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.