# CHAPTER 7

# WEIGHTED LEAST SQUARES

## 7.1 INTRODUCTION

So far in our discussion of regression analysis it has been assumed that the underlying regression model is of the form

where the *ε _{i}*'s are random errors that are independent and identically distributed (i.i.d.) with mean zero and variance

*σ*

^{2}. Various residual plots have been used to check these assumptions (Chapter 4). If the residuals are not consistent with the assumptions, the equation form may be inadequate, additional variables may be required, or some of the observations in the data may be outliers.

There has been one exception to this line of analysis. In the example based on the Supervisor Data of Section 6.5, it is argued that the underlying model does not have residuals that are i.i.d. In particular, the residuals do not have constant variance. For these data, a transformation was applied to correct the situation so that better estimates of the original model parameters could be obtained (better than the ordinary least squares (OLS) method).

In this chapter and in Chapter 8 we investigate situations where the underlying process implies that the errors are not i.i.d. The present chapter deals with the *heteroscedasticity* problem, where the residuals do not have the same variance, and Chapter 8 treats the *autocorrelation* problem, where the residuals are not independent.

In Chapter 6 heteroscedasticity was handled by transforming ...

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