Suppose ????1, … , ????????, with ???? = 60, is a sample from a bivariate VAR(1) process, with sample covariance matrices obtained as

????̂(0) = [

1.0 1.0 1.0 2.0

]

????̂(1) = [

0.6 0.4 0.7 1.2

]

????̂(2) = [

0.30 0.10 0.42 0.64]

(a) Obtain the corresponding estimated correlation matrices ????̂(0), ̂????(1), and

????̂(2).

(b) Find the sample Yule--Walker estimates for ???? and ???? in the VAR(1) model, and find an estimate for the approximate covariance matrix of the estimator ????̂ , that is, for the covariance matrix of vec[????̂ ′

].

(c) Based on the results in (b), test whether the matrix ???? has a lower triangular structure; that is, test whether ????12 = 0.