# An Invariance Property of the Common Trends under Linear Transformations of the Data

*Soren Johansen* () and
*Katarina Juselius*

No 10-30, Discussion Papers from University of Copenhagen. Department of Economics

**Abstract:**
It is well known that if X(t) is a nonstationary process and Y(t) is a linear function of X(t), then cointegration of Y(t) implies cointegration of X(t). We want to find an analogous result for common trends if X(t) is generated by a finite order VAR. We first show that Y(t) has an infinite order VAR representation in terms of its prediction errors, which are a linear process in the prediction error for X(t). We then apply this result to show that the limit of the common trends for Y(t) are linear functions of the common trends for X(t). We illustrate the findings with a small analysis of the term structure of interest rates.

**Keywords:** cointegration vectors; common trends; prediction errors (search for similar items in EconPapers)

**JEL-codes:** C32 (search for similar items in EconPapers)

**Date:** 2010-10

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http://www.econ.ku.dk/english/research/publications/wp/dp_2010/1030.pdf/ (application/pdf)

**Related works:**

Working Paper: An invariance property of the common trends under linear transformations of the data (2010)

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**Persistent link:** http://EconPapers.repec.org/RePEc:kud:kuiedp:1030

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