1

“NONPARAMETRIC NONLINEAR CAUSALITY TESTING WITH

STEPWISE MULTIVARIATE FILTERING”

Stelios D. Bekiros 1

European University Institute, Via Delle Fontanelle 10, 50014

San Domenico di Fiesole (FI), Italy

Cees G.H Diks

Center for Nonlinear Dynamics in Economics and Finance (CeNDEF)

Department of Quantitative Economics, University of Amsterdam, Roetersstraat 11,

1018 WB Amsterdam, The Netherlands

Abstract

The present study investigates the linear and nonlinear causal linkages in exchange, equity and derivatives markets. Specifically, in case of exchange markets, among six currencies denoted relative to United States dollar (USD), namely EUR, GBP, JPY, CHF, AUD and CAD. The prime motivation for choosing these exchange rates comes from the fact that they are the most liquid and widely traded, covering about 90% of total FX trading worldwide. The data spans two periods (PI: 3/20/1991 – 3/20/1997, PII: 3/20/2003 – 3/20/2007) before and after the structural break of the Asian financial crisis, which set a platform for departure for causality testing. In case of derivatives markets, between daily spot and futures prices for maturities of 1, 2, 3 and 4 months of West Texas Intermediate (WTI) crude oil. The data cover two periods October 1991-October 1999 and November 1999-October 2007, with the latter being significantly more turbulent. Finally in case of stock markets, among NIKKEI 255, FTSE100 and NYSE. The data spans four periods (PI: 3/20/1991 – 3/20/2007, PII: 3/20/1991 – 3/20/1997, PIII: 3/20/2003 – 3/20/2007, PIV: 3/20/2000 – 3/22/2004), namely the total period, a pre- and post-bubble period and an in-bubble period. The structural break point is again the Asian financial crisis.

We apply a new nonparametric test for Granger non-causality as well as the conventional linear Granger test on the return time series after controlling for cointegration. In addition to the traditional pairwise analysis, we test for causality while correcting for the effects of the other variables in a multi-variate formulation. To check if any of the observed causality is strictly nonlinear in nature, we also examine the nonlinear causal relationships of VAR or VECM filtered residuals. Finally, we investigate the hypothesis of nonlinear non-causality after controlling for conditional heteroskedasticity in the data using a GARCH-BEKK model. Our approach allows the entire variance-covariance structure of the time series interrelationship to be incorporated in order to explicitly capture the volatility spillover mechanism. In case of exchange markets, whilst the nonparametric test statistics are smaller in some cases, significant nonlinear causal linkages persisted even after GARCH filtering during both the pre- and post-Asian crisis period. This indicates that currency returns may exhibit asymmetries and statistically significant higher-order moments. Similar

1 E-mail addresses: Stelios.Bekiros@eui.eu , S.Bekiros@uva.nl (corresponding author); Tel.: +39-055-4685-698.

This study was conducted at the Center for Nonlinear Dynamics in Economics and Finance (CeNDEF), Department of Quantitative Economics, University of Amsterdam.

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results are presented for stock markets. Finally, in oil derivatives markets whereas the linear causal relationships disappear after VECM cointegration filtering, nonlinear causal linkages in some cases persist even after GARCH filtering in both periods. This indicates that spot and futures returns may exhibit asymmetries and statistically significant higher-order moments. Moreover, the results imply that if nonlinear effects are accounted for, neither market leads or lags the other consistently, videlicet the pattern of leads and lags changes over time.

JEL classification: C14; C51; F31; Q40

Keywords: Nonparametric nonlinear causality; VAR/VECM filtering; Cointegration; GARCH-BEKK filtering; foreign exchange; oil futures; stock markets.

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A. Foreign Exchange Markets

1A. Introduction

In the nineties the gradual abolition of capital controls and trade barriers provided

the foundation for liberalized and deregulated emerging financial markets. This less

restrictive environment created a systematic interrelationship between and within the stock

and currency markets. Specifically, foreign exchange markets have grown fiscally and

monetarily (i.e. by achieving lower inflation and/or interest rate differentials) more similar,

which generally led to lower exchange rate volatility and caused asymmetry in reactions

toward macroeconomic developments to significantly decrease (Laopodis, 1998). This growing

similarity may also reflect a temporary, or long-term, causal relationship between the major

currency markets. A rich empirical literature exists on the propagation mechanism

(spillovers) of US currency volatility across other foreign exchange markets and on “stylized

facts” like leptokurtosis and volatility clustering. These studies focus on the investigation of

the stochastic behavior of the US dollar, mostly employing the autoregressive conditional

heteroskedastic (ARCH) methodology of Engle (1982) (Engle and Bollerslev, 1986; Boothe

and Glassman, 1987; Hsieh, 1989; Baillie and Bollerslev, 1989, 1990; Engle et al., 1990). The

nature of the volatility transmission mechanism as well as the degree of price information

efficiency was investigated in the beginning of the higher integration of foreign exchange

rates vis-à-vis the US dollar (Hogan and Sharpe, 1984; Ito and Roley, 1987). Some empirical

evidence by Koutmos and Booth (1995) and Laopodis (1997) suggests that the size and/or

sign of an innovation in US exchange rates may seriously affect the extent of dependence and

spillovers across markets. Given the status of the USD as the anchor currency, it should be

interesting to examine its volatility transfers and more general the nature of causal linkages

with other major currencies. If these exist, it would suggest that on a global scale, news

originating in a specific market is fully and efficiently transmitted to other foreign markets,

thereby providing support to the “meteor shower” notion coined by Engle et al. (1990). This

term applies to a situation where volatility originating in one market flows over other

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markets, as opposed to the term “heat wave” also introduced by Engle et al. (1990) which

suggests that volatility would continue in the originating market the next day or that it is

country-specific.

The nature of causality in currency markets, i.e. linear or nonlinear is also a matter

for investigation. Ever since the influential work of Meese and Rogoff (1983) in which they

examined the failure of some linear exchange rate models, several more recent studies have

provided further evidence of the empirical failure of the linear models. The theoretical

extension of the linear exchange rate framework to nonlinear models has been growing in the

literature. According to Ma and Kanas (2000) these nonlinear extensions include the concept

of bubbles with self-fulfilling expectations (Blanchard and Watson, 1982; Froot and Obstfeld,

1991), target zone models (Krugman, 1991), models of micro-foundation of trading behaviour

(Krugman and Miller, 1993), nonlinear monetary policies (Flood and Isard, 1989), and noise

trading (Shiller, 1984; Kyle, 1985; Black, 1986; Frankel and Froot, 1986; Summers, 1986; De

Long et al., 1990). Empirical studies have mainly tested for nonlinearities due to target zones,

and have failed to support such nonlinearities (Flood et al., 1991; Lindberg and Soderlind,

1994). It still remains an open question whether these or other types of nonlinear

interdependencies across currency markets exist.

The recent empirical evidence is invariably based on the linear Granger causality

test (Granger, 1969). The conventional approach of testing for Granger causality is to assume

a parametric, linear time series model for the conditional mean. This approach is appealing,

since the test reduces to determining whether the lags of one variable enter into the equation

for another variable, although it requires the linearity assumption. Moreover, tests based on

residuals will be sensitive only to causality in the conditional mean while covariables may

influence the conditional distribution of the response in nonlinear ways. Additionally, Baek

and Brock (1992) noted that parametric linear Granger causality tests have low power

against certain nonlinear alternatives. In view of this, nonparametric techniques are

appealing because they place direct emphasis on prediction without imposing a linear

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functional form. Various nonparametric causality tests have been proposed in the literature.

The test by Hiemstra and Jones (1994) which is a modified version of the Baek and Brock

(1992) test is regarded as a test for a nonlinear dynamic relationship. This test can detect the

nonlinear Granger-causal relationship between variables by testing whether the past values

influence present and future values. However, Diks et. al. (2006) demonstrate that the

relationship tested by Hiemstra and Jones test is not generally compatible with Granger

causality, leading to the possibility of spurious rejections of the null hypothesis. As an

alternative we developed a new test statistic that overcomes these limitations.

The aim of the current paper is to test for the existence of both linear and nonlinear

causal relationships among six currencies denoted relative to United States dollar (USD),

namely Euro (EUR), Great Britain Pound (GBP), Japanese Yen (JPY), Swiss Frank (CHF),

Australian Dollar (AUD) and Canadian Dollar (CAD). The prime motivation for choosing

these exchange rates (also known as “FX majors”) comes from the fact that they are the most

liquid and widely traded exchange rates in the world. Trades involving “majors” make up

about 90% of total Forex trading worldwide. The data cover a pre- and a post-Asian crisis

period. The Asian crisis started with a 15 - 20% devaluation of Thailand’s Bath which took

place on July 2, 1997. Subsequently it was followed by devaluations of the Philippine Peso,

the Malaysian Ringgit, the Indonesian Rupiah and the Singaporean Dollar. In addition, the

currencies of South Korea and Taiwan suffered. Further in October, 1997 the Hong Kong

stock market collapsed with a 40% loss. In January 1998, the currencies of most South-East

Asian countries regained parts of the earlier losses. In that context, it is worth investigating

whether the time period after the 1997 Asian financial crisis may have changed the direction

and strength of the causal relationships among the currencies under study.

In the present study we apply a three-step empirical framework for examining

dynamic relationships among foreign exchange markets. First, we explore linear and

nonlinear dynamic linkages between exchange rates, applying both a parametric Granger

causality test and the new nonparametric causality test. Then, after filtering return series

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pairwise, as well as in a six-variate formulation for linear Vector AutoRegressive (VAR)

structure, the series of residuals are examined pairwise by the nonparametric causality test.

This step ensures that any remaining causality is strictly nonlinear in nature, as the VAR

model has already purged the residuals of linear causality. Finally, in the last step, we

investigate the hypothesis of nonlinear non-causality after controlling for conditional

heteroskedasticity in the data using a GARCH-BEKK model again, both pairwise and in a

six-variate representation. Our approach allows the entire variance-covariance structure of

the currency interrelationship to be incorporated. The empirical methodology employed with

the multivariate GARCH-BEKK model can help not only to understand the short-run

movements but also explicitly capture the volatility spillover mechanism. The method’s

advantage rests with its ability to examine all markets concurrently and paired, assuming

that spillovers are realizations of a process of international news affecting the examined

markets. Improved knowledge of the direction and nature of causality and interdependency

between the currency markets and consequently the degree of their integration will expand

the information set available to international portfolio managers, multinational corporations,

and policymakers for decision-making.

The remainder of the 1st part of the paper is organized as follows. Section 2A briefly

reviews the linear Granger causality framework and provides a description of the new

nonparametric test for nonlinear Granger causality. Section 3A describes the data used and

Section 4A presents the results. Section 5A concludes with suggestions for future research.

2A. The Nonparametric Causality Test

Granger (1969) non-causality has turned out to be a useful notion for characterizing

dependence relations between time series in economics and econometrics. Assume

that { }≥, ; 1t tX Y t are two scalar-valued strictly stationary time series. Intuitively { }tX is a

strictly Granger cause of { }tY if past and current values of X contain additional information

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on future values of Y that is not contained only in the past and current tY values. Let ,X tF and

,Y tF denote the information sets consisting of past observations of tX and tY up to and including

time t, and let ‘~’ denote equivalence in distribution. Then { }tX is a Granger cause of { }tY if,

for ≥ 1k :

( ) ( )+ +1 , ,,..., ,t t k X t Y tY Y F F ( )+ +1 ,,...,t t k X tY Y F (1)

In practice = 1k is used most often, in which case testing for Granger non-causality amounts

to comparing the one-step-ahead conditional distribution of { }tY with and without past and

current observed values of { }tX . A conventional approach of testing for Granger causality

among stationary time series is to assume a parametric, linear, time series model for the

conditional mean ( )( )+1 , ,,t X t Y tE Y F F . Then, causality can be tested by comparing the residuals

of a fitted autoregressive model of tY with those obtained by regressing tY on past values of

both { }tX and { }tY (Granger, 1969). Now, assume delay vectors

( )− +=ℓ

ℓ 1 ,...,X

t Xt tX XX and ( )− +=ℓ

ℓ 1 ,...,Y

t Yt tY YY , ( )≥ℓ ℓ, 1X Y . In practice the null hypothesis that past

observations of ℓ X

tX contain no additional information (beyond that in

ℓY

tY ) about +1tY is tested,

i.e.:

( )+ +ℓ ℓ ℓ

0 1 1: ; ~X Y Y

t t t t tH Y YX Y Y (2)

For a strictly stationary bivariate time series Eq. (2) comes down to a statement about the

invariant distribution of the ( + +ℓ ℓ 1X Y )-dimensional vector ( )= ℓ ℓ, ,X X

t tt tZW X Y where += 1t tZ Y .

To keep the notation compact, and to bring about the fact that the null hypothesis is a

statement about the invariant distribution of ( )ℓ ℓ, ,X X

t t tZX Y we drop the time index and also

= =ℓ ℓ 1X Y is assumed. Hence, under the null, the conditional distribution of Z given (X, Y) =

(x, y) is the same as that of Z given Y = y. Further, Eq. (2) can be restated in terms of ratios

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of joint distributions. Specifically, the joint probability density function , , ( , , )X Y Zf x y z and its

marginals must satisfy the following relationship:

= ⋅, , , ,( , , ) ( , ) ( , )

( ) ( ) ( )X Y Z X Y Y Z

Y Y Y

f x y z f x y f y z

f y f y f y (3)

This explicitly states that X and Z are independent conditionally on Y = y for each fixed value

of y. Diks and Panchenko (2006) show that this reformulated H0 implies:

≡ − = , , , ,( , , ) ( ) ( , ) ( , ) 0X Y Z Y X Y Y Zq E f X Y Z f Y f X Y f Y Z (4)

Let ˆ ( )W if W denote a local density estimator of a dW - variate random vector W at Wi defined by

ε − −

≠= − ∑1 ,

ˆ ( ) (2 ) ( 1)Wd WW i n ijj j if W n I where ( )ε= − <W

ij i j nI I W W with ⋅( )I the indicator function and

ε n the bandwidth, depending on the sample size n. Given this estimator, the test statistic is

a scaled sample version of q in Eq. (4):

ε Ζ

−= ⋅ −

−∑ , , , ,

1 ˆ ˆ ˆ ˆ( ) ( ( , , ) ( ) ( , ) ( , ))( 2)

n n X Z Y i i i Y i X Y i i Y i i

i

nT f X Z Y f Y f X Y f Y Z

n n (5)

For = =ℓ ℓ 1X Y , if βε β−

= > < <

1 1 0,

4 3n Cn C then it is proven under strong mixing that the

test statistic in Eq. (5) satisfies:

( )ε −→

( )(0,1)

Dn n

n

T qn N

S (6)

where →D

denotes convergence in distribution and Sn is an estimator of the asymptotic

variance of ⋅( )nT . We implement a one-tailed version of the test, rejecting the null hypothesis

if the left-hand-side of Eq. (6) is too large.

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3A. Data and preliminary analysis

The data consist of six time series of daily closing foreign exchange rates denoted

relative to United States dollar (USD), namely Euro (EUR), Great Britain Pound (GBP),

Japanese Yen (JPY), Swiss Frank (CHF), Australian Dollar (AUD) and Canadian Dollar

(CAD). The exact ratios represent EUR/USD, GBP/USD, USD/JPY, USD/CHF, AUD/USD

and USD/CAD respectively. These are the most liquid and widely traded currency pairs in

the world and make up about 90% of total Forex trading worldwide. The data cover two

periods, before and after a structural break, which sets a platform for departure for causality

tests. The first period PI spans from March 20, 1991 to March 20, 1997, denoting a pre-Asian

crisis period (1567 observations)2, and the second PII, a post-Asian crisis period, from March

20, 2003 to March 20, 2007 (1044 observations). Recall that the on-set of the Asian financial

crisis started with the devaluation of Thailand’s Bath which took place on July 2, 1997 and

followed by devaluations of the Philippine Peso, the Malaysian Ringgit, the Indonesian

Rupiah, the Singaporean Dollar and in October, 1997 the Hong Kong stock market collapsed

with a 40% loss. In January 1998, the currencies of South-East Asian countries began to

regain part of the earlier losses.

4A. Empirical results

The empirical methodology comprises three steps. In the first pre-filtering step, we

explore the linear and nonlinear dynamic linkages applying both a Granger causality test

and the nonparametric test on the raw log-differenced time series of the currencies. Then, we

implement both bi-variate and six-variate VAR filtering on the return series and the

residuals are examined pairwise by the test. Finally, we investigate the hypothesis of

nonlinear non-causality after controlling for conditional heteroskedasticity using a GARCH-

BEKK filter again pairwise as well as in a six-variate representation.

2 Euro values before 1999 refer to ECU (European Currency Unit). The ECU, a basket of EU currencies, ceased to

exist on 1 January 1999, when it was replaced by the Euro at a rate of 1:1. From that date, the currencies of the Euro-zone became sub-divisions of the Euro at irrevocably fixed rates of conversion.

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Descriptive statistics and Augmented Dickey-Fuller (ADF) test for the logarithmic

levels and log-daily returns are reported in Tables 1A and 2A. The lag lengths which are

consistently zero in all cases were selected using the Schwartz Information Criterion (SIC).

All the variables appear to be nonstationary in log-levels and stationary in log-returns based

on the reported p-values. The results are reported in the corresponding columns of Tables 3A

and 4. In order to overcome the difficulty of presenting large tables with numbers we use the

following simplifying notation: “ ** ” indicating that the corresponding p-value of a particular

causality test is smaller than 1% and “ * ” that the corresponding p-value of a test is in the

range 1-5%; Directional causalities will be denoted by the functional representation →.

[ Insert Table 1A here ]

[ Insert Table 2A here ]

4A.1 Causality testing on raw returns

The linear Granger causality test is usually constructed in the context of a reduced-

form vector autoregression (VAR). Let tY the vector of endogenous variables and ℓ number of

lags. Then the VAR( ℓ ) model is given as follows:

ε−

=

= +∑ℓ

1t s t s t

s

Y A Y (7)

where [ ]=ℓ1 ,...,t t tY YY the ×ℓ 1 vector of endogenous variables, sA the ×ℓ ℓ parameter matrices

and ε t the residual vector, for which εε ε ε =

= = ≠

∑' ( ) 0 , ( )

0 t t s

t sE E

t s. Specifically, in case of

two stationary time series { }tX and { }tY the bivariate VAR model is given by:

ε

ε

= + +=

= + +

ℓ ℓ

ℓ ℓ

,

,

( ) ( ) 1,2,...,

( ) ( )t t t X t

t t t Y t

X A X B Yt N

Y C X D Y (8)

where ℓ ℓ ℓ( ), ( ), ( )A B C and ℓ( )D are all polynomials in the lag operator with all roots outside the

unit circle. The error terms are separate i.i.d. processes with zero mean and constant

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variance. The test whether Y strictly Granger causes X is simply a test of the joint restriction

that all the coefficients of the lag polynomial ℓ( )B are zero, whilst similarly, a test of whether

X strictly Granger causes Y is a test regarding ℓ( )C . In each case, the null hypothesis of no

Granger causality is rejected if the exclusion restriction is rejected. If both ℓ( )B and ℓ( )C joint

tests for significance show that they are different from zero, the series are bi-causally related.

For each of the six raw return series linear causality testing was carried out using the

Granger’s test. The lag lengths of the VAR specification were selected using the Schwartz

Information Criterion (SIC), which are presented in Tables 3A, 4A in parentheses. For the

new test, in what follows we discuss results for lags = =ℓ ℓ 1X Y . To implement the test, the

constant C for the bandwidth ε n was set at 7.5, which is close to the value 8.0 for ARCH

processes suggested by Diks and Panchenko (2006). With the theoretical optimal rate

β = 2 7 this implies a bandwidth value of approximately 1, for both PI and PII. Selected

bandwidth values smaller (larger) than 1 resulted, in general, in larger (smaller) p-values.

The results presented in Tables 3A and 4A allow for the following observations: GBP,

JPY, CHF and AUD linearly cause EUR with small differences in PI and PII as well as

regarding the degree of statistical significance. None of the currencies Granger causes GBP.

Further, there is a strong causal relationship which affects JPY, CHF, AUD and CAD, with

JPY Granger causing the others. Finally, CHF presents a significant unidirectional linear

relationship CHF→CAD in PII. AUD and CAD appear to lack any causal relationship. We

next discuss the results for the nonparametric test. Interestingly starting form the latter

example, in period PI, there is now strong evidence of bi-directional nonlinear relationship

AUD↔CAD. Any of the previously detected linear causality among JPY, CHF, AUD and

CAD has vanished with the exception of a unidirectional nonlinear relationship JPY←CAD

in both periods. Again none of the currencies causes GBP. Only in PI, GBP causes CHF and

the same currency causes AUD in PII. Finally, strong nonlinear causality appears from

others currencies toward EUR but it no longer exists for AUD and CAD.

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[ Insert Table 3A here ]

[ Insert Table 4A here ]

4A.2 Causality testing on VAR-filtered residuals

The results from the previous step suggest that there are some significant and

persistent linear and nonlinear causal linkages between the FX rates. However, even though

we found nonlinear causality, the new test should be reapplied to filtered VAR-residuals to

ensure that any causality found is strictly nonlinear in nature. The lag lengths of the VAR

specification were based on the Schwartz Information Criterion (SIC).

The application of the nonparametric test on the VAR residuals points roughly

towards the preservation of the results reported for the raw returns in the pairwise and six-

variate implementation. Comparing the summary results in Table 3A, it is interesting to see

that they show identical significant causal nonlinear relationships, except for the absence

now of causality GBP→EUR in PII and the emergence of the unidirectional causality

CHF→JPY in PI. In the six-variate representation (Table 4A) the causal relationship

JPY→EUR has vanished and the same applies to the unidirectional linkage CAD→JPY. The

nature and source of the detected nonlinearities are different from that of the linear Granger

causality test and may also imply a temporary, or long-term, causal relationship between the

currencies. For instance, exchange rate volatility might induce nonlinear causality. Given

the status of the USD as the anchor currency, an innovation in US stock markets or an

increase in the interest rates from the Federal Reserve System in the USA etc., may

seriously affect the extent of dependency and volatility spillovers across currencies. The

nature of the volatility transmission mechanism can be investigated after controlling for

conditional heteroskedasticity using a GARCH-BEKK model, pairwise and in a six-variate

representation. This approach allows the entire variance-covariance structure of the

currency interrelationship to be incorporated.

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4A.3 Causality testing on GARCH-BEKK filtered VAR-residuals

The use of the new test on filtered data with a multivariate GARCH model enables

one to determine whether the posited model is sufficient to describe the relationship among

the series. If the statistical evidence of nonlinear Granger causality lies in the conditional

variances and covariances then it would be strongly reduced when the appropriate

multivariate GARCH model is fitted to the raw or linearly filtered data. However, failure to

accept the no-causality null hypothesis may also constitute evidence that the selected

multivariate GARCH model was incorrectly specified. This line of analysis is similar to the

use of the univariate BDS test on raw data and on GARCH models (Brock et al., 1996;

Brooks, 1996; Hsieh, 1989). Many GARCH models can be used for this purpose. In the

present study the GARCH-BEKK model of Engle and Kroner (1995) is used. The BEKK (p,q)

model is defined as:

ε ε− − −= =

′ ′ ′ ′= + +∑ ∑1 1

q p

tj j

jk t j t j jk jk t j jkH C C A A G H G , ε = 1/2t t tH v (9)

where jkC,A and jkG are (NxN) matrices and C is upper triangular. tH is the conditional

covariance matrix of { }ε t with ε −Φ 1| ~ (0 )t t t, H and −Φ 1t the information set at time t − 1. The

residuals are obtained by the whitening matrix transformation ε1/2 tH . Gourieroux (1997)

gives sufficient conditions for tA and tG in order to guarantee that tH is always positive

definite.

Tables 3A and 4A show results before and after GARCH-BEKK (1,1) filtering. The

order parameters were determined for the time series in terms of the minimal SIC. One

conclusion after the nonlinear causality testing is that in some cases the statistical

significance is weaker after filtering than before. These differences in statistical significance

indicate that the nonlinear causality is largely due to simple volatility effects. However, this

is not indicative of a general conclusion. Instead, significant nonlinear interdependencies

remain after the pairwise and six-variate GARCH-BEKK filtering revealing that volatility

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effects and spillovers are probably not the only ones inducing nonlinear causality. This of

course does not apply to all the pairs of FX rates but some main results can be drawn for

specific relationships. These are also depicted graphically in Figure 1A where strong

causality (“ ** ”) is denoted by a “double arrow”. In particular, the pairwise nonlinear

causality reveals the unidirectional linkages JPY→EUR, CHF→EUR, GBP→EUR and

AUD→CAD in PI, while in PII, JPY→EUR and CHF→EUR remain and AUD→ EUR is

added. Thus, there is strong evidence of the influence of the aforementioned currencies on

EUR though some in pre- and others in post-crisis period. This is perhaps an after-effect of

the independent and robust Euro zone behavior against the USD (Bénassy-Quéré et. al.,

2000; Wang et al., 2005). A potential increase/decrease in the US dollar volatility affects the

Euro zone currencies less than (and with a significant delay) the USD closest dependent

economies of Canada and Australia.

Now, incorporating the effect of all the currencies in a six-variate GARCH-BEKK

framework, the “whitened” residuals present different causal relationships than before.

Specifically, in PI two bidirectional linkages exist, namely GBP↔EUR and EUR↔CAD and

two unidirectional relationships CHF→EUR, CAD→JPY. In PII, besides the causal linkage

of GBP→EUR, the influence of CAD is strongly evident in forming the unidirectional

relationships, i.e., CAD→AUD, CAD→CHF, CAD→EUR and CAD→GBP. A possible

explanation is that Canadian market acting as a proxy for the neighboring US stock, bond or

currency market reacts faster to any innovation or volatility jumps (Yang, 2005; Bessler et.

al., 2003; Eun, 1989). Finally, in all results, third moment causality may be also a significant

factor of the remaining interdependence.

5A. Conclusions for currency markets

In the first part of the study we investigated the existence of linear and nonlinear

causal relationships among the most liquid and widely traded currencies in the world

denoted relative to United States dollar (USD), namely Euro (EUR), Great Britain Pound

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(GBP), Japanese Yen (JPY), Swiss Frank (CHF), Australian Dollar (AUD) and Canadian

Dollar (CAD). Several interesting conclusions have already emerged from this study. In

particular, it was shown that almost all FX markets considered here have become more

internationally integrated after the Asian financial crisis. Additionally, whilst the linear

causal relationships detected on the returns have disappeared after proper filtering,

nonlinear causal linkages in some cases emerged and more importantly persisted even after

GARCH filtering during both the pre- and post-Asian financial crisis period. For instance,

there is strong evidence of the influence of the other currencies on EUR and that Canadian

currency market substituting for US financial market nonlinearly causes the other currency

markets beyond the conventional spillover effect. These results, apart from offering a much

better understanding of the dynamic linear and nonlinear relationships underlying the major

currency markets, may have important implications for market efficiency. For instance, they

may be useful in future research to quantify the process of financial integration or may

influence the greater predictability of these markets.

An interesting subject for future research is the nature and source of the nonlinear

causal linkages. As was shown, volatility effects might partly induce nonlinear causality. The

fitted GARCH-BEKK models account for a large part of the nonlinearity in daily exchange

rates, but only in some cases. Perhaps other models of short-term exchange-rate

determination should be developed to explain this stylized fact. Moreover, currency returns

may exhibit statistically significant higher-order moments. This may explain why GARCH

filtering does not capture all the nonlinearity in currency returns. A similar result was

reported in Scheinkman, J., and LeBaron, (1989) for stock returns. Finally, alternative

parameterized asymmetric multivariate GARCH models could be employed. These models

would accommodate the asymmetric impact of unconditional shocks on the conditional

variances.

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B. Oil derivatives Markets

1B. Introduction

The role of futures markets in providing an efficient price discovery mechanism has

been an area of extensive empirical research. Several studies have dealt with the lead–lag

relationships between spot and futures prices of commodities with the objective of

investigating the issue of market efficiency. Garbade and Silber (1983) first presented a

model to examine the price discovery role of futures prices and the effect of arbitrage on price

changes in spot and futures markets of commodities. The Garbade-Silber model was applied

to the feeder cattle market by Oellermann et al. (1989) and to the live hog commodity market

by Schroeder and Goodwin (1991), while a similar study by Silvapulle and Moosa (1999)

examined the oil market. Bopp and Sitzer (1987) tested the hypothesis that futures prices

are good predictors of spot prices in the heating oil market, while Serletis and Banack (1990),

Cologni and Manera (2008) and Chen and Lin (2004) tested for market efficiency using

cointegration analysis. Crowder and Hamed (1993) and Sadorsky (2000) also used

cointegration to test the simple efficiency hypothesis and the arbitrage condition for crude oil

futures. Finally, Schwarz and Szakmary (1994) examined the price discovery process in the

markets of crude and heating oil.

The recent empirical evidence on causality is invariably based on the Granger test

(Granger, 1969). The conventional approach of testing for Granger causality is to assume a

parametric linear, time series model for the conditional mean. Although it requires the

linearity assumption this approach is appealing, since the test reduces to determining

whether the lags of one variable enter into the equation for another variable. Moreover, tests

based on residuals will be sensitive only to causality in the conditional mean while

covariables may influence the conditional distribution of the response in nonlinear ways.

Baek and Brock (1992) noted that parametric linear Granger causality tests have low power

against certain nonlinear alternatives.

17

Recent work has revealed that nonlinear structure indeed exists in spot and futures

returns. These nonlinearities are normally attributed to nonlinear transaction cost functions,

the role of noise traders, and to market microstructure effects (Abhyankar, 1996; Chen and

Lin, 2004; Silvapulle and Moosa, 1999). In view of this, nonparametric techniques are

appealing because they place direct emphasis on prediction without imposing a linear

functional form. Various nonparametric causality tests have been proposed in the literature.

The test by Hiemstra and Jones (1994), which is a modified version of the Baek and Brock

(1992) test, is regarded as a test for a nonlinear dynamic causal relationship between a pair

of variables. The Hiemstra and Jones test relaxes Baek and Brock’s assumption that the

time series to which the test is applied are mutually and individually independent and

identically distributed. Instead, it allows each series to display weak (or short-term)

temporal dependence. When applied to the residuals of vector autoregressions, the Hiemstra

and Jones test is intended to determine whether nonlinear dynamic relations exist between

variables by testing whether the past values influence present and future values. However,

Diks (2005, 2006) demonstrate that the relationship tested by Hiemstra and Jones test is not

generally compatible with Granger causality, leading to the possibility of spurious rejections

of the null hypothesis. As an alternative we developed a new test statistic that overcomes

these limitations.

Empirically it is important to take into account the possible effects of cointegration

on both linear and nonlinear Granger causality tests. Controlling for cointegration is

necessary because it affects the specification of the model used for causality testing. If the

series are cointegrated, then causality testing should be based on a Vector Error Correction

model (VECM) rather than an unrestricted VAR model (Engle and Granger, 1987). When

cointegration is not modelled, evidence may vary significantly towards detecting linear and

nonlinear causality between the predictor variables. Specifically, the absence of cointegration

could mean the violation of the necessary condition for the simple efficiency hypothesis

(Dwyer and Wallace, 1992), which implies that the futures price is not an unbiased predictor

18

of the spot price at maturity. This implies an absence of a long-run relationship between spot

and futures prices, as it was reported in works of Chowdhury (1991), Krehbiel and Adkins

(1993), Crodwer and Hamed (1993). Alternatively, based on the cost-of-carry relationship, a

failure to find cointegration may be attributed to the nonstationarity of the other components

of this relationship such as the interest rate or the convenience yield (Moosa and Al-

Loughani, 1995, and Moosa, 1996).

The aim of the present study is to test for the existence of linear and nonlinear causal

lead–lag relationships between spot and futures prices of West Texas Intermediate (WTI)

crude oil, which is used as an indicator of world oil prices and is the underlying commodity of

New York Mercantile Exchange's (NYMEX) oil futures contracts. We apply a three-step

empirical framework for examining dynamic relationships between spot and futures prices.

First, we explore nonlinear and linear dynamic linkages applying the new nonparametric

causality test, and after controlling for cointegration, a parametric linear Granger causality

test. In the second step, after filtering the return series using the properly specified VAR or

VECM model, the series of residuals are examined by the nonparametric causality test. In

addition to applying the usual bivariate VAR or VECM model to each pair of time series, we

also consider residuals of a full five-variate model to account for the possible effect of the

other variables. This step ensures that any remaining causality is strictly nonlinear in

nature, as the VAR or VECM model has already purged the residuals of linear dependence.

Finally, in the last step, we investigate the null hypothesis of nonlinear non-causality after

controlling for conditional heteroskedasticity in the data using a GARCH-BEKK model,

again both in a bivariate and in a five-variate representation. Our approach incorporates the

entire variance-covariance structure of the spot and future prices interrelationship. The

empirical methodology employed with the multivariate GARCH-BEKK model can not only

help to understand the short-run movements, but also explicitly capture the volatility

persistence mechanism. Improved knowledge of the direction and nature of causality and

interdependence between the spot and futures markets, and consequently the degree of their

19

integration, will expand the information set available to policymakers, international portfolio

managers and multinational corporations for decision-making.

The remainder of the 2nd part of the paper is organized as follows. Section 2B

provides an introduction to the theoretical considerations and existing empirical evidence on

the relationship between spot and futures prices. Section 3B briefly reviews the linear

Granger causality framework and provides a description of the new nonparametric test for

nonlinear Granger causality. Section 4B describes the data used and Section 5B presents the

results. Section 6B concludes with a summary and suggestions for future research.

2B. Theory and Evidence

In theory, since both futures and spot prices “reflect” the same aggregate value of the

underlying asset and considering that instantaneous arbitrage is possible, futures should

neither lead nor lag the spot price. However, the empirical evidence is diverse, although the

majority of studies indicate that futures influence spot prices but not vice versa. The usual

rationalization of this result is that the futures prices respond to new information more

quickly than spot prices, due to lower transaction costs and flexibility of short selling. With

reference to the oil market, if new information indicates that oil prices are likely to rise,

perhaps because of an OPEC decision to restrict production, or an imminent harsh winter, a

speculator has the choice of either buying crude oil futures or spot. Whilst spot purchases

require more initial outlay and may take longer to implement, futures transactions can be

implemented immediately by speculators without an interest in the physical commodity per

se and with little up-front cash. Moreover, hedgers who are interested for the physical

commodity and have storage constraints will buy futures contracts. Therefore, both hedgers

and speculators will react to the new information by preferring futures rather than spot

transactions. Spot prices will react with a lag because spot transactions cannot be executed

so quickly (Silvapulle and Moosa, 1999).

20

Furthermore, the price discovery mechanism, as illustrated by Garbade and Silber

(1983), supports the hypothesis that futures prices lead spot prices. Their study of seven

commodity markets indicated that, although futures markets lead spot markets, the latter do

not just echo the former. Futures trading can also facilitate the allocation of production and

consumption over time, particularly by providing a market scheme in inventory holdings

(Houthakker, 1992). In this case, if futures prices for late deliveries are above those for early

ones, delay of consumption becomes attractive and changes in futures prices result in

subsequent changes in spot prices. According to Newberry (1992) futures markets provide

opportunities for market manipulation by the better informed or larger at the expense of

other market participants. For example, it is profitable for the OPEC to intervene in the

futures market to influence the production decisions of its competitors in the spot market.

Finally, support for the hypothesis that causality runs from futures to spot prices can

also be found in the model of determination of futures prices proposed by Moosa and Al-

Loughani (1995). In their model the futures price is determined by arbitrageurs whose

demand depends on the difference between the arbitrage and actual futures price and by

speculators whose demand for futures contracts depends on the difference between the

expected spot and the actual futures price. The reference point in both cases is the futures

price and not the spot price (Silvapulle and Moosa, 1999).

There is also empirical evidence that spot prices lead futures prices. Specifically, in

the study of Moosa (1996) a spot price change triggers action from all kinds of market

participants and this subsequently changes the futures price. Initially, arbitrageurs will

react to the violation of the cost-of-carry condition3 and then speculators will revise their

expectation of the spot price and respond to the disparity between expected spot and futures

price. Similarly, speculators who act upon the expected futures price will revise their

expectation responding to the disparity between current and expected futures prices. Finally,

3 The relationship between futures and spot prices can be summarized as

−= ( )c y TF Se in terms of what is known as

the cost-of-carry. In that, y is the convenience yield (market’s expectations of the future availability of the commodity), T is the period to maturity, and c the cost-of-carry which equals the storage cost plus the cost of financing a commodity minus the income earned on the commodity (Hull, 2000).

21

in few studies causality is reported to be bi-directional. Kawaller et al. (1988) introduced the

principle that both spot and futures prices are affected by their past history, as well as by

current market information. They argue that potential lead - lag patterns dynamically

change as new information arrives. At any time point each may lead the other, as market

participants filter information relevant to their positions, which may be spot or futures. So

far, the hypothesis that futures prices lead spot prices is stronger in terms of empirical

evidence and more compelling. Thus, further empirical testing is required to infer on this

issue with respect to the crude oil market.

3B. Data and preliminary analysis

The data consist of time series of daily spot and futures prices for maturities of one,

two, three and four months of West Texas Intermediate (WTI), also known as Texas Light

Sweet, which is a type of crude oil used as a benchmark in oil pricing and the underlying

commodity of New York Mercantile Exchange's (NYMEX) oil futures contracts. The NYMEX

futures price for crude oil represents, on a per-barrel basis, the market-determined value of a

futures contract to either buy or sell 1,000 barrels of WTI at a specified time. The NYMEX

market provides important price information to buyers and sellers of crude oil around the

world, although relatively few NYMEX crude oil contracts are actually executed for physical

delivery.

The data cover two equally sampled periods, namely PI which spans October 21,

1991 to October 29, 1999 (2061 observations) and PII November 1, 1999 to October 30, 2007

(2061 observations). The segmentation of the sample corresponds roughly to the reduction in

OPEC spare capacity (defined as the difference between sustainable capacity and current

OPEC crude oil production) and to the increase in the United States’ gasoline consumption

and imports, both of which occurred after 1999. The effect of these events on price dynamics

is evident and it can be summarized in the accelerated rise of the average level of oil prices

and in the increased volatility (Regnier, 2006). Additionally, in PII markets witnessed more

22

occasional spikes in crude prices. The following notation is used: “WTI Spot” is the spot price

and “WTI F1”, “WTI F2”, “WTI F3” and “WTI F4” are the futures prices for maturities of one,

two, three and four months respectively. Descriptive statistics for WTI spot and futures log-

daily returns are reported in Table 1B. Specifically, the returns are defined as

−= − 1ln( ) ln( )t t tr P P , where Pt is the closing price on day t. The differences between the two

periods are quite evident in Table 1B where a significant increase in variance can be

observed as well as a higher dispersion of the returns distribution in Period II reflected in

the lower kurtosis. Additionally, Period II witnessed many occasional negative spikes as it

can be also inferred from the skewness. The results from testing nonstationarity are

presented in Table 2B.

[ Insert Table 1B here ]

[ Insert Table 2B here ]

Specifically, Table 2B reports the Augmented Dickey-Fuller (ADF) test for the logarithmic

levels and log-daily returns. The lag lengths which are consistently zero in all cases were

selected using the Schwartz Information Criterion (SIC). All the variables appear to be

nonstationary in log-levels and stationary in log-returns based on the reported p-values.

Table 1B also reports the correlation matrix at lag 0 (contemporaneous correlation) for both

periods. Significant sample cross-correlations are noted for spot and futures returns

indicating a high interrelationship between the two markets. However, since linear

correlations cannot be expected to fully capture the long-term dynamic linkages in a reliable

way, these results should be interpreted with caution. Consequently, what is needed is a

long-term causality analysis.

4B. Empirical results

The empirical methodology comprises three steps. In the first pre-filtering step, we

explore the linear and nonlinear dynamic linkages applying a Granger causality test based

on a VECM specification on the log-price levels and the new nonparametric test on the log-

23

differenced time series of the spot and futures prices. Then, we implement both pairwise and

five-variate VECM filtering on the log-price series, and the residuals are examined by the

new test. Finally, we investigate the hypothesis of nonlinear Granger non-causality after

controlling for conditional heteroskedasticity using a GARCH-BEKK filter, again in a

bivariate and a five-variate representation. Additionally, in the last two steps we

consistently apply a linear Granger causality test on the “whitened” residuals via a VAR

specification (i.e., no cointegration detected on the residuals) in order to investigate whether

any remaining causality is strictly nonlinear in nature or not.

The results are reported in the corresponding panels of Tables 3B and 4B. In order to

overcome the difficulty of presenting large tables with numbers we use the following

simplifying notation: “ ** ” indicating that the corresponding p-value of a particular causality

test is smaller than 1% and “ * ” that the corresponding p-value of a test is in the range 1-5%;

Directional causalities will be denoted by the functional representation →.

4B.1 Causality testing on raw data

The linear Granger causality test is usually constructed in the context of a reduced-

form vector autoregression (VAR). Let tY the vector of endogenous variables and ℓ number of

lags. Then the VAR( ℓ ) model is given as follows:

ε−

=

= +∑ℓ

1t s t s t

s

Y A Y (10)

where [ ]=ℓ1 ,...,t t tY YY the ×ℓ 1 vector of endogenous variables, sA the ×ℓ ℓ parameter matrices

and ε t the residual vector, for which εε ε ε =

= = ≠

∑' ( ) 0 , ( )

0 t t s

t sE E

t s. Specifically, in case of

two time series { }tX and{ }~ (1)tY I (for example log-prices) the bivariate VAR model is given

by:

24

ε

ε∆

∆

∆ = ∆ + ∆ +=

∆ = ∆ + ∆ +

ℓ ℓ

ℓ ℓ

,

,

( ) ( ) 1,2,...,

( ) ( )t t X t

t t Y t

X A X B Yt N

Y C X D Y (11)

where ∆ tX , ∆ tY the first differences of{ }tX ,{ }tY and ( ), ( ), ( ), ( )A B C Dℓ ℓ ℓ ℓ are all polynomials

in the lag operator with all roots outside the unit circle. The error terms are separate i.i.d.

processes with zero mean and constant variance. The test whether Y strictly Granger causes

X is simply a test of the joint restriction that all the coefficients of the lag polynomial ℓ( )B

are zero, whilst similarly, a test of whether X strictly Granger causes Y is a test

regarding ℓ( )C . In each case, the null hypothesis of no Granger causality is rejected if the

exclusion restriction is rejected. If both joint tests for significance show that ℓ( )B and ℓ( )C are

different from zero, the series are bi-causally related. However, in order to explore effects of

possible cointegration, a VAR in error correction form (Vector Error Correction Model-VECM)

is estimated using the methodology developed by Engle and Granger (1987) and expanded by

Johansen (1988) and Johansen and Juselius (1990). The bi-variate VECM model has the

following form:

[ ]

[ ]

λ ε

λ ε

−

∆

−

−

∆

−

∆ = − − ⋅ + ∆ + ∆ +

=

∆ = − − ⋅ + ∆ + ∆ +

ℓ ℓ

ℓ ℓ

1

1 ,

1

1

2 ,

1

1 ( ) ( )

1,2,...,

1 ( ) ( )

t

t t t X t

t

t

t t t Y t

t

YX p A X B Y

Xt N

YY p C X D Y

X

(12)

where [ ]λ−1 the cointegration vector and λ the cointegration coefficient. Thus, in case of

cointegrated time series, linear Granger causality should be investigated via the VECM

specification.

For the pairwise implementation the linear causality testing was carried out using

the Granger’s test based on a VECM model of the log-prices because all series were found to

be cointegrated. The lag lengths of the VECM specification were set using the Wald exclusion

criterion and for each pair in PI are (in parenthesis): WTI Spot - WTI F1 (3), WTI Spot - WTI

F2 (7), WTI Spot - WTI F3 (3) and WTI Spot - WTI F4 (3). Similarly, in period PII: WTI Spot

25

- WTI F1 (3), WTI Spot - WTI F2 (6), WTI Spot - WTI F3 (6) and WTI Spot - WTI F4 (4). In

addition, in PI for all pairs the Johansen test identified two (2) cointegrating vectors using

the trace statistic and in PII one (1) cointegrating vector. In case of the five-variate

implementation cointegration was also detected and in particular in PI the Johansen test

identified five (5) cointegrating vectors while in PII three (3). The number of lags for the 5x5

system in PI was eleven (11) and in PII nine (9).

For the non-parametric test, in what follows we discuss results for lags = =ℓ ℓ 1X Y .

Moreover, the test was applied directly on log-returns. To implement the test, the constant C

for the bandwidth ε n was set at 7.5, which is close to the value 8.0 for ARCH processes

suggested by Diks and Panchenko (2006). With the theoretical optimal rate β = 2 7 this

implies a bandwidth value of approximately one times the standard deviation of the time

series for both PI and PII. Selecting bandwidth values smaller (larger) than one times the

standard deviation resulted, in general, in larger (smaller) p-values.

The results presented in Tables 3B and 4B allow for the following remarks: In the

pairwise implementation of the linear Granger tests (VECM), strong bi-directional Granger

causality between spot and futures prices was detected in both periods with small differences

regarding the degree of statistical significance. An exception could be that WTI Spot and

WTI F4 present only unidirectional linear relationship WTI Spot→WTI F4. On the contrary,

the linear causality for the five-variate implementation appears to be uni-directional, mainly

in the more volatile and trending period PII and from spot to futures prices regardless of

maturity, providing evidence that spot tend to lead futures prices. This indicates that spot

prices can be useful in the prediction of futures prices under a 5x5 VECM formulation, i.e.,

accounting for the contributions of all maturities in the causality detection. Further, there is

a causal relationship in PI of WTI Spot→WTI F1, WTI F3→WTI Spot and WTI F4→WTI

Spot. The nonlinear causality test revealed a bi-directional nonlinear relationship in PI,

26

whereas in PII only uni-directional causality was detected from Spot to WTI F1, WTI F2 and

WTI F3 returns, excluding WTI F4.

[ Insert Table 3B here ]

[ Insert Table 4B here ]

4B.2 Causality testing on VECM-filtered residuals

The results from the previous step suggest that there are significant and persistent

linear and nonlinear causal linkages between the spot and futures prices. However, even

though we found nonlinear causality, the new test should be reapplied to the filtered VECM-

residuals to ensure that any causality found is strictly nonlinear in nature. The number of

lags and the number of cointegrating vectors identified for the VECM specification were

reported in the previous section. Moreover, a linear Granger test is applied to the filtered

residuals to conclude on a remaining linear structure even after filtering. The causality on

the filtered residuals was investigated with a VAR specification (the null of no cointegration

was not rejected) and the lags were determined using the Schwartz Information Criterion

(SIC).

The pairwise implementation of the Granger tests after VECM filtering shows that

the linear causal relationships detected on the raw returns have now disappeared. In fact

none of the previously mentioned causalities appear or any other new ones have emerged

after linear filtering. Similarly, no causal relationship could be detected after five-variate

filtering. The application of the nonlinear test on the VECM residuals, both in the bivariate

and five-variate implementation, points towards the preservation of the bi-directional

causality reported in PI on the raw log-returns. In PII the nonlinear causal relationships

WTI Spot→WTI F2, WTI Spot→WTI F3 have vanished, while WTI Spot→WTI F1 remains,

albeit statistically less significant. Interestingly, in the same period, a uni-directional

causality from futures to spot returns has now emerged for all maturities.

27

The nature and source of the detected nonlinearities are different from that of the

linear Granger causality and may also imply a temporary, or long-term, causal relationship

between the spot and futures markets. For instance, excess volatility in PII might have

induced nonlinear causality. The nature of the volatility transmission mechanism can be

investigated after controlling for conditional heteroskedasticity using a GARCH-BEKK

model, in a bi-variate and five-variate representation.

4B.3 Causality testing on GARCH-BEKK filtered VECM-residuals

The use of the non-parametric test on filtered data with a multivariate GARCH

model enables one to determine whether the posited model is sufficient to describe the

relationship among the series. If the statistical evidence of nonlinear Granger causality lies

in the conditional variances and covariances then it would be strongly reduced when the

appropriate multivariate GARCH model is fitted to the raw or linearly filtered data.

However, failure to accept the no-causality null hypothesis may also constitute evidence that

the selected multivariate GARCH model was incorrectly specified. This line of analysis is

similar to the use of the univariate BDS test on raw data and on GARCH models (Brock et

al., 1996; Brooks, 1996; Hsieh, 1989). Many GARCH models can be used for this purpose. In

the present study the GARCH-BEKK model of Engle and Kroner (1995) is used. The BEKK

(p,q) model is defined as:

ε ε− − −= =

′ ′ ′ ′= + +∑ ∑1 1

q p

tj j

jk t j t j jk jk t j jkH C C A A G H G , ε = 1/2t t tH v (13)

where jkC,A and jkG are (NxN) matrices and C is upper triangular. tH is the conditional

covariance matrix of { }ε t with ε −Φ 1| ~ (0 )t t t, H and −Φ 1t the information set at time t − 1. The

residuals are obtained by the whitening matrix transformation ε1/2 tH . Gourieroux (1997)

gives sufficient conditions for tA and tG in order to guarantee that tH is positive definite.

Tables 3B and 4B show results before and after GARCH-BEKK (1,1) filtering. The

order parameters were determined for the time series in terms of the minimal SIC. The

28

linear Granger causality interdependencies remain absent, exactly as after VECM filtering

in both periods and for both representations i.e., bivariate and five-variate. After the

nonlinear causality testing in some cases the statistical significance is weaker after filtering,

particularly in the five-variate GARCH-BEKK implementation. These differences in

statistical significance indicate that the nonlinear causality is partially due to simple

volatility effects. However, this is not indicative of a general conclusion. Instead, significant

nonlinear interdependencies remain after the bi-variate and five-variate GARCH-BEKK

filtering, revealing that volatility effects and spillovers are probably not the only ones

inducing nonlinear causality. This of course does not apply to all the pairs of spot and futures

returns but some main results can be drawn for specific relationships. These are also

depicted graphically in Figure 1B where strong causality (“**”) is denoted by a “double

arrow”.

In particular, the pairwise nonlinear causality reveals the bi-directional linkages

WTI Spot↔WTI F1, WTI Spot↔WTI F3 and WTI Spot↔WTI F4 in PI, and WTI Spot→WTI

F1 in PII. In fact, these relationships remain roughly unchanged from the previous VECM

filtering stage. Yet, there are two significant changes; the bi-directional causality WTI

Spot↔WTI F2 in PI is reduced to a weakened WTI Spot→WTI F2 linkage, and most

importantly in PII the uni-directional causality from futures to spot for all maturities, has

now vanished. Thus, there is strong evidence that the latter nonlinear causal relationship

can be attributed to second moment effects.

Now, incorporating the “contribution” of all variables in a five-variate GARCH-BEKK

framework, the whitened residuals present different causal relationships than before.

Specifically, in PI the bi-directional linkages WTI Spot↔WTI F1, WTI Spot↔WTI F3 and

WTI Spot↔WTI F4 are reduced to uni-directional and the WTI Spot→WTI F2 has

disappeared. It seems that the nonlinear causality from futures to spot returns which

persisted even after the five-variate VECM filtering was induced by conditional

heteroskedasticity and thus a five-variate and not a bi-variate GARCH-BEKK filtering of the

29

VECM-residuals is better at “capturing” the volatility transmission mechanism. Instead, in

PII the uni-directional linkages WTI F1→WTI Spot and WTI F4→WTI Spot were not

entirely removed as in the bi-variate GARCH-BEKK filtering of the VECM-residuals.

Eventually, in all results, third or higher-order causality may be a significant factor of the

remaining interdependence.

6B. Conclusions for oil derivatives markets

In the second part of the paper we investigated the existence of linear and nonlinear

causal relationships between the daily spot and futures prices for maturities of one, two,

three and four months of West Texas Intermediate (WTI), which is the underlying

commodity of New York Mercantile Exchange's (NYMEX) oil futures contracts. The data

covered two separate periods, namely PI: 10/21/1991-10/29/1999 and PII: 11/1/1999-

10/30/2007, with the latter being significantly more turbulent. The study contributed to the

literature on the lead–lag relationships between the spot and futures markets in several

ways. In particular, it was shown that the pairwise VECM modeling suggested a strong bi-

directional Granger causality between spot and futures prices in both periods, whereas the

five-variate implementation resulted in a uni-directional causal linkage from spot to futures

prices only in PII. This empirical evidence appears to be in contrast to the results of

Silvapulle and Moosa (1999) on the futures to spot prices uni-directional relationship.

Additionally, whilst the linear causal relationships have disappeared after the cointegration

filtering, nonlinear causal linkages in some cases were revealed and more importantly

persisted even after multivariate GARCH filtering during both periods. Interestingly, it was

shown that the five-variate implementation of the GARCH-BEKK filtering, as opposed to the

bi-variate, captured the volatility transmission mechanism more effectively and removed the

nonlinear causality due to second moment spillover effects.

Moreover, the results imply that if nonlinear effects are accounted for, neither

market leads or lags the other consistently, or in other words the pattern of leads and lags

30

changes over time. Given that causality can vary from one direction to the other at any point

in time, a finding of bi-directional causality over the sample period may be taken to imply a

changing pattern of leads and lags over time, providing support to the Kawaller et al. (1988)

hypothesis. According to that hypothesis, market participants filter information relevant to

their positions as new information arrives and, at any time point, spot may lead futures and

vice versa. Hence it can be safely concluded that, although in theory the futures market play

a bigger role in the price discovery process, the spot market also plays an important role in

this respect. The empirical evidence of a causal linkage from spot to futures prices can be

attributed to the sequence of actions taken from the three market participants following a

spot price change, as described in the Moosa model (1996). In that, arbitrageurs respond to

the violation of the cost-of-carry condition and then speculators acting upon the expected spot

price will revise their expectation. In the same way, speculators who act upon the expected

futures price will revise their expectation and react to the disparity between current and

expected futures prices. These conclusions, apart from offering a much better understanding

of the dynamic linear and nonlinear relationships underlying the crude oil spot and futures

markets, may have important implications for market efficiency. For instance, they may be

useful in future research to quantify the process of market integration or may influence the

greater predictability of these markets.

An interesting subject for future research is the nature and source of the nonlinear

causal linkages. As presented, volatility effects may partly account for nonlinear causality.

The GARCH-BEKK model partially captured the nonlinearity in daily spot and future

returns, but only in some cases. An explanation could be that spot and futures returns may

exhibit statistically significant higher-order moments. A similar result was reported by

Scheinkman and LeBaron, (1989) for stock returns. Alternatively, parameterized asymmetric

multivariate GARCH models could be employed in order to accommodate the asymmetric

impact of unconditional shocks on the conditional variances.

31

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37

Table 1A: Descriptive Statistics

Period I (3/20/1991-3/20/1997)

AUD/USD USD/CAD EUR/USD GBP/USD USD/CHF USD/JPY

Mean 1.47E-05 0.000113 -4.8E-05 -7.1E-05 1.6E-05 -6.9E-05

Standard Deviation 0.004764 0.002721 0.006599 0.006731 0.007806 0.006674

Sample Variance 2.27E-05 7.4E-06 4.35E-05 4.53E-05 6.09E-05 4.45E-05

Kurtosis 1.702 2.265 4.665 4.032 2.422 8.091

Skewness -0.303 0.070 -0.307 -0.274 -0.049 -0.693

Correlation Matrix

AUD/USD USD/CAD EUR/USD GBP/USD USD/CHF USD/JPY

AUD/USD 1

USD/CAD -0.166 1

EUR/USD -0.004 0.010 1

GBP/USD 0.088 -0.008 0.600 1

USD/CHF 0.023 -0.082 -0.659 -0.717 1

USD/JPY 0.108 -0.154 -0.345 -0.408 0.565 1

Period II (3/20/2003-3/20/2007)

AUD/USD USD/CAD EUR/USD GBP/USD USD/CHF USD/JPY

Mean 0.000294 -0.00023 0.000218 0.000216 -0.00013 -2.4E-05

Standard Deviation 0.006549 0.005063 0.005597 0.005368 0.006582 0.005423

Sample Variance 4.29E-05 2.56E-05 3.13E-05 2.88E-05 4.33E-05 2.94E-05

Kurtosis 1.168 0.520 0.399 0.570 0.765 1.016

Skewness -0.430 0.049 0.002 -0.077 -0.183 -0.174

Correlation Matrix

AUD/USD USD/CAD EUR/USD GBP/USD USD/CHF USD/JPY

AUD/USD 1

USD/CAD -0.538 1

EUR/USD 0.567 -0.407 1

GBP/USD 0.647 -0.443 0.647 1

USD/CHF -0.608 0.460 -0.752 -0.757 1

USD/JPY -0.514 0.319 -0.444 -0.487 0.544 1

38

Table 2A: Unit root tests

Variables ADF-statistic (PI) ADF-statistic (PII)

EUR/USD (0) 0.179 0.081

r EUR/USD (0) 0.000** 0.000**

GBP/USD (0) 0.207 0.242

r GBP/USD (0) 0.000** 0.000**

USD/JPY (0) 0.505 0.268

r USD/JPY (0) 0.000** 0.000**

USD/CHF (0) 0.491 0.070

r USD/CHF (0) 0.000** 0.000**

AUD/USD (0) 0.451 0.023

r AUD/USD (0) 0.000** 0.000**

USD/CAD (0) 0.595 0.204

r USD/CAD (0) 0.000** 0.000**

All variables are in logarithms and reported numbers are p-values. The number of lags in parenthesis is selected using the SIC. (**) denotes p-value corresponding to 99% confidence level.

39

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41

Figure 1A: Diagrammatical representation of directional causalities on GARCH-BEKK filtered VAR residuals (nonlinear causality) Notation: denote unidirectional and bi-directional causality corresponding to 5% ≤ p-value < 1%

denote unidirectional and bi-directional causality corresponding to p-value ≤ 1%

42

Table 1B: Descriptive Statistics

Period I (10/21/1991–10/29/1999)

WTI Spot WTI F1 WTI F2 WTI F3 WTI F4

Mean -0.00004 -0.00005 -0.00005 -0.00004 -0.00004

Standard Deviation 0.02060 0.01975 0.01702 0.01542 0.01432

Sample Variance 0.00042 0.00039 0.00029 0.00024 0.00021

Kurtosis 6.14 4.47 4.72 4.29 4.51

Skewness -0.01979 0.17027 0.16022 0.09616 0.10314

Correlation Matrix

WTI Spot WTI F1 WTI F2 WTI F3 WTI F4

WTI Spot 1

WTI F1 0.848 1

WTI F2 0.835 0.955 1

WTI F3 0.824 0.936 0.993 1

WTI F4 0.813 0.917 0.983 0.996 1

Period II (11/1/1999 – 10/30/2007)

WTI Spot WTI F1 WTI F2 WTI F3 WTI F4

Mean 0.00069 0.00069 0.00069 0.00068 0.00069

Standard Deviation 0.02388 0.02276 0.02083 0.01945 0.01879

Sample Variance 0.00057 0.00052 0.00043 0.00038 0.00035

Kurtosis 4.04 3.08 2.65 1.86 3.05

Skewness -0.58017 -0.56054 -0.44623 -0.35836 -0.44760 Correlation Matrix

WTI Spot WTI F1 WTI F2 WTI F3 WTI F4

WTI Spot 1

WTI F1 0.871 1

WTI F2 0.859 0.970 1

WTI F3 0.849 0.957 0.994 1

WTI F4 0.828 0.932 0.973 0.983 1

43

Table 2B: Unit root tests

Variables ADF-statistic (PI) ADF-statistic (PII)

WTI Spot (0) 0.039 0.943

r WTI Spot (0) 0.000 ** 0.000**

WTI F1(0) 0.044 0.967

r WTI F1 (0) 0.000** 0.000**

WTI F2 0) 0.070 0.974

r WTI F2 (0) 0.000** 0.000**

WTI F3 0) 0.085 0.978

r WTI F3 (0) 0.000** 0.000**

WTI F4 0) 0.089 0.979

r WTI F4 (0) 0.000** 0.000**

All variables are in logarithms and reported numbers for the augmented Dickey–Fuller test are p-values. The number of lags in parenthesis is selected using the SIC. (**) denotes p-value corresponding to 99% confidence level. PI: 10/21/1991–10/29/1999; PII: 11/1/1999 – 10/30/2007

44

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TI

F4 (

4).

In

peri

od

PI

for

all

pair

s th

e J

oh

an

sen

test

id

en

tifi

ed t

wo

(2)

coin

tegra

tin

g v

ect

ors

usi

ng t

he

trace

sta

tist

ic a

nd

in

P

II o

ne (

1)

coin

tegra

tin

g v

ecto

r again

for

all

pair

s. T

he

cau

sali

ty o

n t

he V

EC

M r

esi

du

als

was

invest

igate

d w

ith

a V

AR

sp

ecif

icati

on

(th

e n

ull

of

no c

oin

tegra

tion

was

not

reje

cted

) an

d t

he l

ags

wer

e d

ete

rmin

ed

usi

ng t

he S

chw

art

z In

form

ati

on

Cri

teri

on (

SIC

). T

he n

um

ber

of

lags

iden

tifi

ed

for

all

vari

able

s in

all

per

iod

s is

on

e (

1).

Th

e se

con

d

mom

en

t fi

lteri

ng w

as

per

form

ed

wit

h a

GA

RC

H-B

EK

K (

1,1

) m

od

el.

P

an

el

B: N

on

-Lin

ear

Cau

sali

ty

Th

e n

um

ber

of

lags

use

d f

or t

he

non

lin

ear

cau

sali

ty t

est

are

==

ℓℓ1

XY

. T

he

data

use

d a

re l

og-r

etu

rns.

Sin

ce t

he

log-l

evels

wer

e fo

un

d t

o be

coin

tegra

ted (

Pan

el

A)

the

non

lin

ear

cau

sali

ty w

as

inves

tigate

d o

n t

he

VE

CM

resi

du

als

. T

he

nu

mber

of

lags

(Wald

excl

usi

on c

rite

rion

) an

d t

he n

um

ber

of

coin

tegra

tin

g v

ecto

rs (

Joh

an

sen

tra

ce s

tati

stic

te

st)

iden

tifi

ed f

or

the V

EC

M s

pec

ific

ati

on a

re r

ep

ort

ed

in

Pan

el

A.

Th

e se

con

d m

om

en

t fi

lter

ing w

as

perf

orm

ed

wit

h a

GA

RC

H-B

EK

K (

1,1

) m

od

el.

45

Ta

ble

4B

: C

au

sali

ty r

esu

lts

(fiv

e-v

ari

ate

)

Va

ria

ble

s

Pa

ne

l A

: L

ine

ar G

ra

ng

er C

au

sa

lity

P

an

el

B:

No

n-L

inea

r C

au

sa

lity

Ra

w d

ata

V

EC

M

resid

ua

ls

GA

RC

H-B

EK

K

resid

ua

ls

Ra

w d

ata

V

EC

M

resid

ua

ls

GA

RC

H-B

EK

K

resid

ua

ls

X→

Y

Y→

X

X→

Y

Y→

X

X→

Y

Y→

X

X→

Y

Y→

X

X→

Y

Y→

X

X→

Y

Y→

X

X

Y

PI

PII

P

I P

II

PI

PII

P

I P

II

PI

PII

P

I P

II

PI

PII

P

I P

II

PI

PII

P

I P

II

PI

PII

P

I P

II

WT

I S

pot

WT

I F

1

* **

**

**

**

**

* **

*

* *

*

WT

I S

pot

WT

I F

2

**

**

**

**

**

**

*

WT

I S

pot

WT

I F

3

**

**

**

* **

**

**

**

**

WT

I S

pot

WT

I F

4

**

**

**

**

**

**

**

**

*

(*),

(**)

Den

ote

s p

-valu

e s

tati

stic

al

sign

ific

an

ce a

t 5%

an

d 1

%

level.

X→

Y:

r X d

oes

not

Gra

nger

Cau

se r

Y. P

I: 1

0/2

1/1

991–10/2

9/1

999;

PII

: 11/1

/1999 –

10/3

0/2

007.

Pan

el

A: L

inear

Gra

nger

Cau

sali

ty

Th

e 5x5 s

yst

em o

f th

e d

ata

(lo

g-l

evels

) w

as

fou

nd t

o be

coin

tegra

ted

an

d t

he

lag l

en

gth

of

the V

EC

M s

pec

ific

ati

on

was

set

usi

ng t

he

Wald

excl

usi

on c

rite

rion

. T

he

nu

mber

of

lags

iden

tifi

ed

in

PI

is e

leven

(11)

an

d i

n P

II i

s n

ine (

9).

In

peri

od

PI

the

Joh

an

sen

test

id

enti

fied

fiv

e (

5)

coin

tegra

tin

g v

ect

ors

usi

ng t

he t

race

sta

tist

ic a

nd

in

PII

th

ree (

3).

Th

e ca

usa

lity

on

th

e V

EC

M r

esid

uals

was

inves

tigate

d w

ith

a V

AR

sp

eci

fica

tion

(th

e n

ull

of

no c

oin

tegra

tion

was

not

reje

cted

) an

d t

he l

ags

were

dete

rmin

ed

usi

ng t

he S

chw

art

z In

form

ati

on C

rite

rion

(S

IC).

Th

e n

um

ber

of

lags

iden

tifi

ed

for

all

vari

able

s in

all

per

iod

s is

on

e (1

). T

he

seco

nd

mom

en

t fi

lter

ing w

as

per

form

ed

wit

h a

GA

RC

H-B

EK

K (

1,1

) m

odel.

P

an

el

B: N

on

-Lin

ear

Cau

sali

ty

Th

e n

um

ber

of

lags

use

d f

or

the

non

lin

ear

cau

sali

ty t

est

are

==

ℓℓ1

XY

. T

he d

ata

use

d a

re l

og-r

etu

rns.

Sin

ce t

he

5-v

ari

ate

syst

em

of

log-l

evels

was

fou

nd

to

be

coin

tegra

ted

(Pan

el

A)

the n

on

lin

ear

cau

sali

ty w

as

invest

igate

d o

n t

he V

EC

M r

esid

uals

. T

he n

um

ber

of

lags

(Wald

excl

usi

on

cri

teri

on)

an

d t

he n

um

ber

of

coin

tegra

tin

g v

ect

ors

(Joh

an

sen

tr

ace

sta

tist

ic t

est

) id

en

tifi

ed

for

th

e V

EC

M s

peci

fica

tion

are

rep

orte

d i

n P

an

el

A. T

he

seco

nd

mom

en

t fi

lteri

ng w

as

per

form

ed

wit

h a

GA

RC

H-B

EK

K (

1,1

) m

od

el.

46

Figure 1B: Diagrammatical representation of directional causalities on GARCH-BEKK filtered VECM residuals Notation: denote unidirectional and bi-directional causality corresponding to 5% ≤ p-value < 1%

denote unidirectional and bi-directional causality corresponding to p-value ≤ 1%