TimeSeriesBook.pdf

# 22 percent its contribution diminishes with the

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explains more than 50 percent whereas demand shocks account for only 42.22 percent. Its contribution diminishes with the increase of the forecast horizon giving room for price and wage shocks. The variance of the inflation rate is explained in the short-run almost exclusively by price shocks. However, as the the forecast horizon is increased supply and wage shocks become relatively important. The money growth rate does not interact much with the other variables. Its variation is almost exclusively explained by money shocks. 15.5 Identification via Long-Run Restrictions 15.5.1 A Prototypical Example Besides short-run restrictions, essentially zero restrictions on the coefficients of A and/or B , Blanchard and Quah (1989) proposed long-run restrictions as an alternative option. These long-run restrictions have to be seen as complementary to the short-run ones as they can be combined. Long-run restrictions constrain the long-run effect of structural shocks. This technique makes only sense if some integrated variables are involved, because in the stationary case the effects of all shocks vanish eventually. To explain this, we discuss the two-dimensional example given by Blanchard and Quah (1989). They analyze a two-variable system consisting of logged real GDP de- noted by { Y t } and the unemployment rate { U t } . Logged GDP is typically integrated of order one (see Section 7.3.4 for an analysis for Swiss GDP) whereas U t is considered to be stationary. Thus they apply the VAR ap- proach to the stationary process { X t } = { (∆ Y t , U t ) 0 } . Assuming that { X t } is already demeaned and follows a causal VAR process, we have the following representations: X t = Y t U t = Φ 1 X t - 1 + . . . + Φ p X t - p + Z t = Ψ(L) Z t = Z t + Ψ 1 Z t - 1 + Ψ 2 Z t - 2 + . . . . For simplicity, we assume that A = I 2 , so that Z t = BV t = 1 b 12 b 21 1 v dt v st where V t = ( v dt , v st ) 0 WN(0 , Ω) with Ω = diag( ω 2 d , ω 2 s ). Thereby { v dt } and { v st } denote demand and supply shocks, respectively. The causal represen- tation of { X t } implies that the effect of a demand shock in period t on GDP

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302 CHAPTER 15. INTERPRETATION OF VAR MODELS Table 15.1: Forecast error variance decomposition (FEVD) in terms of de- mand, supply, price, wage, and money shocks (percentages) growth rate of real GDP horizon demand supply price wage money 1 99 . 62 0 . 38 0 0 0 2 98 . 13 0 . 94 0 . 02 0 . 87 0 . 04 4 93 . 85 1 . 59 2 . 13 1 . 86 0 . 57 8 88 . 27 4 . 83 3 . 36 2 . 43 0 . 61 40 86 . 13 6 . 11 4 . 29 2 . 58 0 . 89 unemployment rate horizon demand supply price wage money 1 42 . 22 57 . 78 0 0 0 2 52 . 03 47 . 57 0 . 04 0 . 01 0 . 00 4 64 . 74 33 . 17 1 . 80 0 . 13 0 . 16 8 66 . 05 21 . 32 10 . 01 1 . 99 0 . 63 40 39 . 09 16 . 81 31 . 92 10 . 73 0 . 89 inflation rate horizon demand supply price wage money 1 0 . 86 4 . 18 89 . 80 5 . 15 0 2 0 . 63 13 . 12 77 . 24 8 . 56 0 . 45 4 0 . 72 16 . 79 68 . 15 13 . 36 0 . 97 8 1 . 79 19 . 34 60 . 69 16 . 07 2 . 11 40 2 . 83 20 . 48 55 . 84 17 . 12 3 . 74 growth rate of wages horizon demand supply price wage money 1 1 . 18 0 . 10 0 . 97 97 . 75 0 2 1 . 40 0 . 10 4 . 30 93 . 50 0 . 69 4 2 . 18 2 . 75 9 . 78 84 . 49 0 . 80 8 3 . 80 6 . 74 13 . 40 74 . 72 1 . 33 40 5 . 11 8 . 44 14 . 19 70 . 14 2 . 13 growth rate of money stock horizon demand supply price wage money 1 0 . 10 0 . 43 0 . 00 0 . 84 98 . 63 2 1 . 45 0 . 44 0 . 02 1 . 02 97 . 06 4 4 . 22 1 . 09 0 . 04 1 . 90 92 . 75 8 8 . 31 1 . 55 0 . 81 2 . 65 86 . 68 40 8 . 47 2 . 64 5 . 77 4 . 55 78 . 57
15.5. IDENTIFICATION VIA LONG-RUN RESTRICTIONS 303 growth in period t + h is given by: Y t + h ∂v dt = [Ψ h B ] 11 where [Ψ h B ] 11 denotes the upper left hand element of the matrix Ψ h B .

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