Consider calculation of the MLE in the logit regression model when the only regressor is the intercept. Then $\mathrm{E}[y]=1/(1+e^{-\beta })$ and the gradient of the scaled log-likelihood function $g(\beta )=(y-1/(1+e^{-\beta }))$. Suppose a sample yields $\overline{y}=$ 0.8 and the starting value is $\beta =0.0$.

(a) Calculate $\beta$ for the first six iterations of the Newton-Raphson algorithm.

(b) Calculate the first six iterations of a gradient algorithm that sets $A_s=1$ in (10.1), so ${\hat{\beta }}_{s+1}={\hat{\beta }}_s+g_s$.

(c) Compare the performance of the methods in parts

(a) and (b).