Thisquestionconcernsthealternativestate-spacerepresentationofan AR(p)model thatarisesasaspecialcaseofthestate-spacerepresenta tionofARMA(pq)modelswhenq=0.This isalsoeasilyseentobe de nedbyadirectlineartransformationofstatevectorsinthestandard representation,asfollows.

Beginwiththestandardstate-spacerepresentationoftheAR(p)model;

thestatevector isxt=(yt yt 1 yt p+1) andthemodel equations areyt=Fxtandxt=Gxt 1+Ftwhere F=

1 0

0 0

and G=

1 2 p 1 p 1 0 0 0 0 0 0 0 0 0 1 0 withARparameters =( 1 p) andinnovations t N(0v)

De nethep psymmetricmatrixAby A=

1 0 0 0 0 0 0 2 3 4 p 1 p 0 3 4 5 p 0 0 p 1 p 0 0 0 p 0 0 0

(a)VerifythatthematrixproductAGisgivenby AG=

1 2 3 4 p 1 p 2 3 4 5 p 0 3 4 5 5 0 0 p 1 p 0 0 0 p 0 0 0 0 notingthatthisisalsosymmetric.

(b) Showordeducethat:
i. ForaproperAR(p)model inwhich p=0 thenA=0sothat Aisnonsingular.
ii.AGA1=G.
iii.AF=Fand,asaresult,F=FA1 (c)Hence showthatanequivalent state-spaceAR(p) formisgivenby yt=Fzt andzt=Gzt 1+Ft basedonanewp 1statevector zt=AxtandwherethestateevolutionmatrixisG i.e., G= 1 1 0 0 2 0 1 0 p 1 0 0 1 p 0 0 0 (d)What is the interpretationof theelementsof thetransformedstate vectorzt?