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 [1] [2] .

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 [3] [4] , financial [5] and structural models [6] in forecasting crude oil prices. Financial studies involve asset returns [7] - [9] , instead of prices, since, firstly, returns give a complete and scale-free summary of the investment opportunities and secondly, return series have nice statistical pro- perties known as stylised facts. These properties have important implications in model selection. Some of these interesting phenomena include volatility clusters which relate to the observation that the magnitudes of volatili- ties of financial returns tend to cluster together [9] [10] , the leverage effect which relates to the asymmetry of news impact [11] , leptokurticity, calendar effects and non-linear dependence.

In this paper, we regard an observed price series, as a particular realisation of a stochastic process. Standard time series analysis rests on important concepts such as stationarity, ergodicity, auto-

correlation, white noise, innovation, and on a central family of models, the autoregressive moving average (ARMA) models [12] [13] . A stationary time series can be usefully described by it’s mean, variance and autocorrelation or spectral density function [12] [14] , and the joint probability distribution of any two observations, say, , and is the same for any two time periods t and separated by the same lag k [15] . For an ergodic process, we need not observe separate independent replications of the entire process in order to estimate its mean value or other moments [16] .

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 [8] [13] [17] . In order to capture these stylised facts, we fit the autoregressive conditional heteroscedastic (ARCH) model first developed by Engle [18] and later generalized by Bollerslev [19] to generalised autoregressive conditional heteroscedastic (GARCH) model. A GARCH model is a parametric error distribution, which does not only capture volatility clustering but also accommodates some of the thick tails commonly found in 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 [20] . The rest of the paper is organized as follows: Section 2 discuses the methods used in empirical analysis, Section 3 gives a brief discussion of the data used, and the empirical analysis and the results obtained and finally Section 4 gives a summary of the work and the findings.

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. [25] .

Figure 1 shows various series plots for daily spot and futures prices for Cushing OK WTI, RBOB gasoline

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.

Table 1 gives a summary of the descriptive statistics and results for the JB test for the differenced price series. From the measure of skewness, all the series except the crude spot are skewed to the left. These series exhibit positive excess kurtosis, and these are some of the stylised facts observed in financial time series data. Based on the p-values of the JB test, we reject the null hypothesis of normality for all the differenced series.

Figure 2 shows plots of the differenced crude futures and spot prices. The series appear mean stationary but are not variance stationary. Volatility clustering, jumps and spikes are also observed.

We employ the DF test for stationarity on the differenced series. This is a one-sided left tail test and Table 2 gives the summary report of the DF test on the six series. Stationarity is confirmed for all these series, and hence they can be subjected to further time series analysis

Figure 3 and Figure 4, show the ACF and PACF plots of crude futures and spot price series. From these plots, it is not easy to tell the form of, significant lags are noticed up to lag 39 of the 40 plotted. The same behaviour is exhibited by the other four series. The ACF and PACF fail to identify the form of the model. We proceed to fit several models for and, and the one with the smallest AIC is picked. Figure 5 shows a plot of AICs for various models fit for, from which we see that the best model is an with an AIC of −4684.1. This model has the parameters estimated via the maximum

likelihood, summarized in Table 3. For the crude spot data, the best model is an. A similar analysis was also carried out on the differenced gasoline futures and prices and the results were similar with high orders of autoregressive and moving averages.

From Table 3, we can use the values of the t-statistic to check the significance of the parameters. The greater the absolute value of t, the greater the evidence against the null hypothesis that there is no significance dif- ference between the parameter estimate and 0. We can use the general rule of thumb that we reject the null

hypothesis whenever.

After fitting the model, we standardized the inferred residuals and checked for normality.

Figure 6 shows two plots. The first one is the residual series which shows a lot of clustering, suggesting non- normality and non-constant variance. The second plot of Figure 6 is a histogram of the residuals also showing evidence of non-normality. Figure 7 shows a quantile-quantile plot and a box plot of the residuals and from these, we can also deduce non-normality of the residuals largely because of the extreme values.

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

D denotes differenced series.

found to be <0.001 implying the residuals are not normally distributed. In this case the does not conform to the assumption normality in the innovations.

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.

Figure 8 also shows evidence of serial autocorrelation on the squared residuals. The PACF plot shows autoregression of order one so should capture the 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 model gave the best fit with an AIC of −4.4037. From this analysis, we see that a more appropriate model to use would be one with a constant conditional mean and a conditional variance.

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 Figure 9 and Figure 10, reveal serial correlation of the squared differenced series. Gasoline and heating oil series exhibit the same behaviour.

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 Table 4. As is evident, there is enough evidence to reject the null hypothesis of no ARCH effects as per the LB test since the test indicates that there are significant ARCH effects in the all squared series. However, based on Engle’s ARCH test, which is the LM test, we do not have sufficient evidence to reject the hypothesis of no ARCH effects for the gasoline futures and the heating oil futures series. For all the other series we reject that hypothesis of no ARCH effects.

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 [19] . For this reason, a GARCH model with normal errors can replicate some of the fat tailed behaviour observed in financial time series, though, most often, a GARCH model with a non-normal error distribution is required in order to fully capture the observed fat tail behaviour in financial time series.

We fit several model to the differenced crude futures series for. We estimate the parameters, and of the model and the in-sample estimates of the volatility. The Matlab command garch, models the return series as and estimates the parameters, and via maximum likelihood.

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 Figure 11 which shows the AICs of all the models fitted. The best model according to these fits have the parameters summarized in Table 5.

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 for this data.

The other series also behave in much the same way as the crude futures. Fitting a model on all the six series gives the results in Table 6. From the measure of persistence, we see the highest persistence of 0.9938 in the series, although, all other return series are quite persistent. This indicates that the volatility

() measures the persistence.

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.

The 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 Figure 12, we see one of the possibilities of the forecast prices. The RMSFE for this model is 0.2072, an indication that the forecasts are generally good.

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 an for the crude spot it is an among others, based on their AICs. These resultant models however contravene the Gaussian innovation assumption.

We then propose a combined model to capture the ARCH effects in the variance, then find that the best model under these circumstances is the, implying a constant mean conditional variance equation. We finally find that for example, for the crude futures return series, the is the best model based on the AIC, although a does quite well for

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