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# VAR Models Yankun Wang, Cornell University, Oct 2009.

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VAR Models Yankun Wang, Cornell University, Oct 2009

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

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

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.

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

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.

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?

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.

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

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,

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

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

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

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

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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