# Cointegration $p$-value and VECM loading forces show low correlation

After having read about VEC model (VECM), I thought that cointegration and VECM loading forces were strongly, not to say numerically correlated. I thought that the more we have cointegration, the less is the coint. $$p$$-value, the higher are the loading forces $$\alpha_i$$.

To confirm that, I plotted a point cloud of 100 pairs of stocks showing one of their 2 cointegration $$p$$-values < 0.03. There are 2 cointegration tests per pair because it depends on the variable you set as as endogenous. Each point in this plot is the couple (x_axis=avg(coint p-value), y_axis=average(abs(vecm_forces))

We see a little bit of a relationship: the higher the $$p$$-value, the less is likely the pair is to be cointegrated, the smaller are the $$\alpha_i$$s.

However, I find the correlation very low compared to what I expected. Especially, those 2 things mean basically the same thing, right? Whether or not the pairs are returning fast to the equilibrum. With this said, how to interpret a pair that has a low coint $$p$$-value but a low force? Do I do well when I average the 2 loading forces and the 2 coint $$p$$-values? Maybe I should take respectively max and the min of them? Should I compare the loading forces themselves, or would their $$t$$-values be more easy to interpret ($$t$$-value of the loading force shows a stronger correlation with coint. $$p$$-value)?

I want basically to find a strong correlation between both things so that the second dimension disappear, because they mean the same thing (unless I'm wrong and I'm missing some interpretation?). The goal is in the end to sort the pairs with a certain homogenous unity that represent the "speed" at which they go back to their equilibrium in a context of pair trading strategy.

I thought that the more we have cointegration, the less is the coint. $$p$$-value, the higher are the loading forces $$\alpha_i$$

First, cointegration is a discrete YES/NO phenomenon. Either a system of integrated variables is cointegrated or it is not. It is not a gradual phenomenon, thus we cannot have more or less of it.

Second, there is a distinction between effect size and statistical significance. You can have cointegrated systems with their VECM representations with fast error correction (a large loading coefficient, a short half-life) or slow error correction (a small loading coefficient, a long half-life). Depending on the sample size and the error variance, you could have small or large $$p$$-values in each case. All else being equal, faster error correction would also yield smaller $$p$$-values* for a given sample size, but "all else" is not always equal.

*As also more generally, larger effect sizes yield smaller $$p$$-values, ceteris paribus.

• okay looks clear thank you ! So if I may precise, (1) yes indeed, the p-value is associated not a "degree of cointegration" but more precisely to a probability that we see such sample not cointegrated due to sample error. If we see a low probability, we have enough evidence to reject H0 which is "there is no cointegration relashionship" (2) I'm still trying to see visually what would be a sample that shows a high probability to be cointegrated and yet having low loading forces. Apr 24, 2023 at 14:39
• Is it something like a sinusoidal that always falls to 0 periodically and predictably (and then showing a high probability of cointegration), but whose period is long in the sample ? And, vice versa, is a low probable cointegration with a high ECM force would be a sample much less periodical, showing sometime during a pretty long time non stationary behavior, but, when entering some periods, it would be coming back very fast to 0 ? I'm trying to interpret visually the difference between those two things Apr 24, 2023 at 14:42
• @JeremLachkar, regarding (2), take a really long sample of a pair of series that have slow error correction. That will yield a low p-value. Also, take a really short sample of a series that has fast error correction. That will yield a high p-value. Alternatively, instead of sample length, vary the error variance (long sample ~ low variance, short sample ~ high variance). Apr 24, 2023 at 17:02
• Okay this looks much more clear :) So what influences the p-value is also the sample size and the residual variance If there is a lot of ECM residual variance the time series are less likely to be cointegrated is this correct ? Apr 25, 2023 at 8:30
• that makes sense indeed, thank you! Apr 25, 2023 at 8:50