Advanced Multivariate and Time Series Methods in Financial Econometrics

The importance of multivariate statistical methods, for which Chapter 2 provides an introduction, has been demonstrated in Chapters 3 and 4 in connection with portfolio optimization. As noted in Section 4.4, more advanced methods for high-dimensional multivariate data are needed to handle large portfolios. This chapter introduces several advanced multivariate statistical techniques. Section 9.1 begins with canonical correlation analysis, which analyzes the correlation structure between two random vectors x and y, not necessarily of the same dimension, through canonical variate pairs of the form (α^Tx, β^Ty), where the linear combinations α^Tx and β^Ty are chosen in a way similar to how principal components are defined. Section 9.2 generalizes regression analysis to the case where output variables are k × 1 vectors y _t . The multivariate regression model is of the form y _t = Bx _t + ε _t , in which the regressors x _t are p × 1 vectors, as in Chapter 1. The k × p coefficient matrix B involves kp parameters, which may be too many to estimate well for the sample size n often used in empirical studies. Of particular importance is a technique, called reduced-rank regression, that addresses this problem by assuming that rank(B) = r ≤ min(p, k) and applying canonical correlation analysis of x _i , y _i to find a reduced-rank regression model. Section 9.3 introduces modified Cholesky decompositions of covariance matrices and their applications to the analysis of high-dimensional covariance matrices associated with large portfolios.