Навіщо використовувати векторну модель виправлення помилок?

Мене бентежить модель вектора виправлення помилок ( VECM) ).

Технічна інформація:
VECM пропонує можливість застосувати векторну авторегресивну модель ( VAR ) до інтегрованого багатовимірного часового ряду. У підручниках вони називають деякі проблеми із застосуванням VAR до інтегрованих часових рядів, найважливішою з яких є так звана хибна регресія (t-статистика є дуже значною, а R ^ 2 - високою, хоча між змінними не існує зв'язку).

Процес оцінки VECM складається приблизно з трьох наступних етапів, заплутаний один з яких для мене перший:

1. Специфікація та оцінка моделі VAR для інтегрованого багатовимірного часового ряду

2. Обчисліть коефіцієнти ймовірності, щоб визначити кількість співінтеграційних відносин

3. Визначивши кількість коінтеграцій, оцініть VECM

На першому етапі оцінюється модель VAR з відповідною кількістю затримок (використовуючи звичайну корисність критеріїв придатності), а потім перевіряє, чи відповідають залишки припущенням моделі, а саме відсутність послідовного кореляційного зв’язку та гетероскедастичності та що залишки нормально розподіляються . Отже, людина перевіряє, чи відповідна модель VAR належним чином описує багатоваріантний часовий ряд, і переходить до подальших кроків, лише якщо це відбувається.

А тепер до мого запитання: Якщо модель VAR добре описує дані, навіщо мені взагалі потрібен VECM ? Якщо моя мета - генерувати прогнози , чи недостатньо оцінити VAR та перевірити припущення, а якщо вони виконані, то просто використовувати цю модель?

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.

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...

The literature (without a clear consensus) would start with:

• 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.

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.

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.

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.

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)