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S&P 500 Daily Returns Analysis
| Attachment | Size | |
|---|---|---|
| White-paper | ||
| Excel Workbook |
Dataset
In the study, we used the adjusted closing prices of S&P 500 index between Jan 1st, 2000 and May 9th, 2009 (2351 observations)
The data plot above shows the log daily-returns, the 20 days weighted-moving average (WMAi) and the exponential weighted volatility (EWV/EWMAi).
Analysis
The summary statistics above describes a symmetric fat-tailed (leptokurtic) probability distribution for daily log returns
We conducted few additional statistical tests: (1) White-noise (Ljung-Box), (2) Normality Test and (3) ARCHi effect. As one may expect, the probability distribution is not normally distribution, and the log-returns exhibit serial correlation and ARCH effect
Modeling
We are ready to examine different models, compare them, and select the one that fits data best. We begin with considering GARCHi model, and move on to GARCH-MGARCH in the mean and EGARCHExponential general autoregressive conditional heteroskedastic models. In each case, we evaluate the model assuming normal and non-normal distributed innovations. Finally, we summarize those models properties and recommend the one that fits the data best.
In the following table, we summarize the log-likelihood function for the selected models:
EGARCH(1,1) with GEDi innovations has the best fit for the data.
The EGARCH(1,1) conditional volatilities moves together with EWMAExponential weighted moving average, but it is consistently lower and, in many cases, time-lead the change
Forecast
The EGARCH model is used next to forecast out-of-sample conditional volatilities. The data sample ends on Friday May 8th, 2009, and each step is simply a workday.
Conclusion
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 assumption are largely satistifed. Neverthess, the daily log-returns exhibits serial correlation that EGARCH does not capture.
The volatility as of EODi 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 illustrates in details the process of examining each model, calibration, and validate its assumption (i.e. diagnosis). To download the white-paper or the associated spreadsheet model, please login or register with us.