It is interesting to compare these formulas with those obtained when the intercept term is included in the model:
The differences between the two sets of formulas should be obvious: In the model with the intercept term absent, we use raw sums of squares and cross products but in the intercept-present model, we use adjusted (from mean) sums of squares and cross products. Second, the df for computing a2 is (n — 1) in the first case and (n — 2) in the second case. (Why?)
Although the interceptless or zero intercept model may be appropriate on occasions, there are some features of this model that need to be noted. First, , which is always zero for the model with the intercept term (the conventional model), need not be zero when that term is absent. In short, Y^ u need not be zero for the regression through the origin. Second, r2, the coefficient of determination introduced in Chapter 3, which is always nonnegative for the conventional model, can on occasions turn out to be negative for the interceptless model! This anomalous result arises because the r2 introduced in Chapter 3 explicitly assumes that the intercept is included in the model. Therefore, the conventionally computed r2 may not be appropriate for regression-through-the-origin models.3
r2 for Regression-through-Origin Model
As just noted, and as further discussed in Appendix 6A, Section 6A.1, the conventional r2 given in Chapter 3 is not appropriate for regressions that do not contain the intercept. But one can compute what is known as the raw r2 for such models, which is defined as
3For additional discussion, see Dennis J. Aigner, Basic Econometrics, Prentice Hall, Englewood Cliffs, N.J., 1971, pp. 85-88.
168 PART ONE: SINGLE-EQUATION REGRESSION MODELS
Note: These are raw (i.e., not mean-corrected) sums of squares and cross products.
Although this raw r2 satisfies the relation 0 < r2 < 1, it is not directly comparable to the conventional r2 value. For this reason some authors do not report the r2 value for zero intercept regression models.
Because of these special features of this model, one needs to exercise great caution in using the zero intercept regression model. Unless there is very strong a priori expectation, one would be well advised to stick to the conventional, intercept-present model. This has a dual advantage. First, if the intercept term is included in the model but it turns out to be statistically insignificant (i.e., statistically equal to zero), for all practical purposes we have a regression through the origin.4 Second, and more important, if in fact there is an intercept in the model but we insist on fitting a regression through the origin, we would be committing a specification error, thus violating Assumption 9 of the classical linear regression model.
AN ILLUSTRATIVE EXAMPLE:
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