Consider estimation of the fixed effects linear regression model \(y_{i t}=\alpha_{i}+\mathbf{x}_{i t}^{\prime} \beta+\) \(\varepsilon_{i t}\), where \(\alpha_{i}\) are fixed effects possibly correlated with \(\mathbf{x}_{i t}\). Stacking all \(T\) observations for individual \(i\) yields \(\mathbf{y}_{i}=\alpha_{i} \mathbf{e}+\mathbf{X}_{i} \beta+\varepsilon_{i}\) (see (21.29) for definitions). Consider the estimator \(\widehat{\boldsymbol{\beta}}=\left[\sum_{i=1}^{N} \mathbf{X}_{i}^{\prime} \mathbf{J}^{\prime} \mathbf{J} \mathbf{X}_{i}\right]^{-1} \times \sum_{i=1}^{N} \mathbf{X}_{i}^{\prime} \mathbf{J}^{\prime} \mathbf{J}_{i}\), where \(\mathbf{J}\) is a \(T \times T\) matrix of known constants such that \(\mathbf{J e}=\mathbf{0}\). [Note that an example of \(\mathbf{J}\) is \(\left.\mathbf{Q}=\mathbf{I}_{T}-T^{-1} \mathbf{e e}^{\prime}.\right]\)

(a) Provide a motivation for the estimator \(\widehat{\boldsymbol{\beta}}\).

(b) Find \(\mathrm{E}[\widehat{\boldsymbol{\beta}}]\). For simplicity assume that \(\mathbf{X}_{i}\) are fixed regressors and that \(\varepsilon_{i t}\) are iid \(\left[0, \sigma^{2}\right]\). Is \(\widehat{\beta}\) unbiased for \(\beta\) ?

(c) Find \(\mathrm{V}[\widehat{\boldsymbol{\beta}}]\). For simplicity assume that \(\mathbf{X}_{i}\) are fixed regressors and that \(\varepsilon_{i t}\) are iid \(\left[0, \sigma^{2}\right]\).

(d) Now suppose \(\varepsilon_{i t}\) are independent over \(i\) but correlated over \(t\) with \(\mathrm{V}\left[\varepsilon_{i}\right]=\Omega_{i}\). Give \(\mathrm{V}[\widehat{\beta}]\).

(e) Suppose that the effects \(\alpha_{i}\) are random \(\left(0, \sigma_{\alpha}^{2}\right)\) rather than fixed. Would the estimator in this exercise be consistent?

yea; +X+, i = 1,..., N, = (21.29)