I'm familiar with a traditional Error Correction Model (ECM) of the form
$$Y_{t}-Y_{t-1}=\Pi Y_{t-1}+\sum_{i=1}^{p-1}\Gamma_{i}\left(Y_{t-i}-Y_{t-i-1}\right)+\varepsilon_{t}
$$
where $\Pi=\alpha\beta'$, $\alpha$ is the long-run adjustment, $\beta$ is the matrix of cointegrating vectors, and $\beta'Y_{t-1}$ is stationary.
For the data set I am looking at (includes economic variables, interest rates, and stock prices), I was testing cointegrating vectors and found that some are stationary and some are trend stationary. I think I should include the trend stationary ones, but I'm not entirely sure what's the best way to proceed.
For instance, one approach is to estimate the model $$y_{t,i}=a_{i}+b_{i}t+x_{t,i}
$$ to de-trend each variable and then apply the above ECM to $X_{t}$. However, I wonder if it is better to keep everything in terms of $Y_{t}$.
Hence, I was thinking about estimating
$$Y_{t}-Y_{t-1}=A+Bt+\alpha F_{t-1}+\sum_{i=1}^{p-1}\Gamma_{i}\left(Y_{t-i}-Y_{t-i-1}\right)+\varepsilon_{t}
$$ where $F_{t}=\beta' Y_{t}$ and $\beta$ includes all significant cointegrating vectors (including trend stationary ones). Whereas the traditional ECM is stationary since you are adding stationary series, this is a linear combination of a deterministic time trend, trend stationary series, and stationary series. Is this what should be done? Should I de-trend $F_{t}$ if I include the deterministic time trend, and vice-versa if left out?
Best Answer
I was able to find the answer in Pitfalls and Opportunities: What Macroeconomists Should Know About Unit Roots by Campbell and Perron. They suggest a model of the form
$$ Y_{t}-Y_{t-1}=A+\alpha Y_{t-1}^{*}+\sum_{i=1}^{p-1}\Gamma_{i}\left(Y_{t-i}-Y_{t-i-1}\right) $$ where $Y_{t}^{*}=\beta' Y_{t}-\delta t$.