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

],,,[ _ probcrisist

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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?