S&P 500 Daily Returns Analysis


In this study, we used the adjusted closing prices of the S&P 500 Index between Jan 1st, 2000 and May 9th, 2009 (2351 observations).

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The data plot above shows the log daily-returns, the 20 day weighted-moving average (WMA) and the exponential weighted volatility (EWV/EWMA).


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The summary statistics above describe a symmetric fat-tailed (leptokurtic) probability distribution for daily log returns.

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We conduct a few additional statistical tests: (1) White-noise (Ljung-Box), (2) Normality Test and (3) ARCH effect. As one may expect, the probability distribution is not normally distributed, and the log-returns exhibit serial correlation and ARCH effect.


We are ready to examine and compare different models and select the one that best fits the data. We begin with considering the GARCH model and move on to the GARCH-M and EGARCH models. In each case, we evaluate the model assuming both normal and non-normal distributed innovations. Finally, we summarize those model properties and recommend the one that best fits the data.

In the following table, we summarize the log-likelihood function for the selected models:

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EGARCH(1,1) with GED innovations has the best fit for the data.

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The EGARCH(1,1) conditional volatility move together with EWMA, but it is consistently lower, and in many cases it time-leads the change.


The EGARCH model is used next to forecast out-of-sample conditional volatility. The data sample ends on Friday, May 8th, 2009, and each step is simply a workday.

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The EGARCH(1,1) with GED innovations seems like a reasonable model for the S&P 500 daily-log returns; it has the highest log-likelihood value and the model assumptions are largely satisfied. Nevertheless, the daily log-returns exhibit serial correlations that EGARCH does not capture.

The volatility as of EOD May 9th, 2009 is higher than its long-run value, so the EGARCH model forecasts a steady decline (reversion to the mean) over the next 3 months.

The attached white-paper document and Excel workbook illustrate in detail the process of examining each model, calibration, and validation of the assumption (i.e. diagnosis). To download the white-paper or the associated spreadsheet model, please login or register with us.

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