A Study on S&P 500 Index Stock Return and Volatility using ARIMA and GARCH Modeling
Kaiyuan Song, Di Wu Summary
In this project we first checked consistency and seasonality of S&P500 index stock performance by splitting its recent twenty years historical data into ten two year data and built ARIMA and GARCH models for each sub-period. We found that the models are considerably consistent before 2007-2008 sub-period, and there exists some minor seasonality in several subperiods, but no particular pattern can be identified for the whole period. We then tried to predict future return, volatility and VaR using the model we built for the last sub-period based on rolling forecast procedure. Though the fitted values of 10th sub-period model are very acceptable, the predicted values are reasonable yet far from satisfactory. Only some future volatility can be predicted using one-step ahead rolling forecast, and return prediction is not much better than just using historical mean, which is almost 0, to predict. These results suggest that external variables are needed for more accurate predictions, time series models alone are not sufficient.

Data
S&P500 index daily closing price from 1993 to 2012 are obtained from yahoo finance website. It is one of the best measures of current state of U.S. domestic economy, therefore by studying its fluctuations, consistency, seasonality and make predictions, one can determine if it is a good time to invest in U.S. stock market.

Methodology
We first examined the time series plots, ACF plots and PACF plots of whole period and each sub-period. Then ARIMA models for log-returns mean part and GARCH models for the conditional volatility of the log-returns are fitted. After selecting our final models for each subperiod, the final models were compared in order to check for model consistency and potential market seasonality. Finally the model for last period was used to forecast future daily return, volatility and corresponding Value at Risk.

Analysis
By inspecting the time series plot of both the stock price and log-return of entire period, we found that the series is not suitable to analysis as a whole. The underlying trend changed several times throughout the period and the variances were not constant. Therefore we broke the data into ten sub-periods with equal length and fitted models for each period, and tried to build tentative ARIMA+GARCH models for each sub-period. All of our tentative models were selected via standard ARIMA and GARCH model building procedures, all of the ARIMA models were suggested by either ACF, PACF or EACF of log-return data for corresponding time periods. Next we eliminate some models through model diagnosis such as residual ACF, PACF, EACF and Ljung-Box test result checking. Our final models were selected via AIC comparison of remaining models. The GARCH models were built on the residuals of our final ARIMA models, or simply on the averaged log-return if the log-return data for a sub-period is a white noise. All of GARCH(1,1), GARCH(2,1), GARCH(1,2) and ARCH(p), if suggested by PACF of squared residuals, are tried. We kept the model that passed most diagnosis checks as our final GARCH model for each sub-period. The table below shows the order of our final models.

From the table above, we observed that there were strong evidence suggesting that our models were consistent before 2007-2008 period. Most of the mean parts were suggested by ARMA(0,0) and GARCH(1,1) seemed like a common choice for each sub-period. However, after 2007-2008 year, the ARIMA models for mean part became distinguished, especially for the period of 2007- 2008, where ARIMA model goes to an unusual high order. This high order may be caused by the major financial crisis, which interrupted the model consistency as well. At the same time, weak seasonal pattern for several sub-period could be identified, but due to differences in seasonal orders, no seasonal pattern could be confirmed for the whole period. As for goodness of fit of our models, all actual return vs fitted return with fitted volatility plots were checked. All of the fitted returns are very close to zero as expected and all fitted volatilities vary according to fluctuations in actual returns: it goes up when there were large fluctuations and vice versa. Two sample plots are shown below:

Our prediction model, the tenth period model, fitted the data especially well as illustrated below.

Not only the volatility prediction were accurate, the mean part also provides considerably nice fit. With the excellent fit from our prediction model, we expect our predictions to be fairly dependable. Nonetheless, when comparing actual future return with our predicted return and volatility obtained from 5-day ahead rolling forecast procedure, the results were rather unsatisfactory. All of the predicted volatilities were considerably high and did not move along with real fluctuations in return series, which resulted in very significant value at risk. In addition, the return predictions were no much better than just using sample means, which were all very close to zero, to predict future return. The prediction vs actual return plot for 60 days is shown below.

To improve our predictions, particularly for volatility part, one-step ahead rolling predictions were computed, and its prediction vs actual return plot is illustrated below:

Due to the return predictions made by ARIMA were similar to one-step results and not much better than sample mean prediction, we focused on volatility part and found that one-step ahead rolling prediction provided slightly better volatility predictions. In this case the predicted volatility increases a little bit when the fluctuations in actual return series is big.

Conclusions and discussions
Our analysis shows that ARIMA and GARCH models for S&P500 index stock return was indeed consistent through the first seven sub-periods, but no particular seasonal pattern can be found. This result indicates that when the stock market is at its normal state, there are regularities exist and can be traced with one model for a relatively long time. As for predictions, our result exemplifies that even a very well fitted model cannot grant reliable forecasts. This may be caused by ARIMA+GARCH model limitations, since often an ARIMA+GARCH model presents excellent fit to a dataset, while provides very poor out of sample predictions. We expect better prediction performance if external factors such as quantified anticipated events are added to our model, as the sudden changes can then be better explained by such factors and therefore error reduction.

References 1. “A Hybrid ARIMA and Support Vector Machines Model in Stock Price Forecasting” in Omega, the International Journal of Management Science. vol. 33, no. 3, 2005.

2. Ruey S. Tsay, “Analysis of Financial Time Series”, Wiley, 2002. ISBN: 0-471-41544-8