Supposetheq vectoryt=(yt1 ytq) timeseries isVARq(1)with yt= yt 1+ tand t N(0V) independentlyovertime.

(a)Takeanynonsingularq qmatrixAandconsiderthetransformation toxt=Ayt ShowthatthisyieldsaVARq(1)processxtandgivethe expressions fortheVARcoe cientmatrixandinnovationsvariance matrix.

(b) Showthatyoucan ndanuppertriangular,unitdiagonalq qmatrix Aandapositivediagonalq qmatrixU i.e., A=

1 12 13 14 1q 0 1 23 24 2q 0 0 1 34 3q 0 0 1 q 1q 0 0 1 andU=diag(u1 u2 uq)witheachui>0 suchthattheimplied VARq(1)processxt=AythasinnovationsvariancematrixU

(c)DiscussinterpretationsofthismatrixA

(d) Showthatthisresults inasetof qequations forthe individual ytj seriesoftheform ytj=Ft j+ tj+ tj tj N(0uj)

over time t, where:
Ft =yt 1;
j is row j of A ;
tq = 0 and, for j = 1 q 1 tj = jytpa(j)
where the index set pa(j) = j + 1 ytpa(j) = (ytj+1 ytq)
q so that the vector Identify the vectors j here.

(e) Discuss the above result, and consider how it might enable alternative analyses of VARq(1) models that allow alternative, and possible more exible/richer classes of priors on model parameters.

(f) Suppose you perform such an analysis. A colleague points out that you could reorder the elements of yt and redo the analysis, but would get di erent results without some serious thought about how to make the priors assumed compatible across the analyses. Comment on this point, particularly imagining problems that involve large values of q that are increasingly common.
(g) Suppose now you realize that the applied context of interest requires a time-varying model, i.e., the starting VARq(1) is to be replaced with a TV-VARq(1) model yt = tyt 1 + t and t N(0Vt). How would you modify the above discussion?