Suppose (y=mathbf{x}^{prime} boldsymbol{beta}+u), where (u sim mathcal{N}left[0, sigma^{2} ight]), with parameter vector (theta=left[boldsymbol{beta}^{prime}, sigma^{2} ight]^{prime}) and density

Question:

Suppose \(y=\mathbf{x}^{\prime} \boldsymbol{\beta}+u\), where \(u \sim \mathcal{N}\left[0, \sigma^{2}\right]\), with parameter vector \(\theta=\left[\boldsymbol{\beta}^{\prime}, \sigma^{2}\right]^{\prime}\) and density \(f(y \mid \theta)=(1 / \sqrt{2 \pi} \sigma) \exp \left[-\left(y-\mathbf{x}^{\prime} \boldsymbol{\beta}\right)^{2} / 2 \sigma^{2}\right]\). We have a sample of \(N\) independent observations.

(a) Explain why a test of the moment condition \(\mathrm{E}\left[\mathbf{x}\left(y-\mathbf{x}^{\prime} \boldsymbol{\beta}\right)^{3}\right]\) is a test of the assumption of normally distributed errors.

(b) Give the expressions for \(\widehat{\mathbf{m}}_{i}\) and \(\widehat{\mathbf{s}}_{i}\) given in (8.5) necessary to implement the \(\mathrm{m}\)-test based on the moment condition in part (a).

(c) Suppose \(\operatorname{dim}[\mathbf{x}]=10, N=100\), and the auxiliary regression in (8.5) yields an uncentered \(R^{2}\) of 0.2 . What do you conclude at level \(0.05 ?\)

(d) For this example give the moment conditions tested by White's information matrix test.