Probability and Statistics for Finance

Chapter 20. Estimates and Diagnostics for Multivariate Linear Regression Analysis

It is often the case in finance that we find it necessary to analyze more than one variable simultaneously. In Chapter 5, for example, we considered the association between two variables through correlation in order to express linear dependence and in Chapter 6 explained how to estimate a linear dependence between two variables using the linear regression method. When there is only one independent variable, the regression model is said to be a simple linear regression or a univariate regression.

Univariate modeling in many cases is not sufficient to deal with real problems in finance. The behavior of a certain variable of interest sometimes needs to be explained by two or more variables. For example, suppose that we want to determine the financial or macroeconomic variables that affect the monthly return on the Standard & Poor's 500 (S&P 500) index. Let's suppose that economic and financial theory suggest that there are 10 such explanatory variables. Thus we have a multivariate setting of 11 dimensions—the return on the S&P 500 and the 10 explanatory variables.

In this chapter and in the next, we explain the multivariate linear regression model (also called the multiple linear regression model) to explain the linear relationship between several independent variables and some dependent variable we observe. As in the univariate case (i.e., simple linear regression) discussed in Chapter 6, the relationship ...