### Discussion Paper

Testing for Near I(2) Trends When the Signal to Noise Ratio Is Small

## Abstract

Researchers seldom find evidence of I(2) in exchange rates, prices, and other macroeconomics time series when they test the order of integration using univariate Dickey-Fuller tests. In contrast, when using the multivariate ML trace test we frequently find double unit roots in the data. The paper demonstrates by simulations that this often happens when the signal-to-noise-ratio is small.

## Comments and Questions

Interesting and worthy contribution

This is an interesting paper that makes two important contributions:

1. it helps to explain why the "average practitioner" who apply ADF-type tests on individual variables (starting from levels and moving to differences successively) may fail to detect a double unit root in a class of time series whose ...[more]

... data generating process is near-I(2) with small signal-to-noise ratio;

2. it suggests that deviations from the rational expectations hypothesis, and in particular the Imperfect Knowledge Economics (IKE) paradigm discussed by Frydman and Goldberg (2007, 2011), may potentially be used to explain the long swings we observe in variables like e.g. exchange rates and asset prices.

The paper emphasizes the benefits of treating near-I(2) systems as I(2) (to my knowledge, a recent paper which provides a similar idea by exploring a different route is Di Iorio et al. (2013), who devote attention to testing issues). We all know that "the problem" with I(2) processes is that aside from some special cases, many macroeconomists do not easily manage "polynomial cointegration", because the equilibria they have in mind do not generally feature levels and differences of the variables jointly (Katarina has several empirical papers which show how this issue can be takled; Bacchiocchi and Fanelli (2005) show how polynomial cointegrating PPP-like relationships can be justified by appealing to models of deviations from the law-of-one-price). This paper delivers some original insights about the mechanics underlying long swings and suggests an implicit modelling approach based on "our" cornerstore, the cointegrated VAR.

I have two considerations. The first concerns the empirical results in Table 8. It would be nice to explore, in future research, whether using suitable designed bootstrap versions of the LR test would change some of the conclusions. For instance, the fact that the real exchange (bond rate differential) rate is perceived as I(2) by the LR test with a p-value of 0.03 (0.02), might reflect the over-rejection phenomenon we observe when testing cointegration restrictions in samples of length typically available to macroeconomists. Secondly, it would be nice to compare/contrast, always in future research, the viewpoint provided by Katarina about near-I(2) time-series with the results stemming from fractionally integrated/cointegrated VARs (Johansen and Nielsen, 2010, 2012).

Luca

REFERENCES

- Bacchiocchi, E., Fanelli, L. (2005), Testing the purchasing power parity through I(2) cointegration techniques, Journal of Applied Econometrics 20, 749-770.

- Di Iorio, F., Fachin, S., Lucchetti, R. (2013), Can you do the wrong thing and still be right ? Hypothesis testing in I(2) and near-I(2) cointegrated VARs, Quaderni di Ricerca n. 395, Dipartimento di Scienze Economiche e Sociali, Università Politecnica delle Marche.

- Frydman, R. and Goldberg, M. (2007), Imperfect knowledge economics: Exchange rates and risk, Princeton, NJ: Priceton University Press.

- Frydman, R. and Goldberg, M. (2011), Beyond mechanical markets: risk and the role of asset price swings, Priceton University Press.

- Johansen, S. and Nielsen, M.Ø. (2010), Likelihood inference for a nonstationary fractional autoregressive model, Journal of Econometrics 58, 51-66

- Johansen, S. and Nielsen, M.Ø. (2012), Likelihood inference for a fractionally cointegrated vector autoregressive model, Econometrica 80, 2667-2732.

Thanks for useful comments! I agree that the properties of the LR tests need to be further studied in the near I(2) case. There is a study in progress where the authors are presently investigating a procedure where the Ksi Square tables are corrected with a coefficient which depends on ...[more]

... how close the rho parameter is to the unit circle and the sample size. This seems to produce a good size of the test but the power properties are not yet studied. A correction like this might provide a good alternative to bootstrapping.

In the present application the variables could be considered to be a near I(2) or a very persistent I(1) process without really changing the basic idea of associating variables/relations with a similar persistency profile. This is discussed at more length in my reply to the two referees. Whether fractional cointegration will add insight in this case is a future issue that is hard to say anything about at the present stage.

Katarina Juselius

see attached file

see attached file

See the enclosed pdf file!