Answer Key: Problem Set 4 - Weebly

ECON 482 / WH Hong

Answer Key: Problem Set 4

Answer Key

1. Consider the following estimated equation, which can be used to study the effects of

skipping class on college GPA: cn olGPA = 1.39 + 0.412 hsGPA + 0.015ACT - 0.083skipped

(0.33) (0.094)

(0.011) (0.026)

n = 64 , R2 = 0.234 i. Using the standard normal approximation, find the 95% significance interval for

hsGPA . (Ans) 412 ? 1.96(.094), or about .228 to .596

ii. Can you reject the hypothesis H0 : hsGPA = 0.4 against the tow-side alternative at the 5% level? (Ans) No, because the value 0.4 is well inside the 95% CI.

iii. Can you reject the hypothesis H0 : hsGPA = 1 against the tow-side alternative at the 5% level? (Ans) Yes, because 1 is well outside the 95% CI.

2. Consider the multiple regression model with three independent variables, under the classical linear model assumptions MLR.1. through MLR.6: y = 0 + 1x1 + 2 x2 + 3x3 + u You would like to test the null hypothesis H0 : 1 - 32 = 1.

( ) i. Let ^1 and ^2 denote the OLS estimators of 1 and 2 . Find var ^1 - 3^2 in

terms of the variance of ^1 and ^2 and the covariance between them. What is the standard error of ^1 - 3^2 . (Ans)

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ECON 482 / WH Hong

Answer Key

We use Property VAR.3 from Appendix B: Var( ^1 - 3 ^2 ) = Var ( ^1 ) + 9 Var ( ^2 ) ? 6 Cov ( ^1 , ^2 ).

ii. Write the t-statistic for testing H0 : 1 - 32 = 1. (Ans) t = ( ^1 - 3 ^2 - 1)/se( ^1 - 3 ^2 ), so we need the standard error of ^1 - 3 ^2 .

iii. Define 1 = 1 - 32 and ^1 = ^1 - 3^2 . Write a regression equation involving 0 , 1 , 2 , and 3 that allows you to directly obtain ^1 and its standard error. (Ans) Because 1 = 1 ? 32, we can write 1 = 1 + 32. Plugging this into the population model gives y = 0 + (1 + 32)x1 + 2 x2 + 3 x3 + u = 0 + 1 x1 + 2 (3x1 + x2) + 3 x3 + u. This last equation is what we would estimate by regressing y on x1, 3x1 + x2, and x3. The coefficient and standard error on x1 are what we want.

3. Regression analysis can be used to test whether the market efficiently uses information in

valuing stocks. For concreteness, let return be the total return from holding a firm's

stock over the four-year period from the end of 1990 and 1994. The efficient market

hypothesis says that these returns should not be systematically related to information

known in 1990. If firm characteristics known at the beginning of the period help to

predict stock returns, then we could use this information in choosing stocks.

For 1990, let dkr be a firm's debt to capital ratio, let eps denote the earnings per

share, let netinc denote net income, and let salary denote total compensation for the

CEO.

i. Using the data in RETURN.DTA , the following equation was estimated:

rn eturn = -14.37 + 0.321 dkr + 0.043 eps - 0.0051 netinc + 0.0035 salary

(6.89) (0.201) (0.078) (0.0047)

(0.0022)

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ECON 482 / WH Hong

Answer Key

n = 142 , R2 = 0.0395 Test whether the explanatory variables are jointly significant at the 5% significance level. Is any explanatory variable individually significant? (Ans) We need to compute the F statistic for the overall significance of the regression with n = 142 and k = 4: F = [.0395/(1 ? .0395)](137/4) 1.41. The 5% critical value with 4 numerator df and using 120 for the numerator df, is 2.45, which is well above the value of F. Therefore, we fail to reject H0: 1 = 2 = 3 = 4 = 0 at the 10% level. No explanatory variable is individually significant at the 5% level. The largest absolute t statistic is on dkr, tdkr 1.60, which is not significant at the 5% level against a two-sided alternative.

ii. Now, reestimate the model using the log form of netinc and salary :

rn eturn = -36.30 + 0.327 dkr + 0.069 eps - 4.74 netinc + 7.24 salary

(39.37) (0.203) (0.080) (3.39)

(6.31)

n = 142 , R2 = 0.0330 Do any of your conclusions from part (i) change? (Ans) The F statistic (with the same df) is now [.0330/(1 ? .0330)](137/4) 1.17, which is even lower than in part (i). None of the t statistics is significant at a reasonable level.

iii. In this sample, some firms have zero debt and others have negative earnings. Should

we try to use log (dkr ) or log (eps) in the model to see if these improve the fit?

Explain (Ans) We probably should not use the logs, as the logarithm is not defined for firms that have zero for dkr or eps. Therefore, we would lose some firms in the regression.

iv. Overall, is the evidence for predictability of stock returns strong or weak? (Ans) It seems very weak. There are no significant t statistics at the 5% level (against a

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ECON 482 / WH Hong

Answer Key

two-sided alternative), and the F statistics are insignificant in both cases. Plus, less than 4% of the variation in return is explained by the independent variables.

Computer Exercises

4. Use the data in MLB1.DTA for this exercise. In class, you have seen the following

estimation results:

lmog ( salary) = 11.19 + 0.0689 years + 0.0126gamesyr

(0.29) (0.0121)

(0.0026)

+ 0.00098bavg + 0.0144hrunsyr + 0.0108rbisyr

(0.00110) (0.0161)

(0.0072)

n = 353 , SSR = 183.186 , R2 = 0.6278

i. Use the estimated equation above, and drop the variable rbisyr and estimate the

new model. What happens to the statistical significance of hrunsyr ? What about the

size of the coefficient on hrunsyr ?

(Ans)

If we drop rbisyr the estimated equation becomes

ln og(salary) = 11.02 + .0677 years + .0158 gamesyr

(0.27) (.0121)

(.0016)

+ .0014 bavg + .0359 hrunsyr

(.0011) n = 353, R2 = .625.

(.0072)

Now hrunsyr is very statistically significant (t statistic 4.99), and its coefficient

has increased by about two and one-half times.

ii. Add the variables runsyr (run per year), fldperc (fielding percentage), and sbasesyr (stolen bases per year) to the model from part i. Which of these factors are individually significant? (Ans) The equation with runsyr, fldperc, and sbasesyr added is

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ECON 482 / WH Hong

Answer Key

ln og(salary) = 10.41 + .0700 years + .0079 gamesyr

(2.00) (.0120)

(.0027)

+ .00053 bavg + .0232 hrunsyr

(.00110)

(.0086)

+ .0174 runsyr (.0051)

n = 353, R2 = .639.

+ .0010 fldperc ? .0064 sbasesyr

(.0020)

(.0052)

Of the three additional independent variables, only runsyr is statistically significant (t statistic = .0174/.0051 3.41). The estimate implies that one more run per year,

other factors fixed, increases predicted salary by about 1.74%, a substantial increase.

The stolen bases variable even has the "wrong" sign with a t statistic of about ?1.23,

while fldperc has a t statistic of only .5. Most major league baseball players are

pretty good fielders; in fact, the smallest fldperc is 800 (which means .800). With

relatively little variation in fldperc, it is perhaps not surprising that its effect is hard to

estimate.

iii. In the model from part (ii), test the joint significance of bavg , fldperc , and sbasesyr . (DO NOT use the Stata command test. Follow the steps you learned in class and use the formula for F-statistic) (Ans) From their t statistics, bavg, fldperc, and sbasesyr are individually insignificant. The F statistic for their joint significance (with 3 and 345 df) is about .69 with pvalue .56. Therefore, these variables are jointly very insignificant.

5. Use the data in HTV.DTA to answer this question. i. Estimate the regression model educ = 0 + 1motheduc + 2 fatheduc + 3abil + 4abil2 + u by OLS and report the results in the usual form. Test the null hypothesis that educ is linearly related to abil against the alternative that the relationship is quadratic. (Ans) The estimated equation, with standard errors in parentheses below coefficient estimates, is

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