Introduction to Econometrics
Home assignment # 4
(Suggested solutions)
1. For this exercise, use the data in fertil.gdt ﬁle. It contains information about a sample
of women in Botswana: their education (variable educ), their age (variable age), and
the number of children they have (variable children).
(a) Estimate the model
children = β0 + β1educ + ε ,
interpret the coeﬃcient β1.
(b) Conduct the Breusch-Pagan test for heteroskedasticity. Does its result justify
the use of robust standard errors? If yes, reestimate the model using these
robust standard errors and comment on the diﬀerence with respect to the model
estimated by simple OLS.
(c) Redeﬁne the model as
children = β0 + β1educ + β2age + β3age2
+ ε
and estimate it.
i. Does age have a signiﬁcant impact on number of children of Botswana
women? State the hypothesis, and compute the test statistics by hand.
Interpret the results and compare it to the results of the test in Gretl.
ii. Do you ﬁnd justiﬁcation for the inclusion of age in quadratic form? [Hint:
State the four speciﬁcation criteria and argue if they are satisﬁed for the
quadratic in age.]
iii. How does the coeﬃcient β1 change when compared to part (a)? Does this
signal any bias in the model from part (a)? Where does it come from?
Explain the sign of this bias.
(d) Do you think the coeﬃcient β1 from the model in part (c) may suﬀer from some
omitted variable bias? What variable(s) could be missing in the model and how
would the coeﬃcient β1 change if they were included in the regression? (What
is the sign of the potential bias in coeﬃcient β1?)
(e) Conduct the RESET test of the model from part (c) and interpret the results.
1
Solution:
(a) The estimation result is:













The meaning of the coeﬃcient β1 is that an additional year of education will
reduce the number of children a woman has by 0.2. (Obviously, children is not
a continuous variable, so this interpretation sounds a little bit odd, but we have
to realize we talk here in averages.)
(b) The result of the test is:











The p-value equal to zero indicates that we should reject the null hypothesis
of homoskedasticity. It means that we have a heteroskedastic error term in the
model and we should use robust standard errors.
When we reestimate the model using robust standard errors, we get:
2














As expected, we have the same coeﬃcients here, but the standard errors are
larger, showing that they were underestimated by OLS in part (a).
(c) The estimation result of the redeﬁned model is:
















i. To test for a signiﬁcant impact of age on number of children of Botswana
women, we need to use an F-test:
H0 : β2 = 0&β3 = 0 vs HA : β2 = 0 ∨ β3 = 0
F =
(SSER − SSEU )/J
SSEU /(n − k)
∼ FJ,n−k ,
F =
(18571.78 − 9284.147)/2
9284.147/(4361 − 4)
= 2179 ,
Fcrit(2,4357;0.95) = 3.00
Therefore, we reject the null hypothesis of insigniﬁcant coeﬃcients. The
quadratic form of age has a positive impact on number of children. The
results of the test in Gretl conﬁrm our conclusion:
3
ii. The speciﬁcation criteria: test of signiﬁcance of the coeﬃcient, change in R2
,
bias in other coeﬃcients, and theory. The signiﬁcant coeﬃcients for the age
and age2
variables, a sharp rise in R2
and a dramatic change of coeﬃcient
for variable educ signal that this model performs much better than the
one in part (a) and age in quadratic form should indeed be included in
the model. Moreover, theory (intuition) also suggests that age and age2
should be included in the model: obviously, very young women have less
children than older women, because they have just started their families.
The impact of age will be smaller for older women, because after some age, it
is biologically impossible to have more children: this justiﬁes the quadratic
speciﬁcation in age, reﬂecting the concave relationship.
iii. The coeﬃcient β1 decreases signiﬁcantly in absolute value when compared
to part (a), which signals that there was an omitted variable bias given by
the omission of the age variable. This bias was negative and was given by
the fact that age is a relevant variable for the model (and its correlation
with the dependent variable is positive - see the coeﬃcient in the estimated
equation) and that it is correlated with the variable educ (this correlation is
negative, which we can verify in Gretl and explain by possible development
of education in Botswana, leading to higher education of new generations as
compared to the old generations). The bias is proportional to the product
of these two correlations and this is why we should expect it to be negative.
(d) We can hypothesize that the coeﬃcient β1 from the model in part (c) may
still suﬀer from some omitted variable bias. A possible variable that could
have an impact on the number of children a woman has may be some variable
representing socio-economic characteristics of the woman’s family. Since we
can suppose that women in better social and economic situation may have less
children (negative correlation) and more education (positive correlation), we
would expect the bias to be negative. Hence, if we had such variable and we
included it in the equation, we should expect the coeﬃcient to become even less
negative than in part (c) (smaller in absolute value).
(e) The result of the RESET test of the model from part (c) is:
4














The very low p-value signals that we should reject the null hypothesis that the model
is correctly speciﬁed. This signals that there are still some variables which are omitted
from our estimation or that our speciﬁcation has an incorrect functional form.
2. Suppose following investment model was estimated with quarterly data from 1997-
2009 (standard errors in parenthesis):
It = 7.70
1.10)
+ 0.55
0.23)
Yt + 0.63
0.12)
Qt2 + 1.55
1.03)
Qt3 + 2.13
0.74)
Qt4 ,
n = 64 , R2
= 0.72 ,
where It is the investment in period t, Yt is the GDP in period t, and dummy variables
Qti are equal to 1 in the i-th quarter and zero otherwise (i = 2, 3, 4). Denote the
coeﬃcients associated with the dummies δ2, δ3 and δ4.
(a) What restriction on these parameters would lead to the model:
It = β0 + βY Yt + δqt + εt ,
where qt = 0, 1, 2, 3 in the ﬁrst, second, third and fourth quarters respectively?
Brieﬂy discuss this restriction. [Hint: To ﬁnd the restrictions, compare the
coeﬃcients of the two models (restricted and unrestricted) for each quarter.]
(b) Test the restriction if the regression R2
of the restricted model was 0.68.
(c) Explain how would you test for presence of AR(4) autocorrelation of the error
term in this model. Describe all steps that you need to take to conduct the test,
the null and alternative hypothesis, and the test statistics.
Solution:
5
(a) To ﬁnd the restrictions, we will compare the expected values of It in each quarter
for both models:
It = α + βY Yt + δ2Qt2 + δ3Qt3 + δ4Qt4 + ηt
It = β0 + βY Yt + δqt + εt .
1st
quarter:
E[It] = α + βY Yt
E[It] = β0 + βY Yt
2nd
quarter:
E[It] = α + δ2 + βY Yt
E[It] = β0 + δ + βY Yt
3rd
quarter:
E[It] = α + δ3 + βY Yt
E[It] = β0 + 2δ + βY Yt
4th
quarter:
E[It] = α + δ4 + βY Yt
E[It] = β0 + 3δ + βY Yt
The comparison gives us the following results:
1st
quarter ⇒ α = β0
2nd
quarter ⇒ α + δ2 = β0 + δ
3rd
quarter ⇒ α + δ3 = β0 + 2δ
4th
quarter ⇒ α + δ4 = β0 + 3δ
which reduces ﬁnally to the two restrictions
δ3 = 2δ2
δ4 = 3δ2 .
These restrictions assume constant diﬀerence between the quarters accross the
year (meaning that there is e.g. the same diﬀerence between the ﬁrst an the
second quarter as between the second and the third one).
6
(b) We will test the null hypothesis that these restiction are valid using the standard
F-test over restricted and unrestricted models. As given in the setup, R2
U = 0.72
and R2
R = 0.68. Number of restrictions J = 2, number of observations n = 64,
number of parameters of unrestricted model k = 5. We construct the F-statistic
F =
(R2
U − R2
R)/J
(1 − R2
U )/(n − k)
=
(0.72 − 0.68)/2
(1 − 0.72)/(64 − 5)
= 4.2143
and when we compare it to the corresponding critical value F2,59 = 3.1531, we
see that we can reject the null hypothesis that the restrictions are valid.
(c) AR(4) autocorrelation of the error term implies that the error term has the
following structure:
εt = ρ1εt−1 + ρ2εt−2 + ρ3εt−3 + ρ4εt−4 + ut
We can test for AR(4) autocorrelation of the error term using analysis of residuals.
Since OLS is consistent even under autocorrelation, the residuals are
consistent estimates of the stochastic error term and we can thus use them to
test for autocorrelation of the error term. The test proceeds as follows:
i. Estimate the original model by OLS, save the residuals et = It − It.
ii. Estimate the model et = α + ρ1et−1 + ρ2et−2 + ρ3et−3 + ρ4et−4 + ut by OLS.
iii. Test if ρ1 = ρ2 = ρ3 = ρ4 = 0 using the standard F-test:
H0 : ρ1 = 0&ρ2 = 0&ρ3 = 0&ρ4 = 0 vs HA : ρ1 = 0∨ρ2 = 0∨ρ3 = 0∨ρ4 = 0
F =
(SSER − SSEU )/J
SSEU /(n − k)
∼ FJ,n−k .
7