Maximum Likelihood Estimation using Newton-Raphson Gauss file(s) max_nr.g, max_iter.g Matlab file(s) max_nr.m, max_iter.m Consider the regression model set out in Example 1.

(a) Simulate the model for T = 5 observations using the parameter values θ = {β = 1, σ2 = 4}.

(b) Find the log-likelihood function, ln LT (θ), the gradient, GT (θ), and the Hessian, HT (θ).

(c) Evaluate ln LT (θ), GT (θ) and HT (θ) at θ(0) = {1, 4}.

(d) Update the value of the parameter vector using the Newton-Raphson update scheme θ(1) = θ(0) − H −1 (0)

G(0) , and recompute ln LT (θ) at θ(1). Compare this value with that obtained in part (c).

(e) Continue the iterations in

(d) until convergence and compare these values to those obtained from the maximum likelihood estimators βb = PT t=1 xtyt PT t=1 x 2 t , σb 2 = 1 T X T t=1 (yt − βxb t) 2 .