VAR Models - PowerPoint Presentation

Slide1

VAR Models

Yankun

Wang, Cornell University, Oct 2009Slide2

What is VAR?

A

var

(p) model is: with andOriginally proposed by Sims (1980)Efficient way of summarizing information contained in the dataUseful for forecastingConduct economically interesting analysis under meaningful identification restrictionsSlide3

Outline:

Reduced form VAR

Wold

TheoremSpecificationEstimationPresentation of ResultsStructural VAR

IdentificationPotential extension to “Evaluation of Currency Regimes: the Unique Role of Sudden Stops”by

Assaf Razin and Yona RubinsteinSlide4

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. Slide5

Specification

What is the appropriate lag length in the VAR?

Three criterions:

Akaike information criterion (AIC)Schwarz criterion (SIC)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 criterionAIC is inconsistent

For T>20, SIC and HQC will always choose smaller models than AICSlide6

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. Slide7

Presentation of Results

It is rare to report estimated VAR coefficients.

Instead:

Impulse responsesForecast error variance decomposition: assess the relative contribution of different shocks to fluctuations in varablesHistorical Decomposition: given the path of one specific shock, how will the variables evolve?Slide8

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. Slide9

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 variablesSlide10

Long-run Impact Restrictions

Classical example: Blanchard and

Quah

( 1989)Suppose two variable system: output growth and unemploymentTotal long run impact matrix: Assume: accumulated long-run effect of demand shocks on is zero, Slide11

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 explicitMore justifiable by theoretical model: DSGEs seldom deliver all zero restrictions, but lots of sign restrictions usableSlide12

Example:

Uhlig

(2005)

Contractionary Policy: Responses of prices and nonborrowed reserves are not positive and those of the federal funds rate are not negativeSlide13

Razin and Rubinstein:

Output

Growth Rate

Prob

of Sudden Stop/Currency

Crisis

Flexible Exchange Rate Regime

Capital Account Liberalization

-

-

-

+

+Slide14

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 variablesSlide15

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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