In the approximate factor model of asset returns developed by Chamberlain and Rothschild (1983), the random return on each of n assets is a linear combination of k common factors plus an asset specific random return, where n is large and k is small. The asset-specific returns are only weakly correlated, in the sense that the largest eigenvalue of the covariance matrix of asset-specific returns is bounded above for all n. This implies that the risk in portfolios with holdings spread thinly over many assets comes only from the common factor returns, not from the asset-specific returns. The factor returns capture nondiversifiable risks, which arise from economy-wide shocks, whereas the asset-specific returns capture diversifiable risks, which arise from the idiosyncratic movements of individual security prices.
Connor and Korajczyk (1986, 1988) develop and apply the asymptotic principal components (APC) method to estimate approximate factor models. They show that, given that the average variance of asset-specific returns is constant through time, the first k eigenvectors of the cross-product matrix of asset returns are a consistent estimate of the k common factors. Scott (1988) and Jones (2001) provide evidence that the cross-sectional average asset specific variance has considerable time variation. They generalize the APC technique to allow for time-series heteroskedasticity in asset specific returns.
Neither Scott (1988) nor Jones (2001) model the source or nature of the heteroskedasticity in returns since an explicit model is not required for application of their techniques. In this paper, we develop such a model, estimate it, and examine the implications of our findings for asset pricing theory.
We describe a dynamic approximate factor model which includes a three-component model of the dynamic heteroskedasticity in asset returns. One component comes from the dynamic heteroskedasticity in factor returns, one from common heteroskedasticity in asset specific returns, and one from purely asset-specific heteroskedasticity in asset-specific returns. We use the model to decompose the dynamic heteroskedasticity of individual asset returns into factor-related, common asset-specific, and purely asset-specific components. We apply the techniques to monthly US equity returns for the 900-month period January 1926 to December 2000.
Download
The Common and Specific Components of Dynamic Volatility
