By Elhanan Helpman

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**Extra resources for A Theory of International Trade under Uncertainty (Economic theory, econometrics, and mathematical economics)**

**Example text**

37 Stable Vector Autoregressive Processes Furthermore, instead of optimal predictors, optimal linear predictors are compared. ls $; t}) is replaced by the linear minimum MSE h-step predictor based on {zsls $; t}. In the following, when the terms "Granger-causality" and "instantaneous causality" are used, these restrictive assumptions are implicitly used if not otherwise noted. 4) where U t is a white noise process with nonsingular covariance matrix tu. Suppose that Yt consists of the M-dimensional process Zt and the (K - M)-dimensional process x t and the MA representation is partitioned accordingly, Yt = [Zt] = [Jll] xt Jl2 + [tP ll (L) tP 12 (L)] [u lt ].

F/Ji = J AiJ' with J := [IK 0 ... 0] a (K x Kp) matrix. i, the jk-th element of f/Ji represents the reaction of the j-th variable of the system to a unit shock of variable k, i periods ago, provided, of course, the effect is not contaminated by other shocks to the system. Since the u, are just the 1-step ahead forecast errors of the VAR process the shocks considered here may be regarded as forecast errors. The response of variable j to a unit shock (forecast error) in variable k is sometimes depicted graphically to get a visual impression of the dynamic inter- Stahle Vcctor Autoregressive Processes 45 relationships within the system.

0 inflation -+ interest 10 5 1S interest -+ interest 1\ \ 11)1 / or / I ",,- \ " " C::l 00 6f\ ',~ 5 10 15 C::I 00 \" 5 ==+- 10 15 Fig. 6. Impulse responses of the inflation/interest rate system (impulse -+ response).