Impulse responses analysis with unit roots

[MC128]

Dear Tom and everyone,

I have a question about impulse responses analysis.

In the presence of unit roots and cointegration, Phillips (1998) "Impulse Response and Forecast Error Variance Asymptotics in Nonstationary VAR's", shows that impulse responses of a VAR in level in the long horizons are inconsistent i.e. they tend to random variables instead of the true responses. May I ask how erroneous it is, if the impulse responses analysis is restricted to a short time horizon? Also, how short should the time horizon be?

Thanks.

MC128

[TomDoan]

I generally recommend no more than about five years. Of course, if you do error bands, it usually becomes obvious when you run the IRF out too far, because the bands start to get quite wide.

Note that this is a problem only when you run an unconstrained VAR, since the largest root in a big system will generally be slightly above 1.0. If the largest roots are constrained to 1.0 by running a VECM, the "randomness" due to that dominant root won't be present.

[MC128]

Dear Tom,

Thank you so much. So if I run a VAR in level instead of a VECM, and restrict the impulse response analysis to with 5 years, then there shouldn't be many problem (even if the VAR is estimated with quarterly data)?

MC

[TomDoan]

That's usually OK. If you have some concern, you can check the magnitude of the dominant root with:

Code: Select all

eigen(cvalues=cv) %modelcompanion(mymodel)
disp %cabs(cv(1))

It's usually slightly above 1. For instance, in the Y-M model from Sims and Zha(1999), "Error Bands for Impulse Responses",
Econometrica, vol 67, no. 5, pp 1113-1156, the largest root is 1.00237 with quarterly data. Raised to the 20th power, that's still just 1.048 so it won't look all that much different from 1. Now, if you try to do 50 years, raise it to the 200th power, and it's 1.61, which will clearly dominate the roots that are 1.0 or less.