where (02 + 03) = 1 under the null hypothesis and where the denominator is the standard error of (f)1 + 03). Then following Section 8.6, if the t value computed from (8.7.4) exceeds the critical t value at the chosen level of significance, we reject the hypothesis of constant returns to scale; otherwise we do not reject it.
The F-Test Approach: Restricted Least Squares
The preceding t test is a kind of postmortem examination because we try to find out whether the linear restriction is satisfied after estimating the "unrestricted" regression. A direct approach would be to incorporate the restriction (8.7.3) into the estimating procedure at the outset. In the present example, this procedure can be done easily. From (8.7.3) we see that
Therefore, using either of these equalities, we can eliminate one of the coefficients in (8.7.1) and estimate the resulting equation. Thus, if we use (8.7.5), we can write the Cobb-Douglas production function as ln Yi = 0o + (1 - 03)lnXn + 03 ln ^ + u = 0o + ln X1i + 03(ln X3i - ln X^ ) + Ui
12If we had ß2 + ß3 < 1, this relation would be an example of a linear inequality restriction. To handle such restrictions, one needs to use mathematical programming techniques.
268 PART ONE: SINGLE-EQUATION REGRESSION MODELS
where (Yi /X2i ) = output/labor ratio and (X3i /X2i ) = capital labor ratio, quantities of great economic importance.
Notice how the original equation (8.7.2) is transformed. Once we estimate 03 from (8.7.7) or (8.7.8), 02 can be easily estimated from the relation (8.7.5). Needless to say, this procedure will guarantee that the sum of the estimated coefficients of the two inputs will equal 1. The procedure outlined in (8.7.7) or (8.7.8) is known as restricted least squares (RLS). This procedure can be generalized to models containing any number of explanatory variables and more than one linear equality restriction. The generalization can be found in Theil.13 (See also general F testing below.)
How do we compare the unrestricted and restricted least-squares regressions? In other words, how do we know that, say, the restriction (8.7.3) is valid? This question can be answered by applying the F test as follows. Let
RSS of the unrestricted regression (8.7.2)
RSS of the restricted regression (8.7.7) number of linear restrictions (1 in the present example) number of parameters in the unrestricted regression number of observations
follows the F distribution with m, (n — k) df. (Note: UR and R stand for unrestricted and restricted, respectively.)
The F test above can also be expressed in terms of R2 as follows:
where and RR are, respectively, the R2 values obtained from the unrestricted and restricted regressions, that is, from the regressions (8.7.2) and
13Henri Theil, Principles of Econometrics, John Wiley & Sons, New York, 1971, pp. 43-45.
CHAPTER EIGHT: MULTIPLE REGRESSION ANALYSIS: THE PROBLEM OF INFERENCE 269
In exercise 8.4 you are asked to justify these statements.
A Cautionary Note: In using (8.7.10) keep in mind that if the dependent variable in the restricted and unrestricted models is not the same, Rur and RR are not directly comparable. In that case, use the procedure described in Chapter 7 to render the two R2 values comparable (see Example 8.3 below) or use the F test given in (8.7.9).
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