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VAR Models Yankun Wang, Cornell University, Oct 2009

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What is VAR? A var (p) model is: with and Originally proposed by Sims (1980) Efficient way of summarizing information contained in the data Useful for forecasting Conduct economically interesting analysis under meaningful identification restrictions

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Outline: Reduced form VAR Wold Theorem Specification Estimation Presentation of Results Structural VAR Identification Potential extension to “Evaluation of Currency Regimes: the Unique Role of Sudden Stops”by Assaf Razin and Yona Rubinstein

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The Wold Theorem Wold Theorem: Every stationary process can be written as the sum of two components: a deterministic part and an MA(∞) part. As a result: Every stationary process can be written as a VAR process of infinite order. Potential Problem: In reality, we can only deal with finite order.

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Specification What is the appropriate lag length in the VAR? Three criterions: i. Akaike information criterion (AIC) ii. Schwarz criterion (SIC) iii. Hannan-Quinn criterion (HQC) ( all functions of m, T, and variance-covariance matrix) In practice: Fix an upper bound of lag length q (12), choose the q which minimizes one of the information criterion AIC is inconsistent For T>20, SIC and HQC will always choose smaller models than AIC

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Estimation Multivariate GLS estimates are the same as equation by equation OLS estimates. For unrestricted VAR models: ML estimates and equation by equation OLS estimates coincide. When a VAR is estimated under some restrictions, ML estimates are different from OLS estimates; ML estimates are consistent and efficient if the restrictions are true.

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Presentation of Results It is rare to report estimated VAR coefficients. Instead: Impulse responses Forecast error variance decomposition: assess the relative contribution of different shocks to fluctuations in varables Historical Decomposition: given the path of one specific shock, how will the variables evolve?

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Structural VARs Suppose we have estimated the following reduced form VAR: with. ! : u is just reduced form residuals, no economic meaning. Solution: Assume, where is the vector of fundamental shocks, then naturally: Lack m(m-1)/2 restrictions to exactly identify D.

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Short-Run Timing Restrictions Example: Suppose m=3: output, inflation and interest rate: Criticism: hard to justify from theoretical foundations In practice: try to switch the ordering the variables

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Long-run Impact Restrictions Classical example: Blanchard and Quah ( 1989) Suppose two variable system: output growth and unemployment Total long run impact matrix: Assume: accumulated long-run effect of demand shocks on is zero,

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Sign Restrictions Restricting the sign (and/or shape) of structural responses. Faust (1998), Canova and De Nicolo (2002) and Uhlig(2005) Informally used in research ( e.g. monetary shocks must generate a liquidity effect): this approach makes it explicit More justifiable by theoretical model: DSGEs seldom deliver all zero restrictions, but lots of sign restrictions usable

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Example: Uhlig (2005) Contractionary Policy: Responses of prices and nonborrowed reserves are not positive and those of the federal funds rate are not negative

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Razin and Rubinstein: Output Growth Rate Prob of Sudden Stop/Currency Crisis Flexible Exchange Rate Regime Capital Account Liberalization - - - + +

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Could we extend this framework to a dynamic analysis? What are the variables to include? [growth rate of output; change/level of exchange rate regime; change/level of capital account liberalization; probability of crisis] What are the shocks we want to identify? One choice: shocks interpreted according to variables

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How to Identify the Structural Shocks? Shock run restriction? Long run restriction? Sign restriction? Available convention: Exchange rate shock from flexible to peg should increase crisis probability; Capital Account Liberalization shock from less to more free capital flow should increase crisis probability What are their effects on output?

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