Bimodal Likelihood Gauss file(s) prop_binormal.g Matlab file(s) prop_binormal.m

(a) Simulate a sample of size T = 4 from a bivariate normal distribution with zero means, unit variances and correlation ρ0 = 0.6. Plot the log-likelihood function lnLT (ρ) = − ln 2π − 1 2 ln(1 − ρ 2 ) − 1 2(1 − ρ 2)  1 T X T t=1 y 2 1,t − 2ρ 1 T X T t=1 y1,ty2,t + 1 T X T t=1 y 2 2,t ,

and the scaled gradient function GT (ρ) = ρ(1−ρ 2 )+(1+ρ 2 ) 1 T X T t=1 y1,ty2,t−ρ  1 T X T t=1 y 2 1,t+ 1 T X T t=1 y 2 2,t , for values of ρ = {−0.99, −0.98, · · · , 0.99}. Interpret the result and compare the graphs of ln LT (ρ) and GT (ρ) with Figure 2.9.

(b) Repeat part

(a) for T = {10, 50, 100}, and compare the results with part

(a) for the case of T = 4. Hence demonstrate that for the case of multiple roots, the likelihood converges to a global maximum resulting in the maximum likelihood estimator being unique (see Stuart, Ord and Arnold, 1999, pp. 50-52, for a more formal treatment of this property).