Introductory Econometrics
Endogeneity
Suggested Solution
by Hieu Nguyen
Fall 2024
1.
Suppose that you wish to estimate the effect of class attendance on student performance. A model
to explain standardized outcome on a final exam (stndfnl) in terms of percentage of classes attended
(attnd), prior college Grade Point Average (priGPA), and American College Testing score (ACT) is:
stndfnl = β0 + β1attnd + β2priGPA + β3ACT + ϵ.
(a) Why might attnd be suspected to be endogenous in the model?
(b) Let dist be the distance from the students’ living quarters to the lecture hall. Do you think dist is
uncorrelated with ϵ?
(c) Assuming that dist and ϵ are uncorrelated, what other assumption must dist satisfy in order to be
a good instrument for attnd?
(d) Suppose we add the interaction term priGPA · attnd to the model:
stndfnl = β0 + β1attnd + β2priGPA + β3ACT + β4priGPA · attnd + u.
If attnd is correlated with ϵ, then, in general, so is priGPA · attnd. What might be a good
instrument candidate for priGPA · attnd?
Solution:
(a) We might be worried that attnd is correlated with other factors in ϵ. There are various potential
reasons why. E.g., highly motivated students might attend more classes (omitted variable bias).
Or some students need to have a part-time job (a piece of information not collected—thus unobservable—by
the University) to cover their living expenses, so they might not have enough time to
attend classes as well as to study sufficiently for the final exam (omitted influence bias). Hence the
OLS regression of stndfnl on attnd may give us a poor estimate of the causal impact of attended
classes.
(b) In a multiple regression model, various factors potentially correlated with dist (e.g., students from
low-income families or exchange/foreign students may live off-campus) affecting stndfnl can be
included directly in the model so that we can control for their potential impact. Then, dist can
be reasonably (as it generally is as a typical textbook example) expected to be uncorrelated with
ϵ, primarily if the campus rooms are assigned randomly (as is often a best practice at many
universities).
(c) It needs to be correlated with attnd. Especially at large universities, some students commute to
campus, which may increase the likelihood of missing lectures (bad weather, oversleeping, etc.). Or
they may be lazy to commute, or they decide to study efficiently ‘at home’ instead of commuting.
Thus attnd is expected to be negatively correlated with dist. This can be checked by regressing
attnd on dist via a t-test.
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(d) The idea is that class attendance might have a different effect on students who have performed
differently in the past, as measured by priGPA. Thus we need an instrument that is not correlated
with attnd but correlated with priGPA. As priGPA, ACT, and dist are assumed exogenous, then
(technically) any function of these three variables is also exogenous so that we may think of, e.g.,
priGPA · dist. Or we may try to find a real-world variable correlated with priGPA. Or what about
the level of education of one’s mother and father (see seminar #8)?
2.
The data in fertil2.gdt includes, for a sample of women in Botswana during 1988, information on the
number of children, years of education, age, and religious and economic status variables.
(a) Estimate this model by OLS and briefly comment on results:
children = β0 + β1educ + β2age + β3age2
+ ϵ.
If 100 women receive another year of education, how many fewer children are they expected to
have?
(b) In lecture #10, we discussed why we might suspect educ to be endogenous in this model. We also
suggested frsthalf (a dummy variable equal to one if the woman was born in the first six months
of a year) to be a good candidate for an instrument for educ. Show its relevance via a first stage
regression. Assume that frsthalf is uncorrelated with the error term ϵ. Now estimate the model
from part (a) by using frsthalf as an instrument for educ (= IV estimator, 2SLS). Compare the
estimated effect of education with the OLS estimate. Which of the estimators is consistent?
(c) Add the binary explanatory variables electric, tv, and bicycle to the model and assume these
are exogenous as well. Estimate the equation by 2SLS directly in Gretl and compare the estimated
coefficient of educ with part (b) and with the OLS estimate. Interpret the output of the Hausman
test.
Solution:
(a) The model estimated by OLS is:
Model 1: OLS, using observations 1{4361
Dependent variable: children
Coefficient Std. Error t-ratio p-value
-------------------------------------------------
const -4.13831 0.240594 -17.2004 0.0000
educ -0.0905755 0.00592069 -15.2981 0.0000
age 0.332449 0.0165495 20.0882 0.0000
sq age -0.00263082 0.000272592 -9.6511 0.0000
Mean dependent var 2.267828 S.D. dependent var 2.222032
Sum squared resid 9284.147 S.E. of regression 1.459746
R^2 0.568724 Adjusted R^2 0.568427
F (3, 4357) 1915.196 P-value(F ) 0.000000
Log-likelihood -7835.592 Akaike criterion 15679.18
Schwarz criterion 15704.71 Hannan{Quinn 15688.19
Following our expectation, educ has a significantly negative effect, and estimated coefficients of
age and age2
support its polynomial functional form. Interpretation of the model was discussed in
detail during the seminar.
If we interpret the estimated effect of one additional year of educ literally, it suggests reducing the
estimated number of children by 0.09. But this is impossible for any particular woman (children is
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a typical example of a discrete count variable). A standard economic interpretation is that average
fertility decreases by 0.09 children given one more year of education. More reasonably worded: if
100 women receive another year of education, the estimated model suggests that there will be nine
fewer children among this group in the future.
(b) To check the relevance of the frsthalf instrument and to run the 2SLS manually; we start with
the first stage regression:
Model 2: OLS, using observations 1{4361
Dependent variable: educ
Coefficient Std. Error t-ratio p-value
-------------------------------------------------
const 9.69286 0.598069 16.2069 0.0000
age -0.107950 0.0420402 -2.5678 0.0103
sq age -0.000505567 0.000692940 -0.7296 0.4657
frsthalf -0.852285 0.112830 -7.5537 0.0000
Mean dependent var 5.855996 S.D. dependent var 3.927075
Sum squared resid 60001.14 S.E. of regression 3.710957
R^2 0.107651 Adjusted R^2 0.107037
F (3, 4357) 175.2068 P-value(F ) 3.0e{107
Log-likelihood -11904.53 Akaike criterion 23817.05
Schwarz criterion 23842.57 Hannan{Quinn 23826.06
We can see that frsthalf is a relevant instrument. It is correlated with the endogenous explanatory
variable educ: Cov(educ, frsthalf) ̸= 0 because it is statistically strongly significant in the first
stage regression (t-test). But note that the R2
of the model is rather small. Now save fitted values
as educ hat2 (the fitted value from model 2), follow Save—Fitted values in the Gretl Model 1
menu.
We continue with the second stage regression:
Model 3: OLS, using observations 1{4361
Dependent variable: children
Coefficient Std. Error t-ratio p-value
-------------------------------------------------
const -3.38781 0.550340 -6.1558 0.0000
educ hat2 -0.171499 0.0533921 -3.2121 0.0013
age 0.323605 0.0179310 18.0473 0.0000
sq age -0.00267228 0.000280805 -9.5165 0.0000
Mean dependent var 2.267828 S.D. dependent var 2.222032
Sum squared resid 9759.726 S.E. of regression 1.496667
R^2 0.546632 Adjusted R^2 0.546320
F (3, 4357) 1751.100 P-value(F ) 0.000000
Log-likelihood -7944.521 Akaike criterion 15897.04
Schwarz criterion 15922.56 Hannan{Quinn 15906.05
Contrasting the OLS and 2SLS estimated models, we observe a potential reduction of the positive
OLS bias of educ—the estimated effect is now much more extensive (in the expected negative
direction). The SE of the IV estimate for educ is also much bigger, about nine times! This
produces a relatively wide 95
But be aware that the SEs and test statistics obtained this way are generally invalid. The reason
is that the theoretical composite error term of the second stage model also includes the error
term from the first stage. Still, the second stage SEs, if estimated ‘manually,’ are based on only
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residuals of the second stage (OLS does not know that we estimated the first stage before). It is
thus preferred to run 2SLS jointly to account for residuals from both stages. This is an automated
standard option in any regression package:
Model 4: TSLS, using observations 1{4361
Dependent variable: children
Instrumented: educ
Instruments: const frsthalf age sq age
Coefficient Std. Error z p-value
-------------------------------------------------
const -3.38781 0.548150 -6.1804 0.0000
educ -0.171499 0.0531796 -3.2249 0.0013
age 0.323605 0.0178596 18.1194 0.0000
sq age -0.00267228 0.000279687 -9.5545 0.0000
Mean dependent var 2.267828 S.D. dependent var 2.222032
Sum squared resid 9682.216 S.E. of regression 1.490712
R^2 0.552676 Adjusted R^2 0.552368
F (3, 4357) 1765.119 P-value(F ) 0.000000
Log-likelihood -47917.57 Akaike criterion 95843.15
Schwarz criterion 95868.67 Hannan{Quinn 95852.15
Not much has changed, but now the SEs are valid, and we can use them for hypothesis testing.
To answer which of the estimators is consistent, we need to test whether educ is an endogenous
variable, more generally, whether there is evidence of endogeneity in the data. We know that 2SLS
is consistent in both cases but inefficient if there is no endogeneity. On the other hand, OLS is
inconsistent under endogeneity. We use the Hausman test, which output is automatically attached
to the automated Gretl 2SLS regression:
Hausman test {
Null hypothesis: OLS estimates are consistent
Asymptotic test statistic: 2(1) = 2.4501
with p-value = 0.117517
Hypotheses vaguely:
H0 : OLS estimator is consistent vs HA : OLS estimator is inconsistent,
H0 : no endogeneity vs HA : endogeneity.
Hausman (Wald) test statistic:
H(orW) = (ˆβ2SLS − ˆβOLS)′
(Var(ˆβ2SLS − ˆβOLS))−1
(ˆβ2SLS − ˆβOLS) ∼ χ2
k+1.
The critical value for the ‘default’ 5% significance level and four d.f.: 9.49 (would be 3.84 for one
d.f.) ¿ 2.45 ⇒ in any case, we do not reject the H0 ⇒ the consistency of the OLS estimator is not
rejected at 5%. Hence, it should be used preferably because it is efficient.
This result does not necessarily mean (and definitely not prove) that educ is exogenous. The p-value
suggests that we would reject H0 at the 12% significance level, so our conclusion in the previous
paragraph is not very statistically strong. Because the sample size is large, we might rather suspect
insufficient explanatory power of the instrument (remind the not very large R2
of the first stage
model and see also sample correlation between educ and frsthalf) to cause this situation. There
might be other not yet considered omitted variables/influences in the model causing additional
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bias, maybe in the opposite direction, thus decreasing the power of the Hausman test to recognize
endogeneity. Finally, a linear model might not be the correct functional form for the discrete count
nature of the dependent variable. Finally, the discrete count nature of the dependent variable is
likely to be a problematic issue for linear regression (will be discussed in lecture #11). All these
aspects might decrease the overall power of the test.
Advice about what to do when there is uncertainty as to whether an explanatory variable is endogenous
or not is somewhat mixed. The prevailing attitude is probably summarized by Wooldridge
(2010) who suggests: “We find evidence of endogeneity of ... at the 10% significance level against
a two-sided alternative, and so 2SLS is probably a good idea (assuming that we trust the instruments.)”.
Moreover, Guggenberger (2010) advises that if testing the coefficient of the endogenous
explanatory variable is the objective, we should avoid considering the Hausman test result and use
2SLS.
(c) Model estimated by the 2SLS routine:
Model 5: TSLS, using observations 1{4361 (n = 4356)
Missing or incomplete observations dropped: 5
Dependent variable: children
Instrumented: educ
Instruments: const frsthalf age sq age electric tv bicycle
Coefficient Std. Error z p-value
-------------------------------------------------
const -3.59133 0.645089 -5.5672 0.0000
educ -0.163981 0.0655269 -2.5025 0.0123
age 0.328145 0.0190587 17.2176 0.0000
sq age -0.00272216 0.000276559 -9.8430 0.0000
electric -0.106531 0.165965 -0.6419 0.5209
tv -0.00255501 0.209230 -0.0122 0.9903
bicycle 0.332072 0.0515264 6.4447 0.0000
Mean dependent var 2.268365 S.D. dependent var 2.222073
Sum squared resid 9511.714 S.E. of regression 1.478886
R^2 0.559569 Adjusted R^2 0.558961
F (6, 4349) 921.7086 P-value(F ) 0.000000
Log-likelihood -47487.53 Akaike criterion 94989.05
Schwarz criterion 95033.71 Hannan{Quinn 95004.82
Hausman test – Null hypothesis: OLS estimates are consistent Asymptotic test statistic: 2(1) =
1.87295 with p-value = 0.171137
The interpretation of the Hausman test is precisely the same as in the previous part, with similar
caveats. ‘No endogeneity’ H0 is technically not rejected. We thus compare the results with a model
estimated by OLS:
Model 6: OLS, using observations 1{4361 (n = 4356)
Missing or incomplete observations dropped: 5
Dependent variable: children
Coefficient Std. Error t-ratio p-value
-------------------------------------------------
const -4.38978 0.240317 -18.2666 0.0000
educ -0.0767093 0.00635259 -12.0753 0.0000
age 0.340204 0.0164417 20.6915 0.0000
sq age -0.00270808 0.000270551 -10.0095 0.0000
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electric -0.302729 0.0761869 -3.9735 0.0001
tv -0.253144 0.0914374 -2.7685 0.0057
bicycle 0.317895 0.0493661 6.4395 0.0000
Mean dependent var 2.268365 S.D. dependent var 2.222073
Sum squared resid 9116.101 S.E. of regression 1.447804
R^2 0.576060 Adjusted R^2 0.575475
F (6, 4349) 984.9211 P-value(F ) 0.000000
Log-likelihood -7789.323 Akaike criterion 15592.65
Schwarz criterion 15637.30 Hannan{Quinn 15608.41
Adding electric, tv, and bicycle to the model slightly reduces the estimated effect of educ in
both cases. In the model estimated by OLS, the coefficient on tv implies that other factors fixed,
four families that own a television will have about one fewer child than four families without a TV.
A causal interpretation is that TV provides an alternative form of recreation.
A comparison of the models and a general idea behind the inclusion of new variables was discussed
in detail during the seminar (e.g., only 14% of women have electricity in their homes, which can
be considered as a proxy for good economic background of the family; television ownership can
be a proxy for different things, including income and perhaps geographic location, what about an
intuitive logic of the bicycle, which is strongly statistically significant in both models?).
Interestingly, the effects of electric and tv substantially drop in the model estimated by 2SLS.
This supports our suspicion about the functional form of the model.
3.
A researcher estimated by OLS two specifications of a regression model:
y = α + βx1 + ϵ,
y = ˜α + ˜βx1 + ˜γx2 + ˜ϵ
Explain theoretically under what circumstances the following will be true. If some case cannot be
true, explain why.
(a) ˆβ =
ˆ˜β.
(b) β is statistically significant (at the 5
(c) ˜β is statistically significant (at the 5% level) but β is not.
(d) If ˆϵi and ˆ˜iϵ are the estimated residuals from the two equations,
n
i=1
ˆϵi
2
≥
n
i=1
ˆ˜iϵ2
.
Solution:
(a) The estimated coefficients from the two models will be the same if, by omitting x2, we do not cause
any bias of the estimator of β. This happens either if γ = 0 (x2 is an irrelevant variable) or when
x1 and x2 are not correlated. Hence,
ˆβ = ˆβ ⇔ Cov(x1, x2) = 0 or γ = 0.
(b) This will typically happen when x1 and x2 are highly correlated. When x2 is not included in the
model, most of its explanatory power is attributed to x1, and its significance can be overestimated.
When x2 is included, the explanatory power is diluted between the two variables, and x1 can lose
its significance (multicollinearity problem).
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(c) This can happen in the situation when the second model is correct (x2 should be included, i.e.,
γ ̸= 0). In this case, x2 is an omitted variable in the first model, and we experience an omitted
variable bias. If the omission of x2 biases the OLS estimator of β enough in the direction towards
zero, we are likely to observe the impact of the bias in the numerical value of ˆβ and β can thus
become statistically insignificant.
(d) We can consider the two models as a restricted and an unrestricted version of the second model.
We know from lectures that the inequality below always holds:
RSSR ≥ RSSU .
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