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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.

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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.

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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.

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  • $\begingroup$ The graph is obviously growing but the results does not match the graph. In the first pic when I used the P_F as the observed variable, the trend is negative which implies that the gap is shrinking. However when I used P_R as the observed variable (which cannot be used since the error correction term is non negative in this case) the trend is positive which implies that the gap is somehow growing. The diagnostics shows that the model is correctly specified but how can it be interpreted if it appears to be this way? $\endgroup$
    – Ashlley
    Commented Nov 14, 2022 at 10:08
  • $\begingroup$ @Ashlley, I might have been tricked by your plot where the $Y$ axis does not contain zero. Obtain the linear combination of your two series and plot it. You will see whether the trend is positive or negative. Other than that, is it possible that you have mixed up the signs? This happens pretty often in VECM; see e.g. these threads (there are likely more). $\endgroup$
    – Richard Hardy
    Commented Nov 14, 2022 at 10:21
  • $\begingroup$ I tried using Ordinary Least squares to check if I misread the signs but it seems that it shows the same results (as seen on the edited post). Also, how is the linear combination computed manually? Thank you very much for this input. $\endgroup$
    – Ashlley
    Commented Nov 14, 2022 at 10:51
  • $\begingroup$ @Ashlley, $P_F - (0.175170P_R-0.066801t+58.77431)$ will be stationary. That means, $P_F - 0.175170P_R-58.77431$ will yield a slightly positive linear time trend. $\endgroup$
    – Richard Hardy
    Commented Nov 14, 2022 at 11:05
  • $\begingroup$ How did it become a positive time trend? Also if it follows, then it also implies that when the P_R is the response variable it will yield a negative time trend. Is it even possible to have different trends using the same variables without the relationship changing between the two variables? Sorry if I did not fully understand it, I'm quite new to this. $\endgroup$
    – Ashlley
    Commented Nov 14, 2022 at 11:39

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