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The reader can verify that for 6 df (why?) the observed t value exceeds the critical t value even at the 0.002 (or 0.2 percent) level of significance (two-tail test); the p value is extremely small, 0.000006. Hence we can reject the hypothesis that the coefficients of X2 and X3 in the cubic cost function are identical.

8.7 RESTRICTED LEAST SQUARES: TESTING LINEAR EQUALITY RESTRICTIONS

There are occasions where economic theory may suggest that the coefficients in a regression model satisfy some linear equality restrictions. For instance, consider the Cobb-Douglas production function:

where Y = output, X2 = labor input, and X3 = capital input. Written in log form, the equation becomes ln Yi = fa + fa ln X2i + fa ln X3î- + Ui (8.7.2)

CHAPTER EIGHT: MULTIPLE REGRESSION ANALYSIS: THE PROBLEM OF INFERENCE 267

Now if there are constant returns to scale (equiproportional change in output for an equiproportional change in the inputs), economic theory would suggest that

which is an example of a linear equality restriction.11

How does one find out if there are constant returns to scale, that is, if the restriction (8.7.3) is valid? There are two approaches.

### The i-Test Approach

The simplest procedure is to estimate (8.7.1) in the usual manner without taking into account the restriction (8.7.3) explicitly. This is called the unrestricted or unconstrained regression. Having estimated 01 and 03 (say, by OLS method), a test of the hypothesis or restriction (8.7.3) can be conducted by the t test of (8.6.3), namely,

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