a threshold cointegration test with increased power

Mathematics and Computers in Simulation 73 (2007) 386–392

A threshold cointegration test with increased power

Steven Cook ∗Department of Economics, University of Wales Swansea, Singleton Park, Swansea SA2 8PP, United Kingdom

Received 7 March 2006; received in revised form 12 August 2006; accepted 14 August 2006Available online 5 October 2006

Abstract

The low power of threshold, or asymmetric, cointegration tests is addressed. A new test is developed which combines momentum-threshold autoregression (MTAR) and local-to-unity detrending via generalised least squares (GLS). Critical values for the newlyproposed GLS-MTAR threshold cointegration test are provided under alternative decisions regarding the deterministic termsemployed when implementing the test. Simulation analysis of the test shows it to provide a substantial increase in power rela-tive to the previously proposed MTAR threshold cointegration test of Enders and Siklos [W. Enders, P. Siklos, Cointegration andthreshold adjustment, J. Business Econ. Statist. 19 (2001) 166–176].© 2006 IMACS. Published by Elsevier B.V. All rights reserved.

Keywords: Cointegration; Threshold adjustment; Asymmetry; Monte Carlo simulation; Test power

1. Introduction

The distinction between trend stationary (TS) processes and difference stationary (DS) processes is a central issuein the analysis of economic and financial time series. While TS processes exhibit controlled fluctuations about anunderlying deterministic component, DS processes are instead more unpredictable, exhibiting a trend in varianceand, possibly, mean. This distinction has important consequences, as while the bulk of statistical theory is basedupon the assumption that the series under examination are stationary, non-standard distribution theory is requiredto analyse DS processes. Although the distinction between TS and DS processes has been of particular prominencein statistical analysis in economics and finance, it is also of importance in a range of alternative disciplines such asbiology, engineering and meteorology where DS processes, or integrated time series, are analysed (see, inter alia[26,18,20]). Indeed, the issue of integrated time series is apparent in all modelling based upon, for example, the use ofARIMA (autoregressive, integrated, moving average) models of Box and Jenkins [3] which have received widespreadapplication across a range of disciplines.

A particular issue of concern when examining integrated processes is the problem of spurious regression. Due totheir underlying properties, independent integrated processes may exhibit a spurious close relationship when regressedupon each other. As noted by Aldrich [1], this issue of spurious regression has a long history in statistics (see [30]),although it was only formalised in terms of integrated processes following the studies of Granger and Newbold[17] and Hendry [19]. To overcome the problem of misleading or spurious results arising from the examination ofpotential relationships between integrated processes, the notion of cointegration was proposed in the work of Granger

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S. Cook / Mathematics and Computers in Simulation 73 (2007) 386–392 387

[16] and Engle and Granger [14]. Cointegration refers to the presence of a genuine, long-run relationship betweenintegrated time series processes. That is, while series might diverge in the short-run, over the long-run they sharean equilibrium relationship to which they return. Following the research of Engle and Granger [14], examination ofpotential cointegration has become a standard feature in the empirical analysis of economic and financial data. Underthe Engle–Granger procedure, a simple two-step approach is adopted to test the null hypothesis of no cointegrationbetween variables. While this procedure has become a cornerstone in the empirical examination of time series datain economics, finance and time series analysis, it has obvious value in the analysis of genuine relationships betweenintegrated time series in alternative disciplines. However, a drawback of the Engle–Granger approach is the implicitassumption that if an underlying long-run, or cointegrating, relationship exists, it is symmetric. That is, reversion toa defined attractor or equilibrium relationship is presumed to occur at a single speed at all times. This is in contrastto a growing literature which suggests that many economic and financial series may exhibit non-linear or asymmetricbehaviour (see, inter alia [2,10,15,21]). In light of this, more recent research by Enders and Siklos [13] has extendedthe familiar Engle–Granger approach to incorporate the possibility of asymmetry in the cointegrating relationshipbetween time series, with allowance made for differing speeds of reversion under different circumstances.1 Enders andSiklos [13] propose two broad classes of cointegration test incorporating asymmetry based upon alternative decisionrules concerning differing speeds of adjustment about an underlying threshold. As a consequence of the use of anunderlying threshold, these tests are frequently referred to as threshold cointegration tests. Via the application ofalternative decision rules, Enders and Siklos [13] derive threshold autoregressive (TAR) and momentum-threshold(MTAR) tests. However, while the extension of the standard Engle–Granger procedure to allow for asymmetry is tobe welcomed, the simulation results of Enders and Siklos [13] showed their TAR and MTAR threshold cointegrationtests to lack power. In particular, in the presence of threshold, or asymmetric, cointegration, the implicitly symmetricEngle–Granger test was found to dominate the TAR test. While the MTAR test was found to exhibit greater power, ittoo was often dominated by the Engle–Granger test. In this paper, the lack of power of threshold cointegration tests isaddressed, with a new MTAR-based test developed. The proposed test modifies the existing MTAR test of Enders andSiklos [13] via the use of local-to-unity detrending, an extension which is found to result in a substantial gain in testpower.

2. Cointegration and threshold cointegration

To examine possible cointegration between two unit root processes, xt and yt, the popularly employed Engle–Grangerapproach involves a two-step procedure. In the first stage, a static cointegrating regression is performed as below

yt = α + βxt + εt (1)

Potential cointegration between xt and yt is then examined via consideration of the integrated nature of the residuals{εt} from (1) using a Dickey–Fuller [9] test:

�εt = ρεt−1 + ηt (2)

The null of no cointegration (H0: ρ = 0) is tested against an alternative of cointegration (H0: ρ < 0) via the t-ratio forρ using specifically derived non-standard critical values. The resulting test is denoted here as τEG. However, as notedabove, a limitation of the Engle–Granger procedure is that only a single speed of reversion to equilibrium is permitted,this being given by the value of ρ. To incorporate possible asymmetry in the long-run relationship between xt and yt,Enders and Siklos [13] draw upon the research of Tong [28,29] and introduce the Heaviside indicator function (It) topartition the lagged residual εt−1 in (2). The revised version of (2) is then given as

�εt = Itρ1εt−1 + (1 − It)ρ2εt−1 + ηt (3)

The null of no cointegration is now given as H0: ρ1 = ρ2 = 0 in (3). The results of Petrucelli and Woolford [24]show ρ1 < 0, ρ2 < 0 and (1 + ρ1)(1 + ρ2) < 1 to be necessary and sufficient conditions for the stationarity of εt . As notedabove, Enders and Siklos [13] propose two classes of threshold cointegration test, based upon different specifications

1 It should be noted that similar developments concerning the incorporation of asymmetry have occurred in the analysis of the unit root hypothesis(see, inter alia [12,4,22]).

388 S. Cook / Mathematics and Computers in Simulation 73 (2007) 386–392

of the indicator function (It) in (3). The more powerful of these tests, the MTAR threshold cointegration test, employsthe following specification for the indicator function:

It ={

1 if �εt−1 ≥ 0

0 if �εt−1 < 0(4)

Disappointingly, the Monte Carlo analysis of Enders and Siklos [13] found that in the presence of asymmetry theMTAR test is frequently dominated in terms of power by the implicitly symmetric Engle–Granger τEG test.2,3 It istherefore apparent that the estimation of an additional adjustment parameter under the MTAR specification relativeto the τEG test acts as a penalty which more than offsets the misspecification of the Engle–Granger test arising fromneglected asymmetry. It should be noted also that Enders and Siklos [13] further develop their analysis by drawingupon the methods of Chan [5] to endogenise selection of the threshold about which partitioning occurs in the Heavisideindicator function. This grid-search based procedure considers a range of non-zero values for the threshold in (4), thesevalues being drawn from the central 70% of the ordered values of �εt . The selected threshold is then specified as thevalue of �εt delivering the lowest residual sum of squares for the estimated MTAR testing equation. This approachis referred to as consistent-threshold estimation. However, in recent research Cook [8] has examined the propertiesof threshold cointegration testing under consistent-threshold estimation, finding that this procedure does not result inan increase test power. In this paper, the MTAR threshold cointegration test, denoted as Φ*, is examined to see if, incontrast to consistent-threshold estimation, local-to-unity detrending can lead to an increase in test power. The requiredanalysis and resulting test are presented in the following section.

3. The GLS-MTAR threshold cointegration test

3.1. Derivation of the GLS-MTAR test

In previous research, Elliott et al. [11] have employed local-to-unity detrending via generalised least squares (GLS)as a means of increasing the power of the Dickey–Fuller unit root test.4 Following this, Perron and Rodriguez [23]and Cook [7] have utilised the same procedure to increase the powers of the Engle–Granger cointegration test andthe MTAR unit root test of Enders and Granger [12], respectively. To implement GLS detrending for the MTARthreshold cointegration test of Enders and Siklos [13], the following approach is adopted. Given two series, yt andxt, an initial decision is required regarding whether the data should be demeaned or detrended on the basis of theirobserved properties. If the series are clearly trending, GLS detrending requires the use of both a constant and trendterm, in which case the deterministic term to be employed under the GLS-MTAR test is defined as zt = (1, t)′, wheret is a linear trend. If the series are not trending, they are instead demeaned via use of a constant term with zt = 1. Withthe relevant deterministic term zt decided upon, quasi-differenced data are generated as

yα = [y1, y2 − αy1, . . . , yT − αyT−1]′, xα = [x1, x2 − αx1, . . . , xT − αxT−1]′,

zα = [z1, z2 − αz1, . . . , zT − αzT−1]′ (5)

where T denotes the sample size employed, α = 1 + cT−1 and c is a constant determining the extent of local-to-unity detrending. The GLS detrended series yα

t is then derived as yαt = yt − β0 when zt = 1, and yα

t = yt = β0 − β1t

when zt = (1, t)′, with the βi coefficients obtained from the regression of yα upon zα. The GLS detrended versionof xt, denoted as xα

t , is derived in an identical manner.5 The GLS detrended data are therefore dependent upon thedeterministic terms used to detrend the data and the choice of c. Following Elliott et al. [11] and Perron and Rodriquez

2 As noted above, an alternative TAR test exhibited far less power, being overwhelmingly dominated by the Engle–Granger test.3 This finding is of particular concern as following Pippenger and Goering [25] symmetric unit root and cointegration tests are themselves

recognised to suffer from low power in the presence of neglected asymmetry.4 As will be noted later, it can be questioned whether use of the commonly employed term ‘generalised least squares’ is appropriate in the present

context.5 It should be noted that while GLS is employed in the econometrics literature to describe the present approach, the nature of Eq. (6) might suggest

weighted least squares to be a more appropriate description, particularly from the perspective of statistical theory. Indeed, as noted by Perron andRodriguez [23], the suitability of the term GLS in the present context has been questioned in the econometrics literature.


[23] c = −7 is imposed when zt = 1, while c = −13.5 when zt = (1, t)′. The detrended series are then employed inthe following static regression:

yαt = γxα

t + ηt (6)

with possible threshold cointegration examined using the following testing equation:

�ηt = ρ1Itηt−1 + ρ2(1 − It)ηt−1 + et (7)

where

It ={

1 if �ηt−1 ≥ 0

0 if �ηt−1 < 0(8)

with the relevant null of hypothesis of no cointegration given as H0: ρ1 = ρ2 = 0 in (8). The GLS-MTAR test is denotedhere as Φ∗

GLS.

3.2. Critical values for the GLS-MTAR threshold cointegration test

To derive critical values for the GLS-MTAR test, a standard procedure is adopted with the following DGP utilised:

yt = yt−1 + u1,t , t = 1, . . . , T (9)

xt = xt−1 + u2,t (10)

where the innovation series {u1,t, u2,t} are generated as pseudo i.i.d. N(0, 1) random numbers using the RNDNSprocedure in GAUSS, with the initial conditions of the series {yt, xt} set to zero (y0 = x0 = 0). Using 50,000 replications,critical values for the GLS-MTAR test are derived under GLS demeaning (c = −7) and detrending (c = −13.5) foralternative sample sizes. The resulting critical values are reported in Table 1 for eight levels of significance {1%, 2.5%,5%, 10%, 50%, 90%, 95%, 99%} and six sample sizes: T = 50, 100, 200, 400, 1000, 2500.

Table 1Critical values for the GLS-MTAR threshold cointegration test

T Significance level (%)

99 95 90 50 10 5 2.5 1

(a) Intercept model (c = −7)50 0.056 0.265 0.472 1.735 4.191 5.151 6.077 7.386

100 0.046 0.218 0.400 1.601 3.939 4.853 5.748 6.873200 0.042 0.193 0.367 1.560 3.836 4.741 5.587 6.685400 0.041 0.190 0.360 1.546 3.804 4.671 5.530 6.621

1000 0.040 0.185 0.350 1.540 3.795 4.663 5.520 6.6012500 0.036 0.183 0.348 1.530 3.784 4.631 5.482 6.545

(b) Trend model (c = −13.5)50 0.534 1.029 1.378 3.104 6.129 7.285 8.450 9.899

100 0.423 0.906 1.230 2.908 5.697 6.770 7.803 9.168200 0.368 0.828 1.149 2.796 5.551 6.570 7.491 8.671400 0.344 0.788 1.103 2.735 5.421 6.374 7.355 8.566

1000 0.329 0.773 1.089 2.706 5.391 6.350 7.332 8.5182500 0.313 01.060 1.060 2.705 5.369 6.350 7.330 8.486

Notes: the above figures are critical values for the GLS-MTAR (Φ∗GLS) test under demeaning (c = −7) and detrending (c = −13.5) obtained via the

application of (6)–(8) to the data generation process of (9)–(10) over 50,000 replications.


4. A Monte Carlo analysis of test power

To explore the relative powers of the alternative threshold cointegration tests, the following modified version of theDGP of Engle and Granger [14] is employed:

yt + xt = u1,t (11)

yt + 2xt = u2,t (12)

u1,t = u1,t−1 + e1,t (13)

u2,t = ρ1Itu2,t−1 + ρ2(1 − It)u2,t−1 + e2,t (14)

It ={

1 if �u2,t−1 ≥ 0

0 if �u2,t−1 < 0(15)

t = 1, . . . , T (16)

where the innovation series {e1,t, e2,t} are generated as pseudo i.i.d. N(0, 1) random numbers using the RNDNSprocedure in GAUSS, with the initial conditions of the series {u1,t, u2,t} set to zero (u1,0 = u2,0 = 0). The modification ofthe Engle–Granger DGP relates to (14) and (15) where asymmetric behaviour is introduced. Threshold cointegrationbetween the series {xt, yt} is ensured in the DGP via the imposition of |ρ1| < 1, |ρ2| < 1 and ρ1 �= ρ2. The exact valuesemployed are reported in Tables 2–4, but it should be noted that the chosen values of the adjustment parameters ρi areselected to mimic the values reported previously in empirical research for a number of economic and financial series(see, inter alia [6,13,27]). The powers of the alternative τEG, Φ* and Φ∗

GLS tests are calculated over 50,000 simulationsof the above DGP at 5% level of significance for representative samples of 100, 200 and 400 observations, respectively.It should be noted that in line with the τEG and Φ∗

GLS tests which employ an intercept as the sole deterministic term,the Φ* test is considered under demeaning (c = −7).

The empirical powers of the tests are reported in Tables 2–4 for the three sample sizes of 100, 200 and 400observations, respectively. From inspection of the results, it is apparent that as would be expected, the powers of thetests are positively related to the degree of stationarity present, as indicated by the values of the asymmetric adjustmentparameters {ρ1, ρ2}, and the sample size considered. A further feature of the results is the disappointing performanceof the Φ* test relative to the τEG test, a property present also in the simulation results of Enders and Siklos [13].Indeed for all of the 35 experimental designs considered, the τEG test outperforms the Φ* test despite the former notincorporating the asymmetry present in the DGP. However, the tabulated figures show this not to be the case for theΦ* test. Indeed the power of this test always exceeds that of the Φ* and τEG tests. To illustrate this, the final columnsof Tables 2–4 reports the power advantage, or power gain, of the Φ∗

GLS test relative to the alternative Φ∗GLS test. From

Table 2Empirical powers of alternative cointegration tests (T = 100)

ρ1 ρ2 Φ* τEG Φ∗GLS Power gain

−0.10 −0.12 15.68 16.84 24.39 55.5−0.10 −0.14 18.18 19.49 28.20 55.1−0.10 −0.16 21.28 22.65 32.41 52.3−0.10 −0.18 24.80 26.15 37.12 49.7−0.10 −0.20 28.47 30.05 42.21 48.3

−0.08 −0.10 11.85 12.55 18.09 52.7−0.08 −0.12 13.80 14.58 21.41 55.1−0.08 −0.14 16.08 16.97 25.06 55.8−0.08 −0.16 19.03 19.72 29.14 53.1−0.08 −0.18 22.24 22.91 33.80 52.0

Notes: the figures under headings Φ*, τEG and Φ∗GLS represent empirical rejection frequencies for the respective cointegration tests expressed

in percentage terms at the nominal 5% level of significance. The final column reports the increased power of the Φ∗GLS test relative to the Φ*

test expressed in percentage terms. The alternative tests are calculated over 50,000 replications using the DGP of (11)–(16) for a sample of 100observations.




−0.08 −0.10 38.58 42.47 57.00 47.7−0.08 −0.12 47.66 51.88 66.42 39.4−0.08 −0.14 57.36 61.30 74.83 30.5−0.08 −0.16 66.95 70.21 81.83 22.2−0.08 −0.18 75.47 78.15 87.24 15.6

−0.06 −0.08 24.34 26.23 38.47 58.1−0.06 −0.10 31.62 34.07 48.47 53.3−0.06 −0.12 40.43 42.93 58.91 45.7−0.06 −0.14 50.12 52.60 68.81 37.3−0.06 −0.16 60.33 62.18 77.10 27.8

−0.04 −0.06 14.27 14.86 22.44 57.3−0.04 −0.08 19.25 20.03 30.78 59.9−0.04 −0.10 25.90 26.61 40.87 57.8−0.04 −0.12 34.13 34.69 51.73 51.6−0.04 −0.14 43.70 43.77 62.50 43.0






−0.04 −0.06 46.87 50.70 69.36 48.0−0.04 −0.08 65.64 69.09 84.65 29.0−0.04 −0.10 81.45 83.62 93.34 14.6−0.04 −0.12 91.87 92.89 97.28 5.9−0.04 −0.14 97.11 97.45 98.87 1.8

−0.02 −0.04 18.92 19.54 32.30 70.7−0.02 −0.06 33.14 33.53 53.98 62.9−0.02 −0.08 52.28 51.63 74.43 42.4−0.02 −0.10 71.18 70.37 88.40 24.2−0.02 −0.12 85.85 84.74 95.16 10.8




inspection of these figures it can be seen that for some designs, the power of the MTAR threshold cointegration test canbe increased by almost 60% via incorporation of GLS detrending. It should be noted also that instances of a relativelylow power gain occur only for experimental designs where a large sample size and high degree of stationarity result inall tests exhibiting relatively high power.

5. Conclusion

In this paper a new threshold cointegration has been proposed to overcome the noted low power of existing tests.Combining momentum-threshold autoregression and GLS-detrending, the resulting GLS-MTAR test was shown toexhibit much greater power than the previously proposed MTAR test of Enders and Siklos [13]. The results of thepower analysis indicate that the test will be of use to practitioners interested in the examining potential asymmetry orthreshold adjustment in the long-run relationships between integrated time series in economics, finance and a range ofother disciplines.


Acknowledgements

I am grateful to Professors Robert Beauwens and Michael McAleer for comments which have led to an improvementin the presentation and content of this paper.

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a threshold cointegration test with increased power

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