Residual-Based Tests for Fractional Cointegration: Testing the Term Structure of Interest Rates

Abstract

Campbell and Shiller (1987) and Hall et al. (1992) suggest that the term spread of long-term and short-term interest rates should be a stationary I(0) process. However, an empirically nonstationary term spread or rejection of cointegration between long and short term interest rates need not to be considered an empirical rejection of this theoretical relationship. It is likely that the dichotomy between I(1) or I(0) and/or integer values of cointegration are environments which are too restrictive to model the term structure. To overcome this problem, we propose a residual-based approach to test for the null of no cointegration against a fractional alternative which relies on the Exact Local Whittle Estimator (Shimotsu and Philllips, 2005, 2006). We compare its performance to other residual-based tests for fractional cointegration, and then we use it to investigate the term structure in the U.K and the U.S.

Keywords:

• Fractional cointegration
• Local Whittle estimator
• Term-structure

JEL Classification:

• C22
• E43

ACKNOWLEDGMENTS

We thank the Editor, anonymous referees, Qi Li, Li Gan, and the workshop participants at Southwestern University of Finance and Economics for their helpful comments.

Notes

¹See Phillips and Shimotsu (2006).

²Hall et al. (1992) propose a more general relationship between yields with different maturities. They assume that R(k, t) is a continuously compounded yield to maturity of a k period pure discount bond, and F(j, t) is the forward rate, defined as the rate of return from contracting at time t to buy a one-period pure discount bond that matures at time t + k.

³Notice that for comparison, we also estimate d using the GPH and the simple LW estimators.

⁴It turns out that this is a crucial issue in our case because, as it will be shown later, the residuals from our OLS regression will be I(1) under the null.

⁵Notice that our critical values are very similar to those of Cheung and Lai (1993), with the different sample size used not affecting the critical values that much.

⁶The MZ _t test, is a modified DF test statistics in the sense that the time series to be tested is demeaned or detrended by applying a GLS estimator. This turns out to improve the power of the test when there is a large AR root and reduces size distortions when there is a large negative MA root in the differenced series. Further improvement in the size properties is achieve by using modified information criteria to choose the lag truncation of the auxiliary regression.

⁷It should be also noticed that we have tested the term spread for a unit root using standard unit root tests and we have also run standard cointegration tests between the series. All the tests conducted, which are not reported here but can be made available on request, did not reject the null of a unit root on the term spread and or the null of no cointegration between the interest rates series.