# VAR in levels for cointegrated data

I have read some paper that expresses that "recent works" show we can use a VAR model with raw data I(1) but there has to be cointegration. This means that there is no reason to difference the data for VAR modelling. Any paper reference about this?

• I don't think this is recent work. It must be known at least since Engle & Granger "Co-integration and error correction: representation, estimation, and testing" (1987). The first stage regression of the E & G procedure does just that. Or do you have anything else in mind? – Richard Hardy Aug 31 '16 at 5:27
• Not recent. The classical reference paper is: Sims, Stock and Watson. 1990. Inference in Linear Time Series Models with Some Unit Roots (princeton.edu/~mwatson/papers/Sims_Stock_Watson_Ecta_1990.pdf) which show exactly what you mention. This is a very common approach in monteary economics, to simply estimate the VAR in levels. For an application see: Christiano, Eichenbaum and Evans. 2005. Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy (benoitmojon.com/pdf/…) – Plissken Aug 31 '16 at 20:39
• Finally, you could look at this paper: Phillips. 1998. Impulse response and forecast error variance asymptotics in nonstationary VARs (sciencedirect.com/science/article/pii/S030440769700064X) – Plissken Aug 31 '16 at 20:41
• @Plissken, why don't you collect your comments into an answer and post it, then the thread could be done with. – Richard Hardy Mar 17 '17 at 11:06
• I also posted on this topic here: stats.stackexchange.com/questions/191851/…. Like derFuchs mentions, this is something that's been known for a while, but the textbooks never mention it. Very important stuff! – Jacob H Oct 13 '18 at 20:06

It is not recent but many textbooks, video series, etc in Econometrics still do not acknowledge this.

You can have a look into the papers below. The classic reference would be the Sims, Stock and Watson paper. Definetly also look into Lütkepohl, he is an authority when it comes to SVARS.

You are incorrect in stating that "there has to be cointegration" to use VAR in levels. You can also estimate VAR in levels of non-stationary variables when there is no cointegration present! However, the Phillips, Durlauf and Ashley, Vergbugge papers argue for SVARs in levels instead of VECMs if cointegration is present (under certain conditions).

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.

Ashley, R. A., & Verbrugge, R. J. (2009). To difference or not to difference: a Monte Carlo investigation of inference in vector autoregression models. International Journal of Data Analysis Techniques and Strategies, 1(3), 242-274.

Phillips, P. C., & Durlauf, S. N. (1986). Multiple time series regression with integrated processes. The Review of Economic Studies, 53(4), 473-495.

Lütkepohl, H. (2011). Vector autoregressive models. In International Encyclopedia of Statistical Science (pp. 1645-1647). Springer Berlin Heidelberg.

Christiano, L. J., Eichenbaum, M., & Evans, C. (1994). The effects of monetary policy shocks: some evidence from the flow of funds (No. w4699). National Bureau of Economic Research.

Doan, T. A. (1992). RATS: User's manual. Estima.ote

I want to expand on derFuchs post. Further, I feel that too often when a unit root is present, people automatically just first difference their data. It's not always necessary!

## Prediction

We've always known that we can run a VAR in levels when series follow a unit root. For example, assume the two series $$x$$ and $$y$$ follow a unit root. If we regress $$x$$ on $$y$$ (i.e. $$y_t = \alpha + x_{t-1} + \epsilon$$) and they are not cointegrated, we'll obtain spurious results. However, if we include lags of $$y$$ then the results will no longer be spurious. This is because the lags of $$y$$ will guarantee that the residuals will be stationary.

If we regress $$x$$ on $$y$$ and they are cointegrated, we're fine. After all, in the traditional two-step ECM method we estimate this regression in the first stage.

We've only discussed AR models with distributed lags. However, VARs are just a system of AR models with distributed lags, so the above intuition still holds in the VAR context.

The reason why this all works is because unit roots (other than in the spurious regression case) have little impact the coefficients estimates. For example, if $$z$$ follows a unit root and we fit an AR(1), we'll get a coefficient of roughly 1; which is the best estimate of where a random walk will be next period (i.e. where it was last period). However, because $$z$$ follows a stochastic trend, it will not have a tendency to come back to its mean. Loosely speaking, this implies that the variance of our estimates will tend toward infinity as we have more data (i.e. no asymptotic variance). Broadly speaking, a unit root is a problem for estimating variance (i.e. standard errors) and less so for means (i.e. coefficients)

## Inference

As discussed above, the nature of a random walk (i.e. a unit root process) implies that the variance is explosive. You can see this yourself. Estimate prediction intervals after fitting an AR(1) to a unit root process.

As a result of this fact, it is tricky to perform hypothesis testing. Let's again abuse our incorrect, but enlighting, statement from above. If a unit root process has a variance that tends toward infinity, then you will never be able to reject any null hypothesis.

The big breakthrough of Sims, Stock, and Watson is that they showed that under some situations it is possible to perform inference when a process follows a unit root.

Another good paper, that expands on Sims, Stock, and Watson is Toda and Yamamoto (1995). They show that Granger Causality is possible in the presence of a unit root.

Finally, keep in mind that unit roots a still really tricky. They will impact what your VAR in weird ways. For example, a unit root implies that the MA representation of you VAR does not exist, as the coefficients matrix is not invertible. Therefore an IRF will not be accurate (though some people still do it).