1

Non-Linear Dependence in Oil Price Behavior

Semei Coronado Ramirez1, Leonardo Gatica Arreola

2 and Mauricio Ramirez Grajeda

3

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

In recent years, several time series analyses

have aimed to understand the behavior of the

crude oil market, particularly its volatility (see

for example Refs. [1-5]).

The application of time-series methods to

analyze volatility in economic variables was

recently acknowledged by the award of the

2003 Nobel Prize in economics to Robert Engel

and Clive Granger, whose contributions have

been widely employed in financial time-series

models. The simplicity of the linear structures

of these types of models lends itself to the study

of financial asset returns and commodity prices

[6-7].

The autoregressive conditional

heteroskedasticity model (ARCH), and its

generalization GARCH introduced by [8] and

[9] respectively, have been widely applied to

model volatility in time series and particularly

to model oil price volatility.

This issue is extremely important.

Volatility is an essential determinant of the

value of commodity-based contingent claims of

crude oil and of the risk faced by producers and

Corresponding author: Semei Coronado Ramirez,

PhD., Department of Quantitative Methods, University

of Guadalajara, Periférico Norte 799 esq. Av. José

Parres Arias Módulo M 2do. Nivel, Núcleo

Universitario Los Belenes, C.P. 45100, Zapopan,

Jalisco, México. Research fields: time series. E-mail:

semeic@gmail.com.

consumers. Furthermore, volatility impacts

investment behavior in the oil sector. In the

short run, volatility can also affect storage

demand, the value of firms’ operation options,

and, consequently, the marginal cost of

production [1, 2]. Thus, understanding the price

behavior and volatility of this commodity is an

important issue.

Then, a central question is the statistical

adequacy of ARCH/GARCH models to analyze

oil price behavior. If these formulations are not

adequate, then any prediction or conclusion

derived from the analysis can be misleading.

Our goal is to advance in this important

question. Thus, the main aim of this paper is to

explore the oil price behavior and its returns to

analyze the adequacy of ARCH/GARCH

specification to study these series, by the

application of nonlinearity tests.

Since [10] seminal work presented

irrefutable evidence of nonlinear behavior by

the majority of stocks traded on the NYSE,

studies of this type of behavior on economic

and financial variables has become a growing

subfield within econometric analysis (see Refs.

[11-16]).

Despite the growing literature that

documents the existence of nonlinearity in

financial and economic series, most models and

methods used to analyze financial series,

particularly their volatility, are based on highly

restrictive statistical assumptions and do not

2

properly capture the statistical behavior of these

series. This has been the case for most of the

analyses of the crude oil market (see for

example Refs. [3, 4, 7-19]).

In this paper, we use the Hinich

portmanteau bispectrum model to analyze the

nonlinear and asymmetric behavior of the

Mexican Maya crude oil price from 1991 to

2008. We also test for the asymmetric behavior

of the series using the REVERSE test. Our

findings suggest that the oil price behavior

contains nonlinear structures that cannot be

captured by any type of ARCH and GARCH

models. We find four windows in the series that

present nonlinear events. We also reject that the

series is time reversible. This could be because

the underlying model is nonlinear but the

innovations are i.i.d. or because the underlying

innovations are produced by a non-Gaussian

probability distribution, although the model is

linear. Therefore, we cannot conclude whether

the innovations are i.i.d.

Analyzing and predicting the price of oil is

a difficult task due to the random nature of oil

prices. In recent years, studies that attempt to

model oil price behavior have become more

sophisticated. In particular, a growing body of

literature attempts to capture the nonlinear

behavior of the series. [20] use a methodology

called TEI @ I to analyze the series of monthly

crude oil West Texas Intermediate (WTI) prices

from 1970 to 2003. This approach decomposes

the series using a different method to model

each of the components. It uses an

Autoregressive Integrated Moving Average

(ARIMA) for the linear components that

determine the trend, neural networks to

approach the nonlinear behavior incorporated in

the error term, and Web-based Tex Mining

(WTM) techniques and the Rule-based Expert

System (RES) to model the non-frequent

irregular effects. This study examines irregular

events in the series and concludes that the series

has a nonlinear behavior with short nonlinear

periods affecting the oil price behavior.

Because it has been observed that oil price

series present volatility clustering effects, some

analyses use conditional variance models to

parameterize this fact. The relationship between

the nonlinear behavior of the oil price and other

fundamentals has been studied using Smooth

Transition Regression with Generalized

Autoregressive Conditional Heteroskedasticity

(STR-GARCH). This analysis finds that

fluctuations in oil prices may be due to the

nonlinearity of the behavior of different

operators in the market [19]. For the Mexican

case, [18] analyze the volatility of Mexican oil

prices by applying the Generalized Autoregressive Conditional Heteroskedasticity

(GARCH) model to study the conditional

standard deviations and asymmetric effects in

the series.

Comparative analyses of different types of

models are also used to examine oil price

behavior. Autoregressive models with

Conditional Heteroskedasticity (ARCH),

Cointegration, Granger Causality and Vector

Autoregressive (VAR) have been compared

with the Data Mining model to analyze their

suitability and to obtain information about their

statistical structures. The latter method uses a

sophisticated statistical tool of mathematical

algorithms, fractal mechanics methods, neural

networks and decision trees, building on

holistic features to identify variables that

determine the fluctuations in oil prices that are

not captured by other models [17].

Other studies analyze the relationship

between oil prices and other macroeconomic

fundamentals, such as GDP, gas and gasoline

prices, interest rate, exchange rate and inflation.

[21] use a wavelet spectra method to

decompose the oil price series in the time

frequency to study how macroeconomic

changes affect oil price.

[22] studies the relationship between the

volatility of oil prices and the asymmetry of

gasoline prices using a VAR model. He

concludes that there is a negative relationship

between oil price volatility and the asymmetry

of gasoline prices.

Other analyses study the relationship

between oil price and other commodities. [23]

analyze the behavior of oil prices compared

with the prices of sugar and ethanol in Brazil

through a TVEECM (Threshold Vector Error

Correction Models) model. They find evidence

of threshold-type nonlinearity, in which the

three commodities have a threshold behavior.

Sugar and ethanol are linearly cointegrated, and

oil prices are determined by the prices of sugar

and ethanol.

Although many of these studies note the

existence of nonlinear behavior in the series,

they do not identify these episodes, and they

3

base their analyses on highly restrictive

assumptions. However, there is a growing

number of analyses of the nonlinear behavior of

financial data. With the works of [10] and [24],

the statistical tools needed to identify the

presence of nonlinearity in financial data series

have become available [25]. A growing number

of papers analyze episodes of nonlinear

behavior in financial asset markets. Numerous

studies report nonlinearity in the American

market, including [10, 26-32]. Similar findings

have been reported for Asian cases by [14, 33-

37] and for the European markets by [25, 38-

46]. In the case of Latin American financial

assets, [15] and [47] find nonlinear behavior.

[40] test the validity of specifying a

GARCH error structure for financial time-series

data on the pound sterling exchange rate for a

set of ten currencies. Their results demonstrate

that a structure is statistically present in the data

that cannot be captured by a GARCH model or

any of its variants. [34] study of the Taiwan

Stock Exchange and the stock indices of other

exchanges, such as New York, London, Tokyo,

Hong Kong and Singapore, finds support for

nonlinear behavior in the data series. [36]

analyze various international financial indices

to determine the degree of dispersion of the

nonlinearity. They analyze the Taiwan stock

market to determine whether the phenomenon is

truly characteristic of financial time series.

Their results indicate that nonlinearity is, in

fact, universal among such series and is found

in all studied markets and the vast majority of

stocks traded on the Taiwanese exchange. [32]

analyzes 60 stocks on the NYSE that represent

companies with varying market capitalizations

for odd years between 1993 and 2001. The

results show a significant statistical difference

in the level and incidence of nonlinear behavior

among portfolios of different capitalization

categories. Highly capitalized stocks show the

greatest levels and frequency of nonlinearity,

followed by medium and thinly capitalized

stocks. These differences were more

pronounced at the beginning of the 1990s, but

they remain significant for the entire period.

Nonlinear correlation increased over the course

of the decade under study for all portfolios,

whereas linear correlation declined. There were

also cases of sporadic correlation among the

portfolios, suggesting that the relationship is

more dynamic than was previously thought.

These papers test the adequacy of GARCH

models and detect the nonlinear episodes using

the Hinich portmanteau model based on the

bicorrelation of the series. [48] developed a

frequency-dominant test of time reversibility

based on the bispectrum to explore the high-

order spectrum properties. This test provides

information about the time reversibility of the

series; therefore, it is also useful to test the

adequacy of GARCH models. Identifying

nonlinear episodes and asymmetric behavior is

important for understanding the statistical

characteristics of the oil price time series and its

volatility, which is the main issue of this paper.

To our knowledge, this paper is the first to use

these methods to analyze oil price behavior.

2. Materials and Methods

2. 1 The Hinich Portmanteau Test for

Nonlinearity

Our nonlinearity analysis is based on the

Hinich portmanteau model developed by [49].

The model separates the series into small, non-

overlapping frames or windows of equal length

and applies the C statistic and the Hinich

portmanteau statistic, denoted as H, to test

whether the observations in each window are

white noise.

Let x(t)denote the time series where t is

an integer, t =1,2,3,..., which denotes the time

unit. The series is separated into non-

overlapping windows of length n. The kth

window is

x(tk),x(t

k+1),...x(t

k+ n-1){ } and

the next non-overlapping window is

x(t

k+1),x(t

k+1+1),...x(t

k+1+ n-1){ },

where tk+1

= tk+ n. For each window, the null

hypothesis is that x t

k( ) is a stationary pure

noise process with zero bicorrelation, and the

alternative hypothesis is that x t

k( ) is a random

process for each window with correlation not

equal to zero, C

xx(r ) = E x(t)x(t + r )éë ùû , or non

zero bicorrelation,

C

xxx(r ,s) = E x(t)x(t + r )x(t + s)éë ùû , in the

primary domain 0 < r < s< L , where L is the

number of lags defined in each window.

4

We now consider the standardized

observations, z t

k( ), with

z tk( ) =

x tk( ) - m

x

sx

2,

where m

x is the expected value of the process

and s

x

2 is the variance. Then, the sample

correlation is given by the following:

Czz

(r ) =1

n- rZ(t)Z(t + r )

t=1

n-r

å . (1)

Therefore, the C test statistic is as follows:

C = (Czz(r ))2 ~ cL

2

r=1

L

å . (2)

The r ,s( )sample bicorrelation is given by the

following:

Cxxx

(r ,s) =1

n- sZ(t

t=1

n-s

å )Z(t + r )Z(t + s), (3)

for 0 £ r £ s.

The H statistic tests for the existence of

non-zero bicorrelation in the sample windows

and is distributed in the following way:

H = Gzzz

2 (r ,s) ~ c( L-1)( L/2)

2

r=1

s-1

ås=2

L

å (4)

with G(r ,s) = n- sC

zzz(r ,s) . The number of

lags is defined bycL n , with 0 0.5c .

Based on the results of [49], the recommended

value for c is 0.4. A window is significant for

any of the statistical C or H if the null

hypothesis is rejected at a significant threshold

level. For each of the two tests for

autocorrelation and bicorrelation, the for

each window is a =1- (1-a

c)(1-a

H)éë ùû (see

Ref. [34]). In this study, we use a threshold of

0.1 percent.

Examining whether ARCH, GARCH or

any other volatility stochastic model can

adequately characterize the series using the

above test can be done by transforming the

returns into a set of binary data:

x(t)[ ] =1 if z(t) ³ 0

-1 if otherwise

ìíî

. (5)

If z t is generated by an ARCH,

GARCH or stochastic volatility process with

innovation symmetrically distributed around a

zero mean, then the binary transformed data (5) converts into a Bernoulli process [14] with

well-behaved moments with respect to the

asymptotic theory (see Ref. [50]). If the C and

H statistics reject the null for pure noise for the

data generated by (6), then the structure of the

series cannot be modeled by an ARCH,

GARCH or other stochastic volatility model.

2.2 Testing for Reversibility

Our second approach is the analysis of the

statistical structure of the series cycle by testing

for time reversibility. If the time series is i.i.d.

forward and backward, then time is said to be

reversible; otherwise, it is irreversible.

As in the case of the business cycle, we

expect that the oil price cycles will be

asymmetric due to their fundamentals.

Therefore, the impulse response functions

cannot be invariant, and the commonly used

models cannot capture this. [50] developed a

frequency-domain test of time reversibility

based on the bispectrum called the REVERSE

test. Similar to the TR test of [51], the

REVERSE test examines the behavior of

estimated third-order moments; however, it has

a better analysis of variance and higher power

to test against time-irreversible alternatives.

If x(t) represents a third-order stationary

process with mean zero, then the third-order

moment is defined by the following:

cx(r,s) = E x(t)x(t + r )x(t + s)[ ],

s£ r, r = 0,1,2,... (6)

The bispectrum is a double Fourier

transformation of the third-order cumulative

function. If the bispectrum is defined by

frequencies 1f and 2f in the domain,

W= ( f1, f2 ) : 0 < f1 < 0.5, f2 < f1,2 f1 + f2 <1{ } , (7)

then the bispectrum is defined as follows:

Bx( f1, f2 ) = cx(r,s)exp -i2p ( f1r + f2s)[ ]t2 =-¥

¥

åt1=-¥

¥

å . (8)

If x(t) is time reversible, then

cx(r,s) = cx(-r,-s); thus, the imaginary part of

the bispectrum is zero. More elaboration on the

imaginary part can be found in the work of [53].

We divide the sample

x(0),x(1),..., x(T -1){ } within each non-

overlapping window of length Q and define the

discrete Fourier transformation as fk = k /Q. If

T is not divisible by Q, then T is the sample size

of the last window, with some data not used.

5

The number of frames used is equal to

P = T /Q[ ], where the brackets signify the

division of an integer. The resolution bandwidth

() is defined as d = 1/Q.

Let

Bx

fk

1

, fk

2( ) be the smoothing

estimator for B

xf1, f

2( ) , which obtains

B

xfk

1

, fk

2( ) from the average of over values

for

Y fk

1

, fk

2( )

Qacross the P frames, where

Y( fk1, fk2

) = X( fk1)X( fk2

)X*( fk2+ fk2

), (9)

and

X( fk ) = x(t + (p.Q)exp -i2p fk(t + (p.Q))[ ]t=0

Q-1

å (10)

for the pth frames of length Q, for

p = 0,1,..., P-1.

[48] show that if the sequence

fk

1

, fk

2( )

converges to

f1, f

2( ), this is a consistent and

asymptotically normal estimator of the

bispectrum Bx( f1, f2 ) . Then, the large sample

variance of

Bx

fk

1

, fk

2( ) is as follows:

Var =1

d2T( )

æ

è

çç

ö

ø

÷÷× S

xf

k1

( ) Sx

fk

2( ) S

xf

k1

+ fk

2( ) , (11)

where

Sx

f( ) is defined as a consistent

estimator with an asymptotic normal

distribution of the frequency spectrum f, and δ

is the resolution bandwidth set in the

calculation.

The normalized estimator of the

bispectrum is the following:

A( fk1, fk2

) = P /T × Bx( fk1, fk2

) /Var1/2 . (12)

The imaginary part of A( fk1, fk2

) is

denoted by Im A( fk1, fk2

). Then, the statistical

REVERSE is represented below:

REVERSE = Im A( fk1, fk2

)å(k1,k2 )ÎD

å2

(13)

where

D = k

1,k

2( ) : fk

1

, fk

2( )ÎW{ } . (14)

If the imaginary part Im B

xf1, f

2( ) = 0 ,

then the REVERSE statistic is distributed c 2

with M = T 2 /16 degrees of freedom [51].

This test can be also used for nonlinear

time series to detect deviations in the series

under the assumption of Gaussianity [53].

If the null hypothesis of time reversibility

is rejected, then the series may be time

irreversible in two ways. The underlying model

could be nonlinear while the innovations are

symmetrically distributed. The second

alternative is that the underlying innovations

come from a non-Gaussian probability

distribution, and the model is linear. Hence, the

REVERSE is not equivalent to a nonlinearity

test [54].

3. Results and Discussion

The data used in this analysis were

obtained from the Economatica database. The

series is the daily Mexican Maya crude oil price

from 01/01/1991 to 08/28/2008, denominated in

U.S. dollars. The series has a total of 4,607

observations. Figure 1 shows the behavior of

the Maya oil spot price during the analyzed

period.

Figure 1. Maya oil prices for the period 1/01/91-

08/28/08 in U.S. dollars.

0

20

40

60

80

100

120

140

1000 2000 3000 4000

Before applying the different tests in our

analysis, the data were transformed to the

compounded returns series by the following

relationship:

Pt= ln

pt

pt-1

æ

èç

ö

ø÷ ,

6

where tp is the closing price at time t. Figure 2

shows the behavior of the logarithmic returns of

the Mayan oil price for the analyzed period.

Figure 2. Logarithmic returns of Maya oil prices for

the period 01/01/91-08/28/08.

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

1000 2000 3000 4000

3.1 Results

The summary of statistics for the Mexican

Mayan oil price returns series is documented in

Table 1. It is apparent that the return over the

complete series is positive and quite large

because the mean is 1. The median is also 1, but

skewness is positive. Kurtosis is also positive

and extremely large; therefore, the distribution

has a leptokurtic shape. This does not mean that

the shape of the distribution has less variance,

but it is more likely that this distribution offers

larger extreme values than a normal

distribution. The positive skewness and the high

kurtosis values imply deviations from

Gaussianity in the series [56].

Finally, as expected, the Jarque-Bera

normality test statistic is quite large, and the

null hypothesis of normality is rejected.

Table 1. Summary statistics for Maya oil price

returns over the period 01/01/91-08/28/08

Number of Observations 4,607

Mean 1

Median 1

Standard Deviation 0.03

Skewness 7.21

Kurtosis 184.62

Jarque-Bera Test Statistic 6371923

p-Value 0.00

Table 2 presents the C and H statistics

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

they are characterized by nonlinear

dependencies, which contradicts the assumption

of independent and identical distributed

innovations.

Thus, GARCH models are not suitable to

capture the statistical structure of the underlying

process.

Table 2. C, H and REVERSE statistics for the entire

period transformed

Period 01/01/91-08/28/08

Number of observations 4607

Number of lags 29

p-value of C 0.000

p-value of H 0.000

To further explore whether nonlinear

dependence is present throughout the full

sample or within certain sub-periods, we divide

the series into a set of 117 non-overlapping

windows with 30 observations each and analyze

them. This process helps to clarify the nature of

market efficiency over different periods. The

length of the windows should be long enough to

apply statistical C and H but short enough to

capture nonlinear events within each window

[40]. We use a length of 30 observations

because a month lasts 30 days, on average.

For both the C and H statistics, we use a

threshold of 0.1 percent. The results of the C

and H tests are shown in Table 3.

Table 3. Windows test results

Threshold 0.001

# of Windows 135

Length of Window 30

# Windows sig. C 1

# Windows sig. H 19

% Windows C 0.740

% Windows H 14.070

p-value of REVERSE 0.000

Given the chosen threshold of 0.01, the

results show that the C statistic rejects the null

hypothesis of pure noise in a single window.

7

However, with the H statistic, we found 19

significant windows. These results show that

the percentage of significant C and H windows

is low. These significant windows reject the

null hypothesis of pure noise, indicating the

presence of nonlinearity confined to these

windows. Although the tests find a single C

window, it is sufficient to influence the overall

performance of the oil price. This peculiarity

should be studied further. In any case, these

results provide sufficient evidence to conclude

that the oil price series for the Mexican Mayan

presents nonlinear events and therefore violates

the assumptions for GARCH.

To explore the symmetry of the behavior

of oil prices, we use the REVERSE statistic.

The bandwidth for each window was 30, with

an exponent of 0.40. The result rejects the null

hypothesis. Therefore, we have evidence to

conclude that the series is time irreversible.

This result is consistent with the findings

of the nonlinear analysis. However, it is also

possible that the underlying innovations

correspond to a non-Gaussian probability

distribution [54]. Given both results of

nonlinearity and irreversibility, there is strong

evidence to conclude that the series behavior

and its volatility cannot be captured by a

GARCH-type process.

4. Conclusions

Oil price volatility has become an

important issue. Even though concern about

nonlinear dependence has gained importance,

many of the analyses of oil price behavior are

based on the assumption of linear behavior.

This is the case for the Mexican oil price

analyses that use GARCH-type models.

Motivated by this concern, this paper uses the

Hinich portmanteau test to model the behavior

and to test nonlinear dependence in Mexican

oil price behavior.

The results from the Hinich portmanteau

test suggested the presence of nonlinear

dependence within oil price behavior that

questions the GARCH assumption. However,

the windowed Hinich test showed that the

reported nonlinear dependencies were not

consistent throughout the entire period,

suggesting the presence of episodic nonlinear

dependencies in returns series surrounded by

periods of pure noise. To complement this

evidence, the REVERSE test showed that the

series was not time reversible and did not

comply with the property that the innovations

are i.i.d.

Our results indicates that GARCH models

fail to capture the data generating process for

the Mexican oil returns.

References

[1] R. H. Litzemberger, N. Rabinowitz,

Backwardation in oil futures markets: theory and

empirical evidence, The Journal of Finance 50

(1995) 1517-545.

[2] R. S.Pindyck, The dynamics of commodity spot

and future markets: A primer, The Energy Journal

22 (2001) 1-29.

[3] R.S Pindyck, Volatility in natural gas and oil

markets, The Journal of Energy and

Development 30 (2004) 1-19.

[4] M. S. Haigh, M. Holt, Crack spread hedging:

accounting for time varying volatility spillovers in

the energy futures market, Journal of Applied

Econometrics 17 (2002) 269-89.

[5] R. Bacon, M. Kojima, Coping With Oil Price

Volatility. World Bank. Energy Sector

Management Assistance Program, special report,

2008.

[6] W. Sharpe, Capital asset prices: A theory of

market equilibrium under conditions of risk, The

Journal of Finance 19 (1964) 425-442.

[7] R. S Pindyck, Volatility and commodity price

dynamics, Journal of Futures Markets 24 (2004)

1029-1047.

[8] R. Engel, Autoregressive conditional

heteroscedaticity with estimates of the variance of

United King inflation, Econometrica 50 (1982)

987-1007.

[9] T. Bollerslev, Generalized autoregressive

conditional heteroskedasticity, Journal of

Econometrics 31 (1986) 307-327.

[10] M. J. Hinich, M. D. Patterson, Evidence of

nonlinearity in daily stock returns, Journal of

Business & Economic Statistic 3 (1985) 69-77.

[11] R. Tsay, Nonlinearity test for time series,

Biometrika 73 (1986) 461-466.

[12] W. Brock, D. Dechert, J. Scheinkman J. A Test

for Independence Based on The Correlation

Dimension. Department of Economics, University

8

of Wisconsin, University of Houston and

University of Chicago. Revised version, 1991, W.

A. Brock, D. Dechert, J. Scheinkman, B.D.

LeBaron, 1987.

[13] H. White, Connectionist nonparametric

regression: Multilayer feedforward networks can

learn arbitrary mappings, Journal Neural

Networks 3 (1990) 535-549.

[14] K. Lim, M. J. Hinich, Cross-temporal universality

of nonlinear dependencies in Asian stock market,

Economic Bulletin. 7 (2005) 1-6.

[15] C. Bonilla, R, Romero-Meza, M. J. Hinich,

Episodic nonlinearity in Latin American stock

market indices, Applied Economic Letters 13

(2006) 195-199.

[16] R. Romero-Meza, C. Bonilla, M. J. Hinich,

Nonlinear event detection in the Chilean stock

market, Applied Economic Letters 14 (2007) 987-

991.

[17] D. Wong, L. Cao, X. Gao, T. Li, Data mining in

oil price time series analysis, Communications of

The IIMA 6 (2006) 11-116.

[18] D. Pérez, M. J. Núñez, P.A Ruiz, Volatilidad del

precio de la mezcla mexicana de exportación,

Teo. Prác. 25 (2007) 37-52.

[19] S. Reitz, Non linear oil prices dynamics: A tale of

heterogeneous speculators?, German Economic

Review 10 (2008) 270-283.

[20] W. Shouyang, Y. Lean, K. K. Lai, Crude oil price

forecasting with TEI@I methodology, Journal of

Systems Science and Complexity 18 (2005) 145-

166.

[21] C. L. Aguiar, M. Soares, Oil and The

Macroeconomy: New Tools to Analyze Old

Issues, working paper, 2007.

[22] S. Radchenko, S. Oil price volatility and the

asymteric response of gasolina prices to oil

increases and decreases, Energy Economics 27

(2005) 708-730.

[23] G. Rapsomanikis, D. Hallam, Threshold

Cointegration in The Sugar-Ethanol-Oil Price

System in Brazil: Evidence rom Nonlinear Vector

Error Correction Models. Food and Agriculture

Organization (FAO). working paper, 2006.

[24] D. Hsieh, Testing for nonlinear dependence in

daily foreign exchange rates, Journal of Business

62 (1989) 339 - 368.

[25] C. Brooks, Testing for non-linearity in daily

sterling exchange rates, Applied Financial

Economic 6 (1996) 307-17.

[26] J. Scheinkman, B. LeBaron, B. Nonlinear

dynamics and stock returns, Journal of Business

62 (1989) 311-37.

[27] D. Hsieh, Chaos and nonlinear dynamics

application to financial markets, The Journal of

Finance 46 (1991) 1839-1877.

[28] W. Brock, J. Lakonishok, B. LeBaron, Simple

technical trading rules and the stochastic

properties of stock returns, Journal of Finance 47

(1992) 1731-1764.

[29] D. Hsieh, Nonlinear dynamics in financial

markets evidence and implications, Financial

Analysts Journal 51 (1995) 55-62.

[30] T. Kohers, V. Pandey, G. Kohers, Using

Nonlinear dynamics to test for market efficiency

among the major U.S. stock exchanges, The

Quarterly Review of Economics and Finance 37

(1997) 523-45.

[31] D. Patterson, R. Ashley, A Nonlinear Time Series

Workshop: A Toolkit for Detecting and

Identifying Nonlinear Time Series Dependence.,

Kluwer Academic Publishers, United States of

America, 2002.

[32] D. Skaradzinski, The nonlinear behavior of stock

prices the impact of firm size, seasonality, and

trading frequency, Ph.D. Thesis, Virginia

Polytechnic Institute and State University, 2003.

[33] A. Antoniou, N. Ergul, P. Holmes, Market

efficiency, thin trading and nonlinear behavior

evidence from an emerging Market, European

Financial Management 3 (1997) 175-90.

[34] P. Ammerman, Nonlinearity ando verseas capital

markets: evidence from the taiwán stock

exchange. Ph.D Thesis, Virginia Polytechnic

Institute, 1999.

[35] E. Ahmed, J. Barkley, J. Uppal, Evidence of

nonlinear speculative bubble in pacific-rim stock

markets, The Quarterly review of Economics and

Finance 39 (1999) 21-36.

[36] P. Ammermann, D. Patterson, D. The cross-

sectional and cross-temporal universality of

nonlinear serial dependencies evidence from

world stock indices and the Taiwan stock

exchange, Pacific-Basin Finance Journal 11

(2003) 175-95.

[37] K. Lim, M. J. Hinich, V. Liew, Episodic non-

linearity and non-stationarity in ASEAN

exchange rates returns series, Labuan Bulletin of

International Business & Finance 1 (2003) 79-93.

[38] F. Panunzi, N. Ricci, Testing non linearities in

Italian stock exchange, Rivista Internazionale di

9

Scienze Economiche e Commerciali 40 (1993)

559-574.

[39] A. Abhyankar, L. S. Copeland, W. Wong,

Uncovering nonlinear structure in real-time stock-

market indexes the S&P 500, the DAX, the

Nikkei 225, and the FTSE-100, Journal of

Business & Economic Statistic 15 (1997) 1-14.

[40] C. Brooks, M. J. Hinich, Episodic nonstationary

in exchange rates, Applied Economics Letters 5

(1998) 719-722.

[41] A. Afonso, J. Teixeira, Non-linear tests of weakly

efficient markets evidence from Portugal,

Estudios de Economía 19 (1998) 169-87.

[42] K. Opong, G. Mulholland, A. Fox, K. Farahmand,

The behavior of some UK equity indices an

application of Hurst and BDS tests, Journal of

Empirical Finance 6 (1999) 267-82.

[43] C. Brooks, M. J. Hinich, Bicorrelations and cross-

bicorrelations as nonlinearity tests and tools for

exchange rate forecasting, Journal of Forecasting

20 (2001) 181-96.

[44] R. Kosfeld, S. Robé, Testing for nonlinearities in

German bank stock returns, Empirical Economics

26 (2001) 581-97.

[45] J. Fernandez-Serrano, S. Sosvilla-Rivero,

Modelling the linkages between US and Latin

American stock markets, Applied Economics 35

(2003) 1423-1434.

[46] T. Panagiotidis, Market capitalization and

efficiency. Does it matter? Evidence from the

Athens stock exchange, Applied Financial

Economics 15 (2005) 707-713.

[47] R. Romero-Meza, C. Bonilla, M. Hinich M.

Nonlinear event detection in the Chilean stock

market, Applied Economics Letters 14 (2006) 987

– 991.

[48] M. Hinich, P. Rothman, Frecuency-domain test of

time reversibility, Macroeconomic Dynamics 2

(1998) 72-88.

[49] M. J. Hinich, D. Patterson, Detecting Epochs of

Transient Dependence in White Noise. Mimeo,

University of Texas at Austin, 1995.

[50] M. J. Hinich, Testing for guassianity and linearity

of a stationary time series, Journal of Time Series

Analysis 3 (1982) 169-176.

[51] J. Ramsey, P. Rothman, Time irreversibility and

business cycle asymetric, Journal of Money,

Credit and Banking 28 (1996) 1-21.

[52] D. Brillinger, M. Rosenblatt, Asymptotic theory

of estimates of k-th order spectra, Proceedings of

the National Academy of Sciences of the United

States of America 57 (1967) 206-210.

[53] F. J. Belaire, D. Contreras, Tests for time

reversibility: a complementary analysis,

Economics Letters 81 (2003) 187-195.

[54] K. Lim, R. Brooks, M. Hinich, M. [Online],

(2008), Are stocks returns time reversible?

International evidence from frequency domain

test, http://ssrn.com/abstract=1320165

[55] K. Lim, M. J. Hinich, V. Liew, Statistical

inadequacy of GARCH models for Asian stock

markets: evidence and implications, Journal of

Emerging Market Finance 4 (2005) 264-279.