Suppose we wish to choose between two nested parametric models. The relationship between the densities of the two models is that \(g(y \mid x, \beta, \alpha=0)=f(y \mid x, \beta)\), where for simplicity both \(\beta\) and \(\alpha\) are scalars. If \(g\) is the correct density then the MLE of \(\beta\) based on density \(f\) is inconsistent. A test of model \(f\) against model \(g\) is a test of \(H_{0}: \alpha=0\) against \(H_{a}: \alpha eq 0\). Suppose ML estimation yields the following results: (1) model \(f: \widehat{\beta}=5.0, \operatorname{se}[\widehat{\beta}]=0.5\), and \(\ln L=-106\); (2) model g: \(\widehat{\beta}=3.0, \operatorname{se}[\widehat{\beta}]=1.0, \widehat{\alpha}=2.5, \operatorname{se}[\widehat{\alpha}]=1.0\), and \(\ln L=-103\). Not all of the following tests are possible given the preceding information. If there is enough information, perform the tests and state your conclusions. If there is not enough information, then state this.

(a) Perform a Wald test of \(H_{0}\) at level 0.05 .

(b) Perform a Lagrange multiplier test of \(H_{0}\) at level 0.05 .

(c) Perform a likelihood ratio test of \(H_{0}\) at level 0.05 .

(d) Perform a Hausman test of \(H_{0}\) at level 0.05 .