Generalized long memory processes, failure of cointegration tests and exchange rate dynamics
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JOURNAL OF APPLIED ECONOMETRICSJ. Appl. Econ. 21: 409417 (2006)Published online 8 March 2006 in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/jae.857
GENERALIZED LONG MEMORY PROCESSES, FAILUREOF COINTEGRATION TESTS AND EXCHANGE RATE
AARON D. SMALLWOODa AND STEFAN C. NORRBINb*a University of Oklahoma, Norman, OK, USA
b Florida State University, Tallahassee, FL, USA
SUMMARYThis paper presents evidence that the equilibrium relationship in a system of nominal exchange rates is bestdescribed as a stationary GARMA process. The implementation of the GARMA methodology helps explainconflicting and puzzling results from the use of linear cointegration and fractional cointegration methods.Furthermore, we use Monte Carlo analysis to document problems with standard cointegration tests when theattraction process is distributed as a long memory GARMA process. Copyright 2006 John Wiley & Sons,Ltd.
1. INTRODUCTIONThe econometric methodologies used in testing for equilibrium relationships in macro-econometricswere revolutionized with the seminal work by Engle and Granger (1987). According to theirdefinition, two I(d) variables are cointegrated if there exists b > 0, such that a linear combination ofthese variables is integrated of order d b. Although researchers typically assume that d D b D 1,recent research has suggested that the utilization of standard cointegration tests can produceincorrect inference when the cointegrating error does not appear to be I(0). As documented byCheung and Lai (1993) and Andersson and Gredenhoff (1999), the performance of these tests canbe particularly dubious when the cointegrating residual is fractional.
The ongoing debate concerning the stochastic properties of equilibrium relationships is illustratedby the literature devoted to exchange rate dynamics. Diebold et al. (1994) challenged the originalfinding of standard cointegration among a set of nominal exchange rates by Baillie and Bollerslev(1989b). In response, Baillie and Bollerslev (1994) presented evidence supporting an equilibriumrelationship with fractionally integrated residuals. However, fractional integration imposes aninfinite cycle in the cointegrating error, while a shock dissipates so slowly that the cumulativeeffect is infinite.1 In this paper, we present evidence that the equilibrium errors among a set ofnominal exchange rates possess long memory cyclical characteristics absent in fractional processes.A modelling alternative that accommodates cyclical long memory is the GARMA model, whichwas first studied by Gray et al. (1989). We demonstrate that the GARMA model captures thedynamics of the cointegrating residuals among a set of nominal exchange rates, and is often ableto resolve conflicting results including the finding of a non-stationary equilibrium error with an
Correspondence to: Stefan C. Norrbin, Department of Economics, Florida State University, Tallahassee, FL 32309-2180, USA. E-mail: email@example.com Hauser et al. (1999) argue that the infinite cumulative effect makes fractional models inappropriate for analysingpersistence.
Copyright 2006 John Wiley & Sons, Ltd. Received 24 June 2003Revised 17 November 2004
410 A. D. SMALLWOOD AND S. C. NORRBIN
infinite cumulative impulse response (CIR) function. Further, the paper shows that cointegrationand fractional cointegration methods are biased away from finding an equilibrium relationshipwhen the residual is a GARMA process.
The rest of the paper is organized as follows. In Section 2, we discuss the GARMA model andshow how other long memory models are special cases of a GARMA framework. Section 3examines updated results for the exchange rate dynamics that were explored in Baillie andBollerslev (1994). In Section 4, we present the Monte Carlo results examining the power ofcointegration tests when the relationship is an ARFIMA or GARMA process. Finally, someconcluding remarks and ideas for future research are presented.
2. LONG MEMORYA long memory ARFIMA process generalizes ARMA and ARIMA processes by allowing thedifferencing operator applied to achieve a stationary ARMA process to take on any real value.An alternative long memory model is the GARMA model, which generalizes the fractional modelby allowing periodic decay in the autocorrelation function of data. Recently, several authors havesuggested that GARMA models can be useful for modelling processes including inflation rates(Chung, 1996b) and nominal and real interest rates (Barkoulas et al., 2001; Ramachandran andBeaumont, 2001; Smallwood and Norrbin, 2001).
A convenient way to introduce the properties of GARMA models is to show how they nestARIMA, ARFIMA and ARMA models. The GARMA model is defined as follows:
B1 2uB C B2xt D Bt 1where B is the lag operator with autoregressive and moving average parameters given by
B D 1 1B pBp and B D 1 1B qBq
The sequence f1g denotes a martingale difference sequence with variance 2. The parameter uprovides information about the long memory periodic movement in the data, is the mean ofthe process, governs the damping of the autocorrelations with 2 D d. If u D 1 then the modelreduces to an ARFIMA(p, 2, q) model. When u D 1 and D 1/2, the result is an ARIMA modelwith a single unit root, and when D 0 the result is an ARMA model.
In this paper, we employ the conditional sum of squares (CSS) estimator, which conditionalon the assumption that the initial observations x0 D x1 D e0 D e1 D D 0, is asymptoticallyequivalent to MLE. The objective function is given by
CSS, 0, 0, u, , 2 D T2
log 2 122
where 0 and 0 are p- and q-dimensional vectors containing the autoregressive and movingaverage parameters, and T is the total number of observations. Consider the GARMA(p,q) processsatisfying the condition that all roots to z D 0 and z D 0 lie outside the unit circle. TheGARMA model and the CSS estimator of the GARMA model have the following properties:
(a) The model is stationary if juj < 1 and < 1/2 or if juj D 1 and < 1/4. The model is invertibleif juj < 1 and > 1/2 or juj D 1 and > 1/4. (Theorem 1(a), Gray et al., 1989.)
Copyright 2006 John Wiley & Sons, Ltd. J. Appl. Econ. 21: 409417 (2006)
GENERALIZED LONG MEMORY PROCESSES 411
(b) The model is mean reverting if juj < 1 and < 1 or if juj D 1 and < 1/2 (see equation (13),Gray et al., 1989; equation (6), Granger and Joyeux, 1980).
(c) The autocorrelations of the GARMA(0,0) process with juj < 1 and < 1/2 are given bythe approximation cos021, as ! 1, where 0 is the Gegenbauer frequencydefined as 0 D cos1u. (Theorem 3(c), Gray et al., 1989, with correction on p. 561 of Grayet al., 1994.)
(d) For > 0, the spectrum of a GARMA process is unbounded at 0 if u < 1 and at 0 otherwise(an ARFIMA process). (Theorem 2, Gray et al., 1989; equation (1), Granger and Joyeux, 1980.)
(e) The cumulative impulse response function for the stationary GARMA model is given by 1 D 1/1[21 u], which follows from the MA representation.
(f) The GARMA process can be rewritten as Bxt D1
jD0CjBjBt (definition 2,
Gray et al., 1989), where Cj D 2u( 1j C 1
(2 1j C 1
)Cj2, with C0 D 1 and
C1 D 2u (equation (144.13), Rainville, 1960).
Note that the relaxation of the assumption that u D 1 allows for a broader range of stationaryparameterizations, while producing an autocorrelation function that can exhibit periodic decay.
3. EXCHANGE RATE DYNAMICS
We consider the equilibrium relationship among the same nominal exchange rates studied byBaillie and Bollerslev (1994), although we update their sample. In their paper, they point outthat the exchange rates of Canada, Germany, France, Italy, Switzerland, Japan and the UnitedKingdom vis-a`-vis the US dollar appear to have unit roots, although a linear combination foundthrough OLS appears to be distributed as a fractional variable with integration order significantlyless than 1. Our full sample period extends from March 3, 1980 through December 31, 1998. Weconsider several subsamples, the first of which corresponds to the period from March 3, 1980 toFebruary 19, 1985. For this subsample, we employ the original data from Baillie and Bollerslev.2For the three remaining subsamples, data from the St. Louis Federal Reserve Board (FRED) areused, namely daily foreign currency buying rates of the dollar at noon in New York.
A series of unit root tests coupled with estimation of a GARMA model supports the establishedfinding in the literature (for example, Nielson, 2004) of a unit root in each of the nominal exchangerates for every subsample considered. The finding of a unit root in the original series, with u D 1and D 1/2, implies that an equilibrium relationship among the nominal exchange rates exists ifa linear combination of these rates can be found that produces a mean reverting series. However,standard cointegration tests fail to find evidence of an equilibrium relationship, confirming theconcern by Diebold et al. (1994). From the second property of the GARMA model, discussedpreviously, a mean reverting combination exists whenever u falls below unity or falls below1/2. If both parameters decrease relative to these values, then the mean reverting relationship isalso stationary. In the context of fractional dynamics a mean reverting equilibrium relationshipexists if d falls below unity, while in this case the relationship is stationary if, and only if, d fallssignificantly below 1/2.
2 We would like to thank Tim Bollerslev for the original Baillie and Bollerslev (1994) data. The other samples areconstructed to approximate the same number of observations as in the original data set, by roughly evenly splitting thetwo decades through which our sample extends.
Copyright 2006 John Wiley & Sons, Ltd. J. Appl. Econ. 21: 409417 (2006)
412 A. D. SMALLWOOD AND S. C. NORRBIN
Residual ACFLog DM ACFUpper 95% ClLower 95% Cl
-0.40 50 100 150
Lag Length200 250 300 350
Figure 1. Autocorrelations of the log of Deutschemark and cointegrating residuals (January 20, 1994December 31, 1998)
Following Baillie and Bollerslev (1994) we regress the log of the DM on a constant and thelog of the remaining six exchange rates. The first 350 autocorrelations of the residuals fromthis regression for the period from 1994 to 1998 are depicted in Figure 1. The residuals for theremaining periods yield similar patterns. The autocorrelations exhibit cyclical decay typical of theGARMA model, in contrast to the autocorrelations of the nominal DM, thus implying that someform of exchange rate connection exists. The results found below indicate that a frequency ofroughly 0.0655 radians exists in the residuals, suggesting a cycle of roughly 96 days.
To formally test if the exchange rates are connected in a system we perform three tests inTable I. First, we test the hypothesis that the estimated value of u for the residuals is unity versusthe alternative that u < 1.3 Second, we test the hypothesis that D 1/2 versus < 1/2 usinga t-test. The inference using fractional cointegration methods is also of interest, and thereforewe also estimate a fractional model for the residuals, and test the hypothesis that d D 1 versusd < 1. Statistical inference has not been introduced for the GARMA model when it is appliedto residuals, and in the current application we must account for the existing evidence regardingthe finding of conditional heteroskedasticity in exchange rates (Baillie and Bollerslev, 1989a;Bollerslev, 1990). Therefore we simulate probability values based on the estimation procedureemployed here by allowing the residuals from a set of seven random walks to follow a multivariateconstant conditional correlation GARCH specification, which is estimated from the data for the198085 period. A GARMA model and an ARFIMA model are fit to the estimated errors foundthrough OLS and the simulated p-values are then calculated based on 20 000 simulations.
The results for each of the subsamples are reported in Table I, with simulated p-values inbrackets.4 When the GARMA model is used, significant mean reversion occurs for each subsample
3 Chung (1996a) shows that this test can be conducted using the test statistic Ou 1.4 The full period, consisting of 4731 daily observations, was also estimated. The resulting values of u D 0.99995 and D 0.5015 indicate an increasing level of persistence relative to the subsample estimates. Given the dramatic eventsaffecting exchange rates over this extremely long span of data, it is hardly surprising that the level of persistence increases.
Copyright 2006 John Wiley & Sons, Ltd. J. Appl. Econ. 21: 409417 (2006)
GENERALIZED LONG MEMORY PROCESSES 413
Table I. Subsample results for residuals
Subsample u[p-value u D 1]
[p-value D 1/2]
d[p-value d D 1]
March 3, 1980February 19, 1985 0.9991 [0.0792] 0.4346 [0.0012] 15.3946 0.8842 [0.0036]February 20, 1985December 31, 1989 0.9990 [0.0613] 0.4678 [0.2020] 18.2423 0.9530 [0.2775]January 2, 1990January 19, 1994 0.9992 [0.1176] 0.4489 [0.0414] 18.2067 0.9355 [0.1510]January 20, 1994December 31, 1998 0.9979 [0.0042] 0.4387 [0.0068] 10.9313 0.8801 [0.0036]
Notes: For the estimated value of u, we test the null hypothesis that u D 1 against the alternative hypothesis that u < 1.For , we test the null hypothesis that D 1/2 versus the alternative that < 1/2, while for d we test the hypothesis thatd D 1 versus d < 1. In constructing the tests for and d, we use the estimated t-statistic. Numerical standard errors arecalculated using the outer product of the gradient. Asymptotic theory for the CSS estimator of the GARMA process has notbeen developed for the case where the estimation procedure is applied to residuals from a cointegrating regression amongmultivariate heteroskedastic I(1) components. It is therefore necessary to simulate the critical values for this environment.The simulated critical values for each hypothesis are given as follows:
H0 : u D 1 vs u < 1 H0 : D 1/2 vs < 1/2 H0 :d D 1 vs d < 1
10% 0.0008 10% 3.3835 10% 2.94165% 0.0011 5% 3.9374 5% 3.50381% 0.0030 1% 5.0747 1% 4.6661
at the 10% level. For example, in the second subsample, a significant reduction in u occurs atthe 10% level, implying the process is significantly mean reverting even though we are unable toreject the hypothesis that D 1/2. Evidence of a stationary equilibrium relationship is found in thefirst and last subsamples, since a significant reduction in both u and occurs when a 10% test isemployed. In contrast, the ARFIMA model shows a mean reverting equilibrium relationship in onlytwo subsamples (198085 and 199498) at the 10% level, and in no case does the ARFIMA modelindicate a stationary equilibrium relationship as the estimated value of d is always significantlygreater than 1/2. In addition, if the estimated value of d is divided by 2 to make it comparable to, the resulting value exceeds the estimated value from GARMA estimation for every subsample.Chung (1996b) has indicated that the estimate of can...