# Null Recurrent Unit Root Processes

## Author

Listed:
• Myklebust, Terje
• Karlsen, Hans Arnfinn
• TjÃ¸stheim, Dag

## Abstract

The classical nonstationary autoregressive models are both linear and Markov. They include unit root and cointegration models. A possible nonlinear extension is to relax the linearity and at the same time keep general properties such as nonstationarity and the Markov property. A null recurrent Markov chain is nonstationary, and Î²-null recurrence is of vital importance for statistical inference in nonstationary Markov models, such as, e.g., in nonparametric estimation in nonlinear cointegration within the Markov models. The standard random walk is an example of a null recurrent Markov chain.In this paper we suggest that the concept of null recurrence is an appropriate nonlinear generalization of the linear unit root concept and as such it may be a starting point for a nonlinear cointegration concept within the Markov framework. In fact, we establish the link between null recurrent processes and autoregressive unit root models. It turns out that null recurrence is closely related to the location of the roots of the characteristic polynomial of the state space matrix and the associated eigenvectors. Roughly speaking the process is Î²-null recurrent if one root is on the unit circle, null recurrent if two distinct roots are on the unit circle, whereas the others are inside the unit circle. It is transient if there are more than two roots on the unit circle. These results are closely connected to the random walk being null recurrent in one and two dimensions but transient in three dimensions. We also give an example of a process that by appropriate adjustments can be made Î²-null recurrent for any Î² âˆˆ (0, 1) and can also be made null recurrent without being Î²-null recurrent.

## Suggested Citation

• Myklebust, Terje & Karlsen, Hans Arnfinn & TjÃ¸stheim, Dag, 2012. "Null Recurrent Unit Root Processes," Econometric Theory, Cambridge University Press, vol. 28(1), pages 1-41, February.
• Handle: RePEc:cup:etheor:v:28:y:2012:i:01:p:1-41_00
as

File URL: https://www.cambridge.org/core/product/identifier/S0266466611000119/type/journal_article
File Function: link to article abstract page

## Citations

Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
as

Cited by:

1. Gourieroux, Christian & Jasiak, Joann, 2019. "Robust analysis of the martingale hypothesis," Econometrics and Statistics, Elsevier, vol. 9(C), pages 17-41.
2. Biqing Cai & Jiti Gao & Dag Tjøstheim, 2017. "A New Class of Bivariate Threshold Cointegration Models," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 35(2), pages 288-305, April.
3. Gao, Jiti & Tjøstheim, Dag & Yin, Jiying, 2013. "Estimation in threshold autoregressive models with a stationary and a unit root regime," Journal of Econometrics, Elsevier, vol. 172(1), pages 1-13.
4. Dong, Chaohua & Gao, Jiti & Tjøstheim, Dag & Yin, Jiying, 2017. "Specification testing for nonlinear multivariate cointegrating regressions," Journal of Econometrics, Elsevier, vol. 200(1), pages 104-117.
5. Kim, Jihyun & Park, Joon Y., 2017. "Asymptotics for recurrent diffusions with application to high frequency regression," Journal of Econometrics, Elsevier, vol. 196(1), pages 37-54.
6. Li, Degui & Li, Runze, 2016. "Local composite quantile regression smoothing for Harris recurrent Markov processes," Journal of Econometrics, Elsevier, vol. 194(1), pages 44-56.
7. Biqing Cai & Dag Tjøstheim, 2015. "Nonparametric Regression Estimation for Multivariate Null Recurrent Processes," Econometrics, MDPI, Open Access Journal, vol. 3(2), pages 1-24, April.
8. Degui Li & Dag Tjøstheim & Jiti Gao, 2012. "Nonlinear Regression with Harris Recurrent Markov Chains," Monash Econometrics and Business Statistics Working Papers 14/12, Monash University, Department of Econometrics and Business Statistics.

## Corrections

All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:cup:etheor:v:28:y:2012:i:01:p:1-41_00. See general information about how to correct material in RePEc.

For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Keith Waters). General contact details of provider: https://www.cambridge.org/ect .

If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

We have no references for this item. You can help adding them by using this form .

If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

Please note that corrections may take a couple of weeks to filter through the various RePEc services.

IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.