Sunday, January 22, 2012

Cointegration Analysis With I(2) & I(1) Data

Last Friday I went to a great seminar given by Takamitsu Kurita (Fukuoka University, Japan). Taka is currently a visiting scholar in our department, and his paper (here)  dealt with an interesting application of cointegration analysis when we have both I(2) and I(1) data to contend with.

This is a topic in time-series econometrics that's of great practical importance, and (quite rightly) is currently attracting quite a bit of attention.

One of the interesting features of the analysis that Taka presented was that it clearly reflected the difference between the broad approaches to cointegration analysis, in general, that tend to be used by European econometricians on the one hand, and their North American counterparts on the other. This is a point that was raised by Georg in a comment on an earlier post in this blog.

A lot of us approach cointegration analysis in a two-step manner - and I'm not referring to the Engle-Granger two-step cointegration test. What I mean is this. We first test each of the time-series for their order of integration, perhaps using the Augmented Dickey-Fuller test, and/or the KPSS test. Then, if we find that the series are integrated of the same order (greater than zero) we proceed at the second stage to test for the number of cointegrating relationships among them. Here, we'd generally use Johansen's framework.

On the other hand, a lot of our European colleagues avoid this two-step approach, and the "preliminary test" distortions that can be associated with it. Instead, they move directly to Johansen's set-up, and out of this they can infer the order of integration of the time-series, as well as the number of cointegrating vectors.

Although a good deal of progress has been made when it comes to modelling with a mixture of both I(2) and I(1) data, there are still lots of important questions waiting to be addressed. This, of course, is good news for young econometricians looking for an interesting field of research!

Some of the seminal contributions to date include those of Johansen (1992, 1995, 1997, 2006), Paruolo (1996), Haldrup (1998), Paruolo and Rahbek (1999), and Rahbek et al. (1999), among others.

And what of those issues yet to be addressed? I believe they include:
  • An extension of the I(2)/I(1) cointegration analysis to the case where there are unit roots at the seasonal frequencies. Obviously, this is an important issue in the case of quarterly or monthly data.
  • A treatment of the (commonly occurring) situation where the I(2) and I(1) data exhibit structural breaks; thereby extending the analysis discussed in this earlier post.

I think that we're going to see a lot more research on the modelling of (possibly cointegrated) I(2) and I(1) economic time-series data.

Note: The links to the following references will be helpful only if your computer's IP address gives you access to the electronic versions of the publications in question. That's why a written References section is provided.


Haldrup, N., 1998. An econometric analysis of I(2) variables. Journal of Economic Surveys, 12, 595-650.

Johansen, S., 1992. A representation of vector autoregressive processes integrated of order 2. Econometric Theory, 8, 188–202.

Johansen, S., 1995. A statistical analysis of cointegration for I(2) variables. Econometric Theory, 11, 25–59.

Johansen, S., 1997. Likelihood analysis of the I(2) model. Scandinavian Journal of Statistics, 24, 433–62.

Johansen, S., 2006. Statistical analysis of hypotheses on the cointegrating relations in the I(2) model. Journal of Econometrics, 132, 81–115.

Kurita, T., 2012. Modelling time series data of monetary aggregates using I(2) and I(1) cointegration analysis. Bulletin of Economic Research, in press.

Pauolo, P., 1996. On the determination of integration indices in I(2) systems. Journal of Econometrics, 72, 313–56.

Paruolo, P. and A. Rahbek, 1999. Weak exogeneity in I(2) VAR systems. Journal of Econometrics, 93, 281–308.

Rahbek, A., H. C. Kongsted, and C. Jørgensen, 1999. Trend stationarity in the I(2) cointegration model. Journal of Econometrics, 90, 265–289.

© 2012, David E. Giles


  1. Thank you for writing about I(2) analysis. I guess only few students are presented for the possibility that data could be modelled as I(2). The I(2) analysis makes it possible to analyse data without some kind of preliminary detrending with unknown consequences for the statistical analysis. If you want to focus on the effects from aggregated nominal variables, you really need to consider these models.

    However, the drawback is that these models are cumbersome to work with and no statistical packages supports I(2) analysis as they come. There exist the commercial add on package "Cats" for Rats, which has most of what an applied cointegration econometrician would need, including some tools for I(2) analysis. The procedures can of course be programmed in e.g. Ox as Kurita has done or Matlab, but if you want to do hypothesis testing, things get complicated.

    I believe some of the work you are asking for has been done or at least started. Two articles are shown below and actually Kurita coauthers one of them.

    Kurita, Takamitsu, Heino Bohn Nielsen, and Anders Rahbek. “An I(2) Cointegration Model with Piecewise Linear Trends.” The Econometrics Journal 14, no. 2 (July 1, 2011): 131-155.

    Johansen, Søren, Katarina Juselius, Roman Frydman, and Michael Goldberg. “Testing Hypotheses in an Model with Piecewise Linear Trends. An Analysis of the Persistent Long Swings in the Dmk/$ Rate.” Journal of Econometrics 158, no. 1 (September 2010): 117-129.

    1. Andeas: Great comments! Thanks, too, for the references to the work that has started to deal with the structural breaks issue - I;m sure we;ll see more of that.

  2. But is it true that I(2) data is rare?

  3. Anonymous: Thanks for the comment. No - not at all. I(2) data arise with often with stocks (as opposed to flows). This is part of the point of Taka's paper.

  4. thank you for great info

  5. THANK YOU DAVIS GILES for writing on I(2) & I(1) Data analysis. Basically, the series are said to be co-integrated, if two or more series are individually integrated but some linear combination of them has a lower order of integration.

  6. Sorry I just want to ask a quick question:
    In a time series model with a lagged dependent variable, the Durbin Watson statistic is biased towards 2 which is supporting the null, does this make this model useless ?

    1. If you have lagged values of the dependent variable as regressors you shouldn't be using the DW test. These days, we'd use the LM test - keep it mind that it has only asymptotic validity.

  7. Hello professor.

    If I am not mistaken, the Johansen cointegration test cannot be applied when the variables have different orders of integration (for example: GDP and Labour Supply as I(1) variables and Capital Stock as I(2) variable). What, then, should be the correct procedure to test for cointegration between these variables?

    Thank you in advance.

    1. The usual Johansen procedure is not appropriate in this case. Hence the many references I give above to the appropriate way of proceeding when you have a mixture of I(2) and I(1) - check them out.

  8. Dear Professor,

    Can STATA be used to conduct similar tests as mentioned in the article? (cointegration, ECM and Granger causality) The data is of I(2) (wages) and I(1) (CPI) structure.

    Thank you for the most helpful view on the matter!

  9. Dear Professor,

    I am testing two series which them Patent app. and Export figures of some country. I am totally confused about Johansen cointegration test about using variable lag (for instance Pat (-1)). Can I use lag of patent variable while testing johansen cointegration.

    Moreover We are very thankfull sharing your knowledge and experiences with us.

    1. I'm assuming that both series are I(1). In that case you should fit a VAR in the levels of the two original variables, determine the optimal lag length, and then go from there to test for cointegration.

  10. Dear Professor,
    I have two I(2) series and I would like to test if there are cointegrated. I follow the same procedure as if they were I(1) but instead of considering the variables in levels, I consider their first difference. Is this correct?
    More concretely, here is what I do: I estimate a VAR with both variables in first difference and then I determine the optimal lag length on the basis of this equation. Then, I perform the trace and the eigenvalue tests (again on the basis on the first difference of the two variables) in order to test if the two variables (their first difference) are cointegrated. Finally I check that the coefficient of the error correction term is negative but higher that -1 and that it is statistically significant.
    I would be very grateful if you could give me some pieces of advice. I would like to be sure that this procedure is correct.
    Thank you very much for your consideration.

    1. No, this is not the way to proceed - take a look at the references given for this post.

    2. Thank you very much for your reply.


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