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# SolutionsChapter8.pdf

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York University

Economics

ECON 3210

Razvan Sufana

Fall

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CHAPTER 8
Exercise Solutions
271 Chapter 8, Exercise Solutions, Principles of Econometrics, 4e 272
EXERCISE 8.1
2 2
When i
N N N
x 2 22 2 xx
i1i1i i i1 i 2
2 2 2 N
N 2 2 2 2
ix i i xx xx i x
1 i1 i1 i1 Chapter 8, Exercise Solutions, Principles of Econometrics, 4e 273
EXERCISE 8.2
1 2
(a) Multiplying the first normal equation by i i and the second one by i yields
2 1 ˆ ˆ 1 2
i i i 1 i i yx i i i i
i i i i ix 2ˆi i i * x x
Subtracting the first of these two equations from the second yields
2
y 2 xy * ˆ 1 1
i i i i ii ii iii 2
Thus,
2* x y x y 1
i i ii ii i
2 2 2
i i i i x
yx y x
i 22 2i i i
i
2
i ix i i
2
i i
In this last expression, the second lin e is obtained from the first by making the
1 1
substitutions yi ii and xiii x , and by dividing numerator and denominator by
2 2 2 ˆ ˆ 1 ˆ
i . Solving the first normal equationii i ii 1 2 x y for 1
1 1
and making the substitutions yi ii and xiii x , yields
2 2
ˆ i x i i
1 2
i i
2 2 2 2 2 2 2 2
(b) When i for all i, i ii ii yx , i iy i , i i i x ,
2 2 ˆ
and i N . Making these substitutions into the expression fo2 yields
xii yi i
2 22 yii yx
ˆ N NN N
2 2 2 2 2
xi i xi x 2
2 N
N N
ˆ
and that for 1 becomes
2 2y x
ˆ 2 1ˆ 2 i i yx
N N 2
These formulas are equal to those for the least squares estimatorsb and b . See pages 52
1 2
and 83-84 of the text. Chapter 8, Exercise Solutions, Principles of Econometrics, 4e 274
Exercise 8.2 (continued)
(c) The least squares estimator1and b2are functions of the following averages
1 1 1 1 2
x xi yy i xii xi
N N N N
For the generalized least squares estimator 1ornd 2, these unweighted averages are
replaced by the weighted averages
2 2 2 22
i i i i i ii i i
i2 i2 i2 i2
In these weighted averages each observaton is weighted by the inverse of the error
variance. Reliable observations with small error variances are weighted more heavily than
those with higher error variances that make them more unreliable. Chapter 8, Exercise Solutions, Principles of Econometrics, 4e 275
EXERCISE 8.3
For the model e i ii 1 2 where var(ei i 22 , the transformed model that gives a
constant error variance is
yi i x1 i2
* * *
where yi ii x , xi i x , and ei ii . This model can be estimated by least squares
with the usual simple regression formulas, but with and reversed. Thus, the
1 2
generalized least squares estimators for 1 and 2 are
** * *
ˆ ˆNy i i i i and y x *
1 1 2 * 2 * 2
Nx ()i i
Using observations on the transformed variables, we find
y 7 , x 37 12 , xy* 47 8 , (x ) 349 144
i i i i i
With N 5 , the generalized least squares estimates are
ˆ 5(47 8) (37 12)(7)
1 2 2.984
5(349 144) (7 12)
and
ˆ ˆ * * (37 12)
2 1 x (7 5) 2.984 0.44
5 Chapter 8, Exercise Solutions, Principles of Econometrics, 4e 276
EXERCISE 8.4
(a) In the plot of the residuals against income the absolute value of the residuals increases as
income increases, but the same effect is not apparent in the plot of the residuals against
age. In this latter case there is no apparent relationship between the magnitude of the
residuals and age. Thus, the graphs suggest that the error variance depends on income, but
not age.
(b) Since the residual plot shows that the erro r variance may increase when income increases,
and this is a reasonable outcome since greater income implies greater flexibility in travel,
2 2
we set up the null and alternative hypotheses as the one tail test H 0 1 2 versus
2 2 2 2
H 1 1 2 , where 1 and 2 are artificial variance parameters for high and low income
households. The value of the test statistic is
2 (2.9471 10 ) (100 4)
F 2 7 2.8124
2 (1.0479 10 ) (100 4)
The 5% critical value for (96, 96) degrees of freedom is F(0.95,96,96)401 . Thus, we reject
H 0nd conclude that the error variance depends on income.
Remark : An inspection of the file vacation.dat after the observations have been ordered
according to INCOME reveals 7 middle observations with the same value for INCOME,
namely 62. Thus, when the data are ordered only on the basis of INCOME, there is not one
unique ordering, and the values for SSE and SSE will depend on the ordering chosen.
1 2
Those specified in the question were obtained by ordering first by INCOME and then by
AGE.
(c) (i) All three sets of estimates suggest that vacation miles travelled are directly related to
household income and average age of all adu lts members but inversely related to the
number of kids in the household.
(ii) The White standard errors are slightly larger but very similar in magnitude to the
conventional ones from least s quares. Thus, using White’s standard errors leads one
to conclude estimation is less precise, but it does not have a big impact on assessment
of the precision of estimation.
(iii) The generalized least squares standard e rrors are less than the White standard errors
for least squares, suggesting that genera lized least squares is a better estimation
technique. Chapter 8, Exercise Solutions, Principles of Econometrics, 4e 277
EXERCISE 8.5
(a) The table below displays the 95% confidence intervals obtained using the criticalt-value
t(0.975,497)965 and both the least squares standard errors and the White’s standard errors.
After recognizing heteroskedasticity and usin g White’s standard errors, the confidence
intervals for CRIME, AGE and TAX are narrower while the confidence interval for
ROOMS is wider. However, in terms of the magn itudes of the intervals, there is very little

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