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Linear and nonlinear causalitytests in an LSTAR model: waveletdecomposition in a nonlinearenvironmentYushu Li a & Ghazi Shukur ba Center for Labor Market Policy Research (CAFO), Department ofEconomic and Statistics , Linnaeus University , Växjö, Swedenb Jönköping International Business School , Jönköping, SwedenPublished online: 27 Apr 2011.

To cite this article: Yushu Li & Ghazi Shukur (2011) Linear and nonlinear causality tests in an LSTARmodel: wavelet decomposition in a nonlinear environment, Journal of Statistical Computation andSimulation, 81:12, 1913-1925, DOI: 10.1080/00949655.2010.508163

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Journal of Statistical Computation and SimulationVol. 81, No. 12, December 2011, 1913–1925

Linear and nonlinear causality tests in an LSTAR model:wavelet decomposition in a nonlinear environment

Yushu Lia and Ghazi Shukurb*

aCenter for Labor Market Policy Research (CAFO), Department of Economic and Statistics, LinnaeusUniversity, Växjö, Sweden; bJönköping International Business School, Jönköping, Sweden

(Received 19 July 2009; final version received 10 July 2010 )

In this paper, we use simulated data to investigate the power of different causality tests in a two-dimensionalvector autoregressive (VAR) model. The data are presented in a nonlinear environment that is modelledusing a logistic smooth transition autoregressive function. We use both linear and nonlinear causality teststo investigate the unidirection causality relationship and compare the power of these tests. The lineartest is the commonly used Granger causality F test. The nonlinear test is a non-parametric test based onBaek and Brock [A general test for non-linear Granger causality: Bivariate model. Tech. Rep., Iowa StateUniversity and University of Wisconsin, Madison, WI, 1992] and Hiemstra and Jones [Testing for linearand non-linear Granger causality in the stock price–volume relation, J. Finance 49(5) (1994), pp. 1639–1664]. When implementing the nonlinear test, we use separately the original data, the linear VAR filteredresiduals, and the wavelet decomposed series based on wavelet multiresolution analysis. The VAR filteredresiduals and the wavelet decomposition series are used to extract the nonlinear structure of the originaldata. The simulation results show that the non-parametric test based on the wavelet decomposition series(which is a model-free approach) has the highest power to explore the causality relationship in nonlinearmodels.

Keywords: Granger causality; LSTAR model; wavelet multiresolution; Monte Carlo simulation

Mathematics Subject Classification 2000: 62J02, 62J05

1. Introduction

In vector autoregression (VAR) models, the causality or feedback relationship between subsets ofthe variables is always an attractive aspect for further analysis. The causality test is widely appliedto explore this kind of relationship, and the Granger-type test has been the most frequently used inprevious studies. The main idea behind the Granger causality test is to measure whether the pastinformation of a set of variables contains information on changes in another set of variables andhelps to predict them. It is carried out by checking if the variance of the prediction error of one setof variables at the present time is reduced by the incorporation of the past values from the other setof variables. The mean square error (MSE) is commonly used as a measurement for the prediction

*Corresponding author. Email: ghazi.shukur@vxu.se

ISSN 0094-9655 print/ISSN 1563-5163 online© 2011 Taylor & Francishttp://dx.doi.org/10.1080/00949655.2010.508163http://www.tandfonline.com

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error and the Granger causality test is concerned with the minimized MSE predictor, which is theunbiased conditional mean. However, in the Granger causality test, the conditional mean is alwaysset as a linear function of variables. Thus, when we discuss the Granger causality, it always impliesthat the past values of a set of variables help ‘linear predict’ another set of variables. Therefore,although the Granger causality test performs well in linear models, when the data show nonlinearproperties, then factors such as the structural change or heteroscedasticity will affect the forecasterror variance based on the linear model that influences the Granger causality test. In other words,the linear Granger causality test tests the significance of the linear coefficients of the set of pastvalues of variables in the model. If the underlying variables in the VAR system contain nonlinearrelationships and we use the test based on the linear model, the linear coefficients of the modelmay be insignificant, and as a result, the test cannot explore any causality relationship and willloose power in the corresponding nonlinear environment.

Numerous empirical studies have shown that many economic variables display nonlinear fea-tures and can only be modelled with nonlinear models, such as the business cycle, structuralswings, abrupt breaks or time-varying coefficients. In order to investigate the causality relation-ship in nonlinear dynamic models, a large number of studies have appeared in the field. Chenet al. [1] proposed a method that identifies nonlinear dependence according to locally linearapproximations and phase reconstructions, with a linear regression predictor employed for thelocal neighbourhood. The causality test checks if the time index of the neighbourhood points inthe reconstructed space helps in predicting the future dynamics. Later, Ancona et al. [2] pointedout that Chen et al. [1] required adequately high neighbourhood points in the local linear fit-ting. Instead, Ancona et al. [2] first proposed a statistically independent condition that shouldbe satisfied for their extended nonlinear Granger causality method. They also described a classof nonlinear models that satisfies this condition and proposed radial basis functions methods tochoose those models. In addition to the above-mentioned parametric methods, Bell et al. [3]proposed a non-parametric method that, using a back-fitting algorithm, first estimates underlyingsmooth functions to describe the relationship of the response and explanatory variables. Then, theauthors used F statistics based on the residual sum of squares from the restrictive non-causal andthe alternative causal related equations. Another type of non-parametric test was first proposed byBaek and Brock [4] and later modified by Hiemstra and Jones [5]. This non-parametric test canbe viewed as a test of non-causality in density. It is based on the correlation integral that is theestimator of spatial probabilities over time. This non-parametric test has the advantage of simpleimplementation with good size and power properties, and it is robust to the series being tested as itdoes not require a specified a priori model. It is therefore widely applied to exploit the nonlinearcausality relationships in nonlinear vector time series [6,7]. Here in this paper, we will use thistest to test the causality relationship in our nonlinear VAR model.

Among the many varieties of nonlinear models, the logistic smooth transition autoregressive(LSTAR) model allows nonlinear structures between the data regimes to be described by a logisticsmooth regime transition function. This is of particular interest in fields that contain mass ofunits, where even if the decisions leading unit structure break are made discretely, the aggregatedbehaviour shows smooth regime changes [8]. It is natural to extend the univariate LSTAR modelto a VAR system when the causality relationship among the variables varies with time smoothly,where the LSTAR model can be used to capture the time-dependent characteristic, and the seriesin the VAR system are nonlinear. It is also obvious that in this case, the traditional linear Grangercausality test may lose power in exploiting the time-dependent varying dynamics. Thus, in thispaper, we test the time-varying causality relationships that can be modelled by an LSTAR typeof model in a two variable VAR system, and we use both the linear Granger test and the non-parametric nonlinear test proposed by Baek and Brock [4] and Hiemstra and Jones [5] in orderto compare their performances. The investigation of the finite samples properties of the testshas been done by means of the Monte Carlo experiment, where we use 5000 replications for

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the size and power estimations, while for producing the critical values tables we use 10,000replications.

This paper is organized as follows. Section 2 presents the nonlinear bivariate VAR model for thecausality test. Section 3 introduces the simulated model and the data generating process (DGP)together with the size and power property result of the traditional linear causality test. In Section 4,we present and apply the non-parametric nonlinear test, while in Section 5, the non-parametrictest based on the wavelet decomposition series is introduced. The last section contains a shortconclusion.

2. The bivariate VAR model with LSTAR causality

In this paper, we are interested in causality tests in the bivariate VAR system, where the causalityrelationship from series {Yt } to series {Xt } is time-varying, and this time-dependent character iscaptured by an LSTAR model. Thus, the resulting VAR model is as follows:

xt = μ1 + α1xt−1 + F(t, γ, c)(β1 + β2yt−1) + ε1t ,

yt = μ2 + α2yt−1 + ε2t , (1)

where

F(t, γ, c) = 1

(1 + exp{−γ (t − c)}) − 1

2, ε1t , ε2t ∼ i.i.d. N(0, 1).

The transition function F(t, γ, c) implies that the causality relationship from {Yt } to {Xt } changessmoothly as time evolves under the restriction of β2 �= 0, γ �= 0. In F(t, γ, c), γ determines thespeed of transition from one extreme regime to another at time c, and the larger is γ , the steeperthe transition function will be, leading to a faster transition speed. In Figure 1, we set c fixedat halfway with γ = 20, 10, 5. Then, the smooth transition function Y is a bounded continuousnon-decreasing transition function with t from 1 to 44.

It is obvious that the past information of {Yt } will help to predict {Xt } under the restrictionof β2 �= 0, γ �= 0. Moreover, with the causality relationship changing over time according tothe logistic smooth transition function, {Xt } is also nonlinear with a smooth structural change.Thus, in this two variables VAR model, we have a nonlinear series {Xt } and a linear series {Yt },with the nonlinearity of {Xt } caused by the time-varying causality dependence on {Yt }. In thefollowing sections, we use both linear and nonlinear tests to explore the causality relationship inEquation (1).

0 10 20 30 40

0.4

0.2

0.0

–0.4

–0.2

t

y

Figure 1. Logistic smooth transition functions: γ = 20 (dashed-dotted line), γ = 10 (dotted line), γ = 5 (solid line).

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3. The DGP process and the linear causality test

Based on the procedure of the Granger linear test to test causality between {Xt }and {Yt }, we firstregress {Yt } on its own past values and lag values of {Xt } with lag length Lx, then regress {Xt } on itsown past value and lag values of {Yt } with lag length Ly. Then, F -test statistics is used to determinewhether the coefficients of the past values of Xt and Yt are zero, and insignificant coefficientsimply non-causality or feedback relationship. Here for the testing procedure based on data fromEquation (1), as we already know, the DGP, we set Ly = Lx = 1. To test the unidirectionalcausality relationship from {Yt } to {Xt }, the test procedure is based on the following linear VARmodel as follows:

xt = u1 + ϕ1xt−1 + γ1yt−1 + ε1t

yt = u2 + ϕ2yt−1 + γ2xt−1 + ε2t, ε1t , ε2t ∼ i.i.d. N(0, 1). (2)

Furthermore, the null hypothesis of non-causality is H0 : γ1 = 0. The test statistic for the linearcausality test is

Flinear = (SSRR − SSRu)/m

SSRu/(T − K),

where SSRR and SSRu are the sum of squared errors from the restricted and unrestricted models,respectively, m the number of restrictions, T the number of observations, and K the number ofparameters in the unrestricted model. Under H0 of non-causality, the test statistics Flinear willfollow an F(m, T − k) distribution.

The simulated data from Equation (1) are used to investigate the size and power propertiesof the causality test based on different parameter restrictions on the DGP. For both cases, weset μ1 = 0.02, μ2 = 0.03, α1 = 0.5, and α2 = 0.5 in the linear part of the model. In order toinvestigate the size property of the F -test statistic, we generate the DGP under the null hypothesisby setting β1 = β2 = 0, thus the simulated observations {Xt } and {Yt } are mainly two independentseries with non-causality. Here to produce the critical values for the size table, we set the numberof the Monte Carlo replication to 5000. This means that we simulate {Xt } and {Yt } under nullhypothesis 5000 times and carry out the Granger causality F test to find out the proportion ofthe rejection times. Table 1 presents the size property of the linear test for the finite sample sizes:T = 25, 50, 100, 250, 500 at the 5% significant level.

To judge the validity of the results, the estimated size of the test should lay between the approx-imate 95% confidence intervals of the nominal size 5%. With 5000 replications, the confidenceinterval for the estimated size is: 0.05 ± 1.96

√0.05(1 − 0.05)/5000 = (0.0441, 0.0559). Thus

from Table 1, we can see that at the 95% confidence level, the linear Granger causality F -teststatistic has an unbiased size. However, the result is expected when β1 = β2 = 0 because the DGPis a pure linear independent modelled VAR system. What we are more interested in here is thepower property of the linear test if the DGP is in a nonlinear unicausality relation. In this situation,we need a DGP that satisfies the unicausality relationship. Thus, in the linear part of Equation(1), we still set μ1 = 0.02, μ1 = 0.03, α1 = 0.5, and α2 = 0.5, but in the nonlinear part we setγ = 1, c = T/2, β1 = 3, and β2 = 3. With these fixed parameters set, there exists a unicausalityrelationship from {Yt } to {Xt }, and Figure 2 shows the structure of the simulated data {Xt } and{Yt } when T = 200.

Table 1. Size property for the linear Granger causality test.

T 25 50 100 250 500

Size (T ) 0.0495 0.0525 0.0546 0.0530 0.0512

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0 50 100 150 200

–50

5

t

Xt

0 50 100 150 200

–2–1

01

2

t

Yt

Figure 2. VAR(1) system with LSTAR nonlinear structure.

Table 2. Power properties for the linear Granger causality test.

T 25 50 100 250 500

Power (T ) 0.116 0.109 0.102 0.097 0.095

Figure 2 shows that {Xt } has a structural break at the mid-point of the sample making thewhole series show nonlinearity due to the structural change point c in the transition functionwhich is T/2. Thus, the causality relationship from {Yt } to {Xt } shows different patterns in thetwo regimes before and after time point c. The other variable {Yt } is still a linear time seriesover the entire sample, and it does not include any past information of {Xt }, which means thecausality relationship is just from {Yt } to {Xt }. In this section, we will use the above linear GrangerF test to investigate its power properties for the finite sample sizes. Here, with m = 1, K = 6and different sample sizes T , at the 5% significant level, we replicate the test procedure 5000times based on the DGP under the restriction of μ1 = 0.02, μ2 = 0.03, α1 = α2 = 0.5; γ = 1,c = T/2, β1 = β2 = 3; and we get the power table (Table 2) for the linear Granger test.

Table 2 shows that the linear Granger test has very low power in testing the causality relationshipwhen the lag value of {Yt } influences {Xt } according to a nonlinear structure. This is not surprisingas the nonlinear structure will result in an insignificant parameter γ1 in Equation (2), which reducesthe power of the linear Granger F test. It proves that although the linear Granger causality testmay perform well in testing the linear prediction among the variables, if there are some factorsthat cannot be modelled by a linear function, the causality relationship may not be exposed by thelinear causality test because the linear coefficients of the lag value is not significant in the linearmodel. As we can see in our case, when there exists a nonlinear causality relationship, the lineartest has very low power in exploring it, although one of the series {Yt } still shows linearity.

4. The non-parametric nonlinear test

For the STAR model, there is a parametric test for causality proposed by Dimitris and Miguel[9], where they used the Taylor expansion method to test the time-varying causality relationshipbetween variables. It first tests the linearity of the variables, and if linearity is rejected, a F -teststatistic is proposed after the estimation of the nonlinear parameters. Moreover, Li [10] proposeda heteroscedasticity-robust Wald test. This joint test of the threshold and causality leads to anon-standard asymptotic distribution because of the nuisance parameters. These two test methodsbasically follow the logic of the minimal variance error of prediction, as in the linear Grangertest. Here, we switch to the non-parametric test that was first proposed by Baek and Brock [4],and later modified by Hiemstra and Jones [5]. The main idea of the test is illustrated as follows.

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Consider two strictly stationary and weakly dependent time series {Xt } and {Yt } that satisfy theergodicity condition. Denote the m-length lead vector of {Xt } as Xm

t , the Lx-lag length vector of{Xt } as XLx

t−Lx , and the Ly-length of {Yt } as YLy

t−Ly . For given values of m, Lx, Ly ≥ 1, and e > 0,the hypothesis that {Yt } does not strictly Granger cause {Xt } will be expressed as

Pr(‖Xmt − Xm

s ‖ < e|‖XLxt−Lx − XLx

s−Lx‖ < e, ‖YLy

t−Ly − YLy

s−Ly‖ < e)

= Pr(‖Xmt − Xm

s ‖ < e|‖XLxt−Lx − XLx

s−Lx‖ < e), (3)

where Pr(·) measures the probability and ‖ · ‖ measures the vector distance, which is themaximum norm of the two vectors. Thus, Pr(‖Xm

t − Xms ‖ < e|‖XLx

t−Lx − XLxs−Lx‖ < e, ‖YLy

t−Ly −Y

Ly

s−Ly‖ < e) in Equation (3) is the conditional probability that the distance of two arbitrary m-length lead vectors Xm

t , Xms is within the distance e, given that the corresponding Lx- and Ly-length

lag vectors of {Xt } and {Yt } are within distance of e. Pr(‖Xmt − Xm

s ‖ < e|‖XLxt−Lx − XLx

s−Lx‖ < e)

is the conditional probability without taking the lag vector of {Yt } into consideration. The testbased on Equation (3) is a non-causality test that implies for the given criteria measurement whichis represented by the vector distance e, the Ly-length of the number of lag values of {Yt } doesnot help predict the future period values of {Xt }, given the Lx-length of the number of lag valuesof {Xt }. Generally speaking, this non-parametric causality test tests for non-causality in densitywhich is different from the Granger linear test, which tests non-causality in mean. Non-causalityin mean depends on the properties of the prior model, which needs stronger hypothesis. Thus, thisnon-parametric test is more robust to the data and can be used to test the causality relationshipwhen the data structure shows nonlinearity.

For further construction of the test statistic, we can rewrite Equation (3) into the followingequation:

Pr1(‖Xmt − Xm

s ‖ < e, ‖XLxt−Lx − XLx

s−Lx‖ < e, ‖YLy

t−Ly − YLy

s−Ly‖ < e)

Pr2(‖XLxt−Lx − XLx

s−Lx‖ < e, ‖YLy

t−Ly − YLy

s−Ly | < e)

= Pr3(‖Xmt − Xm

s ‖ < e, ‖XLxt−Lx − XLx

s−Lx‖ < e)

Pr4(‖XLxt−Lx − XLx

s−Lx‖ < e). (4)

Moreover, we can use the correlation integral estimators C1, C2, C3, C4 as the estimators ofprobabilities Pr1, Pr2, Pr3, Pr4, where when we set I (Z1, Z2, e) as the index function of vectorsZ1, Z2 within maximum norm distance e, we have

C1 = 2

n(n − 2)

∑

t<s

∑I (xm+Lx

t−Lx , xm+Lxs−Lx , e) ∗ I (y

Ly

t−Ly, yLy

s−Ly, e),

C3 = 2

n(n − 2)

∑

t<s

∑I (xm+Lx

t−Lx , xm+Lxs−Lx , e),

C2 = 2

n(n − 2)

∑

t<s

∑I (xLx

t−Lx, xLxs−Lx, e) ∗ I (y

Ly

t−Ly, yLy

s−Ly, e),

C4 = 2

n(n − 2)

∑

t<s

∑I (xLx

t−Lx, xLxs−Lx, e), (5)

and Equation (4) turns into :

C1(m + Lx, Ly, e)

C2(Lx, Ly, e)= C3(m + Lx, e)

C2(Lx, e). (6)

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Thus, the non-causality relationship implies that {Yt } and {Xt } should satisfy Equation (6), and anatural test statistic is

t = C1(m + Lx, Ly, e)

C2(Lx, Ly, e)− C3(m + Lx, e)

C2(Lx, e).

Under H0 of non-causality between {Yt } and {Xt }, the asymptotic distribution of the test statisticis

√T t → N(0, σ 2(m, Lx, Ly, e)). A detail derivation of the asymptotic distribution of the test

statistic and estimator of σ 2(m, Lx, Ly, e) can be found in Baek and Brock [4] and Hiemstra andJones [5].

Note that this non-parametric test is always implemented using the linear VAR model withfiltered residuals of the original data, while what we are testing is the pure ‘nonlinear predictive’power of the causality. The logic is that the linear predictive element can be extracted by thepre-chosen linear VAR model and the residuals will retain the nonlinear structure of the originalseries. However, this procedure requires caution as the filter processes depend heavily on the rightchoice of the linear VAR model. The reason is that a misspecified VAR model may withdraw ‘toomuch’ nonlinear information or ‘too little’ linear information of the dependence relationship inthe original data and leave the residual not informative enough to illustrate the nonlinear causalityrelationship, which might lead to lost power for this non-parametric test. Hiemstra and Jones [5]showed that the asymptotic distribution of the test statistics is the same when we use the originaldata or the VAR filtered residuals. As the asymptotic distribution needs further strict assumptionsto ensure good power and size properties, here we only discuss the test properties for the finitesamples. We now apply the test to both the original data and the linear VAR filtered residuals andthen compare the power properties. To carry out the tests in small samples, we first use the MonteCarlo experiment to generate the critical value table for the test in both situations using the DGPsystem that satisfies the null hypothesis as follows:

xt = 0.03 + 0.5xt−1 + ε1t

yt = 0.02 + 0.5yt−1 + ε2t, where ε1t , ε2t ∼ i.i.d. N(0, 1). (7)

The above DGP system simulates two independent series {Yt } and {Xt }. Thus, the critical valuesof the test statistic t = (C1(m + Lx, Ly, e)/C2(Lx, Ly, e)) − (C3(m + Lx, e)/C2(Lx, e)) areconstructed from the data generated by this DGP system which satisfies the null hypothesis ofnon-causality. Using a 10,000 Monte Carlo replications, we get the following critical value tablefor the finite sample sizes T = 25, 50, 100, 250, 500.

Table 3 shows that under H0, the test statistic in finite samples is distributed symmetricallyaround 0, which is the expected value based on the relationship from Equation (4). Moreover, theQQ plot (not included here to save space) shows that the distribution of the test approaches to thenormal distribution as the sample size grows. Since the size property of the test here is unbiased,we only need to examine the power property. When we perform the power investigation for thenon-parametric test, we use the same DGP as in the power investigation procedure in Section3, that is, the data are from the nonlinear LSTAR models in Equation (1) under the restrictions

Table 3. Critical values for the non-parametric test based on original data.

T 99% 97.5% 95% 90% 10% 5% 2.5% 1%

25 −0.9354 −0.7559 −0.6156 −0.4585 0.4249 0.5642 0.7002 0.868050 −0.7793 −0.6168 −0.5058 −0.3816 0.3324 0.4347 0.5342 0.6437100 −0.6254 −0.5150 −0.4309 −0.3296 0.2945 0.3880 0.4750 0.5820250 −0.5416 −0.4639 −0.3868 −0.2944 0.2600 0.3375 0.4108 0.4805500 −0.5129 −0.4251 −0.3569 0.2751 0.2552 0.3310 0.3987 0.4770

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γ = 1, c = T/2, β1 = 3, and β2 = 3 which leads to the unidirection causality from {Yt } to {Xt }.A significant positive test statistic suggests that {Yt } helps in predicting {Xt }, whereas a significantnegative value implies that {Yt } confounds the prediction of {Xt }, and hence we only use the right-tailed critical values. Based on the 5000 simulated values from the DGP with the unidirectionGranger causality from {Yt } to {Xt } and the critical values at the 5% significance level fromTable 3, we obtain the power of the non-parametric test based on the original data.

Table 4 shows that for T = 25, 50, and 100, the non-parametric test has low power. However,when T increases to 250 and 500, the power increases significantly. When comparing Table 4with Table 2, the non-parametric test shows an obvious power improvement, which proves thatthe non-parametric test performs better than the traditional linear Granger test when the serieshas a nonlinear structure.

We next apply the non-parametric test to the VAR model filtered residuals. A VAR modelwith one lag is considered as a reasonable pre-chosen model to extract the linear structure of theoriginal data, although the same procedure can be applied to higher order VAR models. The filterprocedure is as follows: we first use a linear VAR model with one lag to model the simulated datafrom the independent model based on Equation (7), and each equation in Equation (7) generatesa series of residuals. Then we rescale the two series of residuals and use them to construct thetest statistic. In the later step, when we carry out the test to investigate the power, we use thelinear filtered residuals as well. Then, based on the VAR model filtered residuals of data fromEquation (7), we get the following critical values table by Monte Carlo simulations with 10,000simulations.

Table 5 shows the same distribution characteristics as Table 3 with the approach to normaldistribution with expected value of zero and shorter interval for larger sample sizes. Furthermore,to investigate the power property, again we use the DGP from Equation (1), under the restrictionsof γ = 1, c = T/2, β1 = 3, and β2 = 3 which leads to the causality relationship from {Yt } to{Xt }, and using the rescaled residual after the linear VAR filtering, we get the following powertable based on 5000 Monte Carlo simulations (Table 6).

Table 4. Power properties for the non-parametric causality test based onoriginal data.

T 25 50 100 250 500

Power 0.142 0.228 0.358 0.6325 0.836

Table 5. Critical values for the non-parametric test based on residuals.

T 99% 97.5% 95% 90% 10% 5% 2.5% 1%

25 −0.7676 −0.6150 −0.5057 −0.3799 0.3702 0.4896 0.5948 0.718950 −0.5341 −0.4422 −0.3733 −0.2886 0.2793 0.3507 0.4244 0.5032100 −0.4233 −0.3592 −0.3039 −0.2361 0.2317 0.2950 0.3548 0.4256250 −0.3726 −0.3090 −0.2630 −0.2049 0.2013 0.2590 0.3117 0.3805500 −0.3575 −0.3051 −0.2514 −0.1975 0.1892 0.2424 0.2898 0.3504

Table 6. Power property for the non-parametric causality test based onresiduals.

T 25 50 100 250 500

Power 0.0645 0.1045 0.167 0.35 0.593

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For small samples T = 25 and 50, the test based on the residuals has even less power than thelinear Granger test. Although the power increases for larger sample sizes and exceeds the powerin the linear test, it is still lower than the test based on the original data. This outcome may bedue to that when the sample is too small, the nonlinear structure in the model is not obvious anda pre-chosen linear VAR model extracts all the dependency structure in the data, which leadsto a low power if we use the residuals for the test. It shows that when we use the residuals inthe test after the linear filter process, the pre-used linear VAR filter model may extract too muchinformation about the dynamics between the variables and destroy their dependency; thus, whenwe run the test based on the residuals, we get very weak power.

5. The non-parametric test based on the wavelet multiresolution

Section 4 presents the non-parametric nonlinear test based on the original data and the linear VARmodel filtered residuals, with the first one performing much better (in terms of higher power)than the latter one for the larger sample size, which reminds us that the residual-based methodsare sensitive to the specification of the pre-chosen linear filter model. However, if the residual isused to maintain the nonlinear relationship of the original data, we can consider another alterna-tive method of achieving this. This alternate method captures the nonlinear characteristics of theoriginal data using the wavelet multiresolution analysis (MRA). This frequency decompositionmethod has been widely used after its theoretical development in the 1980s Mallat [11]. In signalsmoothing and spectrum analysis, Chiann and Morettin [12] showed how the wavelet capturessignals in different scales by the wavelet spectrum decomposition. In economics, Schleicher [13]also mentioned that the wavelet method can catch the macroscopic behaviour as well as the micro-scopic detail in economic areas. In perspective of nonlinear models, the wavelet decompositionis an efficient method due to its ability of localizing the non-stationary structure which dependson time. Moreover, it is of particular use in the identification of nonlinear models, see Coca andBillings [14] who used wavelet multiresolution to process the nonlinear system in NARMAXmodels. Chang and Shi [15] also used this methodology to identify time-varying properties ofhysteretic structures, and a comprehensive elaborate of MRA can be found in Carl [16]. In ourcircumstance, we expect the low-frequency wavelet smooth based on MRA to capture the maintrend of the data, which is the nonlinear LSTAR structure in the VAR system, leading to a betterperformance of the test. We use the maximal overlap discrete wavelet transform (MODWT) as ithas no restriction on the sample size. We first start with a brief introduction of the MODWT andwavelet multiresolution, and more details are to be found in the cited references.

For an N -dimensional vector X = {Xt, t = 0, . . . , N − 1}, the level J MODWT of X con-tains J + 1 vectors W1, . . . ,WJ , VJ with the wavelet coefficients Wj corresponding to changeson scale τj = 2j−1, while the wavelet scaling coefficients VJ corresponds to the average onscale λJ = 2j . The N -dimensional vectors Wj and VJ are computed by Wj = wjX, vJ = vJ X,where wj and vJ are N × N matrices. Then, the MODWT-based MRA of X is defined as:X = ∑J

j=1 wTj Wj + vT

J VJ = ∑Jj=1 Dj + SJ , where Dj is the j th level MODWT details con-

taining the microscopic detail of X and SJ is the J th level MODWT smooth containing landscapecharacteristics of X. Thus, the MODWT transformation and multiresolution can be viewed as aband-pass filter process of X, and based on different transformation matrices wj and vJ , we havedifferent choices of filters. For more information about the MODWT method and how to choosea suitable filter, we refer to Vidakovic [17], Percival and Walden [18], and Gençay et al. [19].An important issue now is how to choose the wavelet filter. A central factor to use a particularwavelet is to match the characteristics of the series under consideration. The number in the nameof the wavelet indicates the width of the filter. In general, the wavelets with small L are narrower

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and less smooth, whereas wavelets with large L are relatively wide and smooth. However, in thispaper, we use the wavelet least asymmetric with L = 8, i.e. LA(8), since it has better band-passcharacteristics.

Thus, by MRA, the wavelet smooth SJ can capture the low-frequency trend of the originaldata, which we use to capture the nonlinear and linear structures of {Xt } and {Yt } for the non-parametric test, instead of using theVAR filtered residuals. Here, and to simplify the procedure, wesimply let J = 1 and get the low-frequency wavelet smoothes {S1t } and {S2t } of the original timeseries {Xt } and {Yt }, respectively: S1t = ∑L−1

l=0 g◦l V1t+l mod T , S2t = ∑L−1

l=0 g◦l V2t+l mod T with the

MODWT scale coefficients V1t = ∑L−1l=0 glXt−l mod T , V2t = ∑L−1

l=0 glYt−l mod T , while {gl} is thescaling filter satisfying

∑l gl = 1,

∑l g

2l = 1/2,

∑l gl gl+2∗n = 0 and {g◦

l } is the periodization of{gl} to circular filters of length N . Since {gl} and {g◦

l } are the low-band-pass filters, the resultingwavelet smooth maintains the low-frequency structure of the original data which is the nonlinearstructure of {Xt } and the linear character of {Yt } in our case. Thus, the corresponding waveletsmoothes of the data presented in Figure 2 are now shown in Figure 3.

Figure 3 shows that the wavelet smoothes (which are the reconstructed signals from the scalecoefficients of each series) are smoother compared with the original data in Figure 2 with a clearernonlinear structural in {Xt }. Then, if we use the wavelet smooth, we can keep the main structureof the original data, regardless of whether it is linear or nonlinear. Especially if the original datahave a nonlinear structure compared with the linear VAR filtered residuals, the wavelet smoothwill maintain the nonlinear information as well as the dependence relationship between {Xt } and{Yt }. Now to carry out the non-parametric causality test, when creating the critical value table,we first simulate the same DGP process as in the last sections (i.e. Equation (7)), which satisfiesthe null hypothesis, that is μ1 = 0.02, μ2 = 0.03, α1 = α2 = 0.5 and β1 = β2 = 0, but insteadof using the original data or the linear filtered residuals, we use the wavelet smoothes S1t and S2t

in the test.The test statistic is now:

ts = C1(m + Ls1, Ls2, e)

C2(Ls1, Ls2, e)− C3(m + Ls1, e)

C2(Ls1, e),

where Ls1 and Ls2 are the lag indices of S1t and S2t , respectively, corresponding to Lx and Ly asthe lag indices of {Xt } and {Yt } in the original test statistic in Section 4. Then based on 10,000Monte Carlo replications, we get the critical values table for the test based on S1t and S2t .

Table 7 also shows symmetric, zero-centred distribution which approaches the normal distribu-tion as the sample size grows. This implies that the asymptotic distribution of the test statistic basedon the wavelet smooth is still a normal distribution. This is obvious because here in the waveletenvironment, we use the same form of the non-parametric test statistic t = C1/C2 − C3/C4 withthe only difference being that we, instead of the original {Xt } and {Yt }, use {S1t } and {S2t } in Ci .

0 50 100 150 200

–50

5

t

S1t

0 50 100 150 200

–2–1

01

2

t

S2t

Figure 3. Wavelet smoothes of the VAR(1) system with LSTAR nonlinear structure.

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Table 7. Critical values for the non-parametric test based on wavelet smooth.

T 99% 97.5% 95% 90% 10% 5% 2.5% 1%

25 −0.6478 −0.5313 −0.4226 −0.3103 0.3657 0.4815 0.5803 0.701950 −0.5425 −0.4473 −0.3593 −0.2696 0.2876 0.3767 0.4558 0.5396100 −0.4574 −0.3734 −0.3115 −0.2378 0.2352 0.3099 0.3889 0.4648250 −0.3844 −0.3087 −0.2590 −0.1983 0.2017 0.2599 0.3126 0.3894500 −0.3501 −0.2964 −0.2476 −0.1952 0.1855 0.2460 0.2914 0.3599

Thus the test statistic is

ts = C1(m + Ls1, Ls2, e)

C2(Ls1, Ls2, e)− C3(m + Ls1, e)

C2(Ls1, e),

where its asymptotic distribution satisfies√

T ts → N(0, σ 2(m, Ls1, Ls2, e)). A detailed expla-nation of σ 2(m, Ls1, Ls2, e) can be found in Baek and Brock [4] and Hiemstra and Jones [5].

For the finite sample size, and to investigate the power property, we still use the same DGPfrom Equation (1) under the restrictions of γ = 1, c = T/2, β1 = 3, and β2 = 3 to generate {Xt }and {Yt } which satisfies the nonlinear causality relationship, and using their wavelet smoothes S1t

and S2t to carry out the test. Based on the critical value at the 5% significant level in Table 7 and2000 Monte Carlo replications, we get the power property of the non-parametric test in waveletenvironment (Table 8).

Although the power is still low when the sample size T = 25, the power for other sample sizesare much higher compared with Tables 4 and 6. When T goes to 250, the power approachesalmost one. Thus based on the power table, the wavelet smooth performs best in extracting thenonlinear characteristics and the dependency relationship of the original data. Moreover, thewavelet method has the advantage of being easy to perform as it saves us the effort of choosingthe correct specification of the VAR linear model compared with the residual-based test. We canalso asses its asymptotic distribution as we only need to put the wavelet smooth, instead of theoriginal data, into the test statistic, and we still retain the asymptotic normal distribution with thezero mean value but with a different variance.

Note that the wavelet smooth is a product of low pass filter which is a kind of weighted movingaverage. Hence, and for the sake of comparison, we also use other smoothing methods, e.g. the localweighted scatter plot smoothing (LOWESS) and the Hodrick–Prescott (HP) filters that are knownfor their robustness to the nonlinear structure of LSTAR series. Unfortunately, when applyingthese two methods, we come across two main problems. First, the simulation results show that thepower properties of the test are very sensitive to the smooth restriction criterion, such as the spanparameter in the LOWESS and the multiplier parameter (λ) in the HP filter, and that ‘misspecified’smooth parameters lead to very low power as well. However, in empirical implications, as wemay know scarce information of the original data, it is always not easy to choose a reasonablesmooth parameter at the first sight and the result may be seriously distorted. Second, as it is noteasy to get an explicate representation of the smoothed series from the LOWESS and HP filters,we cannot derive further asymptotic distributions of the test statistic as when we use the wavelet.In the wavelet case, we find that it is quite straightforward to just choose the first level smooth and

Table 8. Power properties for the non-parametric causality test basedon a wavelet smooth.

T 25 50 100 250 500

Power 0.166 0.358 0.657 0.975 1

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capture the main nonlinear structure as well as maintaining the causality information of the data.Moreover, we also get a power property which actually overmatch all the results from LOWESSand HP filters (The simulation results are not included here to save space, but they are availablefrom the authors on request). In addition, the explicit representation of the wavelet smooth St

leads to the possibility for derivation of the asymptotic distributions of the smoothed series.

6. Conclusion

In this paper, we investigated the power properties of causality tests when the causality relationshipis characterized by an LSTAR model. We compared power values from the linear causality test,the non-parametric test based on the original data, the VAR filtered residuals, and the waveletsmooth based on MRA for which the best power property is obtained. The low power propertyin the residual-based test might be due to that the linear model we used for filtering extract toomuch information and thus destroy the dependency relationship that is supposed to be retained inthe residuals. However, all the tests here use the data for the whole period for testing, althoughthere are data breaks and the causality relationship may vary with time. Li [10] proposed a testthat can test causality before and after breaks, but with a more complicated procedure. Thus, forthe VAR model with LSTAR nonlinear characteristics, the non-parametric test can be used for afirst examination of the causality relationship between the variables. Furthermore, when applyingthe non-parametric test, the simulation results show that when we try to identify the nonlinearcharacteristics, the VAR filtered residuals should be used carefully, while on the other hand thenon-parametric test based on wavelet multiresolution can capture the nonlinear relationship wellwhich leads to the best power property.

Acknowledgements

The authors are grateful for the valuable comments raised by the anonymous reviewers.

References

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[4] E.G. Baek and W.A. Brock, A general test for non-linear Granger causality: Bivariate model. Tech. Rep., Iowa StateUniversity and University of Wisconsin, Madison, WI, 1992.

[5] C. Hiemstra and J. Jones, Testing for linear and non-linear Granger causality in the stock price–volume relation, J.Finance 49 (5) (1994), pp. 1639–1664.

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[9] K.C. Dimitris and A.L. Miguel, Testing for Granger (non-)causality in a time varying coefficient VAR model, J.Forecast. 27(4) (2008), pp. 293–303.

[10] J. Li, Testing Granger causality in the presence of threshold effects, Int. J. Forecast. 22 (2006), pp. 771–780.[11] S.G. Mallat, A theory for multiresolution signal decomposition: The wavelet representation, Pattern Anal. Mach.

Intell. IEEE Trans. 11(7) (1989), pp. 674–693.[12] C. Chiann and P.A. Morettin, A wavelet analysis for time series, J. Nonparametr. Stat. 10 (1998), pp. 1–46.[13] C. Schleicher, An introduction to wavelets for economists, Working Paper, Bank of Canada, Series: 2002–3, 2002.

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