# A GARCH Analysis of Equity Returns in Energy Industry

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2206 A GARCH Analysis of Equity Returns in Energy Industry

Michale Yu

12/9/2012

Introduction
Stock market returns are predictable from a variety of financial and macroeconomic variables and have long been an attraction for equity investors. Recently increasing attention has shifted on how to make time series models to fit the financial data and forecast. Stock return series exhibit certain pattern, which can be captured and quantified using fine tuned model.

The main goal of this project is using ARMA and Garch Model to analyze return series of US energy firms (i.e. Oil firms listed on NYSE) and to catch and quantify the impact of external economical and political shocks and their potential long lasting effect. We focused on two stocks: Suncor Energy Inc. (NYSE: SU) and Chevron Corporation (NYSE: CVX). Suncor Energy Inc. is a Canadian integrated energy company listed on NYSE. It specializes in production of synthetic crude from oil sands. Chevron Corporation is a typical US based multinational oil corporation. Since the energy stock price is highly correlated with crude oil price, our team also analyzed the variance caused by firm specific factors through controlling crude oil as an exogenous variable into the model (ARMAX).
Model Construction and Definitions
2.1ARMA and ARIMA model
2.1.1ARMA model
The process is said to be ARMA (p, q) process if {Ext} is stationary and if for every t,
,
where Zt~WN (0, ).
2.1.2 ARIMA model
Arima model is the modification of ARMA which replacing Xt term with 〖(1-B)〗^pXt, the lag-difference term.

2.2 ARCH and GARCH model
2.2.1 GARCH model
Generalized AutoRegressive Condition Heteroskedasticity models are used to characterize and model time series when it’s believed that the variance of the term is a function of the previous variance and terms. That is Var ( ) are related to and Var ( ).
2.2.2 ARMA-GARCH model
If an ARMA model is assumed that the error terms have conditional variance, the model becomes ARMA-GARCH. In that case, the ARMA(p,q)-GARCH(r, k) model (where r is the order of the GARCH terms and k is the order of the ARCH terms ) is given by

Assumptions and Data Prepartion
Assumptions
3.1.1 Weakly Stationary
Before fitting time series model, we assume that the sequence of data is Weak Stationarity: Generally speaking, the mean and the variance of a stochastic process do not depend on t (i.e. they are constant) and the auto covariance should only depend on the lag p (p is an integer, the quantities also needs to be finite). This assumption can be justified by Dicky-Fuller test.
3.1.2 In the ARMA model, we assume that the innovations are iid White noise distributed as WN (0, ).
3.1.3 In the ARCH/GARCH model, we assume that the data does not have significant autocorrelation but have volatility clustering; in other words, we assume that the variance of the each random variable is dependent on the historical information we have observed.
3.2 Data Preparison and Data package
In this paper, we choosed daily closing price of 2 US oil stocks (NYSE:SU, NYSE:CVX) that spanning ten years from Yahoo Finance to fit the time series model. The data spans from 1/3/2000 to 11/20/2012. This project also controled oil price as an exogenous variable. WTI Crude oil daily closing spot price was extracted from Wikiposit.org for the same time span. Matlab script was used to automate data extraction from Yahoo Finance. Data analysis and model fitting is done in R and Matlab.
3.2.1 Dividend adjustment and Missing missing data
Since all dividends paid to shareholders, dividend ajusted returns were calculated. For missing crude oil spot price due to the different holiday schedule between NYSE and Chicago Commodity exchange, we used previous business day closing price instead.
3.2.2 Detrend
The closing price series are clearly upward trending. In order to detrend and get stationary series, one step difference and log transformation were applied. The result (log return series) is stationary series, which passed our stationary test.
Model Construction and empirical results
4.1 ARMA model
The log return series passed the stationary test (the resulted Augemented DF test is -15.7585, and the p value is 0.01); no further lag difference is required. Therefore an ARMA (p, q) model is suggested. According to the AIC criterian, we arrive an ARMA (2, 2) model.
The fitted model is
.
4.2 ARMAX model
It is clear that oil stock price is highly correlated with crude oil price. Therefore, we decided to run a simple linear regression in order to test how strong this relation is.
4.2.1 Linear Regression against Oil Price
The linear regression result shows that Oil price and Oil Stock Prices have strong positive correlation. This result inspires us to add an exogenous variable into the ARMA model.
The estimated results of linear regression is .
From the fitted model, we found the slope term (valued 0.892) on the regression model is significantly not deviant from zero (t-statistic equals 24.757, and the p-value smaller than 0.001).
4.2.2 ARMAX model
ARMAX offers the ability to control oil price as an exogenous variable. The standard formula of ARMAX is
,
Where X stands for the exogenous variable.
A striking advantage of ARMAX model is that ARMAX model has the capacity to reflect a long-term dependence of Stock Prices on the Oil Prices. The reason is stated as follows:
Since the formula of ARMAX model: can be rephrased as a ARX(∞) model,
, where and are respectively the ith coefficients of the infinite power polynomials ( ).

Hence we observed that in the ARMAX model, if for , the exogenous variable, Xt will have a long time effect on . That is, remains certain correlation to even though p is very big.

The result of ARMAX, after MLE estimation, is

Moreover, ARMAX model slightly improve the fitting results. The following two plots, correspondently showing the residual of ARMAX and ARMA, can support this conclusion:

4.3 GARCH model
After calculating the coefficients of ARMAX model, we plot the acf of the residuals and the square of the residuals. The plots are exhibited below:

The above plot gave us a strong signal that we may nee to apply GARCH model to fit the noise terms of ARMAX.
Therefore our final fitted model is modified to be
ARMAX:
with
and has the autoregressive process:

The coefficients in each model above are estimated by numerically maximizing the likelihood function(Quasi-Newton methods):

Two plots:
The estimated conditional variance (left) and the innovations (Zt) of the GARCH:

The forecasting plot of Garch:
And the forecasting plot data is displayed below:

From the forecasting plot, we found out that the ARMAX-GARCH model gave a very accurate predication, with modest errors. However, this accuracy may be the special case. Because according to economic theory the oil stock price is driven by economic and political news; in most of time, it should deviate from the forecasted value, which is purely based on the historical information. Only during the relative quiet period (rare case), the forecasted value which is based on history will close to true value. More data is required to test on this, which is beyond the scope of this project and left for future study.
Interpretation of Results and Prediction
5.1 Interpretation of Results
Fist of all, from the original return time series plot, the variance is high around the major economic events (such as 2001 “911 attack”, 2008 “Financial Crisis”). The ARMA model’s constant variance assumption is clearly violated. In addtion, the innovations are iid distribution may not be the case. By fitting garch model with the residual, we may loose this assmption to: The innovations have conditional variance .The innovation plot does show difference variances along the time.
Secondly, by controlling exogenous variable (i.e. Crude oil price), the residual and variance decreased significantly. However, we didn’t find different pattern in the residual plot between the Suncor and Chevron, even around the major economics events.
Thirdly, after garch model fitting, the conditional variance series is a better measurement of the variance of the time period.
5.2 Prediction
We used fitted model to forecast for future 10 periods, the model calcuated variance increased quickly along the time. From the plot (page 5), we can also found the variance (95% confidence band) increased quickly when moving forward.
Conclusion
First of all, we demonstrated that fine tuned ARMA, Garch model has forecasting power.
Secondly, ARMA model can increase prediction power and reduce the error variance by including proper exogenous variables.
Last but not least, Autoregressive model (and its modifications) with Garch Noise assumption delivers good fitting performances on oil stock’s prices data. This is justified by the fact that stock’s prices (their residuals after autoregressive) are squared-autocorrelated. The Garch model fitted conditional variance will be good measure of the variation and provides a more convincing forecasting confidence interval.

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