# 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**^{ T }**x**, **β**^{ T }**y**), where the linear combinations **α**^{ T }**x** and **β**^{ T }**y** 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.

## Keywords

Canonical Correlation Analysis Implied Volatility Cholesky Decomposition Stochastic Volatility Model Multivariate Time Series## Preview

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