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could someone kindly explain what I need the ECM for in a bivariate Cointegration test?

I am currently trying to reproduce the results of Rad et al. (2015): "The profitability of pairs trading strategies: distance, cointegration, and copula methods".

For the implementation of the Cointegration pairs trading strategy, the two step Engle Granger procedure is used. I understand that first, I can estimate the cointegration coefficient beta by doing an OLS estimation by regressing y(t) on x(t). In the case of bivariate cointegration, the OLS estimator would be superconsistent. Therefore, I understand that I need to test for Cointegration next, using the ADF test on the residuals. If the ADF test now tells me that the residuals are stationary and thus y(t) and x(t) are cointegrated, I would assume, that I proved that the two variables are cointegrated and I do also already have my coefficient beta. I do not understand why I should still estimate an ECM, or what I should use the estimated ECM for respectively, because in the paper, it seems like they only test for cointegration and use beta fom then. However, they say that they use the Engle Granger two step method and estimate the ECM in the second step.

I could imagine that for some reason, I cannot use the beta estimated with OLS. Is that true? And if so, why can't I use it if it is superconsistent and tends to the true value?

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    $\begingroup$ Hi: If you were implementing a pairs trading strategy and wanted to decide when to enter a trade, the ecm formulation would be one way to decide that. The cointegrating relationship that is estimated doesn't provide that. Note that the cointegrating relationship is estimating the long term relationship between $y_{t}$ and $x_t$ whereas the ecm formulation is giving the dynamics between the short term change in $y_t$ and A) the short term change in $x_t$ and B) the current difference in the cointegrating relationship. $\endgroup$
    – mark leeds
    Jul 26, 2021 at 23:47
  • $\begingroup$ I don't have enough space ( nor the memory ) to write out the exact formula for the ecm but banerjee's text discusses the ecm in great detail. $\endgroup$
    – mark leeds
    Jul 26, 2021 at 23:50
  • $\begingroup$ Hi, thank you very much for our support. The paper I am looking at uses another opening trigger for their trades. They just look at the normalized spread of the pairs and if this spread increases above 2, they open a trade. Therefore, it does not seem to me as if they would use the ecm to decide if they open a trade. Or am I getting something wrong here? $\endgroup$
    – GC2023
    Jul 27, 2021 at 8:03
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    $\begingroup$ Hi and apologies for confusion. One doesn't have to use the ecm to decide when to trade. I was just giving that as an example of why would could find the ecm useful even if one has estimated the cointegration relationship. The cointegration relationship shows the LONG TERM relationship between the two variables. (the levels relationship, namely $y_t = \beta x_t $). There is also a possible short term change in $y_t$ which can be caused by A) changes in $x_t$ and B) the level of dis-equilibrium in the long term relationship, namely $y_t - \beta x_t$. This is what the ECM captures. $\endgroup$
    – mark leeds
    Jul 27, 2021 at 11:10
  • $\begingroup$ Thank you again! I think I understood it now $\endgroup$
    – GC2023
    Jul 27, 2021 at 19:35

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