In Chris Brooks book Introductory Econometrics for Finance on page 292 when discussing the problem with VARS he mentions.
"... many proponents of the VAR approach recommend that differencing to induce stationarity should not be done. They would argue that the purpose of VAR estimation is purely to examine the relationship between the variables, and that differencing will throw away information on any long-run relationship between the series away.."
Who are these "proponents of the VAR approach" and under what circumstances can I use a Non-stationary VAR?
Good answer Thomas. Further, many questions in economics are addressed at levels superficial to the dynamical realities of the phenomena. In other words, we need more physics in economics. More often, economics must go revisit Samuelson's struggles with Lotka-Volterra in the seventies. The fundamental construct of the ergodicity premise in economics should be more widely appreciated.
Chris Brooks was referring to Sims-Stock and Watson (1990) and Yamamoto (1995) lag-augmented model. The argument was whether a simple rearrangment of the levels generates I(0) series.
I guess this largely-debated issue can be resumed as follows:
1. VAR estimates are consistent iff variables are stationary. So, no stationarity no consistency. That is, you could not distinguish spurious correlations from causality. Bad...
Thus, and as a general rule, in a VAR model you need the stationarity of your time series to guarantee the consistency of your estimations, as noted by Thomas and established in a seminal paper in the 1980s by Sims, in which he put forward VAR models (Macroeconomic and reality).
So, if you are interested in checking whether the relationships among the endogenous variables are spurious, or if you want to perform hypothesis testing, then you should absolutely differencing your series to make them stationary (or jointly stationary if they are cointegrated, once a common stochastic trend is washed out). However, this is not the whole story....
In a seminal paper, Sims et al (1993) showed that:
2. If you are interested in the short-run dynamic relationships among your time-series, and you know for the nature of the data that these correlations are not spurious (e.g. a VAR with macro data, GDP, aggregate consumption, investment, inflation, etc.), then the accuracy of model fitting data is more important than consistency and, therefore, you should use log data and NOT differenced data even if they are I(1), given that taking differences implies a loss of estimation power because you are throwing away information (i.e. long-run covariance of the endogenous variables).
I would agree with Thomas here. The only case where you can use a non-stationary VAR is when there is cointegration in the system.
If there is no cointegration, a VAR(p) in levels reduces to a VAR(p-1) in first differences. In that case no information is lost because there are no long-run relationships in the data.
But if there is cointegration, information about long-run relationships will be lost by differencing.
Could we say that, in this context stationarity in the mean and the stationarity in the variance is enough? When you say stationarity you do not mean full stationarity?
Thomas Dimpfl Niklas Ahlgren What Sims and coathors had in mind was VAR estimations with macro series. Think, for example, to GDP, interest rate, inflation, investment, etc. Fo these variables you KNOW that correlation is NOT spurious, and this is why they suggested not to difference series. Their indication is valid for this class of models.
After all, consistency is about the convergence of point estimates to the population moment in the very long run ( t goes to infinite). Since they were interested in the short-run dynamics and cross-dynamics, they recommend to rely on log levels.
Bojan Baskot in VAR model usually assumes normally distributed errors, for which stationarity in mean and variance implies full stationarity...
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