Forecasting With Level VARs Despite Non-Stationarity And/Or Cointegration. Intuition.

I sometimes run into people who are violently opposed to fitting VARs to non-stationary or cointegrated data. For sure, there are problems with frequentist inference and IRFs under these conditions. See Sims, Stock, and Watson (1990). But forecasting ability is not necessarily destroyed.

Here, I am talking about vanilla OLS VARs, to say nothing of Bayesian VARs for which inference and IRFs are still generally good.

A VAR in levels is usually competitive with a VAR in differences (non-stationary but not cointegrated) or a VECM (cointegrated) out-of-sample. Christoffersen and Diebold (1998) find that “nothing is lost by ignoring cointegration” with respect to out-of-sample MSE. They instead introduce new error measurements that account for the cointegrated relationship explicitly and find (not surprisingly) that VECM performs better. But most forecasters today are still mostly concerned with conventional measures, like MSE.

Intuition: Non-Stationary, Not Cointegrated

When the data are non-stationary but not cointegrated, an AR in differences can trivially be re-written as an AR(2) in levels.

y_t - y_{t-1} = \alpha_0 + \beta(y_{t-1}-y_{t-2}) + \varepsilon_t

y_t = \alpha_1 + (1+\beta) y_{t-1} - \beta y_{t-2} + \varepsilon_t

The intercepts may change between the two may change as a function of having different control variables, but the relationship in the slope coefficients actually bares out if you try it with real data (within reasonable OLS margin of error).

The levels-AR(2) has more parameters to estimate, so we might expect that it could perform worse from a variance point-of-view. However, a non-stationary variable is cointegrated with lags of itself, meaning that OLS is super-consistent; i.e. it converges faster than normal to the true coefficients. So, these effects come out in the wash.

Intuition: Non-Stationary, Cointegrated

Yes, cointegration implies non-stationarity, but humor my desire for symmetrical headers. Here, we still have that cointegration implies super-consistency for the VAR in-levels. So whatever information is lost by ignoring the error-correction term is likely to come out in the wash. Similarly, the lagged dependent variable in the VAR in-levels will ensure that the residuals are stationary, which will guard against spurious lagged independent variables; i.e., spuriousness comes from unaccounted non-stationarity, but a lagged dependent variable is an accountant.

I will be following up with some simulation results… eventually.

References

Christoffersen, P. F., & Diebold, F. X. (1998). Cointegration and long-horizon forecasting. Journal of Business & Economic Statistics, 16(4), 450-456.

Sims, C. A., Stock, J. H., & Watson, M. W. (1990). Inference in linear time series models with some unit roots. Econometrica: Journal of the Econometric Society, 113-144.

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