S&P 500 monthly returns

The S&P 500 is a value weighted index published since 1957 of the prices of 500 large cap common stocks actively traded in the United States.Many index funds and exchange-traded funds attempt to replicate (before fees and expenses) the performance of the S&P 500 by holding the same stocks as the index, in the same proportions.

Theoretically, market returns are not serially correlated, and, thus, studying the past returns/prices can't help in predicting future returns. In this study, we'll examine this assumption using S&P 500 as a proxy.

Objective

For our purpose, we try to model the S&P 500 returns using pure econometric techniques.

Data

Source: Yahoo Finance.

Date Range: January 1980 to March 2009.

Preliminary Analysis

Examining the graph for S&P 500 levels above, it becomes obvious that index levels follow a non-stationary process. To proceed with our analysis, we convert levels into logarithmic returns using $$r_{t} =\ln(\frac{P_{t}}{P_{t-1}}) = (1-L)\ln(P_{t})$$
Using NumXL and Excel built-in function, we can generate the required time series (i.e. DIFF(ln(G2:G711),1))

Next, we examine the series descriptive statistic and test the hypothesis whether the log returns are not serially correlated (i.e. white noise):

The log returns is not a white noise, and its probabilistic distribution exhibits fat tails than normal distribution.

 
ACF PACF

Conclusion

The preliminary analysis shows negligible serial correlation or ARCH effect. Furthermore, the returns probability distribution exhibits some excess kurtosis, but negligible skewness. In sum, the S&P 500 monthly returns appears as non-Gaussian white noise.


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