Assignment #7 Hypothesis Tests

**Reading: ** Wooldridge Ch 4

**1.** Consider an equation to explain salaries of CEOs in terms of annual firm sales, return on equity (*roe*, in percentage form), and return on the firm’s stock (*ros*, in percentage form):

(i)

By what percentage is *salary *predicted to increase if *ros *increases by 50 points?

Does *ros *have a practically large effect on *salary*?

(ii) Test the null hypothesis that *ros *has no effect on *salary *against the alternative that

*ros *has a positive effect. Carry out the test at the 10% significance level.

(iii) Would you include *ros *in a final model explaining CEO compensation in terms of

firm performance? Explain.

**2. ** Are rent rates influenced by the student population in a college town? Let *rent *be the aver- age monthly rent paid on rental units in a college town in the United States. Let *pop *denote the total city population, *avginc *the average city income, and *pctstu *the student population as a percentage of the total population. The equation is estimated for 64 college towns:

(i) If population increases by 10%, what is the expected increase in rent?

(ii) Test the following hypothesis at the 1% level: The size of the student body relative to the population has no ceteris paribus effect on monthly rents.

(iii) Using the standard normal approximation, find the 95% confidence interval for

the coefficient on average city income.

(iv) Can you reject the hypothesis H0: the coefficient on average city income equals 0.55, against the two-sided alternative at the

5% level?

**3. **Consider the estimated equation

(i) Is either *educ *or *age *individually significant at the 5% level against a two-sided

alternative? Show your work.

(ii) Dropping *educ *and *age *from the equation gives

Are *educ *and *age *jointly significant in the original equation at the 5% level? Justify

your answer.

(iii) Does including *educ *and *age *in the model greatly affect the estimated tradeoff

between sleeping and working?

(iv) Suppose that the sleep equation contains heteroskedasticity. What does this mean

about the tests computed in parts (i) and (ii)?

**Computer problem**

**4. ** Use the data in DISCRIM to answer this question. These are ZIP code–level data on prices for various items at fast-food restaurants, along with characteristics of the zip code population, in New Jersey and Pennsylvania. The idea is to see whether fast-food restaurants charge higher prices in areas with a larger concentration of blacks.

(i) Estimate this model by OLS. If the dataset contains observations with missing values on your X or Y variables, be sure to exclude those records.

* psoda = β*_{0}* + β*_{1}* prpblck+ β*_{2 }*income + u*

Report the results in equation form, including the sample size and *R*-squared. Interpret the coefficient on *prpblck*. Do you think it is economically large?

(ii) Compare the estimate above with the simple regression estimate from *psoda *on *prpblck*. Is the discrimination effect larger or smaller when you control for income?

(iii) A model with a constant price elasticity with respect to income may be more appropriate. Report estimates of the model

* log(psoda) = β*_{0}* + β*_{1}* prpblck+ β*_{2 }*log(income) + u*

If *prpblck *increases by .20 (20 percentage points), what is the estimated percentage

change in *psoda*? (*Hint*: The answer is 2.*xx*, where you fill in the “*xx*.”)

(iv) Use OLS to estimate the model

* log(psoda) = β*_{0}* + β*_{1}* prpblck+ β*_{2 }*log(income) + β*_{3 }*prppov + u*

Report the results in the usual form. Is *β** _{1 }*statistically different from zero at the

5% level against a two-sided alternative? What about at the 1% level?

(v) What is the correlation between log(*income*) and *prppov*? Is each variable statistically significant in any case? Report the two-sided *p*-values.

Evaluate the following statement: “Because log(*income*) and *prppov *are so highly correlated, they have no business being in the same regression.”

(vi) To the regression in part (iii), add the variable log(*hseval*). Interpret its coefficient

and report the two-sided *p*-value for H0: *β*log(*hseval*) = 0.

(vii) In the regression above, what happens to the individual statistical significance

of log(*income*) and *prppov*? Are these variables jointly significant? (Compute a

*p*-value.) What do you make of your answers?

(viii) Given the results of all your regressions, which one would you report as most reliable in determining whether the racial makeup of a zip code influences local fast-food prices?

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