Abstract
In this chapter we show how to model the long-run relationship between variables in their levels, even if they are integrated. This is possible if two or more variables are “cointegrated.” Two variables are cointegrated is the difference between them is stationary. Or, to put it loosely, they move in parallel. In this chapter we explore the concept of cointegration, error correction mechanisms, and some of the more popular tests of contegration.
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Notes
- 1.
More precisely, two more variables which are integrated of order I(b) are cointegrated if a linear combination of them is integrated of a lower order than b.
- 2.
- 3.
That is, ADF with zero lags.
- 4.
- 5.
Since it is user-written and not an official Stata command, you must install it. You can do this by typing ssc install egranger.
- 6.
There are many features which recommend Johansen’s (1988) approach. For example, Gonzalo (1994) shows that Johansen’s method outperforms four rival methods—asymptotically and in small samples—at estimating cointegrating vectors. This is the case even when the errors are not normal or when the correct number of lags is unknown.
- 7.
We do not consider the I(2) case in this book. A workable but incomplete solution is to difference the I(2) variables once to render them I(1) and then follow the procedures as outlined below.
- 8.
The online help for the Eviews econometric software also warns against using Cases 1 and 5 (http://www.eviews.com/help/helpintro.html#page/content/coint-Johansen_Cointegration_Test.html). Likewise, Zivot and Wang (2007) warn against using Case 1. Sjö (2008, p.18) calls Case 4 “the model of last resort” (since including a time in the vectors might induce stationarity) and Case 5 as “quite unrealistic and should not be considered in applied work.” Thus, we are left with Cases 2 and 3 as reasonable choices.
- 9.
That is, the trend is due to drift from a random walk.
- 10.
Dwyer (2014, p.6) explains that the trace statistic does not refer to the trace of \(\hat {\boldsymbol {\Pi }}\) but refers instead to the “trace of a matrix based on functions of Brownian motion.” It also shares the similarity with the trace of the matrix in that both involve the sum of terms (here the sum of the eigenvalues); more specifically, we sum \(ln(1-\lambda ) \approx \lambda \) when (\(\lambda \approx 0\)).
- 11.
It is unclear to me why Stata opted not to have trace and max options.
- 12.
Cointegration merely requires that a linear combination of the variables is stationary. In practical terms, this means that the two variables can be tilted up or down until their difference is stationary. Two parallel lines are stationary, regardless of the constant difference between them. Or, what we care about is the slopes that establish stationarity; econometrically, we are less concerned with the constant. Economically, the constant term seldom has practical significance.
- 13.
I am indebted to David Giles and his popular “Econometrics Beat” blog for bringing this and the Toda-Yamamoto procedure to my attention. The blog piece can be found at http://davegiles.blogspot.com/2011/10/var-or-vecm-when-testing-for-granger.html. Readers are encouraged to read the cited references in that blog entry, especially the work by Clarke and Mirza (2006).
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Levendis, J.D. (2023). Cointegration and VECMs. In: Time Series Econometrics. Springer Texts in Business and Economics. Springer, Cham. https://doi.org/10.1007/978-3-031-37310-7_12
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