1. Vector Autoregressions (cntd)
Econ 427 lecture 19 slides
Vector Autoregressions (cntd)
Byron Gangnes

2. Analyzing dependence in a VAR system
We looked at Granger Causality tests last time.
We can also use impulse-response functions to see how a shock to the variables affect each other.
We want to know how an innovation in one of the variables will affect itself over time and the other variable(s).
Byron Gangnes

3. Recall the VAR(1) model
A VAR(1) for a system of N=2 variables runs 2 equations where in each case 1 lags of the own and other variables are included.
where
Byron Gangnes

4. Impulse-Response Functions
We can write the VAR in moving average form:
There are a couple difficulties here.
First, we would like to “normalize” the size of a shock so that we can meaningfully compare size of shocks hitting the two variables
and we would like to be able to shock one variable independently of the other and see how that affect both variables in the system.
Byron Gangnes

5. Normalizing by the “Cholesky factors”
Define:
Note that by construction epsilon2-star is orthogonal to epsilon 1.
How would you show that?
Byron Gangnes

6. Proof of orthogonal errors
Byron Gangnes

7. Normalizing by the “Cholesky factors”
Substituting, this gives y1 and y2 as functions of shocks to epsilon1 and epsilon2*:
Byron Gangnes

8. The model for Impulse-Response Analysis
We also normalize both innovations so that they have a unit variance (not shown here—see discussion in book).
The normalized model is:
Note that it may make a difference which order you put the variables in. You can check that out empirically.
Byron Gangnes

9. Forecasting with VARs
Econometric models and interdependence of forecasts
Byron Gangnes