Читать книгу Introduction to Linear Regression Analysis - Douglas C. Montgomery - Страница 27

In addition to estimating β₀ and β₁, an estimate of σ² is required to test hypotheses and construct interval estimates pertinent to the regression model. Ideally we would like this estimate not to depend on the adequacy of the fitted model. This is only possible when there are several observations on y for at least one value of x (see Section 4.5) or when prior information concerning σ² is available. When this approach cannot be used, the estimate of σ² is obtained from the residual or error sum of squares,

(2.16)

A convenient computing formula for SS_Res may be found by substituting into Eq. (2.16) and simplifying, yielding

(2.17)

But

is just the corrected sum of squares of the response observations, so

(2.18)

The residual sum of squares has n − 2 degrees of freedom, because two degrees of freedom are associated with the estimates and involved in obtaining . Section C.3 shows that the expected value of SS_Res is E(SS_Res) = (n − 2)σ², so an unbiased estimator of σ2 is

(2.19)

The quantity MS_Res is called the residual mean square. The square root of is sometimes called the standard error of regression, and it has the same units as the response variable y.

Because depends on the residual sum of squares, any violation of the assumptions on the model errors or any misspecification of the model form may seriously damage the usefulness of as an estimate of σ². Because is computed from the regression model residuals, we say that it is a model-dependent estimate of σ².