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34 CHAPTER 2. SOME USEFUL TIME-SERIES METHODSyou would say that s does not Granger cause m (and therefore m iseconometrically exogenous to s). But when you look at (2.23) and(2.24), m is exogenous in the structural or economic sense when θ =0but this is not implied by π 12 = 0. The failure of s to Granger cause mneed not tell us anything about structural exogeneity.Suppose you orthogonalize the error terms in the VAR. Letδ = Cov(u mt ,u st )/Var(u mt )betheslopecoefficient from the linearprojection of u st onto u mt . Then u st − δu mt is orthogonal to u mt byconstruction. An orthogonalized system is obtained by multiplying(2.25) by δ and subtracting this result from (2.26)m t = π 11 m t−1 + π 12 s t−1 + u mt , (2.27)s t = δm t +(π 21 − δπ 11 )m t−1 +(π 22 − δπ 12 )s t−1 + u st − δu mt . (2.28)The orthogonalized system includes a current value of m t in the s tequation but it does not recover the structure of (2.23) and (2.24). Theorthogonalized innovations areu mt = ² 1t + θ² 2t1 − γθ , (2.29)u st − δu mt = (γ² 1t + ² 2t ) − ³ ´γσ1 2+θσ2 2σ1 2+θ2 σ22 (²1t + θ² 2t ), (2.30)1 − γθwhich allows you to look at shocks that are unambiguously attributableto u mt in an impulse response analysis but the shock is not unambiguouslyattributable to the structural innovation, ² 1t .To summarize, impulse response analysis of unrestricted VARs providesummaries of dynamic correlations between variables but correlationsdo not imply causality. In order to make structural interpretations,you need to make assumptions of the economic environment andbuild them into the econometric model. 66 You’ve no doubt heard the phrase made famous by Milton Friedman, “There’sno such thing as a free lunch.” Michael Mussa’s paraphrasing of that principle indoing economics is “If you don’t make assumptions, you don’t get conclusions.”