I have two I(1) time series and I regressed one against the other and found that it had low to moderate R-squared but my DW statistic is about 0.015. I know the literature says this is the case of spurious regression? Now, upon running co-integration tests on the residuals (I ran an ADF test using updated MacKinnon's p table, used Phillips Ouliaris test, Johansen test and Elliott Rothenberg and stock test). Now, all my tests pass except for Phillips Ouliaris and Johansen test. These are the only tests where I am not getting the residuals from the data. I believe the PO test automatically runs an regression of y against x and uses the Phillips Ouliaris distribution rather than the ADF distribution on residuals.

My main question is, which test do I trust and whether these tests even make any sense considering I have a spurious regression phenomena? I believe if the two series are co-integrated, then the residuals won't be spurious correct? So my main questions are the follows:

  1. Can you have co-integration even with spurious regression?
  2. which test do I ultimately have to chosen from? PO test, Johansen test (both these tests accept the series are not co-integrated). ERS test passes on residuals and so does the R functions adf.test, adfTest, & ur.df.
  3. My time series is from 1998 to 2015. Sometimes, daily gives me co-integration, but monthly doesn't. What time frame is most acceptable?

The cointegrating regression Durbin-Watson statistic of 0.015 is not statistically significantly different from 0. A DW statistic close to 0 is expected of a random walk. This is a red flag that you do not have a co-integrating regression.

The only tests for cointegration that you report as passing are the ADF type tests. These are sensitive to structural breaks in the data, number of lags used, and the quality of the auxiliary regression. You should check the residuals of the ADF auxiliary regression for symptoms that may be inflating t-values.

  1. Yes, you can have statistical co-integration reported even with spurious regression, e.g. a Type I error with ADF or problems with the assumptions of the tests being violated.
  2. Combined with the extremely low DW statistic, I'd be heavily inclined to continue to believe in no co-integration. The ADF-type tests should be evaluated more closely, as described above.
  3. Appropriate frequency depends on the nature of the data. You need to capture the dependency structure of the data (e.g. autocorrelation). For some data this is more evident at daily frequency, others at monthly or quarterly.
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First off, just to comment on terminology. Cointegration, just like correlation, is a relationship between time series. Spuriousness is usually used in the context of a "spurious correlation", and is not a property of the time-series themselves, but rather a property of a test statistic, in this case the correlation. For me it makes more sense to ask: can the correlation between two time-series can be spurious even if the time-series are cointegrated?

1: Despite @A. Webb's answer, which raises some valid points, the short answer to your question is no: according to Granger (2004), if the two time series are co-integrated, even if they both have stochastic trends, correlation significance tests are not going to be spurious. This is why he won the Nobel Prize in 2004. More precisely, testing for co-integration will fix the problem of spurious correlations arising specifically from the initial time series being I(1), i.e. having trends that confound the results. There are of course more general ideas of "spurious relationships", but this is a different idea. When it comes to "spurious correlation", a term that refers specifically to this particular problem, testing for co-integration indeed solves the problem.

2: Generally speaking, the two most common tests are the Johansen test and the Engle-Granger test. However, the Johanson test is usually favored in multi-dimensional situations such as VAR models. In contrast, the engle-granger test was designed for two time series, and due to its relative simplicity, it is often prefered in this situation.

3: Well, what is you time resolution for your regression? You should simply use the same time-resolution. If you are trying to actually do something with you cointegration model, then it is important what time scale you choose as this affects the parameter estimates in your model, as @A. Webb suggests. But if you are simply trying to verify whether a given correlation coefficient is spurious, the answer to you question is simple: just use the same time scale as your regression.

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