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Structural VAR: the AB model
- 1. TIME SERIES ECONOMETRICS STRUCTURAL VAR: THE AB MODEL
- 2. Structural VAR representation Suppose we have the following bivariate VAR(1) model of the following form (Note: we suppress the intercept term for simplicity): In the framework, y and x are specified to be related contemporaneously as well as with lags and u1t and u2t are structural shocks of yt and xt respectively and OLS estimation method may not be used to estimate the above since the error terms are correlated with the right-hand-side contemporaneous variables. ttttt ubxyxay 11111211112 +++−= −− αα ttttt ubxyyax 22212212121 +++−= −− αα 10 01 , 0 0 ~ 2 1 N u u t t
- 3. SVAR and Reduced-Form VAR • Expressing (1) and (2) in matrix notation, we have: • Rearranging terms: + = − − t t t t t t u u b b x y x y a a 2 1 22 11 1 1 2221 1211 21 12 0 0 1 1 αα αα + = − − − − t t t t t t u u b b a a x y a a x y 2 2 22 11 1 21 12 1 1 2221 1211 1 21 12 0 0 1 1 1 1 αα αα + = − − t t t t t t x y x y 2 1 1 1 2221 1211 ε ε ββ ββ Reduced form VAR/Tradition al VAR SVAR or primitive VAR
- 4. Objectives of VAR Analysis To uncover dynamic interactions among the variables examined: Two VAR approaches: - minimal theoretical restrictions: traditional VAR - theoretical identifying restrictions: Structural VAR (Theoretic-dependent) Similar analytical principle – analyzing the effects of structural shocks on the variables under study.
- 5. Main issue in VAR/SVAR The basic issue in VAR modeling is structural shock identification. Note that, the reduced form errors make up of structural errors. Accordingly, a shock to a reduced form “error” cannot be taken as a structural shock to a particular variable: − − − = t t t t u u b b a a aa 2 1 22 11 21 12 21122 1 0 0 1 1 1 1 ε ε
- 6. Structural shock identification Note that: This is AB model proposed by Amisano and Giannini (1997)). Let B be an identity matrix. To just identify the structural shocks, the number of restrictions on the off-diagonal elements of matrix A is n(n- 1)/2 since the estimated variance-covariance matrix of reduced form residuals has n(n+1)/2 unique elements. In general, the number of restrictions for exact identification is 2n2 - n(n+1)/2 on A and B matrices. = − t t t t u u b b a a 2 1 22 11 1 21 12 2 1 0 0 1 1 ε ε = t t t t u u b b a a 2 1 22 11 2 1 21 12 0 0 1 1 ε ε tt BuA =ε
- 7. Case 1: a12 = a21 = 0 • In this case, the contemporaneous value of x will not appear in the y equation and, likewise, the contemporaneous value of y will not be in the x equation. Hence, the reduced form residuals are identical to structural residuals (or shocks). +− − − = tt tt t t uua uau aa 2121 2121 21122 1 1 1 ε ε = t t t t u u 2 1 2 1 ε ε
- 8. Case 2: a12 = 0 This is a recursive identification as suggested by Sims(1980), the so-called Cholesky factorization. In this case, x is affected contemporaneously by structural shock in y but not the reverse (y is also affected by structural shock in x with lag). = t t t t u u a 2 1 2 1 21 1 01 ε ε ttt tt uua u 21212 11 +−= = ε ε
- 9. Theoretical Restrictions In general, the model contains more than two variables. The essence of SVAR is to use theoretical restrictions to identify the shocks. Since these restrictions are theoretical-dependent, illustration need to be given from the theoretical model adopted to assess the relationship among the variables. To be fruitful, the models involving more than two variables will be used.
- 10. Example I: Kozluk and Mehrotra (2009) Kozluk, T., and Mehrotra, A., (2009), The impact of monetary policy shocks on East and South-east Asia. Economics of Transition, 17(1), 121-145.
- 11. Step I: Estimate the VAR • The model contains 6 variable. • The construction of the AB model and its restrictions should be based on theories (see the paper). • Estimate the 6-variable VAR model containing the six variables (i.e. the reduced form VAR) • It is suggested that the information criterion is used to the determine the VAR lag order. • This step is standard in any VAR analysis, i.e. an estimation of the reduced-form VAR. • From the reduced form VAR, a structural identification is made.
- 12. Step II: Construct the A and B Matrices • There are various ways to construct the A and B matrices. • Take note the matrices are square matrices with the size equals to the number of endogenous variables. In our case, it is 6. • There are various ways to construct the matrices. The simple way is to use matrix object. • Click object/new object and choose Matrix- Vector-Coef. • Eviews will bring up the following (next page).
- 14. Specify the size of matrix
- 15. Fill in the restrictions as specified in the model with NA for the value to be estimated
- 16. Close the Matrix Object. MATA (i.e. Matrix A) will appear in the workfile. Construct Matrix B in the same manner.
- 17. Step III: Estimate the structural factorization After the estimation of the VAR in Step I and construction of the A and B matrices in step II, proceed to estimate the AB model by estimating structural factorization. Do not close your VAR result window as the structural factorization needs to be estimated from there. Click Proc/Estimate Structural Factorization. Eviews will bring to…(next page).
- 18. Choose matrix specification Enter the name of matrix A and matrix B previously created.
- 19. Step IV: Generate IRF and VDC • The previous step gives you the estimated of contemporaneous relations as specified in the AB format. • In order the generate the IRF and VDC based on your identification scheme (the SVAR), make sure that the impulse definition is STRUCTURAL DECOMPOSITION. • Shock will be named as Shock 1, Shock 2 and so on corresponding to the order of the variables entered when estimating the VAR in the first step.