Consider the logit model with (operatorname{Pr}left[y=1 mid x_{1}, x_{2} ight]=Lambdaleft(beta_{0}+beta_{1} x_{1 i}+beta_{2} x_{2} ight)), where (Lambda(z)=e^{z} /left(1+e^{2}

Consider the logit model with \(\operatorname{Pr}\left[y=1 \mid x_{1}, x_{2}\right]=\Lambda\left(\beta_{0}+\beta_{1} x_{1 i}+\beta_{2} x_{2}\right)\), where \(\Lambda(z)=e^{z} /\left(1+e^{2}\right) x\).

(a) Write down the likelihood scores and information matrix in an expanded form.

(b) Use these to derive Wald and \(L M\) score tests of \(H_{0}: \beta_{2}=0\).

(c) Explain how you would computationally implement the tests.

(d) In what sense is the logit model intrinsically heteroskedastic?