Simulatethree-dimensionaldatayt fromthefollowingmodel:

yt =

1 1 0 1 0 1 1 1 0 xt+ t with t N(04I3)andxt=(x1t x2t x3t)suchthat x1t = 095x1t 1+ 1t x2t = 2 095 cos(2 18)x2t 1 0952x2t 2+ 2t x3t = 2 095 cos(2 7)x3t 1 0952x3t 2+ 3t where t=( 1t 2t 3t) and t N(0I3) Inotherwords,x1t isan AR(1)processwithcoe cient095xt2isaquasiperiodicAR(2)process withmodulus0.95andwavelength18,andxt3 isaquasiperiodicAR(2)

processwithmodulus0.95andwavelength7.

(a) Fit a VAR(p) model with the ar function in R. Choose the model or der according to Akaikes information criterion (AIC) or the Bayesian information criterion (BIC).

(b) Compute and plot the time series decomposition of yt based on your estimates of the VAR matrices of coe cients.