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I know that for univariate framework, a typical process to deal with seasonality is :

  1. detect
  2. correct (for instance, withdraw seasonal factors)
  3. forecast
  4. re-seasonalize the forecasted series (for instance, incorporate seasonal factors back)

What would be the equivalent of such a pattern for VECM ?

The following crossvalidated thread points out that seasonality can be "handled [...] outside of the model (by seasonally adjusting the series before fitting a VAR)". But the precise steps are fuzzy to me.

Let's say $Y$ is my $I(1)$ target variable for VECM and I go :

  1. check if $Y$ seasonal
  2. correct with seasonal factors
  3. Engle-Granger test and find my cointegrating vector
  4. Forecast Long-Term relationship
  5. Apply seasonal factors back to re-seasonalize the forecast of $Y$
  6. Find the rest of VECM ($\Delta Y = ...$ & short-term)
  7. Forecast short-term relationship and final equation

Is this process correct ?

And what about the case where there is seasonality in the predictors too ?

Let's say, we have seasonality for some of the variables in $ X \;=\; (\;X_1,\;...,\;Xn\;) $ where $X$ is the cointegrating vector. And we come to step 5.

How am I supposed to seasonalize back the forecast ?

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  • $\begingroup$ Steps 4-7 are a bit confusing, especially the division into long and short term stuff. Putting the seasonal factors back in between them does not sound logical. Here is what I would do. Given that you have done steps 1-3 and found that your variables are cointegrated, you should then: 4. Fit a VECM. 5. Make forecasts from it. 6. Add/multiply the seasonal forecasts to the ones you got from the VECM. Done. $\endgroup$ – Richard Hardy Jan 21 at 19:27
  • $\begingroup$ Thanks for the additional details. I was inclined to think that my pattern was flawed, but I could not point where exactly. Seems indeed better practice to leave the ECM inner-sequence untouched and to incorporate seasonality back only after the overall forecast of both long-term and short-term components. $\endgroup$ – Johannes Konrad Jan 22 at 10:41
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How to handle seasonality outside of the VECM and incorporate it into the forecasts? One possible recipy is as follows*:

  1. Seasonaly adjust each seasonal variable separately using e.g. SARIMA, ARIMA with seasonal dummies or Fourier terms, STL decomposition or another method.
    For each seasonal variable, produce forecasts of the seasonal component.
  2. Model the seasonally-adjusted seasonal variables and the nonseasonal variables, if any, with a VECM.
    Produce forecasts from the VECM.
  3. If the seasonal adjustment was additive, add the forecasts from the VECM to the seasonal forecasts. If the seasonal adjustment was multiplicative, multiply them.

You now have forecasts of the original variables.

*This assumes the seasonally-adjusted variables can be adequately modelled by a VECM.

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  • $\begingroup$ Thanks, I got the overall approach now. Your answer leaves me with another puzzle though. What do you mean by "forecast of the seasonal component" ? It seems you're advocating for a forecast in the same sense as for a stationary residual, meaning something quite different from applying seasonal indices at the end of the process. I have no memory of ever reading about this. I have found some litterature about logair (a series of airline passenger dealt with in an article of Box and Jenkins) that seems related to a handling of seasonality I was not aware of, but it may be out of the subject. $\endgroup$ – Johannes Konrad Jan 22 at 10:39
  • $\begingroup$ @JohannesKonrad, in the simplest case of seasonal dummies, you just use the estimated slope coefficient for the dummy for the appropriate season as a forecast. In case of other deterministic seasonal patterns, you extrapolate them into the future and pick the one you need for the corresponding season of interest. I am not sure what you mean by seasonal indices. $\endgroup$ – Richard Hardy Jan 22 at 11:01
  • $\begingroup$ I should probably have said seasonal factors, as in my main post, instead of indices which I though was equivalent. I just mean the values S(t) that are added (multiplied) with other components in additive (multiplicative) schema. From what I grasp of your response I will have to modify the inner equations of long-term and short-term relationships to add seasonal dummies as predictors. Which I'm not sure I know how to do neither in SAS nor in R. Anyway, thanks for your very quick and instructive answers. I understand much better now. $\endgroup$ – Johannes Konrad Jan 22 at 12:10
  • $\begingroup$ @JohannesKonrad, once the variables have been seasonally adjusted in step 1, there is nothing special about VECM in step 2; no need to add any seasonal dummies there. Now, if you want to model seasonality within VECM (which is not what my answer is about), then you add seasonal dummies as exogenous variables in a VECM. Conceptually that is pretty simple. In R it is also simple, as model fitting functions often have the option for exogenous regressors. $\endgroup$ – Richard Hardy Jan 22 at 12:28

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