34
CHAPTER 4
TEACHING NOTES
At the start of this chapter is good time to remind students that a specific error distribution
played no role in the results of Chapter 3. That is because only the first two moments were
derived under the full set of Gauss-Markov assumptions. Nevertheless, normality is needed to
obtain exact normal sampling distributions (conditional on the explanatory variables). I
emphasize that the full set of CLM assumptions are used in this chapter, but that in Chapter 5 we
It is crucial to emphasize that we test hypotheses about unknown population parameters. I tell
my students that they will be punished if they write something like H0:
1
ˆ
= 0 on an exam or,
even worse, H0: .632 = 0.
One useful feature of Chapter 4 is its illustration of how to rewrite a population model so that it
One can use an F test for single linear restrictions on multiple parameters, but this is less
transparent than a t test and does not immediately produce the standard error needed for a
confidence interval or for testing a one-sided alternative. The trick of rewriting the population
model is useful in several instances, including obtaining confidence intervals for predictions in
4.1 (i) and (iii) generally cause the t statistics not to have a t distribution under H0.
4.2 (i) H0:
3
= 0. H1:
3
> 0.
(iii) The 10% critical value for a one-tailed test, using df = , is obtained from Table G.2 as
1.282. The t statistic on ros is .00024/.00054
.44, which is well below the critical value.
Therefore, we fail to reject H0 at the 10% significance level.
4.3 (i) Holding profmarg fixed,
rdintens
= .321 log(sales) = (.321/100)[100
log( )sales
]
.00321(%sales). Therefore, if %sales = 10,
rdintens
.032, or only about 3/100 of a
4.4 (i) H0:
3
= 0. H1:
3
0.
(ii) Other things equal, a larger population increases the demand for rental housing, which
37
4.5 (i) .412 1.96(.094), or about .228 to .596.
4.6 (i) With df = n – 2 = 86, we obtain the 5% critical value from Table G.2 with df = 90.
Because each test is two-tailed, the critical value is 1.987. The t statistic for H0:
0
0
= 0 is about –
(iii) We use the R-squared form of the F statistic. We are testing q = 3 restrictions and there
are 88 – 5 = 83 df in the unrestricted model. The F statistic is [(.829 – .820)/(1 – .829)](83/3)
1.46. The 10% critical value (again using 90 denominator df in Table G.3a) is 2.15, so we fail to
reject H0 at even the 10% level. In fact, the p-value is about .23.
4.7 (i) While the standard error on hrsemp has not changed, the magnitude of the coefficient has
increased by half. The t statistic on hrsemp has gone from about –1.47 to –2.21, so now the
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(ii) If we add and subtract
2
log(employ) from the right-hand-side and collect terms, we
have
(iii) No. We are interested in the coefficient on log(employ), which has a t statistic of .2,
4.8 (i) We use Property VAR.3 from Appendix B: Var(
1
ˆ
− 3
2
ˆ
) = Var (
1
ˆ
) + 9 Var (
2
ˆ
) – 6
Cov (
1
ˆ
,
2
ˆ
).
4.9 (i) With df = 706 – 4 = 702, we use the standard normal critical value (df = in Table G.2),
which is 1.96 for a two-tailed test at the 5% level. Now teduc = −11.13/5.88
−1.89, so |teduc| =
0
2
2
0
2
2
39
(ii) We need to compute the R-squared form of the F statistic for joint significance. But F =
4.10 (i) We need to compute the F statistic for the overall significance of the regression with
(ii) The F statistic (with the same df) is now [.0330/(1 – .0330)](137/4)
1.17, which is
4.11 (i) In columns (2) and (3), the coefficient on profmarg is actually negative, although its t
statistic is only about –1. It appears that, once firm sales and market value have been controlled
40
4.12 (i) If expend increases by 10% then lexpend increases by .10. Multiplying .10 by 11.16
gives 1.116, or about 1.1 percentage points. So a 10% increase in spending is associated with a
1.1 percentage point increase in the math pass rate.
(ii) The low R-squared does not imply that lexpend is uncorrelated with the underlying error
(iii) The coefficient on lexpend falls to 7.75, but its t statistic is 2.55, which is statistically
SOLUTIONS TO COMPUTER EXERCISES
C4.1 (i) Holding other factors fixed,
(ii) The null hypothesis is H0:
2
= –
1
, which means a z% increase in expenditure by A
(iii) The estimated equation (with standard errors in parentheses below estimates) is
The coefficient on log(expendA) is very significant (t statistic
15.92), as is the coefficient on
(iv) Write
1
=
1
+
2
, or
1
=
1
–
2
0
1
2
. Plugging this into the original equation, and
rearranging, gives
C4.2 (i) In the model
(ii) LSAT is not statistically significant (t statistic
1.18) but GPA is very significance (t
(iii) When we add clsize and faculty to the regression we lose five observations. The test of
C4.3 (i) The estimated model is
Therefore,
1
ˆ
= 150(.000379) + .0289 = .0858, which means that an additional 150 square foot
bedroom increases the predicted price by about 8.6%.
C4.4 The R-squared from the regression bwght on cigs, parity, and faminc, using all 1,388
observations, is about .0348. This means that, if we mistakenly use this in place of .0364, which
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C4.5 (i) If we drop rbisyr the estimated equation becomes
(ii) The equation with runsyr, fldperc, and sbasesyr added is
Of the three additional independent variables, only runsyr is statistically significant (t statistic =
(iii) From their t statistics, bavg, fldperc, and sbasesyr are individually insignificant. The F
C4.6 (i) In the model
44
C4.7 (i) The minimum value is 0, the maximum is 99, and the average is about 56.16.
(iii) Adding phsrank makes the t statistic on jc even smaller in absolute value, about 1.33, but
(iv) The variable id is just a worker identification number, which should be randomly
C4.8 (i) There are 2,017 single people in the sample of 9,275.
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(iii) The intercept is not very interesting as it gives the predicted nettfa for inc = 0 and age =
0. Clearly, there is no one with even close to these values in the relevant population.
(iv) The t statistic is (.843 − 1)/.092 −1.71. Against the one-sided alternative H1:
2 < 1,
C4.9 (i) The results from the OLS regression, with standard errors in parentheses, are
(ii) The correlation is about −.84, indicating a strong degree of multicollinearity. Yet each
(iii) The OLS regression results when log(hseval) is added are
(iv) Adding log(hseval) makes log(income) and prppov individually insignificant (at even the
46
C4.10 (i) Using the 1,848 observations, the simple regression estimate of
bs
is about
.795−
.
The 95% confidence interval runs from
1.088 to .502−−
, which includes −1. Therefore, at the
5% level, we cannot reject that
0
H : 1
bs
=−
against the two-sided alternative.
(ii) When lenrol and lstaff are added to the regression, the coefficient on bs becomes about
(iii) The standard error of the simple regression estimate is about .150, and that for the
multiple regression estimate is about .109. When we add extra explanatory variables, two factors
(iv) The variable lstaff is the log of the number of staff per 1,000 students. As lstaff
(v) When lunch is added to the regression, its coefficient is about −.00076, with t = −4.69.
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(vi) Yes, the pattern obtained using ELEM94_95.RAW is very similar to that in Table 4.1,
C4.11 (i) The estimated equation, with standard errors in parentheses below coefficient
estimates, is
(ii) We could rewrite the model by defining, say,
1 1 2
=−
and then substituting in
(iii) I used the test command in Stata to test the joint significance of the tuition variables.
With 2 and 1,223 degrees of freedom I get an F statistic of about .84 with association p-value of
(v) The positive coefficient on avgtuit does not make a lot of sense if we think that, all other