VAR models I Presentation of a Standard VAR model Vector Autoregressive VAR models are a generalization of univariate Autoregressive AR models and can be considered a kind of hybrid between the univariate time series models and simultaneous equations ID: 1028511
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1. Dr. Thomas Kigabo RUSUHUZWAVAR models
2. I. Presentation of a Standard VAR model Vector Autoregressive (VAR) models are a generalization of univariate Autoregressive (AR) models and can be considered a kind of hybrid between the univariate time series models and simultaneous equations models:(1) Structural VAR models and (2) Reduced form
3. Standard VAR model…Vector of innovations have zero means; Variance and covariance matrix: diagonal matrix;Yt: Vector of stationary variables, each of whose current values depend on different combinations of its p previous values and those of other variables.
4. Standard VAR model…The equation ( 1) may also be written as:
5. II. Example Let us consider the following VAR(1): With Y a vector of two stationary variables Y1 and Y2;Structural shocks
6. Example…The error terms (structural shocks) yt and xt are white noise innovations with standard deviations y and x and a zero covariance.The two variables y and x are endogenous Note that shock yt affects y directly and x indirectly.There are 10 parameters to estimate;Premultipication by B-1 allow us to obtain a standard VAR(1):
7. Example…
8. III. Reduced form The last equation is the reduced form whcih can be estimated by OLS equation by equation;Before estimating it, we will present the stability conditions (the roots of some characteristic polynomial have to be outside the unit circle) for a VAR(p)After estimating the reduced form, we will discuss the following:Granger-causality, Impulse Response Function; and variance decompostion
9. To illustrate this
10. IV. The stationarity condition
11. The stationarity condition…The VAR (1) is stable if the two roots are big than unit in absolute value
12. V. Estimation of a standard VAR (p) modelConsider the bivariate VAR(p)
13. V. Determination of the number of lag pUse of information criteria like AIC, SC, Hannan-Quinn (HQ). The multivariate versions of the information criteria are defined as follow:
14. VI. Granger Causality Consider two random variables
15. Test for Granger-causalityAssume a lag length of pEstimate by OLS and test for the following hypothesisUnrestricted sum of squared residualsRestricted sum of squared residuals
16. VII. Impulse Response Function (IRF)Objective: the reaction of the system to a shock
17. Impulse Response Function (IRF)…(multipliersReaction of the i-variable to a unit changein innovation j
18. Impulse Response Function …Impulse-response function: response of to one-time impulse in with all other variables dated t or earlier held constant.
19. Example: IRF for a VAR(1)
20. Reaction of the system(impulse)
21. Another way of explaining this VAR(1):Suppose a shock in the error term
22. CTDThe effect of y1 is in the second period is The effect on Y2 isIn the third period, the effect on y13 isThe effect o y2,3 is In summary:
23. Representation of VAR (p)If the VAR is stable then a representation exists.This representation will be the “key” to study the impulse response function of a given shock
24. Cholesky decompositionThen, the MA representation:
25. CTDOrthogonalized impulse-responseFunction.However, Q is not unique
26. VIII. Variance decompositionVariance decompositions give the proportion of movements in the dependent variables that are due to their own shocks, versus shocks to the other variables A shock to the variable directly affect that variable, but will also be transmitted to all of the other variables in the system through the dynamic structure of the VAR;