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2.1. UNRESTRICTED VECTOR AUTOREGRESSIONS 27Write it out as (7) (see line 2)I = C 0 +(C 1 − C 0 A 1 )L +(C 2 − C 1 A 1 − C 0 A 2 )L 2=+(C 3 − C 2 A 1 − C 1 A 2 − C 0 A 3 )L 3+(C 4 − C 3 A 1 − C 2 A 2 − C 1 A 3 − C 0 A 4 )L 4 + ···⎛⎞∞XjX⎝C j − C j−k A k⎠ L j .j=0k=1Now to equate coefficients on powers of L, Þrst note that C 0 = I andthe rest of the C j follow recursivelyC 1 = A 1 ,C 2 = C 1 A 1 + A 2 ,C 3 = C 2 A 1 + C 1 A 2 + A 3 ,C 4 = C 3 A 1 + C 2 A 2 + C 1 A 3 + A 4 ,.C k =kXj=1C k−j A j .For example if p =2,setA j = 0 for j ≥ 3. Then C 1 = A 1 , C 2 =C 1 A 1 + A 2 , C 3 = C 2 A 1 + C 1 A 2 , C 4 = C 3 A 1 + C 2 A 2 ,andsoon.(8)(formulaeto end of section)⇐(9)Impulse Response AnalysisOnce you get the moving-average representation you will want employimpulse response analysis to evaluate the dynamic effect of innovationsin each of the variables on (q 1t ,q 2t ). When you go to simulate the dynamicresponse of q 1t and q 2t toashockto² 1t ,youareimmediatelyconfronted with two problems. The Þrst one is how big should the ⇐(10)shock be? This becomes an issue because you will want to compare theresponse of q 1t across different shocks. You’ll have to make a normalizationfor the size of the shocks and a popular choice is to considershocks one standard deviation in size. The second problem is to getshocks that can be unambiguously attributed to q 1t and to q 2t .If² 1t and² 2t are contemporaneously correlated, however, you can’t just shock ² 1tand hold ² 2t constant.