# VAR in levels or differences for prediction only?

I have several non-stationary time-series I use as predictor variables for time-series changes in bond market liquidity. I aim to do the forecasting with VAR models. I know that for inferences, the VAR model should be specified in differences when data is non-stationary. However, I read in a comment to this question (VAR forecasting methodology), that a VAR in levels is fine for prediction purposes. Can someone confirm this, maybe even with a reference to literature? Thanks !!

## 1 Answer

In absence of cointegration, running a VAR in levels is not justifiable, because the dependent variables diverge from any possible combination of the regressors (unless in each equation of the model, only the own-lag is present, e.g. $$x_{1,t}=a_{11}x_{1,t-1}+\varepsilon_{1,t},\dots,x_{k,t}=a_{k1}x_{k,t-1}+\varepsilon_{k,t}$$, but that is a very special case).

Under cointegration, evidence is mixed as to whether VECMs yield better forecasts than unrestricted VAR models, especially for short forecast horizons; see e.g. Engle and Yoo (1987), Hoffman and Rasche (1996), Löf and Franses (2001), and Chigira and Yamamoto (2009); in the meantime, Christoffersen and Diebold (1998) find that imposing cointegration does improve forecasting results in long horizons. There, VECMs do better than VARs in levels, because over longer periods, the effects of error correction due to cointegration are sufficiently strong to warrant modelling the error correction mechanism explicitly (as done in VECMs).

See this answer by @Matifou for several of the same references.

References:

• CHIGIRA, H. & YAMAMOTO, T. 2009. Forecasting in large cointegrated processes. Journal of Forecasting, 28, 631-650.
• CHRISTOFFERSEN, P. F. & DIEBOLD, F. X. 1998. Cointegration and long-horizon forecasting. Journal of Business & Economic Statistics, 16, 450-456.
• ENGLE, R. F. & YOO, B. S. 1987. Forecasting and testing in co-integrated systems. Journal of Econometrics, 35, 143-159.
• HOFFMAN, D. L. & RASCHE, R. H. 1996. Assessing forecast performance in a cointegrated system. Journal of Applied Econometrics, 11, 495-517.
• LÖF, M. & FRANSES, P. H. 2001. On forecasting cointegrated seasonal time series. International Journal of Forecasting, 17, 607-621.
• Thanks Richard. That helps! So am I right in concluding that if all time-series data turns out to be stationary after differencing and is hence integrated of order 1, I should use a VAR in differences for my prediction task? (assuming absence of cointegration) – neo_one Nov 21 '17 at 16:49
• @neo, yes, that is correct. – Richard Hardy Nov 21 '17 at 17:02
• @RichardHardy Could you explain your first point a bit more. What do you mean by "dependent variables diverge from any possible combination"? – Jacob H Nov 21 '17 at 21:16
• @JacobH, any combination of integrated but not cointegrated variables diverges from any individual series over time. E.g. two noncointegrated random walks diverge over time. Regressing on on the other is technically possible but will deliver misleading results. – Richard Hardy Nov 22 '17 at 6:17