Covariance selection and estimation and the value/growth spreads as predictors of returns

This thesis is the result of efforts in three separate papers. Due to the nature of each paper, we decide to keep them separately as chapters most of the time. In the first chapter, we propose a nonparametric and data-driven method to identify parsimony and to exploit any such parsimony to produce a statistically efficient estimator of a large covariance matrix. The approach reparameterizes the covariance matrix through the modified Cholesky decomposition of its inverse. The Cholesky factor is likely to have off-diagonal elements that are zero or close to it. Penalized normal likelihood of the new unconstrained parameters with L1 and L2 penalties are shown to be closely related to Tibshirani's (1996) LASSO approach and the ridge regression. Adding either penalty to the likelihood helps produce more stable estimators by introducing shrinkage to the elements in the Cholesky factor, while the L1 penalty can also effectively identify structural zeros. The maximum penalized likelihood estimator and the sample covariance matrix are compared using simulation. In the second chapter, we propose another approach to reparameterize the covariance matrix through the modified Cholesky decomposition of its inverse. A great deal of smoothness is observed in the Cholesky factors when the underlying data set is longitudinal. We use quadratic splines to model the smoothness. This approach provides a simultaneous and unified method of estimation for the mean and covariance of longitudinal data. It also handles missing data naturally. The third chapter is the result of joint work with Professor Lu Zhang. Recent rational theory predicts that the value spread is countercyclical and should be a positive predictor of future returns, and that the growth spread is procyclical and should be a negative predictor of future returns. From January 1927 to December 2001, the value spread predicts positively, and the growth spread predicts negatively future market excess returns and small firm excess returns. The value spread exhibits clearly countercyclical, and the growth spread exhibits clearly procyclical movements. However, both the cyclical properties and the predictive power of the value and growth spreads are substantially weaker in the postwar sample.