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BES Tutorial Sample Solutions, S1/11
WEEK 11 TUTORIAL EXERCISES (To be discussed in the week starting
May 16)
1. Use a calculator to compute the sample least squares regression line for
the model ݕ ൌ ߚ
ߚ
ଵ
ݔ ߝ, given the following six observations.
y
2
8
6
12
9
11
x
1
4
3
10
10
8
ݔҧ ൌ
14310108
6
ൌ6
;ݕ
തൌ
28612911
6
ൌ8
ሺݔ
െݔҧሻሺݕ
െݕ
തሻ ൌ ሺ1 െ 6ሻ ሺ2 െ 8ሻ ڮ ሺ8 െ 6ሻ ሺ11 െ 8ሻ ൌ 62
ሺݔ
െݔҧሻ
ଶ
ൌ ሺ1െ6ሻ
ଶ
ڮ ሺ8െ6ሻ
ଶ
ൌ 74
ܾ
ଵ
ൌ
ݏ
௫௬
ݏ
௫
ଶ
ൌ
∑ሺݔ
െݔҧሻሺݕ
െݕ
തሻ
∑ሺݔ
െݔҧሻ
ଶ
ൌ
62
74
ൎ0
.8378
ܾ
ൌݕ
തെܾ
ଵ
ݔҧ ൌ 8 െ 0.8378 ൈ 6 ൌ 2.9732
Thus the sample regression line is ݕ
ො ൌ 2.9732 0.8378ݔ
2. Suppose the relationship between the dependent variable weekly
household consumption expenditure in dollars (
y
) and the independent
variable weekly household income in dollars (
x
) is represented by the
simple regression model (
i
refers to the
i
th observation or household):
ݕ
ൌߚ
ߚ
ଵ
ݔ
ߝ
Suppose a sample of observations yields least squares estimates of
b
0
= ‐32 and
b
1
= 0.82.
(a)
What does ߝ
represent in the model?
It is the random disturbance term.
It includes any purely random factors or
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(b)
State the basic (classical) assumptions made about the
‘s in this
model. Explain in words what the assumptions mean.
(i)
ܧሺߝ
ݔ
ሻൌ0
for all observations. The conditional mean of the disturbance
does not depend on x and is normalized to zero. Note this is different from
Keller who only mentions the normalization to zero. That the conditional
mean of the disturbances does not depend on x ensures unbiasedness of
the OLS estimator and so is the much more important component of this
assumption. Relating back to the previous part of the question it implies
that omitted factors that might affect expenditure but appear in the
disturbance are assumed to be uncorrelated with x.
(ii)
ሺݕ
,ݔ
ሻ
are drawn by simple random sampling and hence iid.
(iii)
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 Three '11
 DenzilGFiebig

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