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Recently I wanted to check wether there is a cointegration relationship between consumption $C_t$ and income $Y_t$ where there is also some autonomous consumption present (i.e. test wether $C_t = a + Y_t$). In my data both $C_t$ and $Y_t$ are of order I(1). However I am unsure which cointegration vector I should test. Since cointegration can only be present on series which are all of the same order of integration, and since a constant is I(0), I do not know wether I should add the constant in the test for cointegration. Or in other words wether I should test the following vector for cointegration: $$ \begin{bmatrix} U \\ C_t \\ Y_t \end{bmatrix} $$ Where U is a constant (autonomous consumption). Or wether I should only test the vector with the I(1) series: $$ \begin{bmatrix} C_t \\ Y_t \end{bmatrix} $$

So I am wondering how I should test for the presence of a constant.

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There are two different questions you are addressing:

  1. Which cointegrating relationship to test for?
  2. How to test for presence of a constant in a cointegrating relationship?

I will answer the first one, while the answer for the second one should be included in standard textbooks and lecture notes on cointegration (please correct me if I am wrong).

If you have three time series, $U$, $C_t$ and $Y_t$ where $U$ is constant through time, then $U$ has no place in the cointegrating relationship. Cointegration may take place between integrated series, whereas $U$ is stationary. (Even though the true relationship between the series could be $C_t=U+Y_t$, that does not mean $U$ belongs to the cointegrating relationship.)

In your setting, you could test for whether the cointegrating vector is $(1,-1)$ for $(C_t,Y_t)$. You could do that, for example, by examining the stationarity of the residual series $\varepsilon_t=Y_t-C_t$ by the ADF test with the "standard" critical values (the ones used for unit root testing of a single series). This is because the cointegrating vector is not estimated but determined by outside knowledge. Or you could test cointegration with the Johansen procedure allowing for a constant term.

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