Introduction to Linear Regression Analysis, 5th Edition

CHAPTER 4

MODEL ADEQUACY CHECKING

4.1 INTRODUCTION

The major assumptions that we have made thus far in our study of regression analysis are as follows:

1. The relationship between the response y and the regressors is linear, at least approximately.
2. The error term ε has zero mean.
3. The error term ε has constant variance σ².
4. The errors are uncorrelated.
5. The errors are normally distributed.

Taken together, assumptions 4 and 5 imply that the errors are independent random variables. Assumption 5 is required for hypothesis testing and interval estimation.

We should always consider the validity of these assumptions to be doubtful and conduct analyses to examine the adequacy of the model we have tentatively entertained. The types of model inadequacies discussed here have potentially serious consequences. Gross violations of the assumptions may yield an unstable model in the sense that a different sample could lead to a totally different model with opposite conclusions. We usually cannot detect departures from the underlying assumptions by examination of the standard summary statistics, such as the t or F statistics, or R². These are “global” model properties, and as such they do not ensure model adequacy.

In this chapter we present several methods useful for diagnosing violations of the basic regression assumptions. These diagnostic methods are primarily based on study of the model residuals. Methods for dealing with model inadequacies, as well as additional, more sophisticated diagnostics, ...