(This article was originally published at Econometrics Beat: Dave Giles' Blog, and syndicated at StatsBlogs.)
Recently, a reader asked about generating forecasts from an estimated Error Correction Model (ECM). Really, the issues that arise are no different from those associated with any dynamic regression model. I talked about the latter in a previous post in 2013.
Anyway, let's take a look at the specifics.........
Anyway, let's take a look at the specifics.........
For simplicity, suppose that we have just two variables, Y and X, and a single-equation ECM, with Y as the variable that we want to model. The following discussion extends trivially if we have additional variables.
Recall that an ECM is used when all of the variables are I(1), and cointegrated. We'll assume that both of these features of the data have been established by previous testing. For instance, the non-stationarity of the series may have determined by applying augmented Dickey-Fuller tests; and the presence of cointegration may have been determined by using the Engle-Granger two-step procedure.
Because we have just two variables, we can't have more than one cointegrating relationship between them; and any cointegrating relationship is unique. (This situation will change if there are more than two I(1) variables.)
The purpose of an ECM is to enable us to model the short-run dynamics between X and Y. The cointegrating equation measures the long-run relationship.
It will be helpful to think of the construction of the ECM in the following way.
- The second step in the Engle-Granger cointegration testing procedure involves estimating the following "cointegrating regression" relating Y and X, using OLS:
Yt = a + bXt + ut (1)
- The lagged residual from (1) is Zt-1 = (Yt-1 - a* - b*Xt-1), where a* and b* are the OLS estimates of a and b. Zt-1 is the so-called "error correction" term.
- The ECM is then formulated as
or,
ΔYt = α + βΔXt + γ (Yt-1 - a* - b*Xt-1) + εt (3)
(In fact, we may also have lags of either or both of ΔXt and ΔYt as additional regressors in (3). Only the latter lags will have any effect on the following discussion, and this will be taken up below.)
Suppose that we estimate the ECM, (3) by OLS, yielding parameter estimates α*, β*, and γ*.
Re-arranging the estimated equation (3), we have:
Yt = (α* - a*γ*) + β*ΔXt - γ*b*Xt-1 + (1 + γ*)Yt-1 + residual (4)
This equation is a "dynamic" regression - it predicts Yt, but Yt-1 appears as a regressor on the RHS. (In addition, certain restrictions apply to the estimated coefficients as a result of the inclusion of the error correction term in the ECM. However, that's not the important point here.)
To use (4) to obtain a forecast, Y*t, for Yt, we would set the residual to zero and use the estimated coefficients and the data for ΔXt, Xt-1, and Yt-1. (The latter value is known at time t.) However, when it comes to forecasting Yt+1, we have to distinguish between "static" and "dynamic" forecasting. If these terms aren't familiar, this is the time to read my earlier post.
To forecast Yt+1 we can use (4), with a shift of one time-period, in one of two ways.
We can use the actual value for Yt on the RHS:
Y*t+1 = (α* - a*γ*) + β*ΔXt+1 - γ*b*Xt + (1 + γ*)Yt (5)
or, we can use the previous forecast value, Y*t on the RHS:
Y*t+1 = (α* - a*γ*) + β*ΔXt+1 - γ*b*Xt + (1 + γ*)Y*t (6)
Equation (5) generates "static" forecasts; while equation (6) generates "dynamic" forecasts.
When we are doing genuine ex ante forecasting into the future, we have to use dynamic forecasting. My earlier post illustrated all of this, using EViews.
If our ECM includes lags of ΔYt as regressors, as will often be the case, the story changes in a pretty obvious way. For instance, suppose that (2) is generalized to:
ΔYt = α + βΔXt + γZt-1 + δΔYt-1 + εt (7)
Then the forecasting equation, (4), becomes:
Yt = (α* - a*γ*) + β*ΔXt - γ*b*Xt-1 + (1 + γ* + δ*)Yt-1 - δ*Yt-2 + residual (8)
Again, we have a (restricted) dynamic model - this time there are two lags of Y on the RHS. We can again distinguish between static and dynamic forecasts, as above.
That's all that there is to it.
[Postscript: Can you see where an example of a "pre-testing" problem arises in the discussion above?]
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