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Non-Linear Dependence in Oil Price BehaviorSemei Coronado Ramirez1, Leonardo Gatica Arreola2 and Mauricio Ramirez Grajeda31. Department of Quantitative Methods, University of Guadalajara, Zapopan, Jalisco, México2. Department of Economics, University of Guadalajara, Zapopan, Jalisco, México3. Department of Quantitative Methods, University of Guadalajara, Zapopan, Jalisco, MéxicoAbstract: In this paper, we analyze the adequacy of GARCH-type models to analyze oil price behavior by applying twotypes of non-parametric tests, the Hinich portmanteau test for non-linear dependence and a frequency-dominant test oftime reversibility, the REVERSE test based on the bispectrum, to explore the high-order spectrum properties of theMexican oil price series. The results suggest strong evidence of a non-linear structure and time irreversibility. Therefore,it does not comply with the i.i.d (independent and identically distributed) property. The non-linear dependence, however,is not consistent throughout the sample period, as indicated by a windowed test, suggesting episodic nonlineardependence. The results imply that GARCH models cannot capture the series structure.Keywords: Bispectrum, time reversibility, nonlinearity, asymmetry, oil price. 1. Introduction consumers. Furthermore, volatility impacts In recent years, several time series analyses investment behavior in the oil sector. In thehave aimed to understand the behavior of the short run, volatility can also affect storagecrude oil market, particularly its volatility (see demand, the value of firms’ operation options,for example Refs. [1-5]). and, consequently, the marginal cost of The application of time-series methods to production [1, 2]. Thus, understanding the priceanalyze volatility in economic variables was behavior and volatility of this commodity is anrecently acknowledged by the award of the important issue.2003 Nobel Prize in economics to Robert Engel Then, a central question is the statisticaland Clive Granger, whose contributions have adequacy of ARCH/GARCH models to analyzebeen widely employed in financial time-series oil price behavior. If these formulations are notmodels. The simplicity of the linear structures adequate, then any prediction or conclusionof these types of models lends itself to the study derived from the analysis can be misleading.of financial asset returns and commodity prices Our goal is to advance in this important[6-7]. question. Thus, the main aim of this paper is to The autoregressive conditional explore the oil price behavior and its returns toheteroskedasticity model (ARCH), and its analyze the adequacy of ARCH/GARCHgeneralization GARCH introduced by [8] and specification to study these series, by the[9] respectively, have been widely applied to application of nonlinearity tests.model volatility in time series and particularly Since [10] seminal work presentedto model oil price volatility. irrefutable evidence of nonlinear behavior by This issue is extremely important. the majority of stocks traded on the NYSE,Volatility is an essential determinant of the studies of this type of behavior on economicvalue of commodity-based contingent claims of and financial variables has become a growingcrude oil and of the risk faced by producers and subfield within econometric analysis (see Refs. [11-16]). Despite the growing literature that Corresponding author: Semei Coronado Ramirez, documents the existence of nonlinearity inPhD., Department of Quantitative Methods, Universityof Guadalajara, Periférico Norte 799 esq. Av. José financial and economic series, most models andParres Arias Módulo M 2do. Nivel, Núcleo methods used to analyze financial series,Universitario Los Belenes, C.P. 45100, Zapopan, particularly their volatility, are based on highlyJalisco, México. Research fields: time series. E-mail: restrictive statistical assumptions and do notsemeic@gmail.com. 1
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properly capture the statistical behavior of these (STR-GARCH). This analysis finds thatseries. This has been the case for most of the fluctuations in oil prices may be due to theanalyses of the crude oil market (see for nonlinearity of the behavior of differentexample Refs. [3, 4, 7-19]). operators in the market [19]. For the Mexican In this paper, we use the Hinich case, [18] analyze the volatility of Mexican oilportmanteau bispectrum model to analyze the prices by applying the Generalizednonlinear and asymmetric behavior of the Autoregressive Conditional HeteroskedasticityMexican Maya crude oil price from 1991 to (GARCH) model to study the conditional2008. We also test for the asymmetric behavior standard deviations and asymmetric effects inof the series using the REVERSE test. Our the series.findings suggest that the oil price behavior Comparative analyses of different types ofcontains nonlinear structures that cannot be models are also used to examine oil pricecaptured by any type of ARCH and GARCH behavior. Autoregressive models withmodels. We find four windows in the series that Conditional Heteroskedasticity (ARCH),present nonlinear events. We also reject that the Cointegration, Granger Causality and Vectorseries is time reversible. This could be because Autoregressive (VAR) have been comparedthe underlying model is nonlinear but the with the Data Mining model to analyze theirinnovations are i.i.d. or because the underlying suitability and to obtain information about theirinnovations are produced by a non-Gaussian statistical structures. The latter method uses aprobability distribution, although the model is sophisticated statistical tool of mathematicallinear. Therefore, we cannot conclude whether algorithms, fractal mechanics methods, neuralthe innovations are i.i.d. networks and decision trees, building on Analyzing and predicting the price of oil is holistic features to identify variables thata difficult task due to the random nature of oil determine the fluctuations in oil prices that areprices. In recent years, studies that attempt to not captured by other models [17].model oil price behavior have become more Other studies analyze the relationshipsophisticated. In particular, a growing body of between oil prices and other macroeconomicliterature attempts to capture the nonlinear fundamentals, such as GDP, gas and gasolinebehavior of the series. [20] use a methodology prices, interest rate, exchange rate and inflation.called TEI @ I to analyze the series of monthly [21] use a wavelet spectra method tocrude oil West Texas Intermediate (WTI) prices decompose the oil price series in the timefrom 1970 to 2003. This approach decomposes frequency to study how macroeconomicthe series using a different method to model changes affect oil price.each of the components. It uses an [22] studies the relationship between theAutoregressive Integrated Moving Average volatility of oil prices and the asymmetry of(ARIMA) for the linear components that gasoline prices using a VAR model. Hedetermine the trend, neural networks to concludes that there is a negative relationshipapproach the nonlinear behavior incorporated in between oil price volatility and the asymmetrythe error term, and Web-based Tex Mining of gasoline prices.(WTM) techniques and the Rule-based Expert Other analyses study the relationshipSystem (RES) to model the non-frequent between oil price and other commodities. [23]irregular effects. This study examines irregular analyze the behavior of oil prices comparedevents in the series and concludes that the series with the prices of sugar and ethanol in Brazilhas a nonlinear behavior with short nonlinear through a TVEECM (Threshold Vector Errorperiods affecting the oil price behavior. Correction Models) model. They find evidence Because it has been observed that oil price of threshold-type nonlinearity, in which theseries present volatility clustering effects, some three commodities have a threshold behavior.analyses use conditional variance models to Sugar and ethanol are linearly cointegrated, andparameterize this fact. The relationship between oil prices are determined by the prices of sugarthe nonlinear behavior of the oil price and other and ethanol.fundamentals has been studied using Smooth Although many of these studies note theTransition Regression with Generalized existence of nonlinear behavior in the series,Autoregressive Conditional Heteroskedasticity they do not identify these episodes, and they 2
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base their analyses on highly restrictive These papers test the adequacy of GARCHassumptions. However, there is a growing models and detect the nonlinear episodes usingnumber of analyses of the nonlinear behavior of the Hinich portmanteau model based on thefinancial data. With the works of [10] and [24], bicorrelation of the series. [48] developed athe statistical tools needed to identify the frequency-dominant test of time reversibilitypresence of nonlinearity in financial data series based on the bispectrum to explore the high-have become available [25]. A growing number order spectrum properties. This test providesof papers analyze episodes of nonlinear information about the time reversibility of thebehavior in financial asset markets. Numerous series; therefore, it is also useful to test thestudies report nonlinearity in the American adequacy of GARCH models. Identifyingmarket, including [10, 26-32]. Similar findings nonlinear episodes and asymmetric behavior ishave been reported for Asian cases by [14, 33- important for understanding the statistical37] and for the European markets by [25, 38- characteristics of the oil price time series and its46]. In the case of Latin American financial volatility, which is the main issue of this paper.assets, [15] and [47] find nonlinear behavior. To our knowledge, this paper is the first to use [40] test the validity of specifying a these methods to analyze oil price behavior.GARCH error structure for financial time-seriesdata on the pound sterling exchange rate for a 2. Materials and Methodsset of ten currencies. Their results demonstratethat a structure is statistically present in the data 2. 1 The Hinich Portmanteau Test forthat cannot be captured by a GARCH model or Nonlinearityany of its variants. [34] study of the TaiwanStock Exchange and the stock indices of other Our nonlinearity analysis is based on theexchanges, such as New York, London, Tokyo, Hinich portmanteau model developed by [49].Hong Kong and Singapore, finds support for The model separates the series into small, non-nonlinear behavior in the data series. [36] overlapping frames or windows of equal lengthanalyze various international financial indices and applies the C statistic and the Hinichto determine the degree of dispersion of the portmanteau statistic, denoted as H, to testnonlinearity. They analyze the Taiwan stock whether the observations in each window aremarket to determine whether the phenomenon is white noise.truly characteristic of financial time series.Their results indicate that nonlinearity is, in Let x(t) denote the time series where t isfact, universal among such series and is found an integer, t = 1,2,3,..., which denotes the timein all studied markets and the vast majority of unit. The series is separated into non-stocks traded on the Taiwanese exchange. [32] overlapping windows of length n. The kthanalyzes 60 stocks on the NYSE that represent { window is x(tk ),x(tk +1),...x(tk + n-1) and }companies with varying market capitalizations the next non-overlapping window is { x(t ),x(tk+1 +1),...x(tk+1 + n-1)} ,for odd years between 1993 and 2001. Theresults show a significant statistical difference k+1in the level and incidence of nonlinear behavior where tk+1 = tk + n . For each window, the null ( )among portfolios of different capitalizationcategories. Highly capitalized stocks show the hypothesis is that x tk is a stationary puregreatest levels and frequency of nonlinearity, noise process with zero bicorrelation, and thefollowed by medium and thinly capitalizedstocks. These differences were more ( ) alternative hypothesis is that x tk is a randompronounced at the beginning of the 1990s, but process for each window with correlation notthey remain significant for the entire period. equal to zero, Cxx (r ) = E é x(t)x(t + r ) ù , or non ë ûNonlinear correlation increased over the course zero bicorrelation,of the decade under study for all portfolios, Cxxx (r ,s) = E é x(t)x(t + r )x(t + s) ù , in thewhereas linear correlation declined. There were ë ûalso cases of sporadic correlation among the primary domain 0 < r < s< L , where L is theportfolios, suggesting that the relationship is number of lags defined in each window.more dynamic than was previously thought. 3
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We now consider the standardized asymptotic theory (see Ref. [50]). If the C and ( )observations, z tk , with z tk = ( ) ( ) x tk - m x , H statistics reject the null for pure noise for the data generated by (6), then the structure of the 2 sx series cannot be modeled by an ARCH,where m x is the expected value of the process GARCH or other stochastic volatility model.and s x is the variance. Then, the sample 2 2.2 Testing for Reversibilitycorrelation is given by the following: 1 n-r å Z(t)Z(t + r ) . Our second approach is the analysis of theCzz (r ) = (1) n- r t=1 statistical structure of the series cycle by testingTherefore, the C test statistic is as follows: for time reversibility. If the time series is i.i.d. L forward and backward, then time is said to beC = å (Czz (r ))2 ~ c L . 2 (2) reversible; otherwise, it is irreversible. r =1 As in the case of the business cycle, we ( )The r ,s sample bicorrelation is given by the expect that the oil price cycles will be asymmetric due to their fundamentals.following: Therefore, the impulse response functions 1 n-sCxxx (r ,s) = å Z(t )Z(t + r )Z(t + s) , (3) n- s t=1 cannot be invariant, and the commonly used models cannot capture this. [50] developed afor 0 £ r £ s. frequency-domain test of time reversibility The H statistic tests for the existence of based on the bispectrum called the REVERSEnon-zero bicorrelation in the sample windows test. Similar to the TR test of [51], theand is distributed in the following way: REVERSE test examines the behavior of L s-1 estimated third-order moments; however, it hasH = å å Gzzz (r ,s) ~ c (2L-1)( L/2) 2 (4) a better analysis of variance and higher power s=2 r =1 to test against time-irreversible alternatives.with G(r ,s) = n- sCzzz (r ,s) . The number of If x(t) represents a third-order stationarylags is defined by L n , with 0 c 0.5 . c process with mean zero, then the third-orderBased on the results of [49], the recommended moment is defined by the following:value for c is 0.4. A window is significant for cx (r, s) = E[ x(t)x(t + r )x(t + s)],any of the statistical C or H if the null (6)hypothesis is rejected at a significant threshold s £ r, r = 0,1,2,...level. For each of the two tests for The bispectrum is a double Fourierautocorrelation and bicorrelation, the for transformation of the third-order cumulativeeach window is a = 1- é(1- a c )(1- a H ) ù (see ë û function. If the bispectrum is defined byRef. [34]). In this study, we use a threshold of frequencies f 1 and f 2 in the domain,0.1 percent. W = {( f1, f2 ) : 0 < f1 < 0.5, f2 < f1,2 f1 + f2 <1} , (7) Examining whether ARCH, GARCH or then the bispectrum is defined as follows:any other volatility stochastic model can ¥ ¥adequately characterize the series using theabove test can be done by transforming the Bx ( f1, f2 ) = å å c (r,s)exp[ -i2p ( f r + f s)] . x 1 2 (8) t1 =-¥ t2 =-¥returns into a set of binary data: If x(t) is time reversible, then ì1 if z(t) ³ 0 cx (r,s) = cx (-r,-s) ; thus, the imaginary part of[ x(t)] = í-1 if otherwise . (5) î the bispectrum is zero. More elaboration on the imaginary part can be found in the work of [53]. If z t is generated by an ARCH, We divide the sampleGARCH or stochastic volatility process with {x(0), x(1),..., x(T -1)} within each non-innovation symmetrically distributed around a overlapping window of length Q and define thezero mean, then the binary transformed data (5) discrete Fourier transformation as fk = k / Q . Ifconverts into a Bernoulli process [14] withwell-behaved moments with respect to the T is not divisible by Q, then T is the sample size of the last window, with some data not used. 4
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The number of frames used is equal to P = [T / Q] , where the brackets signify the ( If the imaginary part Im Bx f1 , f2 = 0 , ) then the REVERSE statistic is distributed c 2division of an integer. The resolution bandwidth() is defined as d = 1/ Q. with M = T 2 /16 degrees of freedom [51]. ( ) This test can be also used for nonlinear Let Bx f k , f k be the smoothing time series to detect deviations in the series 1 2 Bx ( f1 , f2 ) , under the assumption of Gaussianity [53].estimator for which obtains If the null hypothesis of time reversibility ( ) from the average of over values Bx f k , f k 1 2 is rejected, then the series may be time irreversible in two ways. The underlying model Y( f , f ) could be nonlinear while the innovations arefor k1 k2 across the P frames, where symmetrically distributed. The second Q alternative is that the underlying innovationsY( fk1 , fk2 ) = X( fk1 )X( fk2 )X *( fk2 + fk2 ), (9) come from a non-Gaussian probability distribution, and the model is linear. Hence, theand Q-1 REVERSE is not equivalent to a nonlinearityX( fk ) = å x(t + (p.Q)exp [-i 2p fk (t + (p.Q))] (10) test [54]. t=0for the pth frames of length Q, for 3. Results and Discussion p = 0,1,..., P-1. [48] show that if the sequence (f ,f ) k1 k2 The data used in this analysis were obtained from the Economatica database. Theconverges to ( f , f ), this is a consistent and 1 2 series is the daily Mexican Maya crude oil priceasymptotically normal estimator of the from 01/01/1991 to 08/28/2008, denominated inbispectrum Bx ( f1, f2 ) . Then, the large sample U.S. dollars. The series has a total of 4,607 observations. Figure 1 shows the behavior ofvariance of Bx f k , f k ( 1 2 ) is as follows: the Maya oil spot price during the analyzed period. æ öVar = ç 2 ÷ × Sx f k 1 ( ) S (f ) S (f + fk ) , (11) è ( ) çdT ÷ ø 1 x k2 x k1 2 Figure 1. Maya oil prices for the period 1/01/91- 08/28/08 in U.S. dollars.where Sx ( f ) is defined as a consistent 140estimator with an asymptotic normal 120distribution of the frequency spectrum f, and δ 100is the resolution bandwidth set in thecalculation. 80 The normalized estimator of the 60bispectrum is the following: 40 A( fk1 , fk2 ) = P /T × Bx ( fk1 , fk2 ) /Var 1/2 . (12) 20 The imaginary part of A( fk1 , fk2 ) is 0denoted by Im A( fk1 , fk2 ) . Then, the statistical 1000 2000 3000 4000REVERSE is represented below: Before applying the different tests in our å å Im A( f , f 2REVERSE = k1 k2 ) (13) analysis, the data were transformed to the compounded returns series by the following (k1 ,k2 )ÎD relationship:where æ p öD= {( k ,k ) : ( f , f ) ÎW} . 1 2 k1 k2 (14) P = ln ç t ÷ , t è pt-1 ø 5
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where pt is the closing price at time t. Figure 2 shows the behavior of the logarithmic returns of Table 2 presents the C and H statistics the Mayan oil price for the analyzed period. results for the binary transformation of the full range. A 0.1% threshold was used for the p- values of the Hinich portmanteau test. The null hypothesis of pure noise is clearly rejected. In both cases, for statistics C and H, the p-value is practically zero. Thus, it may be inferred that Figure 2. Logarithmic returns of Maya oil prices for they are characterized by nonlinear the period 01/01/91-08/28/08. dependencies, which contradicts the assumption 2.0 of independent and identical distributed innovations. 1.8 Thus, GARCH models are not suitable to 1.6 capture the statistical structure of the underlying 1.4 process. 1.2 1.0 Table 2. C, H and REVERSE statistics for the entire period transformed 0.8 Period 01/01/91-08/28/08 0.6 Number of observations 4607 0.4 Number of lags 29 1000 2000 3000 4000 p-value of C 0.000 p-value of H 0.000 3.1 Results The summary of statistics for the Mexican To further explore whether nonlinear Mayan oil price returns series is documented in dependence is present throughout the full Table 1. It is apparent that the return over the sample or within certain sub-periods, we divide complete series is positive and quite large the series into a set of 117 non-overlapping because the mean is 1. The median is also 1, but windows with 30 observations each and analyze skewness is positive. Kurtosis is also positive them. This process helps to clarify the nature of and extremely large; therefore, the distribution market efficiency over different periods. The has a leptokurtic shape. This does not mean that length of the windows should be long enough to the shape of the distribution has less variance, apply statistical C and H but short enough to but it is more likely that this distribution offers capture nonlinear events within each window larger extreme values than a normal [40]. We use a length of 30 observations distribution. The positive skewness and the high because a month lasts 30 days, on average. kurtosis values imply deviations from For both the C and H statistics, we use a Gaussianity in the series [56]. threshold of 0.1 percent. The results of the C Finally, as expected, the Jarque-Bera and H tests are shown in Table 3. normality test statistic is quite large, and the Table 3. Windows test results null hypothesis of normality is rejected. Threshold 0.001 Table 1. Summary statistics for Maya oil price # of Windows 135 returns over the period 01/01/91-08/28/08 Length of Window 30Number of Observations 4,607 # Windows sig. C 1 # Windows sig. H 19Mean 1 % Windows C 0.740Median 1 % Windows H 14.070Standard Deviation 0.03Skewness 7.21 p-value of REVERSE 0.000Kurtosis 184.62Jarque-Bera Test Statistic 6371923 Given the chosen threshold of 0.01, the p-Value 0.00 results show that the C statistic rejects the null hypothesis of pure noise in a single window. 6
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However, with the H statistic, we found 19 periods of pure noise. To complement thissignificant windows. These results show that evidence, the REVERSE test showed that thethe percentage of significant C and H windows series was not time reversible and did notis low. These significant windows reject the comply with the property that the innovationsnull hypothesis of pure noise, indicating the are i.i.d.presence of nonlinearity confined to thesewindows. Although the tests find a single C Our results indicates that GARCH modelswindow, it is sufficient to influence the overall fail to capture the data generating process forperformance of the oil price. This peculiarity the Mexican oil returns.should be studied further. In any case, theseresults provide sufficient evidence to conclude Referencesthat the oil price series for the Mexican Mayan [1] R. H. Litzemberger, N. Rabinowitz,presents nonlinear events and therefore violates Backwardation in oil futures markets: theory andthe assumptions for GARCH. empirical evidence, The Journal of Finance 50 To explore the symmetry of the behavior (1995) 1517-545.of oil prices, we use the REVERSE statistic.The bandwidth for each window was 30, with [2] R. S.Pindyck, The dynamics of commodity spotan exponent of 0.40. The result rejects the null and future markets: A primer, The Energy Journal 22 (2001) 1-29.hypothesis. Therefore, we have evidence toconclude that the series is time irreversible. [3] R.S Pindyck, Volatility in natural gas and oil This result is consistent with the findings markets, The Journal of Energy andof the nonlinear analysis. However, it is also Development 30 (2004) 1-19.possible that the underlying innovations [4] M. S. Haigh, M. Holt, Crack spread hedging:correspond to a non-Gaussian probability accounting for time varying volatility spillovers indistribution [54]. Given both results of the energy futures market, Journal of Appliednonlinearity and irreversibility, there is strong Econometrics 17 (2002) 269-89.evidence to conclude that the series behaviorand its volatility cannot be captured by a [5] R. Bacon, M. Kojima, Coping With Oil PriceGARCH-type process. Volatility. World Bank. Energy Sector Management Assistance Program, special report, 2008.4. Conclusions [6] W. Sharpe, Capital asset prices: A theory of Oil price volatility has become an market equilibrium under conditions of risk, The Journal of Finance 19 (1964) 425-442.important issue. Even though concern aboutnonlinear dependence has gained importance, [7] R. S Pindyck, Volatility and commodity pricemany of the analyses of oil price behavior are dynamics, Journal of Futures Markets 24 (2004)based on the assumption of linear behavior. 1029-1047.This is the case for the Mexican oil price [8] R. Engel, Autoregressive conditionalanalyses that use GARCH-type models. heteroscedaticity with estimates of the variance ofMotivated by this concern, this paper uses the United King inflation, Econometrica 50 (1982)Hinich portmanteau test to model the behavior 987-1007.and to test nonlinear dependence in Mexicanoil price behavior. [9] T. Bollerslev, Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 31 (1986) 307-327. The results from the Hinich portmanteautest suggested the presence of nonlinear [10] M. J. Hinich, M. D. Patterson, Evidence ofdependence within oil price behavior that nonlinearity in daily stock returns, Journal ofquestions the GARCH assumption. However, Business & Economic Statistic 3 (1985) 69-77.the windowed Hinich test showed that thereported nonlinear dependencies were not [11] R. Tsay, Nonlinearity test for time series, Biometrika 73 (1986) 461-466.consistent throughout the entire period,suggesting the presence of episodic nonlinear [12] W. Brock, D. Dechert, J. Scheinkman J. A Testdependencies in returns series surrounded by for Independence Based on The Correlation Dimension. Department of Economics, University 7
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