Testing the Null Hypothesis of Nonstationary Long Memory Against the Alternative Hypothesis of a Nonlinear Ergodic Model

Abstract

Interest in the interface of nonstationarity and nonlinearity has been increasing in the econometric literature. This paper provides a formal method of testing for nonstationary long memory against the alternative of a particular form of nonlinear ergodic processes; namely, exponential smooth transition autoregressive processes. In this regard, the current paper provides a significant generalization to existing unit root tests by allowing the null hypothesis to encompass a much larger class of nonstationary processes. The asymptotic theory associated with the proposed Wald statistic is derived, and Monte Carlo simulation results confirm that the Wald statistics have reasonably correct size and good power in small samples. In an application to real interest rates and the Yen real exchange rates, we find that the tests are able to distinguish between these competing processes in most cases, supporting the long-run Purchasing Power Parity (PPP) and Fisher hypotheses. But, there are a few cases in which long memory and nonlinear ergodic processes display similar characteristics and are thus confused with each other in small samples.

Keywords:

• Long memory I(d) and ESTAR processes
• Monte Carlo simulations
• Real exchange rates
• Real interest rates
• The Wald tests

JEL Classification:

• C12
• C22
• F31

ACKNOWLEDGMENTS

We are grateful to the Editor, the Associated Editor, two anonymous referees, Richard Baillie, Peter Phillips and the seminar participants at Universities of Leeds and Edinburgh for their helpful comments on an earlier version of this paper. The second author gratefully acknowledges partial financial support from the ESRC (Grant No. RES-000-22-3161). The usual disclaimer applies.

Notes

The process given by (2.8) is geometrically ergodic and hence asymptotically stationary and strong mixing if the autoregressive polynomial, δ(L) = β(L) + γ(L), has all its roots outside the unit circle.

In fact, we provide a full treatment of the SETAR case in an earlier working paper version (Kapetanios and Shin, 2003).

Setting ℓ to 1 is a common practice in the nonlinear unit root literature as discussed in Kapetanios et al. (2003) and Kapetanios and Shin (2006), who find that tests behave well under this choice.

To derive (3.2), we first subtract z _t from (3.1), and then take the first-order Taylor expansion of in order to resolve the so-called Davies (1987) problem that the nuisance parameters are unidentified under the null.

In practice, we suggest that , for , where is a sufficiently large number, and for . For example, in the Monte Carlo and empirical sections of the paper we use . Our work suggests that there is little sensitivity to that choice as long as is large.

The general case, where u _t is serially correlated, will be considered below.

We provide Monte Carlo evidence in Section 4, suggesting that this is indeed the case.

Selected fractiles of the asymptotic critical values for the 𝒲(d) test have been tabulated using 5,000 replications of a white noise I(d) process with 1,000 observations.

Alternatively, a data dependent procedure such as Akaike's information criterion or Schwarz's Bayesian criterion could be used for selecting p _T . For example, Ng and Perron (1995) examine the finite sample behavior of such criteria, but their setup does not cover the case with long memory processes. Hence, we choose to fix the order of the augmentation in empirical section.

This is equivalent to the nonlinear long memory model considered by Baillie and Kapetanios (2007).

Kapetanios et al. (2003) find that the Augmented Dickey–Fuller (ADF) test rejects the null of unit root in 2 out of 11 cases for real interest rates, and in none out of 10 cases for real exchange rates, both at the 5% significance level. Furthermore, Cheung and Lai (2001) conduct a Monte Carlo experiment to assess the power of the ADF test against the I(d) series simulated from actual data of the real exchange rates for eight countries over the sample period from April 1973 to December 1997, and find that the rejection rates are generally low, and estimated as follows: 26.7% (French franc), 16.8% (German mark), 40.6% (Italian lira), 15.5% (Dutch guilder), 17.2% (Swedish krona), 23.6% (Swiss franc), 15.2% (British pound), and 5.3% (U.S. dollar).

The bilateral Yen real exchange rates against the ith currency are collected from the IMF's International Financial Statistics in CD-ROM, and constructed as , where s _it is the logarithm of nominal exchange rate of ith currency per Yen, p _Jt the logarithm of the consumer price index (CPI) in Japan, and the logarithm of the CPI of the ith country. All data are quarterly, spanning the period 1960Q1 to 2000Q4 and not seasonally adjusted. The bilateral nominal exchange rates against the currencies other than the U.S. dollar are cross-rates computed using the U.S. dollar rates. Next, quarterly data on ex post short term real interest rates for the major economies of the European Union (France, Germany, Italy and the United Kingdom), N. America (Canada and the United States), New Zealand, and Japan are collected from the International Financial Statistics CD-Rom (2001 release date), covering the period 1957Q1 to 2000Q3. The price deflators used throughout are CPI and the nominal interest rate variables used are treasury bill rates for Canada, the United Kingdom and the United States, call money rates for France, Germany, and Japan, and discount rates for Italy and New Zealand.

The long memory parameter d is estimated by the local Whittle estimator proposed by Robinson (1995), denoted . Then, the associated asymptotic null critical values for the test have been tabulated using the white noise process of 1,000 observations and 5,000 replications. In all cases, ^# and * denote significance at 5% and 10% level.

Alternatively, we can impose that , since we know the rank of the probability limit of DX′ XD.

The statistic in (A.14) is different from (A.6) since we need to remove the effects of the lagged u _t 's by regressing both u _t and x _t on them.