Modeling nonlinear Granger causality between the oil ...

Economic Modelling 29 (2012) 1505–1514

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Modeling nonlinear Granger causality between the oil price and U.S. dollar: A wavelet based approach François Benhmad ⁎ LAMETA, Montpellier University, Espace Richter, Avenue de la Mer, 34054 Montpellier Cedex1, France

a r t i c l e

i n f o

Article history: Accepted 9 January 2012 JEL classification: C22 E31 F31 Q43 Keywords: Granger causality Wavelets Real oil price Real effective U.S. Dollar exchange rate

a b s t r a c t In this paper, we use a wavelet approach to study the linear and nonlinear Granger causality between the real oil price and the real effective U.S. Dollar exchange rate. Instead of analyzing the time series at their original level, as it is usually done, we first decompose the two macroeconomic variables at various scales of resolution using wavelet decomposition and then we study the relationships among the decomposed series on a scale by scale basis. A major finding of this paper is that the linear and nonlinear causal relationships between the real oil price and the real effective U.S. Dollar exchange rate vary over frequency bands as it depends on the time scales. Indeed, there is a strong bidirectional causal relationship between the real oil price and the real dollar exchange rate for large time horizons, i.e. corresponding to fundamentalist traders, especially fund managers and institutional investors. But, for the first frequency band which corresponds to a class of traders whom investment horizon is about 3-months and whom trading is principally speculative (noise traders), the causality runs only from the real oil prices to real effective U.S dollar exchange rate. © 2012 Elsevier B.V. All rights reserved.

1. Introduction The issue of the relationship between the real oil price and the U.S. dollar real exchange rate has led to a large literature because they are leading economic variables. On the empirical front, numerous facts have been reported about this relationship. Since U.S. dollar is the major invoicing currency of international crude oil markets, the fluctuations in U.S. dollar exchange rate is believed to underlie the volatility of crude oil price. The crude oil exporting countries may get anxious by a weak dollar. Conversely, the crude oil importing countries will be adversely affected by an overvalued U.S. dollar appreciation: it signifies extra expenditure increase of oil importing countries (such European Union), whereas its depreciation renders a great challenge for oil sales income of main exporting countries (such as OPEC). The econometric results about the causality between the real oil price and the U.S. dollar real exchange rate are mixed. On the one hand, the existing empirical literature in this area generally finds that causality tests indicate that the real oil price Granger-cause the US Dollar real exchange rate (see e.g. Amano and van Norden, 1998a, 1998b; Bénassy-Quéré et al., 2007; Chaudhuri and Daniel, 1998; Chen and Chen, 2007; Coudert et al., 2008; Dibooglu, 1996; Lizardo and Mollick, 2010; Throop, 1993; Zhou, 1995). On the other hand, some authors find that movements in the U.S. dollar may Granger-cause the change of the crude oil price

⁎ Tel.: + 33 4 67 15 84 95; fax: + 33 4 67 15 84 67. E-mail address: [email protected]. 0264-9993/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.econmod.2012.01.003

and contribute to the explanation of oil price movements (see e.g. Sadorsky, 2000; Zhang and Wei, 2010). This paper tries to revisit the relationship between real oil price and U.S. dollar effective real exchange rate, and especially to determine whether there is a nonlinear causality between the two variables. In this respect, the two main features of this paper are as follows: (i) To answer the important issue of the link between real oil price and U.S. dollar real effective exchange rate, we use in this article the nonlinear causality tests introduced by Péguin-Feissolle and Teräsvirta (1999) and Péguin-Feissolle et al. (2008). Indeed, most empirical studies dealing with causality focused on linear relationships; but, given the growing evidence on the nonlinear dynamic of economic and financial time series, there has been an increasing interest in nonlinear causality, based on nonlinear relationships between time series. In contrast with causality studies that rely on the restrictive assumptions of linearity, some authors argue that the traditional Granger causality test, designed to detect linear causality, is ineffective in uncovering certain nonlinear causal relations and recommend the use of nonlinear causality tests (Baek and Brock, 1992; Bell et al., 1996; Chen et al., 2004; Hiemstra and Jones, 1994; Hiemstra and Kramer, 1997; Diks, C., Panchenko, V., 2005; Li, 2006; Péguin-Feissolle and Teräsvirta, 1999; Péguin-Feissolle et al., 2008; Skalin and Teräsvirta, 1999). (ii) Moreover, instead of analyzing the time series at their original level, as it is usually done, we first decompose the data into various time scales and investigate the relationship among

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F. Benhmad / Economic Modelling 29 (2012) 1505–1514

components of the decomposed series on a scale by scale basis. More precisely, the data wavelet decomposition enables us to detect the multi-scale nonlinear causality relationships between these two variables, in contrast to the traditional methodology (i.e. cointegration) which only distinguish two time scales (short-run and long-run scales). Some works applied wavelet analysis to study linear causal relationship between economic and financial variables (see among others Almasri and Shukur, 2003; Cifter and Ozun, 2007; Dalkir, 2004; In and Kim, 2006; Kim and In, 2003; Mitra, 2006; Ramsey and Lampart, 1998; Zhang and Farley, 2004). The major issue of this paper is thus to apply wavelet analysis to the investigation of the relationship between the real oil price and the real effective U.S. dollar exchange rate. It implies that the linear and nonlinear causal relationships between these two variables are not constant over time as it varies depending on the wavelet time scales or frequency bands. More specifically, we find that at the first frequency band – which corresponds to a class of traders whom investment horizon is about 3-months, i.e. mainly to speculative trading (noise traders) – the causality runs from the real oil prices to real dollar effective exchange rate. At the second frequency band which represents an investment horizon of 6-months, there are no causal relationships between oil and dollar. Then, from horizons of 16 to 128 months, which represent fundamentalist traders, especially fund managers and institutional investors, there is a strong bidirectional causal relationship between the real oil and the real effective exchange U.S dollar. The paper is organized as follows. A review of the empirical literature about the relationships between real oil price and the real effective US dollar exchange rate is presented in Section 2. Section 3 describes the wavelet analysis and the causality tests. The econometric results are discussed in Section 4. Section 5 concludes. 2. Oil and dollar: empirical literature review An important empirical literature1 suggests that oil price shocks can be a major source of U.S. macroeconomic variability. For example, Hamilton (1983) finds that major oil price increases have preceded almost all post-WWII recessions in the United States. Gillman and Nakov (2009) study the monetary effects on nominal oil price and revisit the Granger-causality evidence reported by Hamilton (1983): from 1973 to 2009, they report strong evidence that U.S. inflation Granger-caused the nominal price of oil. When allowing for structural breaks, there is a stronger evidence for causality from the oil price to inflation, suggesting that oil price shocks might explain some of the cyclical inflation fluctuations. On the one hand, the existing empirical research in this area generally finds a positive relationship between the two variables, i.e. a rise in the oil price coincides with a dollar appreciation, and, more precisely, that oil price significantly contributes to the explanation of movements in the U.S. dollar. Throop (1993), Zhou (1995), Amano and van Norden (1998a, 1998b) and Dibooglu (1996) have evidenced that the causality runs from oil price variations to exchange-rate changes. Indeed, Amano and van Norden (1998a) empirically studied the relationship between U.S. dollar real effective exchange rate and oil price by means of the cointegration theory and concluded that oil price appeared to be the dominant source of persistent U.S. dollar real exchange rate shocks. They showed that real oil price is weakly exogenous in the sense of Engle et al. (1983) while the real exchange rate is not: in other words, the level of real effective exchange rate adjusts to the oil price in the long-run and not vice versa. Consistently with these weak exogeneity 1 From a theoretical point of view, see Bénassy-Quéré et al. (2007) and Coudert et al. (2008) for a survey, Krugman (1983a, 1983b), McGuirk (1983), Golub (1983) and Rogoff (1991).

results, causality tests indicate that while the oil price Granger-cause the real exchange rate, there is no evidence to support the converse (see also for similar conclusions Burbidge and Harrison, 1984; Hamilton, 1983; Mork, 1989). Amano and van Norden (1998b) considered the relationship between the real domestic price of oil and real effective exchange rates for Germany, Japan and the US. They explained why the real oil price captures exogenous terms-of-trade shocks which impact on the real exchange rate in the long run, and why oil price is thus a main driving factor of the long-term evolution of exchange rates. Chaudhuri and Daniel (1998) used cointegration tests and showed that the nonstationary behavior of U.S. dollar real exchange rates over the post-Bretton Woods period is due to nonstationary behavior of the real oil price. Lizardo and Mollick (2010) studied how oil price shocks affect the value of the U.S. dollar against major currencies from the 1970s to 2008. They concluded that oil price significantly contribute to the explanation of movements in the value of the U.S. dollar in the long-run. Chen and Chen (2007) used a panel cointegration theory to test the long-run relationship between real oil price and real exchange rates of G-7 countries; they suggested that real oil price may have been the dominant source of real exchange rate movements and that real oil price have significant forecasting power. Bénassy-Quéré et al. (2007) and Coudert et al. (2008) provided evidence for a long-term relation (i.e. a cointegration relation) between the two series in real terms and for a causality running from the real price of oil to the real effective exchange rate of the dollar, over the 1974–2004 period; the explanation given by Coudert et al. (2008) is that “an increase in oil price is likely to improve the U.S. net foreign asset position relatively to the rest of the world, and this has a positive impact on the dollar appreciation”. On the other hand, some authors found that movements in the U.S. dollar contribute to the explanation of oil price movements. Sadorsky (2000) examined the cointegration and causal relationship between energy futures price of crude oil, heating oil and unleaded gasoline, and the U.S. dollar effective exchange rates; the results suggested that exchange rates transmit exogenous shocks to energy futures prices, and, therefore, that recent movement in commodity prices may be a response to movements in the U.S. dollar. Zhang et al. (2008), using different econometric techniques, investigated three spillover effects (mean spillover, volatility spillover and risk spillover) of U.S. dollar exchange rate on oil price; different conclusions were given, depending on the spillover effect. For example, considering the mean spillover, they concluded that the U.S. dollar depreciation drives up the international crude oil price. Zhang and Wei (2010), by using cointegration and causality analysis, considered the crude oil market and the gold market; they found that the U.S. dollar index may Granger-cause the change of both the crude oil price and the gold price. 3. Wavelets and causality 3.1. Time-frequency wavelet decomposition Economic time series are often analyzed in time domain. However, this approach does not enable us to get any information regarding the frequency components of the studied time series. Thus, distinction between low frequency, medium and high frequency fluctuations of time series may be better studied by transforming data to frequency domain. Indeed, traditional Fourier spectral analysis can be used to identify the different frequency components of a time series (e.g. trends, cycles, seasonalities, noise) and quantify their respective importance. However, as it is based on a stationary assumption, it seems to be too restrictive because the economic and financial time series are often subject to regime shifts, jumps, volatility clustering, outliers, or long-term trends. In order to overcome this drawback, a short-time Fourier transform (which is also known as the Gabor or windowed Fourier transform) has been developed. In order to estimate the frequencies, it uses a time period (the window) less than the number of observations T, by splitting the sample into subsamples and computing the Fourier transform on


these subsamples. Many other suggestions were made to develop time varying spectra (Priestley, 1965). The problem is the right choice of the window and its constancy over time. The innovation of wavelet transform, which takes its roots from Fourier analysis, is that its window is adjusted automatically to the high or low frequency as it uses short window for high frequency and long window at low frequency by employing time compression or dilatation rather than a variation of frequency in the modulated signal. This is achieved by dividing the time axis into a sequence of successively smaller segments (Percival and Walden, 2000). The discrete wavelet transform (DWT) transforms a time series by dividing it into segments of the time domain called “scales” or frequency “bands”. The scales, from the shortest to the largest, represent progressively high, medium and low frequency fluctuations. The name “wavelet” is reputed to have come from the requirement that admissible functions integrate to zero, that is, they “rise and fall” like ocean waves, above and below the x-axis. The suffix “let” suggests a local function, rather than a global one (Grossman and Morlet, 1984). Thus, a wavelet is a small wave that grows and decays in a limited time frame. Therefore, wavelets overcome the limitations of Fourier analysis as they combine information from both time-domain and frequencydomain, do not require stationary, and allow us to extract the different frequencies driving any macroeconomic variable in the time domain by decomposing into its time scale components, each reflecting the evolution of the signal trough time at a particular frequency. Wavelet analysis has originally been used in signal processing. Percival and Walden (2000) provide a mathematically rigorous and exhaustive introduction to wavelets. The applications of wavelet analysis for specific purpose in economics and finance are relatively recent and provided by Ramsey and Lampart (1998), Gençay et al. (2010) and Gallegati (2005, 2007) Kim and In (2003), Benhmad (2011) among the others. Crowley (2007) and Schleicher (2002) proposed a guide to wavelets for economists. However, the first attempt to apply this methodology to the analysis of long wave has been made by Crowley (2010). 3.1.1. Wavelets The mother wavelet can be denoted by ψ(t). This function is defined on the real axis and must satisfy two conditions, þ∞

þ∞

2

∫−∞ ψðt Þdt ¼ 0; ∫−∞ jψj dt ¼ 1: In order to quantify the change in a function at a particular frequency and at a particular point in time, the mother wavelet ψ(t) is dilated and translated, 1 t−u ψu;s ðt Þ ¼ pffiffi ψ s s where u and s are respectively the time location and scale parameters or frequency ranges. The term p1ffis ensures that the norm of ψu, s(t) is equal to unity. The CWT, W(u, s) which is a function of the two parameters u and s, is then obtained by projecting the original function x(t) onto mother wavelet ψu, s(t), þ∞

W ðu; sÞ ¼ ∫−∞ xðt Þψu;s ðt Þdt: In order to assess the variations of the function on a large scale (i.e. at a low-frequency), a large value for s must be chosen, and vice-versa. Applying the CWT for a continuum of location and scale parameters to a function allows extracting a set of “basic” components. As CWT is computationally complex and contains a high amount of redundant information (Gençay et al., 2002), we will use the discrete variant of the wavelet transform (DWT) grounded on the same concepts as the CWT. The DWT is more parsimonious as it uses a limited number of translated and dilated versions of the mother wavelet to decompose a given signal (Gençay et al., 2002). u and s are chosen in such a way to

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summarize the information contained in the signal in a minimum of wavelet coefficients by setting −j

s¼2

and

−j

u ¼ k2

where j and k are integers representing the set of discrete translations and discrete dilatations (Gençay et al., 2002). Thus, the DWT of a time series with T observations is calculated only at dyadic scales, i.e.at scales 2j and the largest number of scales equals to the integer J such that J = [log 2(T)]. The DWT is based on two discrete wavelet filters which are called mother wavelet hl =(h0,…,hL − 1) and the father wavelet gl =(g0,…,gL − 1). L−1 The mother wavelet integrate (sums) to zero, ∑l¼0 hl ¼ 0, and has 2 unit energy, ∑L−1 h ¼ 1, In addition, the wavelet filter h is orthogonal l¼0 l to its event shifts; ∑L−1 h h ¼ 0 for all integers n ≠ 0. The mother l lþ2n l¼0 wavelet, associated with a difference operator, is a high-pass filter. On the other hand, the father wavelet is a low-pass (scaling) filter. Its coefficients are determined by the quadrature mirror relationship: gl = (−1)l + 1hL − 1 − l for l = 0,…, L − 1 pffiffiffi The basic properties of the scaling filter are: ∑L−1 2, l¼0 g l ¼ 2 L−1 ∑L−1 ∑l¼0 g l glþ2n ¼ 0 for all nonzero integers n, and l¼0 g l ¼ 1, ∑L−1 l¼0 g l hlþ2n ¼ 0 for all nonzero integers n. Therefore, scaling filters satisfy the orthonormality property as they have unit energy and are orthogonal to even shifts. Thus, a wavelet decomposition seems to consist on applying recursively a succession of low-pass and high-pass filters to a given time series, which allows separating its high frequency components from the low frequency ones. The father wavelets represent the smooth or low frequency parts of a signal, and the mother wavelets capture the details or high-frequency components. Thus, father wavelet reconstructs the longest time-scale component of the series (trend) and mother wavelet extracts the cyclical components around the trend. Let w1(t) and v1(t) be respectively the wavelet (high frequency) and the scaling (low frequency) coefficients at the first scale decomposition of a given time series x(t). The frequency-by-frequency decomposition of the time series is done according to what is known as the pyramid algorithm by further decomposing the (low-frequency) scaling coefficients v1(t) into high and low frequency components: w2(t) and v2(t). After two steps, the decomposition looks like x(t) = [w1(t), w2(t), v2(t)]. One can then apply the pyramid algorithm again and again up to scale J = [log 2(T)] to finally obtain (Fig. 1) h i xðt Þ ¼ w1 ðt Þ; w2 ðt Þ; ::::::; wJ ðt Þ; vJ ðt Þ :

3.1.2. Multiresolution analysis (MRA) Multiresolution analysis can be used to reconstruct the original time-series x(t) from the wavelet and scaling coefficients, w and v. In order to achieve this, one has to apply the inverse DWT on vJ and wj, j = 1,… J. the implementation of the inverse DWT is also done according to a pyramid algorithm. Wavelets consist on a two-scale dilatation equation. The dilatation equation of father wavelet ϕ(x) can be expressed as follows: ϕðxÞ ¼

pffiffiffi 2 ∑ lk ϕð2x−kÞ: k

The mother wavelet ψ(x) can be derived from the father wavelet by the following formula:

ψðxÞ ¼

pffiffiffi 2 ∑ hk ϕð2x−kÞ: k

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Fig. 1. Pyramid algorithm.

The coefficients lk and hk are respectively the low-pass and highpass filter coefficients. They can be expressed as:

Péguin-Feissolle and Teräsvirta (1999) suggest two tests; the first one is based on a Taylor series approximation and the second one on artificial neural networks.

1 lk ¼ pffiffiffi ∫ϕðt Þϕð2t−kÞdt 2 1 hk ¼ pffiffiffi ∫ψðt Þϕð2t−kÞdt: 2

3.2.1. Noncausality testing based on a Taylor series approximation The test is based on a Taylor expansion of the nonlinear function:

Thus, a wavelet decomposition of the signal or time series x(t) in L2(R) consists on a sequence of projections onto father and mother wavelets through scaling (stretching and compressing) and translation. The projections give the wavelet coefficients sJ, k, dJ, k,….., d1, k: sJ;k ≈ ∫ϕJ;k ðt Þxðt Þdt dj;k ≈ ∫ψj;k ðt Þxðt Þdt; for j ¼ 1; 2; ……:J: The coefficients sJ, k (smooth) represent the smooth behavior of the signal at the coarse scale 2J (trend) The coefficients dj, k (details) coefficients represent deviations from the trend; dJ, k, dJ − 1, k,....,d1, k capture the deviations from the coarsest to finest scale. The wavelet representation can be expressed as follows: xðt Þ ¼ ∑ sJ;k ϕJ;k ðt Þ þ ∑ dJ;k ψJ;k ðt Þ þ ∑ dJ−1;k ψJ−1;k ðt Þ þ ::::::: k

þ ∑ d1;k ψ1;k ðt Þ

k

yt ¼ f

yt−1 ; …; yt−q ; xt−1 ; …; xt−n ; θ þ εt

ð1Þ

where θ* is a parameter vector and εt ~ nid(0, σ 2); the sequences {xt} and {yt} are weakly stationary and ergodic. The functional form of f* is unknown but we assume that is adequately represents the causal relationship between xt and yt. Moreover, we assume that f* has a convergent Taylor expansion at any arbitrary point of the sample space for every θ* ∈ Θ (the parameter space). In order to apply Eq. (1) for testing noncausality hypothesis, it is stated that xt does not cause yt if the past values of xt does not contain any information about yt that is already contained in the past values of yt itself. More specifically, under the noncausality hypothesis, we have: yt ¼ f yt−1 ; …; yt−q ; θ þ εt :

ð2Þ

k

k

where J is the number of multiresolution levels, and k ranges from 1 to the number of coefficients in each level. Where: SJ ðt Þ ¼ ∑k sJ;k ϕJ;k ðt Þ Dj ðt Þ ¼ ∑k dj;k ψJ;k ðt Þfor j ¼ 1; 2; …; J We can thus reconstruct the original time-series as: xðt Þ ¼ SJ ðt Þ þ DJ ðt Þ þ DJ−1 ðt Þ þ ::::: þ D1 ðt Þ: At scale j, the detail components Dj captures frequencies ≤f ≤1 2j which represent the cycles with periodicity between 2j 2jþ1 and 2j + 1 and the wavelet smooth SJ captures cycles with periodicities greater than 2 J + 1 periods. As each term the later equation represents an orthogonal component of the signal x(t) at different resolutions (scales or frequency ranges), this “reconstruction” is known as multiresolution analysis (MRA) (Mallat, 1989). SJ(t) refers to the decomposed time series using scaling function at scale J and therefore represent the smooth components of x(t) (approximations). Dj(t) refers to the decomposed time series using wavelet function at scales j up to scale J and therefore are related to the detail (or rough) components of x(t) at scale j. According to the preservation of energy property of wavelet transform, the variance for returns series is reconstructed from variance estimates at each scale j. 1

3.2. Testing for nonlinear causality

To test Eq. (2) against Eq. (1), following Péguin-Feissolle and Teräsvirta (1999), we linearize f* in Eq. (1) by expanding the function into a kth-order Taylor series around an arbitrary fixed point in the sample space. After approximating f*, merging terms and reparametrizing, we obtain: yt ¼ β0 þ

q X

βj yt−j þ

n X n X

j1 ¼1j2 ¼j1

þ… þ

γj xt−j þ

q X q X j1 ¼1j2 ¼j1

j¼1

j¼1

þ

n X

γ j1 j2 xt−j1 xt−j2 þ … þ

n X n X j1 ¼1j2 ¼j1

…

n X

βj1 j2 yt−j1 yt−j2 þ

q X q X j1 ¼1j2 ¼j1

…

q X

q X n X

δj1 j2 yt−j1 xt−j2

j1 ¼1j2 ¼1

βj1 …jk yt−j1 …yt−jk

jk ¼jk −1

ð3Þ

γ j1 …jk xt−j1 …xt−jk þ εt

jk ¼jk −1

where εt*=εt +Rt(k)(x,y), Rt(k) being the remainder, and n≤k and q≤k for notational convenience. Expansion (3) contains all possible combinations of lagged values of yt and xt up to order k. The assumption that xt does not cause yt means that all terms involving functions of elements of lagged values of xt in Eq. (3) must have zero coefficients. According to PéguinFeissolle and Teräsvirta (1999), there are two practical difficulties related to Eq. (3). One is numerical and the other one has to do with the amount of information. The numerical problem arises because the regressors in Eq. (3) tend to be highly collinear if both k, q and n are large. The other problem is that the number of regressors increases rapidly with k, so that the number of degrees of freedom may become rather small. A practical solution to both problems consists in replacing some observation matrices by their largest principal components. First divide the regressors in Eq. (3) into two groups: those being the function of lags of yt only and


the rest. Replace the regressors in Eq. (3) by the first p* principal components of each matrix of observations. The null hypothesis is that the principal components of the latter group have zero coefficients. This yields the test statistic: General ¼

ðSSR0 −SSR1 Þ=p SSR1 =ðT−1−2p Þ

ð4Þ

where SSR0 and SSR1 are obtained as follows. Regress yt on 1 and the first p* principal components of the matrix of lags of yt only, form the residuals ε^ t , t=1,…,T, and the corresponding sum of squared residuals SSR0. Then regress ε^t on 1 and all the terms of the two principal components matrices, form the residuals and the corresponding sum of squared residuals SSR1. The test statistic has approximately an F-distribution with p* and T − 1 − 2p* degrees of freedom. The problem of degrees of freedom is less acute if we can assume that the general model is “semi-additive”: yt ¼ g yt−1 ; …; yt−q ; θg þ f xt−1 ; …; xt−n ; θf þ εt

ð5Þ

where θ′ = (θ′g, θ′f)′ is the parameter vector; in this case, xt does not cause yt if f(xt − 1, …, xt − n, θf) = constant. We linearize both functions into a kth-order Taylor series as before and we obtain the statistic called Additive. 3.2.2. Noncausality test based on artificial neural networks The ANN-based noncausality tests are characterized by a single hidden layer network with a logistic neural function and related to model (5), that is, semi-additivity of the functional form has to be assumed before applying the test; f(xt − 1,…,xt − n,θf) in Eq. (5) can be approximated by ′

~t α þ θ0 þ w

p X

βj

j¼1

1

ð6Þ

0

1 þ e−γj wt

′ ~ t ¼ ðxt−1 ; …; xt−n Þ′ , ~ t ′ is a (n + 1) × 1 vector, ω where θ0 ∈R, wt ¼ 1; w α = (α1, …, αn)′ are n × 1 vectors, and the γj = (γj0, …, γjn)′, for j = 1,…, p, are (n + 1) × 1 vectors. The sequences {xt} and {yt} are weakly stationary and ergodic. The null hypothesis of Granger noncausality, i.e. that xt does not cause yt, can be formulated as H02 : α = 0 and β = 0 where β = (β1, …, βp)′ is a p × 1 vector. The identification problem the γj under the null hypothesis is solved by generating γj, j = 1, …, p, randomly from a uniform distribution, following Lee et al. (1993). Implementing a Lagrange multiplier type version of the test requires the computation of the T × (n + p + m) matrix R = [Z F] where Z is a T × m matrix containing all variables due to the k-th order Taylor expansion of g, and the t-th row of F has the form ½F t ¼

~ ′t ; w

1 −γ1 ′ ωt

1þe

; …;

1

−γp ′ ωt

1þe

ð7Þ

where γj*, j = 1,.., p, contain the randomly drawn values of the corresponding unidentified parameter vectors. As Lee et al. (1993) pointed out, the elements of the second submatrix of F tend to be collinear with themselves and with the first part of F. Thus we conduct the test using the first principal components of the second submatrix of F. This leads to the test statistic called Neural where we generate the hidden unit weights, i.e. the different elements of the vectors γj, for j = 1,…, p, randomly from the uniform distribution. 3.3. Nonlinear causality tests in the wavelet domain Although wavelet transform showed strong robustness in dealing with trend, non-stationary and nonlinearities, we decide

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to use log difference transformation of the data and not to take the data “as is” for wavelet decomposition. Therefore, we can ensure the variables stationary and ergodicity assumptions for carrying out the linear and nonlinear causality tests. Moreover, the first differenced data allows us to make a comparison with empirical studies which investigated Granger causality between oil and dollar in cointegration, VECM or unrestricted VAR approach. It is worth noting that the comparison with the empirical findings of many authors will not only consists on carrying out the nonlinearity causality tests introduced by in the time domain, but we will also use a novel methodology which combines the wavelet analysis and the nonlinear causality tests introduced. Indeed, Granger causality is a well established technique for measuring causality and applying GC over the spectrum may prove to be useful in measuring the strength and direction of the causality, which could vary over the frequencies. Granger (1969, 1980) was the first to suggest that a spectral-density approach would give a richer and more comprehensive picture than a one-shot Granger causality measure that is supposed to apply across all periodicities. Thus measuring the bivariate Granger causality over the spectrum has merit compared to the one-shot test. Therefore, it is important to address how the causality changes with frequency and to show if the significance and/or direction of the usual Granger causality tests in time domain can change after adopting the causality test the causality tests across frequency bands. Economic theory followed the convenience of representing the time in just two scales. However, there is more time scales between the short run and long run. Thus, the causality links must be investigated under many different time scales. As time series consist of both high and low-frequency components and the extent and direction of causality can differ between frequency bands (Granger and Lin, 1995). Spectral Granger-causality tests enable us to determine at which frequencies one time series causes (or helps predict) another so that we may distinguish between short-run and medium (business cycle, and long-run impacts (secular trend) of various components of oil prices on US dollar and vice versa). The fact that a stationary series is effectively the sum of uncorrelated components, each of which is associated with a single frequency ordinate, allows the full causal relationship to be decomposed by frequency (Dhamala et al., 2008). The wavelet transform, as a band-pass filter, enables us to make a distinction on the behavior of different frequencies through time. Lower frequencies will represent a periodic structure to cover years or decades, higher frequencies seasons, and the highest frequency band represents noise. In this paper, the causality relationship between oil price and US dollar will be analyzed in different time scales. We would be able to see the causal relationship between these two variables in terms of their month-by-month, quarter-by quarter, and year-by-year behavior. In the spirit of the Granger causality in the frequency domain, we can use the same methodology but taking into account the drawbacks of Granger causality test. Relying on the restrictive assumptions of linearity, it is ineffective in uncovering certain nonlinear causal generated by nonlinear process. To resolve this low power in detecting nonlinear causal relations, Péguin-Feissolle and Teräsvirta (1999) and Péguin-Feissolle et al. (2008), proposed a statistical method for uncovering nonlinear causal relations. Although these causality tests were constructed in the time domain, they can without any doubt nor ambiguity be applied to frequency domain, i.e to the time series generated by the wavelet transformation which are stationary and ergodic. We have just to choose the variables lags, the Taylor expansion order, and hidden units number. Therefore, the combination of wavelet-transformation with the aforementioned nonlinear causality tests which nested the linear case have the double advantage of enabling us to carry out the causality test between the oil price and US dollar in the frequency domain

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Fig. 2. U.S. dollar real effective exchange rate and real oil price.

without the restrictive assumption of linearity of theses links. I will thus reveal to us the full causality relationships dynamics along the frequency bands.

4. Empirical evidence

4.2. Empirical study

4.1. Data description The data set contains monthly observations of the real U.S. dollar effective exchange rate and U.S. real price of oil over the 1970:2–2010:2 time period. The real U.S. dollar effective exchange rate is a trade weighted average of bilateral real exchange rates of the U.S. dollar against the currencies of major developed trade partners (Source BIS, 2005 = 100). The crude oil price variable is expressed in real terms, i.e. deflated by U.S. consumer price index (CPI, 2005 = 100, source OECD). It consists on the simple average of three spot prices; Dated Brent, West Texas Intermediate, and the Dubai Fateh, US Dollars per Barrel, (Source: International Monetary Fund). Both series are taken in logarithms. Fig. 2 shows the real effective U.S. dollar exchange rate (LCH) and the real price of oil (LOIL). According to the results of the augmented-Dickey–Fuller (ADF) and Phillips–Perron (PP) tests reported in Table 1, it turns out that both series are integrated of order one. In order to remove nonstationarity of the series, the monthly returns were calculated as the differences of the two variables natural logarithms of successive months. Fig. 3 shows the U.S. dollar real effective exchange rate returns (DLCH) and the U.S. real oil price returns (DLOIL). The descriptive statistics of monthly oil and US dollar returns data are reported in Table 2. The sample means of oil price returns are positive. For the US dollar returns, sample means are mostly negative. The measures of skewness and excess kurtosis indicate that distributions of returns

Table 1 Augmented Dickey–Fuller (ADF) and Phillips–Perron (PP) unit root tests.

ADF PP

for oil prices and US dollar are mostly positively skewed and leptokurtic relative to a normal distribution. The Jarque–Bera normality test rejects normality of all series at any level of statistical significance.

LCH

DLCH

LOIL

DLOIL

0.9260 − 1.0110

− 15.9650⁎ − 15.9650⁎

0.8596 0.9074

− 16.7111⁎ − 16.7111⁎

Note: All models are without constant nor trend. ⁎ Rejection of the null hypothesis of a unit root at the 5% significance level.

We carry out a wavelet decomposition of the oil and US dollar returns time series into a set of six orthogonal components D1, D2, …, D6, that stand for different frequency components of the original series (details), and a final component (A6), which stands for the long run trend in the returns series, as shown in Fig. 4. This decomposition provides a representation of the original series in the time-frequency domain. The time scale interpretation of different MRA levels is given in Table 3. The time scale Dl represents the highest frequency that occurs at a time horizon of 2–4 months, D2 stands for the next finest level in the series and represents the 4–8 months, …. Finally, the component A6 stands for a time horizon more than 128 months (more than 10 years). The decomposition level was set to 6 levels in order to represent the causal relationship dynamics between oil and US dollar from 2 months up to 128 months which represents the frequency of US business cycle (up to 32 quarters or 96 months). The wavelet decompositions in this paper are made with respect to the Symmlet basis. Figs. 5 and 6 (see Appendix A) show the two variables multiresolution analysis of order J = 6 based on the Symmlet of length 8. The results of the causality tests between the oil and returns time series on a frequency band by frequency band basis are reported in Table 4. As we shown in Section 3.2, tests based on Taylor expansion approximation and artificial neural networks require a huge number of crossproducts and are very data-demanding, thereby causing a dramatic decrease in the degrees of freedom when the lag length increases. However, for the authors, it is not necessary to take a large number of crossproducts or lags on endogenous or exogenous variables to build the test since, as shown by the authors, simulation generally give appreciable results even for low lag values. Therefore, in order to compute the tests statistics, we choose to take two lags on the endogenous variables (q= 2), three lags on the exogenous (n= 3) and a three order for the Taylor expansion (k= 3). For the neural network test, following Lee et al. (1993), we choose to take twenty hidden units (p = 20) and we generate the different elements of the vectors γj, for j = 1,…, p, randomly from the uniform [− μ,μ] distribution with μ = 2. Moreover, the number


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Fig. 3. US dollar and oil price returns.

of principal components is determined automatically for all the tests by including the largest principal components that can explain at least 80% of the variation in the corresponding matrix variation. The empirical results of the all causality tests on the two return time series in the time domain clearly show that the null hypothesis of no causality from oil price returns to U.S. dollar exchange return is never rejected: real oil prices have no significant effect on real effective U.S. dollar exchange rate. However, the causality runs from U.S. dollar returns to real oil prices returns. This conclusion is the opposite of the conclusion of Bénassy-Quéré et al. (2007). From a frequency domain perspective, it is worth noting that the causal relationship between the real oil prices returns and real effective US dollar returns varies depending on the time scale or frequency band. In the short run, the causality is unidirectional running from oil price returns and US dollar returns in the short run, and there is feedback relation between the two variables in the medium to long run. Thus, the strength of causality varies over frequencies. The nonlinear causality tests give different results in comparison with the standard Granger causality test. For instance, the null hypothesis of no causality from real oil prices to real effective U.S. dollar exchange rate is accepted by the linear causality test and is rejected by the nonlinear tests for the frequency band D6. The same conclusion can be drawn the from U.S. dollar real effective exchange rate to real oil prices causal relationship for the frequency bands D3 and A6. Therefore, the nonlinear causality tests may detect causality that would be ignored by the linear Granger test. According to these results, there is far more evidence of causality now than when using the linear causality test. It seems that nonlinearities are an important feature of the data in terms of explaining the causal relationship linking the US dollar returns cycles and the cyclical components of oil

Table 2 Returns descriptive statistics.

Number of obs. Mean Standard dev. Skewness Kurtosis Jarque–Bera (p-Value)

Fig. 4. Wavelet tree decomposition.

price returns. It thus confirms the theory modeling behind the nonlinear causality tests. At the first frequency band (D1) which corresponds to a class of traders whom investment horizon is about 2–4 months (we will say 3-months), which corresponds principally to speculative trading (high frequency trading), the causality runs from the real oil prices to real effective US dollar exchange rate. This nonlinear causal relationship can be explained by three channels. First, the economic reliance of oil exporters on the US has declined over the years. Specifically, they have a lower marginal propensity to consume US products and to invest in USD assets. Therefore, Petrodollars are less recycled in dollar. Thus, the higher the oil price, the more diversification takes place, and the weaker the dollar is. Second, a higher USD price of oil, leading to expectations on high inflation, will induce an interest rates hike by the ECB reputed to

Table 3 Frequency interpretation of MRA scale levels. DLCH

DLOIL

483 − 0.0008 0.0145 0.0759 4.2376 31.2895 (0.00)

483 0.0086 0.0901 1.6511 18.9373 5331.1660 (0.00)

Scale crystals

Monthly frequency

D1 D2 D3 D4 D5 D6

2–4 months 4–8 months 8–16 months 16–32 months 32–64 months 64–128 months

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Table 4 Linear and nonlinear Granger causality tests p-values. Tests

Time domain

Frequency bands (months)

All

D1

D2

D3

D4

D5

D6

A6

Returns

2–4 M

4–8 M

8–16 M

16–32 M

32–64 M

64–128 M

(> 128 M)

H0: DLOIL does not cause DLCH Linear 0.2105 General 0.1653 Additive 0.1653 Neural 0.3210

0.0468 0.0662 0.0662 0.0952

0.1805 0.3853 0.3853 0.1413

0.0008 0.0000 0.0000 0.0000

0.7256 0.2846 0.2846 0.0006

0.0008 0.0000 0.0000 0.0000

0.1164 0.0013 0.0013 0.0000

0.0000 0.0000 0.0000 0.0000

H0: DLCH does not cause DLOIL Linear 0.0379 General 0.0292 Additive 0.2721 Neural 0.0279

0.1223 0.1037 0.1036 0.1933

0.7237 0.5197 0.5199 0.8119

0.1279 0.0000 0.0000 0.0000

0.7929 0.5748 0.5750 0.0007

0.0026 0.0521 0.0520 0.0000

0.3382 0.0881 0.0881 0.0000

0.3659 0.0000 0.0000 0.0000

Note: Linear is the classical linear Granger causality test, General and Additive are the nonlinear causality tests based on a Taylor series approximation, and Neural is the nonlinear causality test based on artificial neural networks. p-Values for the F-test are reported. Rejection of the null of no causality at the 10% significance level. (if p-value b 0.10 → reject H0 (H0: non causality) → accept causality).

have a more hawkish reaction to oil shocks than Fed. Such a monetary tightening will further depressed the dollar, thereby exacerbating the dollar fall. At last, as the United States is the world's largest oil importer, the U.S. economy is more susceptible to damage from rising oil prices than other economies. Thus, it suffers from a large current account deficit — especially the oil trade balance. It turns out that this structural deterioration in the US terms of trade induces a falling dollar (Roach, 2008). At the second frequency band (D2) which represents an investment horizon of 5–8 months (we will say 6-months), there are no causal relationships between oil and dollar. It is probably due to strong seasonality observed in oil markets. From the third frequency band up the sixth one (D3, D4, D5, D6) representing the spectrum of time horizons running from 16 months to 128 months (approximating US business cycle of duration 6–32 quarters), and corresponding to fundamentalist traders especially fund managers and institutional investors, there is a strong bidirectional causal relationship between the real oil price returns and the real effective US dollar exchange rate returns. Moreover, in the long-term trend A6 (more than 128 months), both variables show a bidirectional relationship. There are at least three links through which the dollar can nonlinearly causes oil prices. First, the institutional investors (pension funds, portfolio managers…) began to treat commodities as an asset class. The falling confidence in the U.S. dollar as an investment has led to the increased popularity of other forms of investment including commodities. Moreover, commodities are seen as a hedge against inflation, expectations of higher US inflation will drive the dollar down and oil prices up. The Fed's interest rate cuts in order to stabilize the U.S. Economy have raised widespreads concerning the return of stagflation, making it more attractive for ivestors to reduce the share of their portfolio allocated to fixed-income securities while boosting the amount allocated to commodities. The physical commodity markets, being roiled by traders and investors looking for protection against inflation, attracted huge flow of funds which has contributed to the oil price rise (Verleger, 2008). Second, as many South-east Asian and emergent currencies are “pegged” to the dollar, dollar depreciation effectively makes them more competitive. Their high economic growth rate lead to high consumption of oil. Thus, this feedback through the de facto dollar zone will contribute to an increase the world's demand for oil. Therefore, the weakness of the dollar translates to a higher oil price, even if oil prices. Finally, as Euro is a strong currency, managed according the doctrine of “benign neglect” against the US dollar, had appreciated to offset the impact

of oil price invoiced in US dollar. Therefore, it provided an implicit subsidy on oil which caused a lack of an oil demand destruction. As a result, oil prices continue to rise. To conclude, and as a reported by the US Commodity Fututes Trading Commision (2008) on crude oil, the causality can run from both exchange rates to oil prices and oil prices to exchange rates. Indeed, the evidenced nonlinear unidirectional Granger causality in the short run, and the bidirectional one in the medium to long run confirm the complex dynamics of the relationship between the dollar and oil. The strong evidence of a nonlinear feedback shows that our testing strategy based on combining wavelet analysis and nonlinear Granger causality tests outperforms the classical linear approach which cannot capture such a very complex link between oil and the dollar. Our results are very promising, as we show, in contrast to Bénassy-Quéré et al. (2007) who claim that the emergence of China could strengthen the positive causality from the oil price to the US dollar in the short run and reverse the causality sign in the long run, that the causal relationship is nonlinear and strongly bidirectional between the two variables at medium and long run. Not only the causality runs from oil to US dollar (a high frequency phenomenon), but the two variables may feed on each other producing a vicious cycle. Therefore, the two variables are endogenous, not only the US dollar exchange rate can explain the oil price movements, but the US dollar exchange rate dynamics can be caused by the oil price fluctuations. 5. Conclusion In this paper, we investigate the causal relationship between the real oil price and the real effective U.S. dollar exchange rate returns, using a combination of nonlinear causality tests and wavelet analysis. Indeed, the wavelet analysis enables us to decompose the data into various time scales and to investigate the causal links on a time scale by time scale basis. On the whole, it is found that the relationship between the dollar exchange rate and oil prices is very complex: the relationship is not constant depending on the time scale or frequency ranges. For example, at the first frequency band (3 months) which corresponds to speculative trading (high frequency or noise trading), the causality runs from the real oil prices returns to real effective US dollar exchange rate returns. However, for time horizons longer than 16 months (fundamentalists), especially fund managers and institutional investors, there is a strong bidirectional causal relationship between the real oil and the real effective exchange rate US dollar returns.


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Appendix A

Fig. 6. Wavelet decomposition of real oil price returns.

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Fig. 5. Wavelet decomposition of real US dollar returns.

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