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 !!
Solved – VAR in levels or differences for prediction only
forecastingstationaritytime seriesvector-autoregression
Related Solutions
Regarding your use of ur.df: setting lags=5 may be fine -- but I do not think it could be motivated by the fact that you have daily data with 5-day weeks. So the way you do it is perhaps fine, but the motivation is not. Also, think whether type=drift is the most appropriate specification (use subject-matter knowledge).
Regarding differencing of Vstoxx: if Vstoxx is actually stationary, then by differencing you would induce a MA(1) term with coefficient $\theta=-1$ which makes the series non-invertible; this is kind of nasty and should be avoided.
Now if I look at the acf of Vstoxx there is no sign that Vstoxx should be stationary in level- form. Or am I missing something? Yes, it does look quite persistent. It may have a near-unit root (rather than a unit root), though. So I(0) versus I(1) is not quite clear.
This is counterintuitive and against the existing literature. Are you sure the right-hand-side variables are exogenous? If they are not (so that EurOis3 may be influencing one or more of them) you have a problem and the OLS estimates will be ill-behaved (read about simultaneous equation models (SEM); techniques like 2SLS and 3SLS may be used in these kinds of situations). Also check whether the model residuals are well-behaved (no autocorrelation etc.).
If I just difference all the other variables except for log(Open.Market.Operations) the result looks better. But I am not sure if this is allowed? Unfortunately, this is not allowed. Having a stationary dependent variable, a bunch of stationary regressors and one integrated regressor does not make sense and is known as unbalanced regression; in the long run, you would expect the integrated variable to diverge so it cannot be used as a regressor to explain a stationary variable that is varying around its mean. In other words, the right hand side of the equation would diverge from the left hand side of the equation, and that does not make sense.
If you are interested in forecasting (as you state in the beginning and repeat multiple times) rather than making inference (which you mention once), then
- estimating a VECM,
- transforming it into a VAR model and
- forecasting using the VAR model
is fine. The point forecasts and the confidence intervals will be fine. Note that VECM and the corresponding VAR model are two equivalent representations of the same model. The model outcomes do not change just because you manipulate the equations algebraically a little bit. The equations constituting the VECM and the corresponding VAR model are representing one and the same phenomenon.
Technically you could forecast directly from the VECM; the model equations give you the increments in the dependent variables for the next period, so it is straightforward. But probably it was computationally convenient for the authors of the R packages "urca" and/or "vars" to program a forecasting routine for VAR models only; so then a VECM needs to be converted to a VAR model.
Best 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: