Forecasting Crude Oil Price (Revisited)

Imad Haidar* and Rodney C. Wolff * PhD student

The University of Queensland, Brisbane, QLD 4072, Australia

E-mail: i.haidar@uq.edu.au, rodney.wolff@uq.edu.au

This paper attempts to answer the following questions:

• What type of dynamics is governing crude oil prices and returns? – Specifically, we investigate if there is any non-linear deterministic

dynamics (chaos) which could be miss specified as a random walk.• From a statistical point of view have the dynamics of crude oil

returns changed significantly during the past twenty years?• Do we have strong empirical evidence that crude oil spot

returns are predictable in the short-term? • Can we forecast the direction of crude oil return for multi-

steps ahead?

Data• Crude oil daily spot prices/ returns for West Texas

Intermediate (WTI),• official closing price are from 2 January 1986 to 2 March 2010

(6194 daily observation).• The data were retrieved on 11 March 2010 from the Energy

Information Administration (EIA).

Diagnostic Tests

• The autocorrelation (ACF) is much more evident in the squared log-returns, especially for return II;

• AFC was significantly over the upper confidence level. This could present evidence of heteroskedasticity.

• The Augmented Dicky-Fuller and Phillips–Perron test for crude oil price and return at 1% significant level are as follows:– Crude oil price for the whole series from January 1986 to March 2010

is integrated of the first order, or I(1). – Crude oil price for the first subsection from January 1986 to January

1998 is I(0).– Crude oil price for the second subsection from end of January 1998-

March 2010 is I(1),– The returns for all subsections are I(0).

Testing for non-linearityThree tests:

• The Brock, Dechert and Scheinkman (BDS) test (Brock et al. 1996);

• The Fuzzy Classification System (FCS), by Kaboudan (1999);• The Time Domain Test for Non-linearity, by Barnett and Wolff

(2005);

The BDS test

• The Correlation Integral measures how often a temporal pattern appears in the data.

• The null hypothesis is that the data are pure whiteness (iid).

m

nmBnmW

),,(),,(

]),,1(),,([),,( mnCnmCnnmB

where

The BDS testReturn I

ε =0.5 ε =1 ε =1.5 ε =2m W SIG W SIG W SIG W SIG2 63.28235 0 32.41485 0 28.294 0 24.21126 05 277.9648 0 63.43841 0 41.301 0 34.10912 08 1569.915 0 113.0694 0 51.747 0 38.16579 0

Return IIε =0.5 ε =1 ε =1.5 ε =2

m W SIG W SIG W SIG W SIG2 69.901 0 30.065 0 21.751 0 19.484 05 370.17 0 80.356 0 37.184 0 30.265 08 2574.9 0 194.74 0 57.465 0 36.404 0

The BDS testReturn III

m W SIG2 16.08305 05 30.51802 08 47.82609 0

The FCS test

M

m ss

yy

emCemCemCemC

M2 21

211

),(ln),(ln),(ln),(ln

)1(

2RARIMACreate fuzzy Membership rules

75.085.075.095.085.0

195.01

0)1.0/)85.0(1(

1)05.0/)95.0(

0

Class Membership class Degree of Membership Rule no.

3938373635

Source: Kaboudan (1999)

90.0

90.070.0

70.0

2

2

2

R

R

R

1)2.0/)90.0(1(

02R

321

SL

HNNL

The FCS results

Data set Fitted ARIMA R2 θ DecisionOil price (all) Simple 0.99 0.96 SL-NL-HNOil price I ARIMA(3,1,3) 0.90 0.94 SL-NL-HNOil price II Simple 0.90 0.87 SL-NL-HNOil return (all) ARIMA(0,0,5) 0.005 0.96 NL-NL-HNOil return I ARIMA(3,0,3) 0.01 0.93 NL-NL-HNOil return II ARIMA(2,0,0) 0.02 0.99 NL-WN3-MA return II* ARIMA (2,0,6) 0.77 0.00 SL-NLWavelet return II**

ARIMA (3, 0, 3) 0.018 0.29 WL-NL

SL: strongly linear; NL: non-linear; HN: high noise; WN: white noise. * is smoothed return II with a simple three days moving average. ** is filtered return II using a wavelet filter.

The FCS over time

Testing for Chaos

• Chaos is characterize by sensitivity of a time series to the changes in the initial condition.

• Lyapunov exponents (LE) is a quantitative measure of the existence of chaos.

– Let be a dynamical system where are the data, e is a random error and f is a non-linear function. Also, let be the Jacobian matrix of f, and

– Lyapunov exponents λ are estimated as – – where represents the largest eigenvalue of the Jacobian matrix

Lyapunov exponents Return Lyapunov Exponent 99% Confidence level

1000 times bootstrapm λ Highest Lowest

d 1 0.0924 1.45E-18 0d 2 0.1951 0.063865 -2.20E-20d 3 3.48E-19 0.078794 -1.10E-19d 4 0.0204 0.176158 0.021725d 5 0.0318 1.875162 -1.30E-18d 6 9.74E-20 0.648368 -0.0502

3MA return Lyapunov Exponent 99% Confidence level 1000 time bootstrap

m λ1 Highest Lowestd 1 0.07 1.32E-18 0d 2 0.3405 0.062429 -3.10E-20d 3 0.1101 0.046871 -8.40E-20d 4 0.0766 0.127102 -1.40E-19d 5 0.1353 2.08148 -9.30E-19d 6 2.82E-19 0.594717 -0.03211

Lyapunov exponents

• We cautiously conclude that, the dynamics of crude oil returns series are non-linear deterministic, possibly chaotic.

• This conclusion contradicts the findings of Moshiri and Foroutan (2006) in which they found no evidence of chaos in crude oil futures price.

• It is important to note that Moshiri and Foroutan (2006) were testing LE using raw price of crude oil futures contracts and not the spot return.

FORECASTING

We use three types of Models:• ARIMA• EGARCH• ANN

ANN

• ANN were designed in an attempt to imitate the human brain functionality;

• the fundamental idea of ANN is to learn the desirable behaviour from the data with no a priori assumptions.

• From an econometrics view, ANN falls in the non-linear, non-parametric, and multivariate group of models (Grothmann 2002).

• This makes it a suitable approach to model non-linear relationship in high dimensional space.

Input layer Hidden layer Output layer

ANN (cont.) ji

j

iji wxbu

)( ii ufy

Σ

ws1

wsn

ws3

ws2

xn

x3

x2

X1

b

Input

Summing junction

Base

OutputActivation function

weights

y

b

• Training neural network in the backpropagation paradigm involves continuous change to the values of the network parameters in the direction that reduces the error between the input and the target

• • • • where w is the network weights, t is

the current step, T the number of iterations, λ is the learning rate (step size), is the gradients of the error surface, E the global error

ANN resultsOut-of-sample Benchmark Squared Wav1 Wav2 3 MA

Hit rate (%) 48.82 55.12605 61.99 74.97 79.67

RMSE 0.0356 0.032827 0.0337 0.0196 0.0131

R2 0.0182 0.004447 0.0293 0.0837 0.6219

IC 0.7529 0.965628 0.7554 0.9693 0.806

MSE 0.0013 0.001164 0.0011 0.0004 0.0002

MAE 0.0253 0.02222 0.0233 0.0114 0.0093

SSE 0.3756 0.27694 0.3208 0.1127 0.0829

DA -0.4304 1.51874 4.011 8.4024 13.45

P-value 0.6665 0.091106 0.0008 0 0

ANN results (cont.)

Metric 3 days MA 5 days MA

ANN RW ANN RW

Hit rate 79.67 72.40265 83.52 80.97166

RMSE 0.0131 0.012578 0.0066 0.011199

R2 0.6219 0.408819 0.7302 0.611197

Multi-steps Forecast

Input layer Hidden layer Output layer

Input layer Hidden layer Output layer

)(

)(

)1(ˆ

qt

t

t

y

y

y

Input layer Hidden layer Output layer

)(ˆ ty

)1(

)1(

)2(

ˆ

ˆ

qt

t

t

y

y

y

)(ty

where q is the number of lagsand is the forecast horizon

Multi-steps Forecast

Forecast horizon Hit rate renege Confidence limit Mean hit rate for 1000 tests

19 days 52-60% 95% 56%

20 days 52-60% 95% 56%

21 days 52-59% 95% 56%

22 days 52-58% 95% 55%

23 days 51-57% 95% 54%

24 days 52-58% 95% 54%

25 days 52-58% 95% 55%

Conclusion

• The BDS statistic indicates the existence of non-linear behaviour in all crude oil prices and returns subseries .

• The FCS test also suggests that the dynamics of crude oil series are non-linear stochastic.

• Finally, the Lyapunov exponents for crude oil returns (and smoothed returns) highlights the possibility of low dimensional deterministic dynamics, i.e., chaos.

• The Lyapunov exponent results could explain the random-walk like behaviour of the crude oil return.

Conclusion (cont.)

• Several data transformations and smoothing with the hope that we could reduce the noise and highlight certain dynamics, such as mean reversion.

• Our empirical results showed that some of these measures are effective in improving the forecast accuracy.

• We show that for smoothed data multi-step forecasting is possible (for 19-25 steps ahead) with reasonable accuracy.

Thank you