Narrow Results

This chapter introduces regression analysis, the most popular statistical tool to explore the dependence of one variable (say Y) on others (say X). The variable Y is called the dependent variable or response variable, and X is called the independent variable or explanatory variable. The regression relationship between X and Y can be used to study the effect of X on Y or to predict Y using X. We motivate the importance of the regression function from both the economic and statistical perspectives, and characterize the condition for correct specification of a linear model for the regression function, which is shown to be crucial for a valid economic interpretation of model parameters.

The following sections are included:

• INTRODUCTION

• THE MODEL AND ITS ASSUMPTIONS
  • The Multiple Regression Model
  • The Regression Plane for Two Explanatory Variables
  • Assumptions for the Multiple Regression Model

• ESTIMATING MULTIPLE REGRESSION PARAMETERS
  • Annual Salary, Years of Education, and Years of Work Experience

• THE RESIDUAL STANDARD ERROR AND THE COEFFICIENT OF DETERMINATION
  • The Residual Standard Error
  • The Coefficient of Determination

• TESTS ON SETS AND INDIVIDUAL REGRESSION COEFFICIENTS
  • Test on Sets of Regression Coefficients
  • Hypothesis Tests for Individual Regression Coefficients

• CONFIDENCE INTERVAL FOR THE MEAN RESPONSE AND PREDICTION INTERVAL FOR THE INDIVIDUAL RESPONSE
  • Point Estimates of the Mean and the Individual Responses
  • Interval Estimates of Forecasts

• BUSINESS AND ECONOMIC APPLICATIONS
  • Overall Job-Worth of Performance for Certain Army Jobs
  • The Relationship Between Individual Stock Rates of Return, Payout Ratio, and Market Rates of Return
  • Analyzing the Determination of Price per Share
  • Multiple Regression Approach to Evaluating Real Estate Property
  • Multiple Regression Approach to Doing Cost Analysis

• USING COMPUTER PROGRAMS TO DO MULTIPLE REGRESSION ANALYSES
  • SAS Program for Multiple Regression Analysis
  • MINITAB Program for Multiple Regression Prediction
  • Stepwise Regression Analysis

• Summary

• Appendix 15A Derivation of the Sampling Variance of the Least-Squares Slope Estimations

• Appendix 15B Derivation of Equation 15.30

• Questions and Problems

In this chapter we study the relationships among random variables, which will be characterized by the joint probability distribution of random variables. Most insights into multivariate distributions can be gained by focusing on bivariate distributions. We first introduce the joint probability distribution of a bivariate random vector (X, Y ) via the characterization of the joint cumulative distribution function, the joint probability mass function (when (X, Y ) are discrete), and the joint probability density function (when (X, Y ) are continuous) respectively. We then characterize various aspects of the relationship between X and Y using the conditional distributions, correlation, and conditional expectations. The concept of independence and its implications on the joint distributions, conditional distributions and correlation are also discussed. We also introduce a class of bivariate normal distributions.

In this chapter, we shall build on the fundamental notions of probability distribution and statistics in the last chapter, and extend consideration to a sequence of random variables. In financial application, it is mostly the case that the sequence is indexed by time, hence a stochastic process. Interesting statistical laws or mathematical theories result when we look at the relationships within a stochastic process. We introduce an application of the Central Limit Theorem to the study of stock return distributions.