1. Non-Linear Dependence in Oil Price Behavior
Semei Coronado Ramirez1, Leonardo Gatica Arreola2 and Mauricio Ramirez Grajeda3
1. Department of Quantitative Methods, University of Guadalajara, Zapopan, Jalisco, México
2. Department of Economics, University of Guadalajara, Zapopan, Jalisco, México
3. Department of Quantitative Methods, University of Guadalajara, Zapopan, Jalisco, México
Abstract: In this paper, we analyze the adequacy of GARCH-type models to analyze oil price behavior by applying two
types of non-parametric tests, the Hinich portmanteau test for non-linear dependence and a frequency-dominant test of
time reversibility, the REVERSE test based on the bispectrum, to explore the high-order spectrum properties of the
Mexican 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 nonlinear
dependence. 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 the
have aimed to understand the behavior of the short run, volatility can also affect storage
crude 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 price
analyze volatility in economic variables was behavior and volatility of this commodity is an
recently acknowledged by the award of the important issue.
2003 Nobel Prize in economics to Robert Engel Then, a central question is the statistical
and Clive Granger, whose contributions have adequacy of ARCH/GARCH models to analyze
been widely employed in financial time-series oil price behavior. If these formulations are not
models. The simplicity of the linear structures adequate, then any prediction or conclusion
of 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 to
heteroskedasticity model (ARCH), and its analyze the adequacy of ARCH/GARCH
generalization 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 presented
to 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 economic
value of commodity-based contingent claims of and financial variables has become a growing
crude 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 in
PhD., Department of Quantitative Methods, University
of Guadalajara, Periférico Norte 799 esq. Av. José
financial and economic series, most models and
Parres 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 highly
Jalisco, México. Research fields: time series. E-mail: restrictive statistical assumptions and do not
semeic@gmail.com.
1

2. properly capture the statistical behavior of these (STR-GARCH). This analysis finds that
series. This has been the case for most of the fluctuations in oil prices may be due to the
analyses of the crude oil market (see for nonlinearity of the behavior of different
example 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 oil
portmanteau bispectrum model to analyze the prices by applying the Generalized
nonlinear and asymmetric behavior of the Autoregressive Conditional Heteroskedasticity
Mexican Maya crude oil price from 1991 to (GARCH) model to study the conditional
2008. We also test for the asymmetric behavior standard deviations and asymmetric effects in
of the series using the REVERSE test. Our the series.
findings suggest that the oil price behavior Comparative analyses of different types of
contains nonlinear structures that cannot be models are also used to examine oil price
captured by any type of ARCH and GARCH behavior. Autoregressive models with
models. We find four windows in the series that Conditional Heteroskedasticity (ARCH),
present nonlinear events. We also reject that the Cointegration, Granger Causality and Vector
series is time reversible. This could be because Autoregressive (VAR) have been compared
the underlying model is nonlinear but the with the Data Mining model to analyze their
innovations are i.i.d. or because the underlying suitability and to obtain information about their
innovations are produced by a non-Gaussian statistical structures. The latter method uses a
probability distribution, although the model is sophisticated statistical tool of mathematical
linear. Therefore, we cannot conclude whether algorithms, fractal mechanics methods, neural
the innovations are i.i.d. networks and decision trees, building on
Analyzing and predicting the price of oil is holistic features to identify variables that
a difficult task due to the random nature of oil determine the fluctuations in oil prices that are
prices. In recent years, studies that attempt to not captured by other models [17].
model oil price behavior have become more Other studies analyze the relationship
sophisticated. In particular, a growing body of between oil prices and other macroeconomic
literature attempts to capture the nonlinear fundamentals, such as GDP, gas and gasoline
behavior 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 to
crude oil West Texas Intermediate (WTI) prices decompose the oil price series in the time
from 1970 to 2003. This approach decomposes frequency to study how macroeconomic
the series using a different method to model changes affect oil price.
each of the components. It uses an [22] studies the relationship between the
Autoregressive Integrated Moving Average volatility of oil prices and the asymmetry of
(ARIMA) for the linear components that gasoline prices using a VAR model. He
determine the trend, neural networks to concludes that there is a negative relationship
approach the nonlinear behavior incorporated in between oil price volatility and the asymmetry
the error term, and Web-based Tex Mining of gasoline prices.
(WTM) techniques and the Rule-based Expert Other analyses study the relationship
System (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 compared
events in the series and concludes that the series with the prices of sugar and ethanol in Brazil
has a nonlinear behavior with short nonlinear through a TVEECM (Threshold Vector Error
periods 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 the
series present volatility clustering effects, some three commodities have a threshold behavior.
analyses use conditional variance models to Sugar and ethanol are linearly cointegrated, and
parameterize this fact. The relationship between oil prices are determined by the prices of sugar
the nonlinear behavior of the oil price and other and ethanol.
fundamentals has been studied using Smooth Although many of these studies note the
Transition Regression with Generalized existence of nonlinear behavior in the series,
Autoregressive Conditional Heteroskedasticity they do not identify these episodes, and they
2

3. base their analyses on highly restrictive These papers test the adequacy of GARCH
assumptions. However, there is a growing models and detect the nonlinear episodes using
number of analyses of the nonlinear behavior of the Hinich portmanteau model based on the
financial data. With the works of [10] and [24], bicorrelation of the series. [48] developed a
the statistical tools needed to identify the frequency-dominant test of time reversibility
presence 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 provides
of papers analyze episodes of nonlinear information about the time reversibility of the
behavior in financial asset markets. Numerous series; therefore, it is also useful to test the
studies report nonlinearity in the American adequacy of GARCH models. Identifying
market, including [10, 26-32]. Similar findings nonlinear episodes and asymmetric behavior is
have been reported for Asian cases by [14, 33- important for understanding the statistical
37] and for the European markets by [25, 38- characteristics of the oil price time series and its
46]. 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-series
data on the pound sterling exchange rate for a 2. Materials and Methods
set of ten currencies. Their results demonstrate
that a structure is statistically present in the data 2. 1 The Hinich Portmanteau Test for
that cannot be captured by a GARCH model or Nonlinearity
any of its variants. [34] study of the Taiwan
Stock Exchange and the stock indices of other
Our nonlinearity analysis is based on the
exchanges, 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 length
analyze various international financial indices
and applies the C statistic and the Hinich
to determine the degree of dispersion of the
portmanteau statistic, denoted as H, to test
nonlinearity. They analyze the Taiwan stock
whether the observations in each window are
market 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 is
fact, universal among such series and is found an integer, t = 1,2,3,..., which denotes the time
in 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 kth
analyzes 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. The
results show a significant statistical difference k+1
in the level and incidence of nonlinear behavior where tk+1 = tk + n . For each window, the null
( )
among portfolios of different capitalization
categories. Highly capitalized stocks show the hypothesis is that x tk is a stationary pure
greatest levels and frequency of nonlinearity, noise process with zero bicorrelation, and the
followed by medium and thinly capitalized
stocks. These differences were more
( )
alternative hypothesis is that x tk is a random
pronounced at the beginning of the 1990s, but process for each window with correlation not
they remain significant for the entire period. equal to zero, Cxx (r ) = E é x(t)x(t + r ) ù , or non
ë û
Nonlinear 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 the
whereas linear correlation declined. There were ë û
also cases of sporadic correlation among the primary domain 0 < r < s< L , where L is the
portfolios, suggesting that the relationship is number of lags defined in each window.
more dynamic than was previously thought.
3

4. 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 Reversibility
correlation is given by the following:
1 n-r
å Z(t)Z(t + r ) .
Our second approach is the analysis of the
Czz (r ) = (1)
n- r t=1 statistical structure of the series cycle by testing
Therefore, 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 be
C = 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-s
Cxxx (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 a
for 0 £ r £ s. frequency-domain test of time reversibility
The H statistic tests for the existence of based on the bispectrum called the REVERSE
non-zero bicorrelation in the sample windows test. Similar to the TR test of [51], the
and is distributed in the following way: REVERSE test examines the behavior of
L s-1 estimated third-order moments; however, it has
H = 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 stationary
lags is defined by L  n , with 0  c  0.5 .
c
process with mean zero, then the third-order
Based 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 Fourier
autocorrelation and bicorrelation, the  for transformation of the third-order cumulative
each window is a = 1- é(1- a c )(1- a H ) ù (see
ë û function. If the bispectrum is defined by
Ref. [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 the
above 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 sample
GARCH 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 the
zero mean, then the binary transformed data (5) discrete Fourier transformation as fk = k / Q . If
converts into a Bernoulli process [14] with
well-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

5. 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 2
division 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 are
for
k1 k2
across the P frames, where symmetrically distributed. The second
Q alternative is that the underlying innovations
Y( fk1 , fk2 ) = X( fk1 )X( fk2 )X *( fk2 + fk2 ), (9) come from a non-Gaussian probability
distribution, and the model is linear. Hence, the
and
Q-1
REVERSE is not equivalent to a nonlinearity
X( fk ) = å x(t + (p.Q)exp [-i 2p fk (t + (p.Q))] (10) test [54].
t=0
for 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. The
converges to ( f , f ), this is a consistent and
1 2
series is the daily Mexican Maya crude oil price
asymptotically normal estimator of the from 01/01/1991 to 08/28/2008, denominated in
bispectrum Bx ( f1, f2 ) . Then, the large sample U.S. dollars. The series has a total of 4,607
observations. Figure 1 shows the behavior of
variance of Bx f k , f k ( 1 2
) is as follows: the Maya oil spot price during the analyzed
period.
æ ö
Var = 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 140
estimator with an asymptotic normal 120
distribution of the frequency spectrum f, and δ 100
is the resolution bandwidth set in the
calculation. 80
The normalized estimator of the 60
bispectrum 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
0
denoted by Im A( fk1 , fk2 ) . Then, the statistical 1000 2000 3000 4000
REVERSE is represented below:
Before applying the different tests in our
å å Im A( f , f
2
REVERSE = k1 k2 ) (13)
analysis, the data were transformed to the
compounded returns series by the following
(k1 ,k2 )ÎD
relationship:
where
æ p ö
D= {( k ,k ) : ( f , f ) ÎW} .
1 2 k1 k2
(14) P = ln ç t ÷ ,
t
è pt-1 ø
5

6. 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 30
Number of Observations 4,607 # Windows sig. C 1
# Windows sig. H 19
Mean 1
% Windows C 0.740
Median 1
% Windows H 14.070
Standard Deviation 0.03
Skewness 7.21 p-value of REVERSE 0.000
Kurtosis 184.62
Jarque-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

7. However, with the H statistic, we found 19 periods of pure noise. To complement this
significant windows. These results show that evidence, the REVERSE test showed that the
the percentage of significant C and H windows series was not time reversible and did not
is low. These significant windows reject the comply with the property that the innovations
null hypothesis of pure noise, indicating the are i.i.d.
presence of nonlinearity confined to these
windows. Although the tests find a single C Our results indicates that GARCH models
window, it is sufficient to influence the overall fail to capture the data generating process for
performance of the oil price. This peculiarity the Mexican oil returns.
should be studied further. In any case, these
results provide sufficient evidence to conclude References
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