1. Inverse Matrix 4[(3 7) (02)] 1[(0 7) (3 2)] Recall that the inverse of A is equal to:

Transcription

1 Rfrncs Brnank, B. and I. Mihov (1998). Masuring monary policy, Quarrly Journal of Economics CXIII, Blanchard, O. R. Proi (00). An mpirical characrizaion of h dynamic ffcs of changs in govrnmn spnding and axs on oupu. Th Quarrly Journal of Economics, pp Eichnbaum, M (199). Commns on Inrpring h macroconomic im sris facs: Th ffcs of monary policy. Europan Economic Rviw, pp Endrs, W. (010). Applid conomric im sris. Wily, 3 rd diion. Favro, C. (001) Applid Macroconomrics. Oxford Univrsiy Prss. Hamilon, J. (1994) Tim Sris Analysis. Princon Univrsiy Prss. Sims, C. (1980), Macroconomics and Raliy, Economrica 48, Sims, C. (199), Inrpring h macroconomic im sris facs: Th ffcs of monary policy. Europan Economic Rviw, pp Sock and Wason (001), Vcor auorgrssions. Journal of Economic Prspcivs, 15(4) pp

2 1. Invrs Marix Rcall ha h invrs of A is qual o: 1 Adjuga of Furhr rcall ha h adjuga is qual o h ranspos of h cofacor marix. Find h invrs of h following marix: drminan: 4[(3 7) (0)] 1[(0 7) (3 )] ( 1) [(0 0) (3 3)]

3 Th cofacor marix of is: 1. Invrs Marix

4 1. Invrs Marix Cofacor marix Transpos of cofacors: Invrs of A: 1 1 adj

5 . SVAR Idnificaion: rsricions Imposing h rsricion maks h numbr of unknown paramrs in h srucural modl qual o h numbr of paramrs known from h sandard VAR simaion: 1. g g0 a g 6. Sinc hn, g1 a111 1 g1 1 g a y u y var( y) uy var( r ) ur a1 uy cov ( ) E ( u )( u a u ) a y r y r 1 y 1 uy Subsiu h simad valus for hs 9 paramrs g 10, g 11, g 1, g 0, g 1, g, y, r and y-r in h 9 quaions, o solv for: a 1, β 10, β 0, β 11, β 1, β 1, β, y, and r.

6 3. Impuls-rsponss In a firs ordr auorgrssiv modl: y g0g1y 1 Th sabiliy condiion is ha <1 g 1 In h cas of a VAR, an quivaln condiion is rquird. I bcoms clar whn h sandard VAR is irad backwards. For a simpl VAR(1): X G GX X 1G0GX 1 1 X G G(G +GX ) 6

7 3. Impuls-rsponss Succssiv iraions gnra: X 1 XG0G(G+G(G+GX ) 1) X (I +G G G )G + G GX X i i i 1 ni 0 i1 As i + X rducs o a sum of rrors or Wold rprsnaion of X: X G i G i 1 assum G i hn i 1 i i1 X i i i1 If X has a Wold rprsnaion hn X is sabl. Th condiion ha guarans i is ha G has ignvalus smallr han 1 in modulus. I is a gnral rsul, no only valid for a VAR (1) bu valid for any VAR(p). 7

8 3. Impuls-rsponss Sinc h VAR is saionary, h simad rducd-form VAR ha a moving avrag: X i i i1 Using h sam old rlaion bwn forcas rrors and srucural shocks w find: c 11,i u 1 1 or mor compacly i i X Ci i i 1 i0 u y u r X A A c 1,i u u 1 A u and ar h rsponss of y and r o a chang and rspcivly. c 11,0 No ha is h ffc a impac, is h ffc of on y on +1, and so on in succssion: y k C11, k u y c Also, h cumulaiv ffc is 11, i. i0 c 11,1 u y

9 4. Varianc Dcomposiion Knowldg of h prdicion rrors can b xrmly valuabl in xamining h rlaionships among h variabls. Assum ha w know h cofficins G 0 and G 1 and wish o projc h valus of X +1 condiional on h obsrvd valus of X. If h quaion X is advancd on priod, w obain G0GX 1 1 X 1G0GX 1 1 and h prdicion rror will b X 1EX1 1 for innovaion in +1. X X G G (G +G(G +GX ) ) X 1 X EX G G

10 4. Varianc Dcomposiion EX (I G +G... G )G G X n n1 n This may also b xprssd in rms of srucural rrors: Th prdicion rror for only GDP gap n sps forward will llb: n n i 1 1 X nga 1 u X n ia u n X n Ciu n i n i n X EX +G G G n 1 n n n 1 n1 1 n 1 1 i0 n i i0 Th varianc of his prdicion rror is: y Ey c u + c u + + c u n n 11,0 yn 11,1 yn1 11, n1 y1 c u + c u c u 1,0 rn 1,1 r n 1 1, n1 rn [ c + c + + c ] yn, y 11,0 11,11 11, n1 [ c + c + + c ] z 1,0 1,1 1, n1 i0

11 4. Varianc Dcomposiion c 11,0 As h valus of ar ncssarily posiiv, h varianc of h rrors incrass wih h projcion horizon. I is possibl o dcompos h prdicion rror n priods forward by h conribuion ach of h wo shocks in our xampl. yn, Th proporions of aribuabl o ach srucural shock ar: [ c + c + + c ] [ c + c + + c ] y 11,0 11,1 11, n1 r 1,0 1,1 1, n1 yn, yn, This is h proporion of h changs in on variabl aribuabl o shocks o i and o shocks in anohr variabl. If fails o xplain any changs in y h lar is xognous. ur i Th rsricion imposd abov rquirs ha h nir varianc in h prdicion rror for y on priod forward b aribuabl bl o u yi