Why use vector error correction model?

I am confused about the Vector Error Correction Model (VECM).

Technical background:
VECM offers a possibility to apply Vector Autoregressive Model (VAR) to integrated multivariate time series. In the textbooks they name some problems in applying a VAR to integrated time series, the most important of which is the so called spurious regression (t-statistics are highly significant and R^2 is high although there is no relation between the variables).

The process of estimating the VECM consists roughly of the three following steps, the confusing one of which is for me the first one:

1. Specification and estimation of a VAR model for the integrated multivariate time series

2. Calculate likelihood ratio tests to determine the number of cointegration relations

3. After determining the number of cointegrations, estimate the VECM

In the first step one estimates a VAR model with appropriate number of lags (using the usual goodness of fit criteria) and then checks if the residuals correspond to the model assumptions, namely the absence of serial correlation and heteroscedasticity and that the residuals are normally distributed. So, one checks if the VAR model appropriately describes the multivariate time series, and one proceeds to further steps only if it does.

And now to my question: If the VAR model describes the data well, why do I need the VECM at all? If my goal is to generate forecasts, isn't it enough to estimate a VAR and check the assumptions, and if they are fulfilled, then just use this model?

• As I understand it, a VECM is a VAR where the dependent variables aren't covariance stationary, but their first differences are. So in your step #1, I don't think your description is complete. Nov 27 '13 at 3:35
• Hello Wayne, right, it is about applying the VAR to difference-stationary data. One estimates a VAR for difference-stationary data, and then checks for possible cointegration applying some tests to the residuals of the estimated VAR. And then, if they are fulfilled, continues the procedure: but I don't understand why not just stop here and use the estimated, valid VAR? Nov 27 '13 at 14:48
• I believe the normality of residuals is not an assumption underlying a VAR model, contrary to what you mention in the second-to-last paragraph. Oct 30 '14 at 20:34
• Difference between VAR and VECM lies in co-integration Mar 1 '19 at 7:19

The foremost advantage of VECM is that it has nice interpretation with long term and short term equations.

In theory VECM is just a representation of cointegrated VAR. This representation is courtesy of Granger's representation theorem. So if you have cointegrated VAR it has VECM representation and vice versa.

In practice you need to determine the number of cointegrating relationships. When you fix that number you restrict certain coefficients of VAR model. So advantage of VECM over VAR (which you estimate ignoring VECM) is that the resulting VAR from VECM representation has more efficient coefficient estimates.

• Great!! Is it your own consideration or are you refering to a book/paper? If the second is the case, can you please provide the source? Nov 28 '13 at 12:18
• Well Granger representation theorem is a classical result. The statement about the efficiency is my own addition, which stems from the fact, that you lose efficiency if you estimate unnecessary coefficients. Nov 28 '13 at 13:17

I agree with mpiktas that the greatest interest of a VECM lies in the interpretation of the result, by introducing concepts such as long-term relationship between variables, and the associated concept of error correction, whereas one studies how deviations from the long-run are "corrected". Besides of this, indeed, if your model is correctly specified, the VECM estimates will be more efficient (as a VECM has a restricted VAR representation, while estimating VAR directly would not take this into account).

However, if you are only interested in forecasting, as seems to be the case, you might not be interested in these aspects of the VECM. Furthermore, determining the appropriate cointegrating rank and estimating these values might induce small sample inaccuracies, so that, even if the true model was a VECM, using a VAR for forecasting might be better. Finally, there is the question of the horizon of the forecast you are interested in, which influences the choice of the model (regardless of which is the "true" model). If I remember well, there are kind of contradictory results from the literature, Hoffman and Rasche saying advantages of VECM appear at a long horizon only, but Christoffersen and Diebold claiming you are fine with a VAR for long term...

• Peter F. Christoffersen and Francis X. Diebold, Cointegration and Long-Horizon Forecasting, Journal of Business & Economic Statistics, Vol. 16, No. 4 (Oct., 1998), pp. 450-458
• Engle, Yoo (1987) Forecasting And Testing In Co-Integrated Systems, Journal of Econometrics 35 (1987) 143-159
• Hoffman, Rasche (1996) Assessing Forecast Performance In A Cointegrated System, Journal Of Applied Econometrics, VOL. 11,495-517 (1996)

Finally, there is thorough treatment (but not very clear in my opinion), discussion of your question in the Handbook of forecasting, chapter 11, Forecasting With Trending Data, Elliott.

One description I've found (http://eco.uc3m.es/~jgonzalo/teaching/timeseriesMA/eviewsvar.pdf) says:

A vector error correction (VEC) model is a restricted VAR that has cointegration restrictions built into the specification, so that it is designed for use with nonstationary series that are known to be cointegrated. The VEC specification restricts the long-run behavior of the endogenous variables to converge to their cointegrating relationships while allowing a wide range of short-run dynamics. The cointegration term is known as the error correction term since the deviation from long-run equilibrium is corrected gradually through a series of partial short-run adjustments.

Which seems to imply that a VEC is more subtle/flexible than simply using a VAR on first-differenced data.

• Could you please provide the source of this quotation?
– whuber
Nov 27 '13 at 22:05
• I have read really a lot about VECM, but still, to my own surprise, I don't know why I need this model if I am just interested in, forecasting, say. What the authors suggest is, that one just rewrites the VECM as VAR using some formula in order to generate forecasts. The resulting VAR is, and should be, the VAR I get just directly applying the OLS procedure to the integrated data. So, why this detour over VECM?? Nov 27 '13 at 22:50
• @whuber: It's a paper I found by Googling: eco.uc3m.es/~jgonzalo/teaching/timeseriesMA/eviewsvar.pdf a class handout by Jesús Gonzalo. (The PDF doesn't have any identifying information in it.) Nov 27 '13 at 23:11
• @whuber, the variation of this citation you can find in any time series textbook dealing with VAR and VECM. Nov 28 '13 at 7:39
• @mpiktas The issue I am concerned about, as a moderator, is to identiy the source of this quotation. (I am not challenging its correctness or questioning its meaning or asking for further material to read). Borrowing materials is acceptable on this site, but using them without attribution is not. The quotation is unusual in that it appears in multiple places on the Web, but (IMHO) does not show up in any authoritative places (only in gray literature) and never with attribution. I wonder what the original source of this quotation is?
– whuber
Nov 29 '13 at 15:46

My understanding may be incorrect, but isn't the first step is just fitting a regression between time series using OLS - and it shows you if time series are really cointegrated (if residuals from this regression are stationary). But then cointegration is kind of a long-term relation between time-series and your residuals although stationary may still have some short-term autocorrelation structure that you may exploit to fit a better model and get better predictions and this "long-term + short term" model is VECM. So if you need only long-term relation, you may stop at the first step and use just cointegration relation.

We can selection time series models based on whether the data are stationary. • For this site, this is considered somewhat short for an answer, it is more of a comment. You should consider adding text explaining your figure! Dec 17 '15 at 15:19
• Welcome to our site! It looks like you are well positioned to make useful contributions. Note, however, that we work a little differently than Q&A or discussion sites. If you would take a few minutes to review our help center, I think you will get a better sense of what we're about and how you can best interact here.
– whuber
Dec 17 '15 at 16:15

You can't use VAR if the dependent variables are not stationary (that would be spurious regression). To solve for these issues, we have to test if the variables are cointegrated. In this case if we have a variable I(1), or all dependent variables are cointegrated at the same level, you can do VECM.

What I observed in VAR was that it is used to capture short-run relationship between the variables employed while VECM tests for the long-run relationship. For instance, in a topic where shock is being applied, I think the appropriate estimation technique should be VAR. Meanwhile, when testing through the process of unit root, co-integration, VAR and VECM, if the unit root confirmed that all the variables were I(1) in nature, you can proceed to co-integration and after tested for co-integration and the result confirmed that the variables are cointegrated meaning there is long-run relationship between the variables then you can proceed to VECM but if other wise you go for VAR.

If someone pops up here with the same question, here is the answer why one needs VECM instead of VAR. If your data is non stationary (finance data + some macro variables) you cannot forecast with VAR because it assume stationarity thus MLE (or OLS in this case) will produce forecasts that mean revert to quickly. VECM can handle this problem. (differenced series would not help)

As has been rightly pointed out in the earlier posts , A VECM enables you to use non stationary data ( but cointegrated) for intepretation. This helps retain the relevant information in the data ( which would otherwise get missed on differencing of the same)