variable. So in- struments are readily available to permit consistent IV estimation in situations where failure of the strong exogeneity assumption leads to inconsistency of the Chapter 21 estimators. This section provides a general presentation of panel GMM estimation, a very use- ful framework for panel IV estimation that is used extensively throughout Sections 22.2-22.5. Then we introduce the use of exogenous variables (regressors or instru- ments) in periods other than the current period as an instrument. Once this ground- work is laid it is a relatively minor adaptation to incorporate fixed or random effects, typically included in panel models. This is deferred to subsequent sections.