Multivariate Time Series Analysis and Applications

5Factor analysis of multivariate time series

Similar to principle component analysis, factor analysis is one of the commonly used dimension reduction methods. It is a statistical technique widely used to explain a m‐dimensional vector with a few underlying factors. After introducing different methods to derive factors, we will illustrate the method with empirical examples. We will also discuss its use in forecasting.

5.1 Introduction

Just like principle component analysis, the purpose of factor analysis is to approximate the covariance relationships among a set of variables. Specifically, it is used to describe the covariance relationships for many variables in terms of a relatively few underlying factors, which are unobservable random quantities. The concept was developed by the researchers in the field of psychometrics in the early twentieth century. It has become a commonly used statistical method in many areas.

5.2 The orthogonal factor model

Given a weakly stationary m‐dimensional random vector at time t, Z_t = [Z_1,t, Z_2,t, …, Z_m,t]′ with mean μ = (μ₁, μ₂, …, μ_m)^′, and covariance matrix Γ, the factor model assumes that Z_t is dependent on a small number of k unobservable factors, F_j,t, j = 1, 2, …, k, known as common factors, and m additional noises ε_i,t, i = 1, 2, …, m, also known as specific factors, that is

(5.1)

More compactly, we can write the system in following ...