## Tuesday, April 30, 2013

### Confidence Intervals for Impulse Response Functions

An impulse response function gives the time-path for a variable explained in a VAR model, when one of the variables in the model is "shocked". We get a "picture" of how the variable in question responds to the shock over several periods of time.

An impulse response function (IRF) is essentially a type of conditional forecast. It's a messy function of the estimated coefficients in the VAR model, and the data. So, it's really just a point estimate, period by period. There's some uncertainty associated with the IRF, of course - this comes from the uncertainty associated with the estimated coefficients in the model. So, we really need to report a confidence band, period by period, to go with the IRF.

This point has come up in previous posts in connection with the (free!) JMulTi and Gretl econometrics packages - see here, and here. Here's the sort of result we want - this one is an IRF with a 90% confidence band, produced using Gretl:

Now, there's more than one way to compute a confidence band for an IRF. This becomes an issue when we are looking at the full time-horizon over which the IRF is computed. We could compute (say) a 90% forecast confidence interval at each point in time, and then take the union of these intervals. However, this won;t guarantee a 90% coverage probability for the confidence band as a whole.

A recent paper by Lütkephol et al. (2013) addresses this question, while considering the relative merits of some different ways of proceeding. The abstract to their paper reads:

"In vector autoregressive analysis conﬁdence intervals for individual impulse responses are typically reported to indicate the sampling uncertainty in the estimation results. A range of methods are reviewed and a new proposal is made for constructing joint conﬁdence bands, given a prespeciﬁed coverage level, for the impulse responses at all horizons considered simultaneously. The methods are compared in a simulation experiment and recommendations for empirical work are provided."

Inter alia, the authors conclude as follows:
"A number of proposals for constructing such bands in a classical setting are reviewed and it is argued that they either may not obtain the desired coverage level, are conservative or lack a theoretical justiﬁcation based on asymptotic theory. We have also proposed an adjustment of the Bonferroni method that at least partly accounts for these deﬁciencies. Our adjusted Bonferroni band is shown to have some practical advantages."
And:
"Also the standard bands that just connect conﬁdence intervals for individual impulse response coeﬃcients may have a very distorted coverage substantially below the nominal level."
There are other important results in this paper that users of VAR models should be aware of.

1. Dear prof.,

In the graph above the effect is statistically nill because the lower band is always below 0...is it correct?
Please answer me, I'm in trouble with the interpretation of graphs like that!

Cris.

1. At the 10% significance level, the response is not significantly different from zero.

2. Dear Dr Giles

Could you please elaborate the procedure to compute IRFs and Variance Decomposition in VARs with some numerical examples in your blog.

Thanks.