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Let's assume we have 2 variables and test each of them for a unit root with the ADF test. When plotting the data, we can see that it has some up/down movements but is overall clearly trending upwards.

At level, variable A is stationary with "no constant", "constant" and "constant+trend" (p<0.05). Variable B is only stationary with "constant+trend" (p<0.05).

At first differences, both variables are stationary at "no constant", "constant" and "constant+trend" (p<0.05).

Is it o.k. to conclude that Variable A and B are both integrated of the same order I(1), or do we have to assume that Variable B is I(0) with respect to a trend and Variable B is I(1)?

I want to conduct a Johansen Cointegration test and after reading the literature and several posts, I'm still not sure if it's okay test for correlation with the data at level (and a trend).

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Variable A seems to be stationary as whatever the specification ("no constant", "constant", "constant+trend") of the test regression you still reject the unit root hypothesis. Therefore you cannot conclude that A~I(1). Since at least one of the two variable is stationary, you cannot proceed to cointegration analysis (recall that it requires both variables to be integrated to begin with).

Note on proceeding to testing stationarity of A at first differences: you should not do that as A is already stationary in levels. If A is stationary in levels, testing for stationarity in first differences is redundant when your question is whether A should be included in cointegration analysis.

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  • $\begingroup$ So there's no way to test the order of cointegration of variable A and B? $\endgroup$ – RaymondD Oct 21 '15 at 13:11
  • $\begingroup$ Lütkepohl and Krätzig, Applied Time Series Econometrics, 2004: "Occasionally it is convenient to consider systems with both I(1) and I(0) variables. Thereby the concept of cointegration is extended by calling any linear combination that is I(0) a cointegration relation, although this terminology is not in the spirit of the original definition because it can happen that a linear combination of I(0)variables is called a cointegration relation." $\endgroup$ – RaymondD Oct 21 '15 at 13:18
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    $\begingroup$ When you only have two variables one of which is stationary and one of which is integrated, there is no way of obtaining a stationary linear combination of the two --> No cointegration. If both of the variables are stationary, again you cannot talk about cointegration. Only in systems of more than two variables is your above quote meaningful. $\endgroup$ – Richard Hardy Oct 21 '15 at 13:26
  • $\begingroup$ So it's no possibility to use the level of the I(0) variable and the first difference of the I(1) variable? $\endgroup$ – RaymondD Oct 21 '15 at 14:27
  • $\begingroup$ If you do that, you are working with two I(0) objects. You can use them for a variety of things but not for cointegration analysis. If the variables are not integrated, they cannot be co-integrated (look at the construction of these words). $\endgroup$ – Richard Hardy Oct 21 '15 at 14:33

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