Lecture 32 – Structural VARs: II Consider the following VAR of y1t, the growth rate of real GDP, and y2t, the inflation rate: yt = A0 + A1yt-1 + εt where εt ~ w.n.(Σεε). We assume that εt = Cvt where vt = [v1t v2t]’ is a serially and contemporaneously uncorrelated vector, i.e., E(vtvt’) = Σvv , a diagonal matrix E(vtvs’) = 0 if t ≠ s WLOG, assume Σvv = I2. v1t and v2t are structural shocks: call v1t the “AS supply shock” and call v2t the “AD shock”.How does the system respond over time to an AS shock? How does the system respond over time to an AD shock? Since v1t is a zero-mean, unit variance white noise process which is uncorrelated at all leads and lags with v2t, the following simulation is a natural way to answer the first question – Set y0 = 0. Set v2t = 0 for t = 1,2,…. Set v11 = 1. Set v1t = 0 for t = 2,3,… Then y1 = A0 + Cv1 , where Cv1 =[C11 C21]’ y2 = A0 + A1y1 = A0(I+A1) + A1Cv1y3 = A0 + A1y2 = A0(I+A1+A12) + A12Cv1… So dy1t+s/dv1t = A1sC11dy2t+s/dv1t = A1sC21Problem – Where does C come from? As we noted last time, C is not identified from the reduced-form VAR – There are 10 free structural parameters and 9 free reduced form parameters. We need to impose at least one more restriction on the structure in order to identify C. Sims (1980) – Assume that C has a triangular form, i.e., C12 = 0 or C21 = 0. What does the restriction C12 = 0 (or C21 = 0) mean? Which is more plausible?Assume C12 = 0. How to identify C11,C21,C22? Note that Σεε = E(εtεt’) = E(Cvtvt’C’) = CE(vtvt’)C’ = CC’ So, we need to decompose the 2x2 (nxn) p.d.matrix Σεε into the product of a triangular matrix and its transpose. This decomposition is called the Cholesky decomposition of Σεε and is a matrix operation commonly available in econometric software (including MATLAB).In practice, we replace Σεε with an estimate: ∑=ΣTttT1'ˆˆ1ˆεεεεInnovation Accounting – 1. Impulse Response Functions (generally provided in graphical form) dyi,t+s/dvj,t , for s = 0,1,…; i,j = 1,…,n (Confidence Intervals: Bootstrap) 2. Forecast Error Variance Decompositions (generally provided in tabular form) What fraction of the s-step-ahead variance in yi is attributable to vj, j = 1,…,n? (What proportion of the 1-step ahead variance in real GDP is attributable to AS shocks? AD shocks?)Let fij,s = dyi,t+s/dvj,t The s-step ahead forecast error variance in yi : ∑=++skkinkiff02,2,1)...( The proportion of the s-step ahead forecast error variance in yi attributable to the vj shocks: ∑∑++=skinkiskkijfff02,2,102,)...(/ Examples
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