Maximum Likelihood Estimation using Graphical Methods Gauss file(s) max_graph.g Matlab file(s) max_graph.m Consider the regression model yt = xt + ut , ut iid

Maximum Likelihood Estimation using Graphical Methods Gauss file(s) max_graph.g Matlab file(s) max_graph.m Consider the regression model yt = βxt + ut , ut ∼ iid N(0, σ2 ), where xt is an explanatory variable given by xt = {1, 2, 4, 5, 8}.

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

(b) Compute the log-likelihood function, ln LT (θ), for: (i) β = {0.0, 0.1, · · · , 1.9, 2.0} and σ 2 = 4; (ii) β = {0.0, 0.1, · · · , 1.9, 2.0} and σ 2 = 3.5; (iii) plot ln LT (θ) against β for parts (i) and (ii).

(c) Compute the log-likelihood function, ln LT (θ), for: (i) β = {1.0} and σ 2 = {1.0, 1.5, · · · , 10.5, 11}; (ii) β = {0.9} and σ 2 = {1.0, 1.5, · · · , 10.5, 11}; (iii) plot ln LT (θ) against σ 2 for parts (i) and (ii).