Interpretation of the trend variable in the cointegrating equation of VECM (VECM Case 4: Unrestricted trend) I am doing VECM of prices of meat. This one result confuses me. This one shows that P_F and P_R has a positive relationship. This can be interpreted at 1% of P_R is 0.18% of P_F. However it also shows that there is a negative trend. Can this mean that the P_F and P_R are both decreasing in the long run? The results are questionable because the graph of the variables obviously shows an increasing trend. Also, when I interchange the variables. It showed that the relationship of the variables are still positive. However the trend now becomes positive too.
How can I interpret these results? The model is already stable so I'm just wondering how am I gonna interpret the results as the diagnostics are already met.



To check if I might got the wrong signs, I tried using it in an Ordinary Least Squares with a trend. Using the P_F as the response variable and the trend is still negative. When I interchange the variables, the trend becomes positive. And yet the relationship between the two remains the same. The diagnostics are still showing good results so I wonder how can I interpret this type of relationship.


 A: If there is a linear time trend in a cointegrating relationship, that means there is a gap that is growing (or shrinking) linearly between the two cointegrated series.* In your case the gap seems to be growing, as can be seen from the plot. (Though it is difficult to tell since you $Y$ axis does not contain zero. Consider plotting it in a way that zero is also visible.)
Your EViews output for the first equation suggests that
$$
P_F - (0.175170P_R-0.066801t+58.77431)=u_t
$$
will be stationary. That means
$$
P_F - 0.175170P_R-58.77431=0.066801t+u_t
$$
will have a slightly positive linear time trend (with slope $0.066801$).
The output for the second equation can be interpreted analogously, and you will discover a negative time trend there.

Aside from your specific example, the bivariate case is
\begin{aligned}
X_t &= X_{t-1}+u_t \\
Y_t &= \beta X_{t-1}+\delta t+v_t \\
\end{aligned}
where $u_t$ and $v_t$ are i.i.d. and orthogonal to each other. Then
$$
Y_t-\beta X_t=\delta t+v_t-u_t,
$$
i.e. a linear combination of two integrated series produces a linear time trend and i.i.d. fluctuations around it. You can flip it around if you like to get the opposite signs everywhere:
$$
X_t-\frac{1}{\beta}Y_t=-\frac{\delta t}{\beta}-\frac{v_t}{\beta}+\frac{u_t}{\beta}.
$$

*If you had more than a pair of series, a linear time trend in a cointegrating relationship would imply the linear combination of the series that does not contain a unit root has a linear time trend.
