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Mathematics and Computers in Simulation 78 (2008) 507–513

Small sample improvements in the threshold cointegrationtest using residual-based moving block bootstrap

Zheng Yang a,∗, Zheng Tian a,b, Zixia Yuan a

a Department of Applied Mathematics, Northwestern Polytechnical University, Xi’an 710072, Chinab National Key Laboratory of Pattern Recognition, Institute of Automation, Chinese Academy of Sciences, Beijing 100080, China

Received 19 April 2007; received in revised form 20 June 2007; accepted 30 June 2007Available online 7 July 2007

Abstract

A residual-based moving block bootstrap procedure for testing the null hypothesis of linear cointegration versus cointegrationwith threshold effects is proposed. When the regressors and errors of the models are serially and contemporaneously correlated, ourtest compares favourably with the Sup LM test proposed by Gonzalo and Pitarakis. Indeed, shortcomings of the former motivatedthe development of our test. The small sample performance of the bootstrap test is investigated by Monte Carlo simulations, andthe results show that the test performs better than the Sup LM test.© 2007 IMACS. Published by Elsevier B.V. All rights reserved.

Keywords: Threshold cointegration; Residual-based moving block bootstrap; Sup LM test; Monte Carlo simulation

1. Introduction

Since the introduction of cointegration in econometrics and statistics by Engle and Granger [10], integration andcointegration tests have undergone considerable development and become a central part of modern economics, finance,and time series analysis. The tests are now routinely applied in empirical studies. However, most of the methods presentlyavailable assume that the cointegrated systems are linear, which according to economic theory, need not to be the case.Recently, econometricians and statisticians are increasingly turning to nonlinear models, which include, among others,(i) Markov-switching error correction models (see [15,22]); (ii) cointegrating smooth transition regression (see, interalia [4,14,23]); and (iii) threshold cointegrating model (see, inter alia [2,9,11,13,19,24]).

The investigation of threshold effect in cointegrated systems has recently assumed great significance. The growingliterature on threshold nonlinearity in cointegrated models can typically be categorized into three strands. The firststrand of the literature focuses on the treatment of threshold effect within a multivariate error correction framework. Themodels define as a threshold adjustment toward the long run equilibrium while assume that the cointegration relationshipis linear (see, inter alia [2,13,19,24]). The second strand of the threshold cointegration explores the regressor–errorbeing subject to threshold autoregressive model, for instance, Ender and Siklos [9] and Cook [5]. Finally, Gonzalo andPitarakis (hereafter GP) [11] extend threshold effect in the cointegrating relationship itself. For the purposes of thispaper, we shall limit our analysis to the last case.

∗ Corresponding author. Tel.: +86 29 88494347; fax: +86 29 88491214.E-mail address: yzeagle@yahoo.com.cn (Z. Yang).

0378-4754/$32.00 © 2007 IMACS. Published by Elsevier B.V. All rights reserved.doi:10.1016/j.matcom.2007.06.005

508 Z. Yang et al. / Mathematics and Computers in Simulation 78 (2008) 507–513

Within a single equation cointegrating regression, Gonzalo and Pitarakis propose the Sup LM test for the nullhypothesis of linear cointegration against cointegration with threshold effects, derive the limiting distribution, andpresent the critical values. Although their Proposition 5 indicates that the limiting distribution of the Sup LM statisticremains free of serial correlation of the regressor–errors, the authors do not undertake a Monte Carlo investigation ofthe small size and power properties of the suggested test. When the sample sizes are indeed very large, particularly infinance (stock and bond markets), one might be comfortable using the asymptotic theory. Many other studies, however,are based on small sample sizes (typically less than 100). Hence, the existing test may ignore the fact that the smallsample performance of the proposed test would be affected because of the loss of potentially valuable informationfrom the serial correlation.

In order to overcome the problem of small sample properties of the Sup LM test we suggest to apply the bootstrapapproach to the case of serial correlation in the errors. In this paper, we propose and study a residual-based movingblock bootstrap procedure to approximate the sampling distribution of the Sup LM statistic, and show how to calculatean asymptotic p-value (see inter alia [3,13]). The small sample properties of the bootstrap test in comparison with thatof the Sup LM test is examined by computer simulation.

2. Threshold cointegration and the Sup LM test

Consider the following cointegrating regression with a threshold,

yt = β0 + β′1xt + (λ0 + λ′

1)xtI(qt−d > γ) + ut (1)

xt = xt−1 + vt (2)

where ut and νt are scalar and p-vector valued stationary disturbance terms. qt–d with d ≥ 1 is the stationary thresholdvariable and I(qt–d > γ) is the usual indicator function take the value one when qt–d > γ and zero otherwise. γ is athreshold parameter and γ ∈ Γ = [γL, γU] where γL and γU are such that P(qt–d ≤ γL) = θ1 and P(qt–d ≤ γU) = 1 − θ1.The choice of θ1 is commonly taken to be 10% or 15%. The threshold variable qt–d allows the cointegrating vector toswitching between (1, −β′

1) and (1, −(β1 + λ1)′) depending on whether qt–d cross the unknown threshold level givenby γ .

Write (1) in matrix notation

Y = Xβ + Xγλ + U (3)

where β = (β0, β′1)′, λ = (λ0, λ

′1)′, and

Y =

⎛⎜⎜⎜⎜⎝

y1

y2

...

yT

⎞⎟⎟⎟⎟⎠ , X =

⎛⎜⎜⎜⎜⎝

1 x′1

1 x′2

......

1 x′T

⎞⎟⎟⎟⎟⎠ , Xγ =

⎛⎜⎜⎜⎜⎝

I{q1−d > γ} x′1I{q1−d > γ}

I{q2−d > γ} x′2I{q2−d > γ}

......

I{qT−d > γ} x′T I{qT−d > γ}

⎞⎟⎟⎟⎟⎠ , U =

⎛⎜⎜⎜⎜⎝

u1

u2

...

uT

⎞⎟⎟⎟⎟⎠

Then, the null hypothesis of linear cointegration H0:λ = 0 against the alternative of threshold cointegration H1:λ �= 0.The testing problem is nonstandard because the nuisance parameter γ is not identified under the null hypothesis. This

is called the Davies problem in the literature following Davies [6,7]. Most solutions to this problem involve integratingout unidentified parameters from the test statistics (see [1,12]). GP use the supremum of the Lagrange multiplier (LM)statistic to test the null hypothesis. The Sup LM statistic can be given by:

Sup LM = supγL≤γ≤γU

LMT (γ) (4)

and

LMT (γ) = 1

σ20

U ′MXγ (X′γMXγ )−1

X′γMU (5)

where M = I − X(X′X)−1X′ and I denotes unit matrix, σ20 as the residual variance from OLS estimation of (1) under the

null hypothesis of linear cointegration.

Z. Yang et al. / Mathematics and Computers in Simulation 78 (2008) 507–513 509

3. Residual-based moving block bootstrap testing

The bootstrap method initiated by Efron [8] is a resampling technique which takes advantage of the powerfulcomputers. The bootstrap methods are often designed for stationary weakly dependent processes, for example, for theblock bootstrap [16,18] and the stationary bootstrap [21]. Under nonstationary, we cannot resample the data {xt, yt}directly because the time series structure of the I(1) processes is destroyed. Here we apply a residual-based movingblock bootstrap (RMBB, for short) procedure to approximate the sampling distribution. The RMBB procedure is basedon the block bootstrap of Li and Maddala [17] and Paparoditis and Politis [20].

Li and Maddala [17] use the RMBB procedure to correct the small sample biases in the estimators and the sizedistortions in the linear cointegrating regressions. Although our RMBB is similar to theirs, it should be noted that thereare some significant difference in resampling schemes such as the centered residuals and the statistics, etc. Furthermore,the calculation of our bootstrapped Sup LM statistic is more complicated due to the involved nonlinearity. Followingthe convention in the literature, we denote the bootstrap quantities such as data and the statistic, etc., with an asterisk.

The RMBB testing procedure is conducted through the following algorithm:

(1) Estimate (1) under the null hypothesis using the original sample, and let β0 and β1 be the OLS estimators of β0and β1, respectively. Calculate the regression residuals as

ut = yt − β0 − β1xt, t = 1, 2, . . . , T (6)

Calculate the associated statistic Sup LM for testing the null H0:λ = 0.(2) Let vt = xt . Calculate centered residuals

ut = ut − 1

T − b

T−b∑i=1

1

b

b∑j=1

ui+j, t = 1, 2, . . . , T (7)

(3) Choose a positive integer b(<T), and let i0, i1, . . ., ik−1 be random variables drawn i.i.d. distribution uniform onthe set {1, 2, . . ., T − b}; here we take k = [(T − 1)/b], where [·] denotes the integer part. Bootstrap the triplets ofnumbers {u∗

t , v∗t , q

∗t−d} and constructs a bootstrap pseudo-series {x∗

1, x∗2, . . . , x

∗T } and {y∗

1, y∗2, . . . , y∗

T } as follows:

x∗t =

{x1 for t = 1,

x∗t−1 + vim+s for t = 2, . . . , T

(8)

y∗t = β0 + β1x

∗t + uim+s, t = 1, . . . , T (9)

where m = [(t − 2)/b] and s = t − mb − 1.(4) Compute the pseudo-statistic Sup LM* as defined in (4) using the generated bootstrap samples

{q∗1−d, q

∗2−d, . . . , q

∗T−d}, {x∗

1, x∗2, . . . , x

∗T } and {y∗

1, y∗2, . . . , y∗

T }.(5) Repeating step 3 and step 4 a great number of times (B times, say), we can obtain an empirical distribution for the

bootstrap statistic Sup LM*. Define the bootstrap p-value function by the quantity

p = B−1B∑

i=1

I(Sup LM∗ ≥ Sup LM) (10)

where I(·) is the indicator function that equals one if the inequality is satisfied and zero otherwise, reject the nullhypothesis if the selected significance level exceeds p.

Remark 3.1. When a null hypothesis is tested, the resampling scheme is performed under the null and hence weconsider for bootstrapping the residuals {ut} instead of using the residuals under the alternative. The basic idea for thebootstrap testing is the same as that of [17].

Remark 3.2. Eq. (7) is commonly observed in the standard block bootstrap literature because the block bootstrappedseries is nonstationary, see [21]. Although the series {ut} has a zero mean both under the null and the alternative, the

510 Z. Yang et al. / Mathematics and Computers in Simulation 78 (2008) 507–513

estimated innovations {ut} will likely have nonzero (sample) mean; this discrepancy has an important effect on thebootstrap distribution effectively in the bootstrap world.

Remark 3.3. The bootstrapped data {u∗t , v

∗t , q

∗t−d} preserve both serial and contemporaneous correlations between

the regressors and the error term of the model. It means that the serial correlation of the disturbance term ut is takeninto account, at the same time, and also the endogeneity of xt is preserved.

4. Monte Carlo investigation

In this section we undertake some Monte Carlo investigations of the small sample size and power performance ofthe suggested test in comparison with the Sup LM test. In the first set of experiments we examine the size propertiesof the tests. We use the following data generating process (DGP):

yt = β0 + β1xt + (λ0 + λ1xt)I(qt−1 > γ) + ut (11)

xt = xt−1 + vt (12)

ut = et + δet−1 (13)

vt = ρvt−1 + εt (14)

qt = φqt−1 + vt (15)

where t = 1, 2, . . ., T and the delay parameter d = 1. The innovation series zt = (et, εt, qt) be generated as pseudo i.i.d.N(0, Σz) random numbers using MATLAB procedure and the covariance matrix

Σz = E[ztz′t] =

⎛⎜⎝

1 σeε σeq

σeε 1 σεq

σeq σεq 1

⎞⎟⎠

Under the null we fix β0 = 0.5, β1 = 1, λ0 = 0, λ1 = 0 and φ = 0.5. We consider parameter values for δ = {−0.5, 0.0,0.5} and ρ = {−0.8, −0.4, 0.0, 0.4, 0.8}. In order to investigate the impact of the endogeneity of regressors, we considertwo cases of the covariance matrix: {σeε, σeq, σεq}= {0.0, 0.0, 0.0} and {σeε, σeq, σεq}= {0.5, 0.0, 0.6}. The formeris strictly exogenous where Σz = I when the latter allows for the endogeneity of the regressors.

For all experiments, 50 initial observations are discarded to minimize the effect of initial conditions. The tests arecalculated setting θ1 = 0.15, using 50 grid points on [γL, γU] for calculation of (4). For the RMBB test, the asymptoticp-value is calculated on each simulated sample with B = 200 bootstrap replications and the block length b = 5.

Table 1 shows the size properties when T = 100. Nominal size is set at 5% and 1000 iterations are performed tocalculate the empirical size of the tests. The results of the simulation in Table 1 are summarized as follows:

Table 1Size of the tests

ρ {σeε, σeq, σεq}= {0.0, 0.0, 0.0} {σeε, σeq, σεq}= {0.5, 0.0, 0.6}δ = −0.5 δ = 0.0 δ = 0.5 δ = −0.5 δ = 0.0 δ = 0.5

−0.8 Sup LM 0.058 0.036 0.018 0.069 0.035 0.012RMBB 0.058 0.059 0.058 0.069 0.070 0.047

−0.4 Sup LM 0.042 0.026 0.016 0.059 0.043 0.013RMBB 0.049 0.054 0.051 0.061 0.065 0.046

0.0 Sup LM 0.026 0.034 0.028 0.037 0.022 0.030RMBB 0.046 0.060 0.049 0.057 0.038 0.058

0.4 Sup LM 0.016 0.029 0.047 0.019 0.036 0.056RMBB 0.050 0.054 0.053 0.048 0.064 0.054

0.8 Sup LM 0.004 0.031 0.126 0.003 0.041 0.126RMBB 0.042 0.058 0.086 0.034 0.062 0.083

Note: The critical values of the Sup LM test are 11.97 and 11.74 at 5% nominal size, respectively, according to [11].

Z. Yang et al. / Mathematics and Computers in Simulation 78 (2008) 507–513 511

• The size of the tests is not clearly influenced by the nonzero correlations σeε and σεq, which imply that there are nosignificant differences when the regressors and regressor–errors of the model is correlated or uncorrelated.

• The RMBB test has no substantial size changes relative to the nominal size under all consideration.• The empirical size of the Sup LM test is affected slightly by the serial correlation in the errors. If the parameter δ

is positive, the size of the Sup LM test increases as the parameter ρ becomes larger. If δ is negative, the size of theSup LM test is close to zero as ρ increases. Similarly, the size of the Sup LM test also is sensitive to δ when we fixthe parameter ρ. If ρ is negative, the empirical size tends to be smaller as δ grows. If ρ is positive, the empirical sizetends to be larger as δ grows. To sum up, when δ and ρ have same minus or positive sign, the Sup LM test keeps thesize properties reasonably well relative to the nominal size. Whereas the empirical size of the Sup LM test tends tobe smaller than the nominal size when δ and ρ have different sign.

The next set of experiments examines the power performance against local alternatives. We consider three cases:

• Case 1: Threshold effect only in the intercept term when λ1 = 0.0, i.e., we control the size of the threshold effect byvarying λ0.

• Case 2: Threshold effect only in the slope parameters when λ0 = 0.0, i.e., the threshold effect is controlled by λ1similarly.

• Case 3: Threshold effect is fixed with λ0 =λ1 = 0.1 and the sample sizes changes.

The threshold parameter γ is set at the mean of the variable xt. The parameter β0, β1, θ1, φ, B and b are maintainedas in the size simulation. The parameter values for the experiments using (11)–(15) are (δ, ρ) = (0.5, −0.4), (0.2, −0.4),(−0.5, −0.4), (0.5, 0.4), (−0.2, 0.4), (−0.5, 0.4). Nominal size is set at 5% and 1000 iterations are performed tocalculate the empirical power of the tests. The results are reported in Tables 2–4 when {σeε, σeq, σεq}= {0.5, 0.0,0.6}. When the regressors are strictly exogenous, i.e., {σeε, σeq, σεq}= {0.0, 0.0, 0.0}, the results for the tests are verysimilar and so are omitted.

We may summarize the simulation results in Tables 2–4 as follows:

• The power of the tests is increasing in the size of the threshold effect in Tables 2 and 3. The power performance ofthe tests is affected more significantly when the threshold effect exists in the slope parameter in contrast to in theintercept term. When only the slope parameter is subject to nonlinearity, the power of the RMBB test reaches 60%when λ1 = 0.2 in Table 3. However, when only the intercept term is subject to threshold cointegration, the power ofthe RMBB test reaches 60% when λ0 = 0.8 in Table 2, similarly to the Sup LM test. This is probably due to the factthat the convergence rates of the slope parameter estimates are faster than the intercept term in linear cointegratingregression. Consequently, threshold cointegration has also the same property as linear cointegration.

• The power of the tests improves as the sample size becomes larger in Table 4.

Table 2Power of the tests for Case 1

(δ, ρ) λ0 = 0.2 λ0 = 0.4 λ0 = 0.6 λ0 = 0.8 λ0 = 1.0

(0.5, −0.4) Sup LM 0.022 0.091 0.314 0.639 0.880RMBB 0.071 0.221 0.470 0.783 0.936

(0.2, −0.4) Sup LM 0.048 0.158 0.410 0.752 0.958RMBB 0.092 0.239 0.533 0.830 0.975

(−0.5, −0.4) Sup LM 0.061 0.205 0.395 0.664 0.863RMBB 0.067 0.209 0.399 0.681 0.868

(0.5, 0.4) Sup LM 0.094 0.157 0.370 0.619 0.853RMBB 0.080 0.168 0.374 0.626 0.853

(−0.2, 0.4) Sup LM 0.051 0.155 0.463 0.768 0.937RMBB 0.104 0.257 0.572 0.852 0.967

(−0.5, 0.4) Sup LM 0.035 0.135 0.355 0.667 0.908RMBB 0.098 0.249 0.538 0.809 0.972

Note: The critical value of the Sup LM test is 11.74 at 5% nominal size according to [11].

512 Z. Yang et al. / Mathematics and Computers in Simulation 78 (2008) 507–513

Table 3Power of the tests for Case 2

(δ, ρ) λ1 = 0.05 λ1 = 0.1 λ1 = 0.15 λ1 = 0.2 λ1 = 0.25

(0.5, −0.4) Sup LM 0.054 0.232 0.444 0.602 0.755RMBB 0.116 0.318 0.542 0.689 0.816

(0.2, −0.4) Sup LM 0.083 0.266 0.517 0.680 0.792RMBB 0.141 0.337 0.586 0.722 0.834

(−0.5, −0.4) Sup LM 0.094 0.270 0.462 0.608 0.733RMBB 0.106 0.278 0.484 0.622 0.739

(0.5, 0.4) Sup LM 0.443 0.777 0.924 0.968 0.989RMBB 0.454 0.778 0.931 0.968 0.990

(−0.2, 0.4) Sup LM 0.350 0.714 0.871 0.956 0.980RMBB 0.433 0.786 0.904 0.969 0.986

(−0.5, 0.4) Sup LM 0.294 0.678 0.866 0.945 0.980RMBB 0.386 0.761 0.910 0.962 0.989

Note: The critical value of the Sup LM test is 11.74 at 5% nominal size according to [11].

Table 4Power of the tests for Case 3

(δ, ρ) T = 50 T = 100 T = 150 T = 200 T = 250

(0.5, −0.4) Sup LM 0.046 0.240 0.482 0.654 0.745RMBB 0.122 0.359 0.554 0.721 0.811

(0.2, −0.4) Sup LM 0.065 0.292 0.490 0.676 0.826RMBB 0.118 0.372 0.547 0.719 0.859

(−0.5, −0.4) Sup LM 0.086 0.288 0.507 0.665 0.760RMBB 0.109 0.299 0.508 0.676 0.826

(0.5, 0.4) Sup LM 0.314 0.698 0.880 0.937 0.987RMBB 0.344 0.703 0.889 0.930 0.988

(−0.2, 0.4) Sup LM 0.305 0.724 0.886 0.974 0.992RMBB 0.412 0.786 0.927 0.979 0.993

(−0.5, 0.4) Sup LM 0.258 0.689 0.864 0.944 0.984RMBB 0.382 0.761 0.902 0.968 0.989

Note: The critical value of the Sup LM test is 11.74 at 5% nominal size according to [11].

• The RMBB test is designed in the hope that it has higher power when the alternative model is subject to thresholdcointegration. As expected, Tables 2–4 show that the power of the RMBB test are higher than that of the Sup LM testfor all alternative. Obviously, the power differences are significant when δ and ρ have different sign. For example,in Table 4, the RMBB test rejects 35.9% at the sample size T = 100 but the Sup LM test only rejects 24.0% when (δ,ρ) = (0.5, −0.4).

To assess robustness with respect to RMBB, we generate the simulated data by varying the parameters δ, ρ, φ, γ

and Σz separately. These results of the tests are very similar to the results presented in Tables 1–4. Detailed results arenot reported here to save space.

Overall, the RMBB test appears to be superior to the Sup LM test if the threshold effect in the alternative hypothesisis undistinguishable and/or the number of observations at the same time very small. When the Sup LM test is affectedsignificantly by the serial correlation in the regressor–errors, the RMBB test maintains the structure of serial correlation.In practice, of course, if one wishes to avoid the waste of the RMBB test in computer time, the Sup LM test generallyperforms faster than the RMBB test.

5. Conclusion

In this paper we have evaluated the RMBB testing procedure for detecting the linearity in context of threshold coin-tegrating regressions. The new method extends the Sup LM test proposed in [11] via the residual-based moving blockbootstrap calculating the asymptotic p-values. The theoretical justification for the RMBB is extremely complicated

Z. Yang et al. / Mathematics and Computers in Simulation 78 (2008) 507–513 513

and has not been investigated here. However, we investigate the small sample performance of the proposed methodthrough simulations and find that it often performs well with respect to the Sup LM test. So, despite the lack of thetheoretical basis, the results of simulations indicate that the new method is strongly suggested for use by practitioners.

Acknowledgments

The authors would like to thank the editor and the anonymous referees for useful suggestions for improving thispaper.

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