VAR Models Yankun Wang, Cornell University, Oct 2009.
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Transcript of VAR Models Yankun Wang, Cornell University, Oct 2009.
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
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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
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.
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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
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
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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?