1. VAR models and cointegration
Dr. Thomas Kigabo RUSUHUZWA

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;
How can we recover the structural parameters

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) model
Consider the bivariate VAR(p)

13. V. Determination of the number of lag p
Use 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-causality
Assume a lag length of p
Estimate by OLS and test for the following hypothesis
Unrestricted sum of squared residuals
Restricted sum of squared residuals

16. VII. Impulse Response Function (IRF)
Objective: the reaction of the system to a shock

17. Impulse Response Function (IRF)…
(multipliers
Reaction of the i-variable to a unit change
in 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. CTD The effect of y1 is in the second period is The effect on Y2 is
In the third period, the effect on y13 is
The 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 decomposition
Then, the MA representation:

25. CTD
Orthogonalized impulse-response
Function.
However, Q is not unique

26. VIII. Variance decomposition
Variance 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.