^{1}

^{*}

^{2}

^{2}

^{2}

Precise recognition of a time series path is important to policy makers, statisticians, economists, traders, hedgers and speculators alike. The correct time series path is also a key ingredient in pricing models. This study uses daily futures prices of crude oil and other distillate fuels. This paper considers the statistical properties of energy futures and spot prices and investigates the trends that underlie the price dynamics in order to gain further insights into possible nuances of price discovery and energy market dynamics. The family of ARMA-GARCH models was explored. The trends depict time varying variability and persistence of oil price shocks. The return series conform to a constant mean model with GARCH variance.

Energy use is behind virtually everything a person comes into contact with. The energy industry has rapidly expanded and become increasingly interdependent. In developed economies, the increase in energy consumption indicates a reliance on energy and its related products for continued and sustainable economic growth and development. Developing economies also rely on the development of energy resources to drive their growth. Energy was once viewed just as a utility, and an enabler with limited consumer interest, but now, it is key in the struggle for sustainable future economic growth [

Energy prices, which are largely linked to oil prices, are a major concern for most economies. The recent financial crises and their ripple effects and after shocks have been largely unprecedented in terms of timing, speed and magnitude of impact on the world economies. Forecasting of crude oil prices is important for better investment and risk management and policy development, and econometric models are the most commonly used. Various authors have employed time series [

In this paper, we regard an observed price series,

correlation, white noise, innovation, and on a central family of models, the autoregressive moving average (ARMA) models [

In addition to the ARMA models, for the analysis of financial time series, we introduce the concept of volatility, which is pivotal in finance. Along side volatility we also consider the main stylised facts concerning financial time series [

This paper details the empirical analysis of financial time series data. We consider the official daily prices from the trading floor of the New York Mercantile Exchange (NYMEX) for a specific delivery month for Cushing Oklahoma West Texas Intermediate (OK WTI), Reformulated Blendstock for Oxygenate Blending (RBOB), and the number 1 heating oil futures contracts at close of business at 2:30 p.m. We begin the analysis from model identification, selection, estimation, diagnostic checking and finally forecasting. The analysis is done using MATLAB [

The full model under consideration falls under the general class of the combined

models with constant variance under the

If

an

When

We start with an exploratory analysis of the price data to establish if the set of properties, common across many stocks, in many markets over time periods, that have been observed by many studies [

To test for stationarity, the Dickey-Fuller (DF) test [

Once the models for the conditional mean and variance have been identified from the differenced series, we employ the maximum likelihood estimation (MLE) method to fit parameters for the specified model of the dif- ferenced series.

The final model for each series is then selected based on Akaike’s Information Criterion (AIC) which measures of goodness of fit or uncertainty for the range of values of the data. It measures the difference between a given model and the “true” underlying model. AIC is a function of the squared sum of errors (SSE), number of

observations n, and the number of independent variables

Model selection does not depend only on the goodness-of-fit of a model to the data, but also on the objective of the analysis. A model that is best in the in-sample fitting will not necessarily provide more accurate forecasts [

The forecasting ability of the model can only be determined by considering how well a model performs on data not used in estimating the model. It is common practice to partition the data into two sets, use the larger portion for estimating the model and the smaller one for testing the model. The test data can be used to measure model accuracy on new data. The size of the test data set should typically be about 20% of the total sample, although this depends on the sample size and the forecast horizon. In an ideal sense, the size of the test sample should be at least as large as the maximum forecast horizon required [

The sample standard deviation,

The k-day volatility forecast can also be found by using the GARCH model. Under the condition that returns are uncorrelated across days, the k-day variance as of

The volatility forecast over the future period from

where

Note that if

which is the unconditional variance.

In-sample and out-of-sample forecasting ability of models can be measured using the mean absolute error (MAE) and root mean squared forecasting errors (RMSFE) which measures the out-of-sample losses. The RMSFE assigns greater weight to large forecast errors. This fact is handled using the MAE which on the contrary assigns equal weights to both over and under predictions of volatility.

The realised volatility at each forecast date is calculated from the expression

Forecasting using GARCH models is obtained recursively [

An alternative way of writing Equation (7) is as shown in Equation (8)

where

From Equation (8), we can see that

This paper investigates the trends and patterns of return series for spot and futures prices of Cushing OK WTI, RBOB and the number 1 heating oil traded in the NYMEX, for the period running from 2nd January 2006 to 22nd May 2015. The data was obtained from the US Energy Information Administration (EIA) website http://www.eia.gov/petroleum/data.cfm, accessed on June 25, 2015. The US EIA is the statistical and analytical agency within the US Department of Energy. It is the principal agency of the US Federal Statistical System responsible for collecting, analysing, and disseminating energy information to promote sound policy-making, efficient markets, and public understanding of energy and its interaction with the economy and the environment. EIA is the premier source of energy information in the US and, by law, its data, analyses, and forecasts are independent of approval by any other officer or employee of the US government. [

and No. 1 heating oil traded at the NYMEX. The data appears non-stationary, with occasional jumps and spikes indicating heteroscedasticity. Testing the price series for stationarity using DF test shows that they all contain unit roots.

We employ the DF test for stationarity on the differenced series. This is a one-sided left tail test and

likelihood, summarized in

From

hypothesis whenever

After fitting the

The Kolmogorov-Smirnov (KS) test was employed to check normality of the residuals and the p-value was

Mean | −0.001 | −0.002 | 0.005 | 0.003 | 0.003 | 0.001 |

Median | 0.035 | 0.040 | 0.042 | 0.042 | 0.000 | 0.000 |

Maximum | 16.370 | 18.560 | 13.020 | 21.336 | 12.306 | 11.508 |

Minimum | −14.310 | −14.760 | −17.724 | −21.756 | −17.304 | −10.668 |

Skewness | −0.051 | 0.042 | −0.456 | −0.173 | −0.467 | −0.090 |

Kurtosis | 10.252 | 12.046 | 8.008 | 8.312 | 8.699 | 5.837 |

JB p-value | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 | 0.001 |

D denotes differenced series.

Series | p-value | Test statistic | Critical value |
---|---|---|---|

0.001 | −51.2762 | −1.9416 | |

0.001 | −50.9918 | −1.9416 | |

0.001 | −49.5626 | −1.9416 | |

0.001 | −50.9792 | −1.9416 | |

0.001 | −50.0207 | −1.9416 | |

0.001 | −50.6481 | −1.9416 |

Parameter | Estimate | Std Error | t-Statistic |
---|---|---|---|

Constant | −0.0003 | 0.0055 | −0.0494 |

AR(1) | 0.1064 | 0.2303 | 0.4622 |

AR(2) | 0.5423 | 0.2234 | 2.4273 |

AR(3) | 0.0277 | 0.0757 | 0.366 |

AR(4) | −0.7031 | 0.0729 | −9.652 |

AR(5) | 0.0806 | 0.2265 | 0.3558 |

AR(6) | 0.8426 | 0.2163 | 3.896 |

MA(1) | −0.1625 | 0.2304 | −0.7054 |

MA(2) | −0.5671 | 0.2374 | −2.3886 |

MA(3) | 0.0127 | 0.0852 | 0.1488 |

MA(4) | 0.7652 | 0.0712 | 10.7484 |

MA(5) | −0.1797 | 0.2365 | −0.7597 |

MA(6) | −0.8514 | 0.2529 | −3.3666 |

MA(7) | 0.0874 | 0.0380 | 2.2975 |

MA(8) | 0.0383 | 0.0189 | 2.0253 |

MA(9) | −0.0402 | 0.0185 | −2.1675 |

MA(10) | <0.001 | 0.0257 | −0.001 |

MA(11) | 0.0432 | 0.0242 | 1.7835 |

Variance | 3.069 | 0.0503 | 61.0522 |

found to be <0.001 implying the residuals are not normally distributed. In this case the

The Ljung-Box (LB) test on the residual series resulted in a p-value of 0.9781 indicating the residuals are not autocorrelated. Based on these results, we test for ARCH effects in the squared residuals using the LB test and obtained a p- value <0.001, indicating significant ARCH effects in the residuals.

We then specify a combined ARMA-GARCH model to capture the ARCH effects. Fitting a ARMA(6,11)- GARCH(1,1) had an AIC of 8810.9. However, a search for the best combined model revealed that an

The analysis shows the differenced series exhibit “stylised facts” typically seen in high frequency financial data. volatility clustering manifests itself as autocorrelation in squared and absolute returns, or in the residuals of the estimated conditional mean equation. Examining the ACF and PACF plots for the squared differenced series shown in

The significance of these autocorrelations at various lags was tested using the LB test and the Lagrange multiplier (LM) test at lags 1, 5 and 10, and the results are summarized in

Serial correlation in squared returns, or conditional heteroscedasticity can be modelled using GARCH models. GARCH models allow for the volatility to evolve with time. A GARCH model can be expressed as an ARMA model of squared residuals and hence many of it’s properties follow easily from those of the corresponding ARMA process. If the fourth order moment of a GARCH (1,1) exists, the kurtosis implied by a GARCH (1,1) process with normal errors is greater than that of the normal distribution which is 3 [

We fit several

For the differenced crude futures series, the model with the smallest AIC was GARCH (19, 16) with an AIC of 8776.4 as shown in

Lag | LM -test | |||||
---|---|---|---|---|---|---|

1 | 5 | 10 | 1 | 5 | 10 | |

348.786 | 677.980 | 821.867 | 431.691 | 564.068 | 608.241 | |

(<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | |

383.810 | 696.811 | 800.000 | 287.646 | 346.920 | 358.553 | |

(<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | |

39.142 | 101.436 | 157.990 | 3.614 | 4.324 | 5.329 | |

(<0.001) | (<0.001) | (<0.001) | (0.0573) | (0.5038) | (0.8681) | |

317.298 | 858.416 | 962.298 | 311.786 | 792.509 | 866.792 | |

(<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | |

45.923 | 180.863 | 271.397 | 1.344 | 5.341 | 5.720 | |

(<0.001) | (<0.001) | (<0.001) | (0.2463) | (0.3758) | (0.8382) | |

79.112 | 251.739 | 379.028 | 23.281 | 38.898 | 46.177 | |

(<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) | (<0.001) |

p-values are indicated in the parenthesis.

From these results, it can be seen that most of the in-between lags are not significant, the most important lags at 16 and 19. An analysis on the residuals of this model show that this model gives a fairly good fit. As per the AIC values, most these models do not seem to be significantly different and so the lags may be reduced to gain parsimony. An examination of the plots of the standardised residuals after fitting the GARCH(1,1) model for the return series indicates that the residuals are, on the overall, stable with some clustering. The standardised residuals exhibit no residual autocorrelation. It therefore makes more sense just to fit a

The other series also behave in much the same way as the crude futures. Fitting a

Parameter | Estimate | Std Error | t-Statistic |
---|---|---|---|

Constant | 0.175 | 0.103 | 1.701 |

GARCH (16) | 0.090 | 0.169 | 0.530 |

GARCH (19) | 0.348 | 0.200 | 1.743 |

ARCH (1) | 0.121 | 0.022 | 5.475 |

ARCH (2) | 0.071 | 0.053 | 1.340 |

ARCH (5) | 0.038 | 0.052 | 0.729 |

ARCH (6) | 0.037 | 0.046 | 0.811 |

ARCH (8) | 0.026 | 0.033 | 0.780 |

ARCH (9) | 0.007 | 0.037 | 0.188 |

ARCH (10) | 0.050 | 0.037 | 1.346 |

ARCH (13) | 0.022 | 0.037 | 0.585 |

ARCH (14) | 0.036 | 0.038 | 0.955 |

ARCH (15) | 0.013 | 0.036 | 0.348 |

ARCH (16) | 0.083 | 0.036 | 2.288 |

Parameter | ||||||
---|---|---|---|---|---|---|

0.0191 | 0.0265 | 0.3354 | 0.5388 | 0.0743 | 0.0428 | |

0.9464 | 0.9388 | 0.8236 | 0.8318 | 0.915 | 0.9429 | |

0.0475 | 0.058 | 0.1072 | 0.1036 | 0.0642 | 0.0451 | |

( | 0.9938 | 0.9916 | 0.9308 | 0.9354 | 0.9792 | 0.9878 |

(

processes return to their means after some time. The implication of this high persistence is that these shocks have a significant effect on the prices in these markets.

ForecastingThe total length of the data under consideration is 2363 data points. This data set was divided into two sets, the first sub-sample, used for training the model contained 2063 points and the second one, used for testing, con- tained 300 points.

In order to regain a forecast of the price series, the differencing on the forecast series obtained from the forecasts of the inferred conditional variances is undone. However because the inferred variances are con- ditioned on a standard normal error, there are several possibilities and, having many different forecasts and averaging would give a stable forecast for the crude futures prices, which can then be used to predict future crude oil futures prices. In

This paper employs statistical and econometric techniques to investigate and model financial time series trends in energy markets. To do this, daily closing prices for a period of about 10 years for Cushing OK WTI, RBOB and number 1 heating oil spot and futures contracts traded in the NYMEX are considered. The paper also investigates the existence of stylised facts in these series, in order to fit an appropriate model that adequately describes the market dynamics.

Price data are tested for stationarity using the DF test and results show the existence of unit roots. Normality is tested for using the JB test and non-normality is established. Return series are then generated from the price series through differencing, and then tested for stationarity using the DF test. The results reveal that return series are indeed mean stationary, but are definitely not variance stationary. Several ARMA models are fit to the return series and the standardised residuals analysed. For the crude futures return series, the best model turns out to be

an

We then propose a combined

this and all the other six series with the residual analysis conforming to the assumption of Gaussian Innovations. GARCH models can therefore adequately model the trends and patterns in the energy markets. The trends also depict time varying variability and high persistence of oil price shocks. These shocks therefore have a significant impact on the prices of these energy prices.

Jane Aduda,Patrick Weke,Philip Ngare,Joseph Mwaniki, (2016) Financial Time Series Modelling of Trends and Patterns in the Energy Markets. Journal of Mathematical Finance,06,324-337. doi: 10.4236/jmf.2016.62027