آنالیز کامپوننت مستقل برای نوسانات تحقق یافته: آنالیز سقوط بازار بورس در سال 2008

This paper investigates the factors that drove the U.S. equity market returns from 2007 to early 2010. The period was highlighted by volatile energy and commodity prices, the collapse of insurance and banking firms, extreme implied volatility and a subsequent rally in the overall market. To extract the driving factors, we decompose the returns of the S&P500 sector ETFs into statistically independent signals using independent component analysis. We find that the generated factors have interesting financial interpretations and are consistent with the major economic themes of the period. We find that there are two sets of general market betas during the period along with a dominant factor for energy and materials sector. In addition, we find that the EGARCH model which accommodates asymmetric responses between returns and volatility can plausibly fit the high levels of variance during the crash. Finally, estimated correlations dropped when commodity prices moved higher, but then spiked when the S&P500 crashed in late 2008.

Independent component analysis (ICA) is a technique for extracting factors from a set of mixed sources of variation. It does so by maximizing the statistical independence of the components. We chose to apply ICA to data on S&P500 index sector exchange traded funds (ETF) to see if ICA could separate out key drivers of performance over the (extremely volatile) period 2007 to early 2010. A priori we did not expect the procedure to produce factors that relate back to market structure. Nevertheless, the extracted signals show commodity prices as the key driving economic signals. Over the period 2007 through early 2010, several macroeconomic shocks hit the global economy, causing the stock market to fall over 50% from its all-time high in late 2007. In addition, implied volatilities – sometimes referred to as fear indices – spiked to levels unseen since the crash of 1987. The volatility indices of both the S&P5003 (VIX) and Nasdaq 100 (VXN) peaked around 80 in the fall of 2009. These extreme levels imply an expectation of daily volatility of around 5% for a full month hence.4 By comparison, the VIX was around 12 (a daily volatility of .75%) at the beginning of 2007, and by the end of February 2010 had dropped back to near 20 (a daily volatility of 1.25%). During this period, commodity prices were also highly volatile relative to their historical levels. Hard red spring wheat rose from around $5 per bushel in 2007 to over $25 in early 2008, only to fall back to the $5 range by the end of 2009. Brent crude reached over $145 US per barrel in July of 2008 then quickly fell back to between $40 and $50 by the end of that year. Natural gas prices spiked near $14 per mmBtu, but by mid-2009 dropped to $3 per mmBtu. Silver fell from over $20 per ounce to less than $10 in late 2008. The price of copper fell over 62%, from over $4 per pound to the $1.50 range. In this paper we also attempt to fit general autoregressive conditional heteroskedasticity (GARCH) models utilizing these ICA factors to replicate the volatility of the period. We find that the standard GARCH(1,1) cannot fit the variances without parameters that imply unstable variance processes. However, Nelson (1991) asymmetric exponential GARCH (EGARCH) model does fit the factor variances while still providing finite stationary estimates. Using the EGARCH estimates for each factor, we are able to estimate the sector volatilities as well as the correlations between the sectors during the sample period. The remainder of this paper is organized into five sections. The first section provides an overview of the ICA technique. The second describes the various GARCH processes. The third section, presents the empirical data used in the study. The fourth section presents fitting results. The fifth section concludes.

In this study we examined the factors produced by independent component analysis for the sectors of the S&P500 from 2007 to early 2010. Although the procedure is purely statistical, we find the factors generated have meaningful economic interpretations. Specifically, we find that the three main factors are an energy and materials factor, a standard market factor, and a financial dominated factor. In addition, we find that the independent factors are time varying and do explain the variance patterns observed in the market. The EGARCH(1,1) model is able fit the factors’ variances while providing plausible stationary variance estimates. For each of the factors the estimated leverage is statistically significant and, in the case of the market factor, the sign of the leverage coefficient agreed with previous studies. Lastly, we have been able to derive estimates of both the dynamic variances of the sectors and the dynamic cross correlations between the sectors. Correlations dropped when the energy based sector factor (EMF) had relatively high variance in the summer of 2008, but when the market crashed in the fall of 2008 the correlations quickly rose. By the end of the sample correlations had either remained steady or moved down from their highs during the crash. Future research paths include investigating the factors in other markets such as sector returns in international markets to investigate how the dynamic correlation patterns evolved. In addition, one attempt would be to determine if the option markets were expressing, in general, the same information that is generated by ICA. For example, one could see what the average implied correlation of the SPX was during this period (as in Driessen, Maenhout Vilkov, 2010) or calculate the beta or covariance for the sectors against the SPX and each other (see Chang, Christoffersen, Jacobs, & Vainberg, 2009).