**7 **In Example 4.7, we used data on nonunionized manufacturing firms to estimate the relationship between the scrap rate and other firm characteristics. We now look at this ex- ample more closely and use all available firms.

(i) The population model estimated in Example 4.7 can be written as log(*scrap*) 5 *b*0 1 *b*1*hrsemp *1 *b*2log(*sales*) 1*b*3log(*employ*) 1 *u*.

Using the 43 observations available for 1987, the estimated equation is

l**og(*scrap*) 5 11.74 2 .042 *hrsemp *2 .951 log(*sales*) 1 .992 log(*employ*)

(4.57) (.019) (.370) (.360)

*n *5 43, *R*2 5 .310.

Compare this equation to that estimated using only the 29 nonunionized firms in the sample.

(ii) Show that the population model can also be written as

log(*scrap*) 5 *b*0 1 *b*1*hrsemp *1 *b*2log(*sales*/*employ*) 1 *u*3log(*employ*) 1 *u*,

where *u*3 5 *b*2 1 *b*3. [*Hint: *Recall that log(*x*2/*x*3) 5 log(*x*2) 2 log(*x*3).] Interpret the hypothesis H0: *u*3 5 0.

(iii) When the equation from part (ii) is estimated, we obtain

l**og(*scrap*) 5 11.74 2 .042 *hrsemp *2 .951 log(*sales*/*employ*) 1 .041 log(*employ*)

(4.57) (.019) (.370) (.205)

*n *5 43, *R*2 5 .310.

Controlling for worker training and for the sales-to-employee ratio, do bigger firms have larger statistically significant scrap rates?

(iv) Test the hypothesis that a 1% increase in *sales*/*employ *is associated with a 1% drop in the scrap rate.

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