VAR Slides

# VAR Slides - NBER Summer Institute Whats New in...

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Revised July 23, 2008 7-1 NBER Summer Institute What’s New in Econometrics: Time Series Lecture 7 July 15, 2008 Recent Developments in Structural VAR Modeling

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Revised July 23, 2008 7-2 Outline 1) VARs, SVARs, and the Identification Problem 2) Identification by Short Run Restrictions 3) Identification by Long Run Restrictions 4) Identification by Sign Restrictions 5) Identification from Heteroskedasticity 6) DSGE Priors 7) Identification from Regional/Multicountry Restrictions 8) Inference: Challenges and Recently Developed Tools
1) VARs, SVARs, and the Identification Problem A classic question in empirical macroeconomics: what is the effect of a policy intervention (interest rate increase, fiscal stimulus) on macroeconomic aggregates of interest – output, inflation, etc? Let Y t be a vector of macro time series, and let r t ε denote an unanticipated monetary policy intervention. We want to know the dynamic causal effect of r t on Y t : th r t Y + , h = 1, 2, 3,…. where the partial derivative holds all other interventions constant. In macro, this dynamic causal effect is called the impulse response function ( IRF ) of Y t to the “shock” (unexpected intervention) r t . The challenge is to estimate r t Y + ⎩⎭ from observational macro data. Revised July 23, 2008 7-3

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Revised July 23, 2008 7-4 Two conceptual approaches to estimating dynamic causal effects ( IRF ) 1) Structural model (Cowles Commission) a) tightly parameterized (many restrictions): FMP,…, DSGE b) Structural vector autoregressions (SVARs) 2) Natural experiments The identification problem Consider a 2 -variable system of linear simultaneous equations: Let ε 1 t and 2 t be uncorrelated structural shocks, where E ( t | Y t –1 , Y t –2 ,…) = 0: Y 1 t = B 0,12 Y 2 t + B 1,12 Y 2 t –1 + … + B p ,12 Y 2 t p + B 1,11 Y 1 t –1 +… + B p ,11 Y 1 t p + 1 t Y 2 t = B 0,21 Y 1 t + B 1,21 Y 1 t –1 + … + B p ,21 Y 1 t p + B 1,22 Y 2 t –1 +… + B p ,22 Y 2 t p + 2 t Given the B ’s, we could compute structural impulse responses from this system (formulas below). But the coefficients of this system are not identified. To identify them, we either need an instrument Z t , or a restriction on the parameters.
Revised July 23, 2008 7-5 VAR background and notation: Y 1 t = B 0,12 Y 2 t + B 1,12 Y 2 t –1 + … + B p ,12 Y 2 t p + B 1,11 Y 1 t –1 +… + B p ,11 Y 1 t p + ε 1 t Y 2 t = B 0,21 Y 1 t + B 1,21 Y 1 t –1 + … + B p ,21 Y 1 t p + B 1,22 Y 2 t –1 +… + B p ,22 Y 2 t p + 2 t This simultaneous equations system can be written, B(L) Y t = t , where B(L) = B 0 B 1 L – B 2 L 2 – … – B p L p and in general B 0 is not diagonal. t are the structural shocks . The system B(L) Y t = t is called a structural VAR ( SVAR ). This SVAR has a reduced form (Sims (1980)), which is identified: Reduced form VAR(p) : Y t = A 1 Y t –1 + … + A p Y t p + u t o r A ( L ) Y t = u t , w h e r e A ( L ) = I A 1 L – A 2 L 2 – … – A p L p innovations : u t = Y t – Proj( Y t | Y t -1 ,…, Y t p ) Eu t u t = Σ u

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Reduced form to structure: Suppose: (i) A(L) is finite order p (known or knowable) (ii) u t spans the space of structural shocks ε t , that is, t = Ru t , where R
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VAR Slides - NBER Summer Institute Whats New in...

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