Estimating and Testing VARMA Models Gauss file(s) stsm_varma.g Matlab file(s) stsm_varma.m Simulate T = 500 observations from the VARMA(1, 1) model y1,t = 1 +

Estimating and Testing VARMA Models Gauss file(s) stsm_varma.g Matlab file(s) stsm_varma.m Simulate T = 500 observations from the VARMA(1, 1) model y1,t = µ1 + φ1,1,1y1,t−1 + v1,t + ψ1,1,1v1,t−1 + ψ1,1,2v2,t−1 y2,t = µ2 + φ1,2,2y2,t−1 + v2,t + ψ1,2,1v1,t−1 + ψ1,2,2v2,t−1, where vt is bivariate normal with zero mean and covariance matrix given by the identity matrix, and the population parameters are given in Table 13.2.

(a) Compute the maximum likelihood estimates using the Newton-Raphson algorithm.

(b) Repeat part

(a) using T = 1000 observations. Discuss the asymptotic properties of the maximum likelihood estimator.

(c) Perform a Wald test of the hypothesis H0 : ψ1,1,1 = ψ1,1,2 = ψ1,2,1 = ψ1,2,2 = 0. Choose T = 500, 1000 and discuss the consistency properties of the Wald statistic.

(d) Perform a LR test of the multivariate moving average part H0 : ψ1,1,1 = ψ1,1,2 = ψ1,2,1 = ψ1,2,2 = 0. Choose T = 500, 1000 and discuss the consistency properties of the LR statistic.