Forecasting Crude Oil Price (Revisited) Imad Haidar* and Rodney C. Wolff * PhD student The University of Queensland, Brisbane, QLD 4072, Australia

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

Testing for non-linearity Three 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). where

The BDS test Return I ε =0.5ε =1ε =1.5ε =2 mWSIGW W W Return II ε =0.5ε =1ε =1.5ε =2 mWSIGW W W

The BDS test Return III mWSIG

The FCS test Create fuzzy Membership rules ClassMembership classDegree of Membership Rule no. Source: Kaboudan (1999)

The FCS results Data setFitted ARIMAR2R2 θDecision Oil price (all)Simple SL-NL-HN Oil price IARIMA(3,1,3) SL-NL-HN Oil price IISimple SL-NL-HN Oil return (all)ARIMA(0,0,5) NL-NL-HN Oil return IARIMA(3,0,3) NL-NL-HN Oil return IIARIMA(2,0,0) NL-WN 3-MA return II*ARIMA (2,0,6) SL-NL Wavelet return II** ARIMA (3, 0, 3) 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

Lyapunov exponents ReturnLyapunov Exponent99% Confidence level 1000 times bootstrap mλHighestLowest d E-180 d E-20 d 33.48E E-19 d d E-18 d 69.74E MA returnLyapunov Exponent99% Confidence level 1000 time bootstrap mλ1λ1 HighestLowest d E-180 d E-20 d E-20 d E-19 d E-19 d 62.82E

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 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.

ANN (cont.) b

ANN results Out-of-sampleBenchmarkSquaredWav1Wav23 MA Hit rate (%) RMSE R2R IC MSE MAE SSE DA P-value

ANN results (cont.) Metric3 days MA5 days MA ANNRWANNRW Hit rate RMSE R2R

Multi-steps Forecast where q is the number of lags and is the forecast horizon

Multi-steps Forecast

Forecast horizonHit rate renegeConfidence limitMean hit rate for 1000 tests 19 days52-60%95%56% 20 days52-60%95%56% 21 days52-59%95%56% 22 days52-58%95%55% 23 days51-57%95%54% 24 days52-58%95%54% 25 days52-58%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 steps ahead) with reasonable accuracy.

Thank you