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2.1. UNRESTRICTED VECTOR AUTOREGRESSIONS 33To relate these ideas to unrestricted VARs, consider the dynamicmodelm t = θs t + β 11 m t−1 + β 12 s t−1 + ² 1t , (2.23)s t = γm t + β 21 m t−1 + β 22 s t−1 + ² 2t , (2.24)iidwhere ² 1t ∼ N(0, σ1), 2 iid² 2t ∼ N(0, σ2), 2 and E(² 1t ² 2s ) = 0 for all t, s.Without additional restrictions, ² 1t and ² 2t are exogenous but both m tand s t are endogenous. Notice also that m t−1 and s t−1 are exogenouswith respect to the current values m t and s t .If θ = 0, then m t is said to be econometrically exogenous withrespect to s t . m t ,m t−1 ,s t−1 would be predetermined in the sense thatan intervention due to a shock to m t can unambiguously be attributedto ² 1t and the effect on the current exchange rate is ds t = γdm t . Ifβ 12 = θ =0,thenm t is strictly exogenous to s t .Eliminate the current value observations from the right side of (2.23)and (2.24) to get the reduced formwherem t = π 11 m t−1 + π 12 s t−1 + u mt , (2.25)s t = π 21 m t−1 + π 22 s t−1 + u st , (2.26)π 11 = (β 11 + θβ 21 )(1 − γθ), π 12 = (β 12 + θβ 22 ),(1 − γθ)π 21 = (β 21 + γβ 11 ), π 22 = (β 22 + γβ 12 )(1 − γθ)(1 − γθ)u mt = (² 1t + θ² 2t )(1 − γθ) , u st = (² 2t + γ² 1t )(1 − γθ) ,Var(u mt )= (σ2 1 + θ 2 σ 2 2)(1 − γθ) 2 , Var(u st) = (γ2 σ 2 1 + σ 2 2)(1 − γθ) 2 ,Cov(u mt ,u st )= (γσ2 1 + θσ2)2(1 − γθ) 2 . ⇐(14) (last 3If you were to apply the VAR methodology to this system, youwould estimate the π coefficients. If you determined that π 12 =0,expressions)