Download - Generalized long memory processes, failure of cointegration tests and exchange rate dynamics

JOURNAL OF APPLIED ECONOMETRICSJ. Appl. Econ. 21: 409–417 (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

DYNAMICS

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. INTRODUCTION

The 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

Ł Correspondence to: Stefan C. Norrbin, Department of Economics, Florida State University, Tallahassee, FL 32309-2180, USA. E-mail: [email protected] 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 MEMORY

A 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:

��B��1 � 2uB C B2��xt � �� D ��B�εt �1�

where B is the lag operator with autoregressive and moving average parameters given by

��B� D 1 � �1B� Ð Ð Ð � �pBp and ��B� D 1 � �1B� Ð Ð Ð � �qB

q

The sequence fε1g 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 x�1 D e0 D e�1 D Ð Ð Ð D 0, is asymptoticallyequivalent to MLE. The objective function is given by

CSS��, �0, �0, u, �, �2� D �T2

log�2�� T

2log �2 � 1

2�2

T∑tD1

ε2t �2�

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: 409–417 (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 � ¾ cos��ω0��2��1, as � ! 1, where ω0 is the Gegenbauer frequencydefined as ω0 D cos�1�u�. (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 ω0 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�[2�1 � u�]��, which follows from the MA representation.

(f) The GARMA process can be rewritten as ��B��xt � �� D ∑1jD0CjB

j��B�εt (definition 2,

Gray et al., 1989), where Cj D 2u(�� 1j C 1

)Cj�1 �

(2�� 1

j C 1)Cj�2, with C0 D 1 and

C1 D 2�u (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.2

For 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.



Residual ACFLog DM ACFUpper 95% ClLower 95% Cl

1

0.8

0.6

0.4

0.2

0

-0.2

-0.40 50 100 150

Lag Length200 250 300 350

Figure 1. Autocorrelations of the log of Deutschemark and cointegrating residuals (January 20, 1994–December 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 the1980–85 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.



Table I. Subsample results for residuals

Subsample u[p-value u D 1]

�[p-value � D 1/2]

Cumulativeimpulse response

d[p-value d D 1]

March 3, 1980–February 19, 1985 0.9991 [0.0792] 0.4346 [0.0012] 15.3946 0.8842 [0.0036]February 20, 1985–December 31, 1989 0.9990 [0.0613] 0.4678 [0.2020] 18.2423 0.9530 [0.2775]January 2, 1990–January 19, 1994 0.9992 [0.1176] 0.4489 [0.0414] 18.2067 0.9355 [0.1510]January 20, 1994–December 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 (1980–85 and 1994–98) 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 be biased when one constrains u D 1 evenwhen it is marginally less than unity. To further examine this result, we found a mean bias of0.0417 for �, based on 2500 simulations of Monte Carlo data using the parameter values for the1980–85 period.

The second to last column of Table I shows that the cumulative response to a shock is finite andsimilar across all subsamples. The effect is larger for the two middle subsamples and is smallest forthe most recent subsample, having a value equal to 10.93. In contrast, even when an equilibriumrelationship is found using the fractional methodology, the CIR is infinite.

4. MONTE CARLO RESULTS

The application of standard cointegration tests to the set of nominal exchange rates consideredabove did not isolate an equilibrium relationship for each of the subsamples considered, althoughwe presented strong evidence of a GARMA cointegrating residual. Thus, we extend the pastliterature by examining the power of cointegration tests when the equilibrium residuals amongI(1) processes fall under a larger class of long memory processes. For this experiment, we generated



xt and yt where xt � yt is a long memory process.5 The selected sample sizes correspond to thecommonly used sample sizes for quarterly, monthly and daily data in economics and finance.

The Monte Carlo results for the Johansen (1988) procedure are reported in Table II.6 The topportion of the table corresponds to cases where the cointegrating residuals from a bivariaterelationship are fractional, while the middle panel corresponds to GARMA residuals from abivariate system. The final panel of the table reports results for a larger system of I(1) variables

Table II. Power of Johansen’s cointegration test with a 5% test

(A) ARFIMA residuals bivariate systemd-Parameter

Observations 0.5 0.6 0.7 0.8 0.9

200 0.9260 0.6698 0.3480 0.1396 0.0548500 0.9862 0.8746 0.5784 0.2120 0.0752

1245 0.9996 0.9580 0.7472 0.3648 0.0944

(B) GARMA residuals bivariate systemu-Parameter

Observations � 0.996 0.997 0.998 0.999 0.9995

200 0.40 0.1680 0.1420 0.1172 0.0994 0.0904500 0.40 0.9120 0.6790 0.5578 0.3766 0.2798

1245 0.40 1.0000 1.0000 1.0000 0.9386 0.8362

200 0.45 0.0740 0.0488 0.0400 0.0314 0.0514500 0.45 0.9270 0.5570 0.1816 0.0754 0.0544

1245 0.45 1.0000 1.0000 1.0000 0.5720 0.2436

200 0.50 0.1618 0.5894 0.0364 0.0706 0.0684500 0.50 0.9708 0.9564 0.2314 0.3456 0.0410

1245 0.50 1.0000 1.0000 1.0000 0.8686 0.3390

(C) GARMA residuals seven-variable system

Observations Model 80–85 Model 85–89 Model 90–94 Model 94–98

(u D 0.9991/ (u D 0.9990/ (u D 0.9992/ (u D 0.9979/� D 0.4346) � D 0.4678) � D 0.4489) � D 0.4387)

1245 0.0652 0.0544 0.0536 0.1104

Notes: The values in the table indicate the rejection rate using the maximum eigenvalue test of the false null hypothesisof zero equilibrium relationships. Trace statistics indicated the same conclusions and are not reported. These results areavailable from the authors. The rates are based on 5000 simulations, and we have allowed the lags in the test VAR tovary from 1 to 4 for the bivariate calculations, and from 1 to 12 for the seven-variable system. The reported values forthe bivariate relationships are similar across lags, and we therefore report the value that yields the highest rejection ratesamong the four lags. The larger system results are based on the actual GARMA estimates from the cointegrating residualsfor the nominal exchange rates. Thus, we fit a VAR model to the log of the nominal exchange rates for each subsampleand select the lags through minimization of the Akaike information criterion. In every case, the number of selected lagsis equal to 1.

5 Following Cheung and Lai (1993) an equilibrium relationship was generated from the bivariate system: xt C 2yt D e1t,xt � yt D e2t, where e1t is defined as a unit root process, and e2t is distributed as the long memory process in questioncalculated using the autocovariances of the process.6 The rejection rates using the Engle–Granger procedure, which are available upon request, tend to be marginally higherthan those of the Johansen procedure, although they are well below the correct value of 95% based on a 5% test.



with the same equilibrium relationships as those estimated for the nominal exchange rates foreach of the subsamples reported above. The rejection rates in Table II for the Johansen procedureare quite comparable to those reported in Andersson and Gredenhoff (1999). When d < 0.60, therejection rates are manageable, but as the value of d increases the empirical power of the testdecreases sharply, ending at only 5.5% when d is equal to 0.90 and for a sample size of 200.

For GARMA residuals, the table focuses on values of u that range from 0.996 to 0.9995, and� ranging from 0.40 to 0.50.7 The results are similar to the ARFIMA findings in that as theparameterizations of the equilibrium errors approach I(1) processes (with u D 1 and � D 0.5),cointegration tests perform poorly with rejection rates of the true null as small as 3%. Also notethat the value of u impacts cointegration results. When � D 0.45 and for a sample size of 500,one can observe that as u increases from 0.996 to 0.9995, the Johansen procedure results incorrect inference at a rate falling from 92.7% to 5.44%. While the test performs better as thesample size increases, the rejection rates can be extremely low for near unit root processes evenfor large samples. For example, when u D 0.999 and � D 0.45 with 1245 observations, whichis parametrically close to the cases in Table I, we find the Johansen test correctly identifies anequilibrium relationship only 57.2% of the time.

Finally, for larger systems, the final panel in Table II considers the results of Johansen tests fora system of seven I(1) variables having equilibrium relationships defined by the estimated valuesfound in Table I.8 For this experiment, the number of lags in the test VAR varies from 1 to 12,and we select the number of lags based on the AIC for the actual systems of nominal exchangerates considered above. Rejection rates of the false null range from 5.4% to 11% for all fourmodels. This result implies that Johansen’s cointegration method would be highly unlikely to findan equilibrium relationship in the exchange rate systems.

5. CONCLUSION

This paper shows that an exchange rate system for several subsamples containing the US dollarexchange rates of Canada, Germany, France, Italy, Switzerland, Japan and the United Kingdomexhibits mean reverting behaviour, although each exchange rate has a unit root. In contrast tostandard and fractional cointegration techniques, the use of the GARMA methodology isolates anequilibrium relationship with a finite CIR in every subsample that is also frequently stationary.The implementation of fractional methods restricts the long memory cycle to be infinite, and ourMonte Carlo analysis demonstrates that this restriction can bias tests away from finding stationaryequilibrium relationships. The Monte Carlo evidence also demonstrates a significant power loss forstandard cointegration tests when the cointegrating residuals are long memory processes similarto the ones analysed in this paper.

The finding of a long memory GARMA connection among exchange rates implies that centralbanks either explicitly co-ordinate in an announced exchange rate system, such as the EuropeanMonetary System, or they implicitly co-ordinate by not allowing the exchange rate to deviate too

7 We focus on results where u ½ 0.996, because the cointegration tests perform well when u < 0.996 for larger samplesizes, and our experience with the GARMA model indicates that values of u < 0.996 are unlikely in macroeconomic data.8 To generate a system of seven non-stationary variables with a stationary GARMA residual, we follow the MonteCarlo setup of Marinucci and Robinson (2001) who consider fractional cointegration with non-stationary regressands andstationary disturbances. Given the computational burden, we limit the number of simulations to 2500 and do not considerheteroskedastic components.



far from some unannounced target rate by adjusting interest rates or intervening in the foreignexchange market. Such a GARMA connection should result in a long memory error correctionprocess with a cycle of at least 3 months. Although it is unlikely this information could be exploitedfor superior forecasting in the short run, it may be possible to improve forecasting accuracy inthe very long run. Thus, future research could consider the implications of GARMA cointegrationon long-run forecasting. Such a study would also shed some light on the type of adjustment thattakes place to reach the long-run equilibrium.

Given the infancy of the GARMA cointegration concept, other extensions are also plausible.It may be possible to extend the test for fractional cointegration in the frequency domain byMarinucci and Robinson (2001) to GARMA processes. In addition, Nielsen (2004) has developedan LM test with standard asymptotics for fractional cointegration that he applied to a set of monthlyexchange rates that could be extended to consider GARMA long memory.

ACKNOWLEDGEMENTS

This paper has been greatly improved thanks to comments from the co-editor Tim Bollerslev, twoanonymous referees and Paul Beaumont. In addition, we are grateful for comments from seminarparticipants at the University of Oklahoma, Lund University, the Swedish Central Bank and the2003 Conference on Computing in Economics and Finance at the University of Washington. Anyremaining errors are our own.

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