Long memory and structural changes in the forward discount: An empirical investigation

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Abstract

We analyze the evidence for long memory and structural changes in the five G7 countries' forward discount. We establish evidence for long memory by estimating the long memory parameter without allowing for structural breaks. We also document evidence for multiple structural breaks in the mean of the forward discount. We then adjust for structural changes in the forward discount rate and re-estimate the long memory parameter. After removing the breaks, we still find evidence of stationary long memory in each country's forward discount.

Introduction

Many empirical studies find a negative correlation between the returns on the nominal spot exchange rate and the lagged forward discount. This forward discount anomaly implies that the current forward rate is a biased predictor of the future spot rate. A large number of studies in the existing literature try to explain this anomaly. Engel (1996) summarized four explanations: (1) existence of a foreign exchange risk premium; (2) a peso problem; (3) irrational expectations; and (4) international financial market inefficiency from various frictions. In two detailed studies, Baillie and Bollerslev, 1994, Baillie and Bollerslev, 2000 focused on the time series properties of the spot rate and forward discount as an explanation for the forward discount anomaly. They argued that the forward discount anomaly is due to the statistical properties of the data, because the forward discount is a fractionally integrated (long memory) process and the rate of return on the spot exchange rate is a stationary process which creates an unbalanced test regression. Maynard and Phillips (2001) provided similar results as Baillie and Bollerslev. They argued that traditional asymptotic theory may not be applicable to test forward rate unbiasedness due to the fractional integration of the forward discount and they propose a new limit theory. Their limit theory for the forward rate unbiased hypothesis (FRUH) test statistics have nonstandard limiting distributions with long left tails, which may explain the forward discount anomaly as a statistical artifact.

A criticism against models of long memory is that the long memory property in the data may be due to the presence of structural breaks or regime switches. This is called “the spurious long memory process”. Several recent works including Granger, 1999, Granger and Hyung, 2004, and Diebold and Inoue (2001) show that structural breaks or regime switching can generate spurious long memory behavior in an observed time series. Indeed, Sakoulis and Zivot (2001) find evidence for structural breaks in the mean and variance of the forward discount, and argue that these breaks could be caused by events like discrete changes in policy and changes in interest rates due to the business cycle. After correcting for multiple structural breaks in the mean of the forward discount, they find the persistence of the forward discount is substantially reduced.

The focus of Sakoulis and Zivot (2001) is different from the focus herein. Sakoulis and Zivot estimate a stationary autoregressive model for the forward discount using a partial structural break model to establish the presence of structural breaks in the forward discount without explicitly considering the possibility of long memory behavior. In contrast, in this paper we exclusively focus on the long memory properties of the forward discount using a pure structural break. In particular, we allow for the joint occurrence of long memory and structural breaks. We complement the analysis of Sakoulis and Zivot by critically evaluating the evidence for long memory and structural breaks in the forward discount.

In practice, the usual method to estimate the long memory parameter ‘d’ characterizing a time series is the nonparametric log-periodogram regression estimator suggested by Geweke and Porter-Hudak (1983). When we estimate the long memory parameter using the log-periodogram regression, we first difference the data. This estimator is appropriate for stationary long memory process with 0.5<d<0.5. However, Agiakloglou et al. (1993) show that the estimator is not invariant to first differencing, so that there might be bias due to over-differencing of the data. Kim and Phillips (2000) suggest that if we have no prior information about the magnitude of the long memory parameter before estimation, we need a more flexible estimation technique and inference for both stationary and nonstationary cases. They propose to estimate d using a modified log-periodogram (MLP) regression estimator that includes the nonstationary range where d0.5.

There is a large literature on structural break models, but there are only a few recent studies that deal with multiple structural breaks, and even fewer dealing with long memory and multiple structural breaks.1 In this paper, we assume that the potential structural break dates are unknown and we follow the methods of Bai and Perron, 1998, Bai and Perron, 2003 to estimate the unknown break dates using the least squares principle. We consider a structural change in mean model that allows the errors to be serially correlated and heteroskedastic.

Our approach is as follows. First, we estimate the long memory parameter for a number of forward discount series using the MLP regression without allowing for structural breaks. Second, we test for and estimate the multiple mean break model using Bai and Perron's method, and then adjust for the structural breaks in the forward discount. Finally, we re-estimate the long memory parameter using the MLP regression on the mean-break adjusted data. To our knowledge there is no formal statistical theory to justify our approach. Consequently, we use Monte Carlo simulations to evaluate our procedure for detecting multiple structural breaks in the presence of long memory and unknown structural breaks.

We show that allowing for structural breaks drastically reduces the persistence of the forward discount. However, after removing the breaks, we still find evidence of stationary long memory behavior in all of the forward discount series. Our results have important implications for understanding the statistical properties of the forward discount, because we confirm not only the presence of long memory in the forward discount but also the importance of structural breaks.

The remainder of this paper is organized as follows. Section 2 provides a brief review of the literature relating the forward discount anomaly, long memory and structural breaks. Section 3 reviews some properties of long memory processes and defines the MLP regression estimator of Kim and Phillips. Section 4 presents the multiple mean break model and reviews Bai and Perron's methodology to test for and estimate multiple structural breaks. Section 5 gives the empirical results, and Section 6 concludes with the implications of our findings.

Section snippets

The forward discount anomaly: long memory and structural breaks

Uncovered interest rate parity and covered interest parity imply that the current forward rate is an unbiased predictor of the future spot rate. Covered interest parity impliesftst=itit*,where ft and st denote the (log) 30-day forward and spot exchange rates in month t, it and it* denote the monthly interest rates on one-month domestic and foreign risk free bonds, respectively. Assuming rational expectations and risk neutrality, uncovered interest rate parity impliesEt(st+1)st=itit=ftst,

A brief review of long memory process

In this section, we review some basic properties of long memory processes, and then discuss the nonparametric estimators of the long memory parameter, d, that are used in our empirical analysis.

Multiple mean break model

In this section, we briefly review the methodology of Bai and Perron, 1998, Bai and Perron, 2003 for estimation and inference in a simple multiple mean break model that is utilized in our empirical analysis.

Bai and Perron, 1998, Bai and Perron, 2003, hereafter BP, consider several methods for the estimation of single and multiple structural breaks in dynamic linear regression models. They estimate the unknown break points given T observations by the least squares principle, and provide general

Data

We consider the same data as Sakoulis and Zivot (2001), which is monthly exchange rate data in terms of U.S. Dollars for five G7 countries: Germany, France, Italy, Canada, and Great Britain. All rates are end-of-month, average of bid and ask rates, and span the period 1976:1–1999:1. The Japanese Yen is not considered since the sample period is different (i.e., from 1978:7–1999:1). We multiplied the natural log of all rates by 100, so that the differences in rates are in percentages.

In Table 1

Conclusion

In this paper, we have analyzed the long memory properties of the monthly forward discount series for five G7 countries with and without allowing for structural breaks in the mean. We used the MLP regression to nonparametrically estimate the long memory parameter of a data series, since it is preferred to the GPH estimator when the data may exhibit nonstationary long memory. We also used the Bai and Perron method to detect and estimate the number of breaks in the mean. We found that multiple

Acknowledgements

We thank Chang-Jin Kim, Wen-Fang Liu, Charles Nelson and Mark Wohar, as well as seminar participants at the Empirical Macro Workshop at the University of Washington for their comments and suggestions. We thank Jushan Bai for providing the GAUSS code to estimate the multiple break models and also thank Chang Sik Kim for providing the code to estimate the MLP regression. The first author gratefully acknowledges research support from the Buechel Fellowship at the University of Washington. The

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