# Forecast encompassing test for cointegrated time series

I am forecasting an integrated time series variable $y_t$. I have two competing forecasts, $f^1_t:=f^1_{t|t-h}$ and $f^2_t:=f^2_{t|t-h}$. I would like to test whether $f^1_t$ forecast-encompasses $f^2_t$.
The length of $f^1_t$ and $f^2_t$ time series (the number of out-of-sample forecasts) is between 80 and 90.
I am following Diebold's Forecasting textbook Chapter 12.1, pages 463-464.

If I had cross-sectional data, I would run a test regression

$$y_t = \beta_1 f^1_t + \beta_2 f^2_t+\varepsilon_t$$

and test $H_0$: $\beta_1=1$ and $\beta_2=0$ using an F-test.

If I had stationary time series $y_t$, $f^1_t$ and $f^2_t$, I would run the same test regression but allowing for autocorrelated errors; that would amount to an ARMAX model with $f^1_t$ and $f^2_t$ as exogenous regressors; I would use a likelihood ratio test to test the null hypothesis.

However, I have time series $y_t$, $f^1_t$ and $f^2_t$ that are integrated and cointegrated. I am afraid I cannot use an ARMAX model as with stationary time series. I have thought of using a vector error correction model (VECM) but I cannot figure out what parameter restrictions would correspond to the hypothesis of interest. The hypothesis could be formulated using a structural VECM (SVECM); however, I am not sure if I could test it without making a bunch of identifying assumptions that might not hold in reality.

I have overviewed a number of journal articles about forecasting in cointegrated systems. Most of them seem to ignore the problem of integrated variables and basically use the equation above when testing for forecast encompassing. I think that is incorrect.

Question: how can forecast encompassing be tested in the spirit of Diebold's Forecasting textbook Chapter 12.1, pages 463-464 (see the equation above), when $y_t$, $f^1_t$ and $f^2_t$ are integrated and cointegrated?

References are welcome.

A related question is here. However, it does not address the case of integrated and cointegrated time series.

I hope that the following resources will be helpful (I ranked them per my fuzzy understanding):

• Paper by Jiang, Zhang and Song (2014) <= describes the encompassing test;

• Paper by Duy and Thoma (1998);

• Book by Pfaff (2008); two of the R packages, listed here, are complementary to the book.

References

Duy, T. A., & Thoma, M. A. (1998). Modeling and forecasting cointegrated variables: Some practical experience. Journal of Economics and Business, 50, 291-307. Retrieved from http://pages.uoregon.edu/mthoma/Papers/Modeling%20and%20Forecasting%20Cointegrated%20Variables.pdf

Jiang, C., Zhang, J., & Song, F. (2014). Selecting single model in combination forecasting based on cointegration test and encompassing test. The Scientific World Journal, 2014, Article ID 621917. doi:10.1155/2014/621917 Retrieved from http://www.hindawi.com/journals/tswj/2014/621917

Pfaff, B. (2008). Analysis of integrated and cointegrated time series with R (2nd ed.). Springer.

• Thanks! These sources were among the ones I had overviewed before with no luck; I did not list them in the OP, though, so thanks for pointing out. Jiang, Zhang and Song (2014) is quite messy and does not seem to provide a solution the problem of cointegrated forecasts and realized values (see p. 6 in their paper). Duy and Thoma (1998) seem to ignore the problem altogether (see the description on p. 296). Pfaff (2008) does not discuss it at all. – Richard Hardy Apr 8 '15 at 14:21
• @RichardHardy: You're welcome! I see. It's a pity I couldn't help. – Aleksandr Blekh Apr 8 '15 at 14:24
• @RichardHardy: No problem at all. Thank you. Upvoted yours. – Aleksandr Blekh Jun 24 '16 at 16:16

I figured out that even though $y_t$, $f^1_{t|t-h}$ and $f^2_{t|t-h}$ are cointegrated, there are cases when the test regression in the original post can be used with first differences instead of levels of variables:

$$\Delta y_t = \beta_1 \Delta f^1_t + \beta_2 \Delta f^2_t + \varepsilon_t$$

allowing $\varepsilon_t$ to follow an ARMA($p,q$) process.

The main argument is that in many practical applications, $y_t$ is not likely to adjust to some long-run equilibrium constituted by $y_t$, $f^1_{t|t-h}$ and $f^2_{t|t-h}$ together. Unless the forecasts $f^1_{t|t-h}$ and $f^2_{t|t-h}$ influence the formation of $y_t$ (which may be more likely in some situations than in other), I do not see much reason for $y_t$ to react to forecasts at all.

For example, if one considers historical data, makes some forecasts (emulating real time forecasting) and wishes to examine the performance of the forecasts, one knows that $y_t$ cannot have been influenced by the (currently made) forecasts in any way -- unless historically the actors were incidentally using the same kind of forecasts which then influenced the development of $y_t$.

This may be a messy formulation, but the idea is that often no error correction is happening in $y_t$. (The contrary is likely true with regards to forecasts themselves; here full error correction in $f^i_{t|t-h}$ towards $y_{t-h}$ over one period could be expected. With each new data point of $y_t$, the forecasts likely fully adjust to the change $\Delta y_t$; this can be reformulated as adjustment through error correction term.)

Using first differences in place of levels of variables is also suggested in Clements and Hendry "A Companion to Economic Forecasting" (2008) (Chapter 12.3, p. 275).

Of course, there may be cases where forecasts do influence the development of $y_t$ considerably, and then the problem outlined in the original post remains unsolved.

According to my understanding, the reference in Clements and Hendry is to Fair and Shiller "The informational context of ex ante forecasts". However, they do not suggest regression in first differences of the competing forecasts, but rather regression of actual changes to predicted changes which is something different:

$y_t - y_{t-1} = \beta_1 \cdot (f^1_t - y_{t-1}) + \beta_2 \cdot (f^2_t - y_{t-1}) + \epsilon_t$