Mean First Passage Time for the ESTAR Model Using the Integral Equation Approach and Its Applications

Abstract

Basak and Ho (2004) derived numerical formulas for the mean first passage time of ARMA models using the integral equation approach. Recently, the ESTAR model has become a popular model for the analysis of economic and financial data. Some researchers used the ESTAR model to analyze real exchange rates, purchasing power parity (PPP) deviations, and arbitrage processes. This article shows that under certain conditions, the integral equation approach can still be used for the ESTAR model. Numerical schemes of the mean first passage time for ESTAR(1) and ESTAR(2) models are proposed in this article. The applications of the schemes for real exchange rates between Australian and New Zealand dollars as well as for pairs trading following the ESTAR(1) model are provided.

AMS Subject Classification:

• 62P05
• 62M99
• 91B60
• 91B82
• 91B84

Keywords:

• ESTAR model
• Exchange rate
• Integral equation approach
• Mean first-passage time
• Pairs trading

Acknowledgments

We thank the editor and two anonymous referees for their very helpful comments. We also acknowledge the careful proofreading provided by Martin Bunder.

Notes

1. Without loss of generality, it is assumed that the mean of is zero.

2. 2. Rothe and Sibbertsen (2006) used data from the IMF International Financial Statistics Online service, while we use data from OECD website (http://stats.oecd.org) for CPI and the RBA website (http://www.rba.gov.au/statistics) for the nominal exchange rate.

3. 3. , where is the sample mean of from 1986q1 to 2004q4.