Introduction to Econometrics, 3e (Stock)
Chapter 12 Instrumental Variables Regression
12.1 Multiple Choice
1) Estimation of the IV regression model
A) requires exact identification.
B) allows only one endogenous regressor, which is typically correlated with the error term.
C) requires exact identification or overidentification.
D) is only possible if the number of instruments is the same as the number of regressors.
2) Two Stage Least Squares is calculated as follows; in the first stage:
A) Y is regressed on the exogenous variables only. The predicted value of Y is then regressed on the
instrumental variables.
B) the unknown coefficients in the reduced form equation are estimated by OLS, and the predicted values
are calculated. In the second stage, Y is regressed on these predicted values and the other exogenous
variables.
C) the exogenous variables are regressed on the instruments. The predicted value of the exogenous
variables is then used in the second stage, together with the instruments, to predict the dependent
variable.
D) the unknown coefficients in the reduced form equation are estimated by weighted least squares, and
the predicted values are calculated. In the second stage, Y is regressed on these predicted values and the
other exogenous variables.
3) The conditions for a valid instruments do not include the following:
A) each instrument must be uncorrelated with the error term.
B) each one of the instrumental variables must be normally distributed.
C) at least one of the instruments must enter the population regression of X on the Zs and the Ws.
D) perfect multicollinearity between the predicted endogenous variables and the exogenous variables
must be ruled out.
4) The IV regression assumptions include all of the following with the exception of
A) the error terms must be normally distributed.
B) E(ui W1i,…, Wri) = 0.
C) Large outliers are unlikely: the X‘s, W‘s, Z‘s, and Y‘s all have nonzero, finite fourth moments.
D) (X1i,…, Xki, W1i,…,Wri, Z1i, … Zmi, Yi) are i.i.d. draws from their joint distribution.
5) The rule–of–thumb for checking for weak instruments is as follows: for the case of a single endogenous
regressor,
A) a first stage F must be statistically significant to indicate a strong instrument.
B) a first stage F > 1.96 indicates that the instruments are weak.
C) the t–statistic on each of the instruments must exceed at least 1.64.
D) a first stage F < 10 indicates that the instruments are weak.
6) The J–statistic
A) tells you if the instruments are exogenous.
B) provides you with a test of the hypothesis that the instruments are exogenous for the case of exact
identification.
C) is distributed where m–k is the degree of overidentification.
D) is distributed where m–k is the number of instruments minus the number of regressors.
7) In the case of the simple regression model Yi = β0 + β1Xi + ui, i = 1, …, n, when X and u are correlated,
then
A) the OLS estimator is biased in small samples only.
B) OLS and TSLS produce the same estimate.
C) X is exogenous.
D) the OLS estimator is inconsistent.
8) The following will not cause correlation between X and u in the simple regression model:
A) simultaneous causality.
B) omitted variables.
C) irrelevance of the regressor.
D) errors in variables.
9) The distinction between endogenous and exogenous variables is
A) that exogenous variables are determined inside the model and endogenous variables are determined
outside the model.
B) dependent on the sample size: for n > 100, endogenous variables become exogenous.
C) depends on the distribution of the variables: when they are normally distributed, they are exogenous,
otherwise they are endogenous.
D) whether or not the variables are correlated with the error term.
10) The two conditions for a valid instrument are
A) corr(Zi, Xi) = 0 and corr(Zi, ui) ≠ 0.
B) corr(Zi, Xi) = 0 and corr(Zi, ui) = 0.
C) corr(Zi, Xi) ≠ 0 and corr(Zi, ui) = 0.
D) corr(Zi, Xi) ≠ 0 and corr(Zi, ui) ≠ 0.
11) Instrument relevance
A) means that the instrument is one of the determinants of the dependent variable.
B) is the same as instrument exogeneity.
C) means that some of the variance in the regressor is related to variation in the instrument.
D) is not possible since X and u are correlated and Z and u are not correlated.
12) Consider a competitive market where the demand and the supply depend on the current price of the
good. Then fitting a line through the quantity–price outcomes will
A) give you an estimate of the demand curve.
B) estimate neither a demand curve nor a supply curve.
C) enable you to calculate the price elasticity of supply.
D) give you the exogenous part of the demand in the first stage of TSLS.
13) When there is a single instrument and single regressor, the TSLS estimator for the slope can be
calculated as follows:
A)
1
ˆTSLS ZY
ZX
S
S
=
.
B)
12
ˆTSLS XY
X
S
S
=
.
C)
1
ˆTSLS ZX
ZY
S
S
=
.
D)
12
ˆTSLS ZY
Z
S
S
=
.
14) The TSLS estimator is
A) consistent and has a normal distribution in large samples.
B) unbiased.
C) efficient in small samples.
D) F–distributed.
15) The reduced form equation for X
A) regresses the endogenous variable X on the smallest possible subset of regressors.
B) relates the endogenous variable X to all the available exogenous variables, both those included in the
regression of interest and the instruments.
C) uses the predicted values of X from the first stage as a regressor in the original equation.
D) uses smaller standard errors, such as homoskedasticity–only standard errors, for inference.
16) When calculating the TSLS standard errors
A) you do not have to worry about heteroskedasticity, since it was eliminated in the first stage
B) you can use the standard errors reported by OLS estimation of the second stage regression.
C) the critical values from the standard normal table should be adjusted for the proper degrees of
freedom.
D) you should use heteroskedasticity–robust standard errors.
17) Having more relevant instruments
A) is a problem because instead of being just identified, the regression now becomes overidentified.
B) is like having a larger sample size in that the more information is available for use in the IV
regressions.
C) typically results in larger standard errors for the TSLS estimator.
D) is not as important for inference as having the same number of endogenous variables as instruments.
18) Weak instruments are a problem because
A) the TSLS estimator may not be normally distributed, even in large samples.
B) they result in the instruments not being exogenous.
C) the TSLS estimator cannot be computed.
D) you cannot predict the endogenous variables any longer in the first stage.
19) (Requires Appendix material) The relationship between the TSLS slope and the corresponding
population parameter is:
A)
( )
( )
( )( )
1
11
1
1
ˆ
1
n
ii
TSLS i
n
ii
i
Z Z u
n
Z Z X X
n
=
=
−
−=
−−
.
B)
( )
( )
( )( )
1
11
1
1
ˆ
1
n
i
TSLS i
n
ii
i
ZZ
n
Z Z X X
n
=
=
−
−=
−−
.
C)
( )
( )
( )
1
11 2
1
1
ˆ
1
n
ii
TSLS i
n
i
i
Z Z u
n
ZZ
n
=
=
−
−=
−
.
D)
( )
( )
( )( )
1
11
1
1
ˆ
1
n
ii
TSLS i
n
ii
i
X X u
n
Z Z X X
n
=
=
−
−=
−−
.
20) If the instruments are not exogenous,
A) you cannot perform the first stage of TSLS.
B) then, in order to conduct proper inference, it is essential that you use heteroskedasticity–robust
standard errors.
C) your model becomes overidentified.
D) then TSLS is inconsistent.
21) In the case of exact identification
A) you can use the J–statistic in a test of overidentifying restrictions.
B) you cannot use TSLS for estimation purposes.
C) you must rely on your personal knowledge of the empirical problem at hand to assess whether the
instruments are exogenous.
D) OLS and TSLS yield the same estimate.
22) To calculate the J–statistic you regress the
A) squared values of the TSLS residuals on all exogenous variables and the instruments. The statistic is
then the number of observations times the regression R2.
B) TSLS residuals on all exogenous variables and the instruments. You then multiply the
homoskedasticity–only F–statistic from that regression by the number of instruments.
C) OLS residuals from the reduced form on the instruments. The F–statistic from this regression is the J–
statistic.
D) TSLS residuals on all exogenous variables and the instruments. You then multiply the
heteroskedasticity–robust F–statistic from that regression by the number of instruments.
23) (Requires Chapter 8) When using panel data and in the presence of endogenous regressors
A) the TSLS does not exist.
B) you do not have to worry about the validity of instruments, since there are so many fixed effects.
C) the OLS estimator is consistent.
D) application of the TSLS estimator is straightforward if you use two time periods and difference the
data.
24) In practice, the most difficult aspect of IV estimation is
A) finding instruments that are both relevant and exogenous.
B) that you have to use two stages in the estimation process.
C) calculating the J–statistic.
D) finding instruments that are exogenous. Relevant instruments are easy to find.
25) Consider a model with one endogenous regressor and two instruments. Then the J–statistic will be
large
A) if the number of observations are very large.
B) if the coefficients are very different when estimating the coefficients using one instrument at a time.
C) if the TSLS estimates are very different from the OLS estimates.
D) when you use homoskedasticity–only standard errors.
26) Let W be the included exogenous variables in a regression function that also has endogenous
regressors (X). The W variables can
A) be control variables
B) have the property E(ui|Wi) = 0
C) make an instrument uncorrelated with u
D) all of the above
27) The logic of control variables in IV regressions
A) parallels the logic of control variables in OLS
B) only applies in the case of homoskedastic errors in the first stage of two stage least squares estimation
C) is different in a substantial way from the logic of control variables in OLS since there are two stages in
estimation
D) implies that the TSLS is efficient
28) For W to be an effective control variable in IV estimation, the following condition must hold
A) E(ui) = 0
B) E(ui|Zi,Wi) = E(ui|Wi)
C) E(uiuj) ≠ 0
D) there must be an intercept in the regression
29) The IV estimator can be used to potentially eliminate bias resulting from
A) multicollinearity.
B) serial correlation.
C) errors in variables.
D) heteroskedasticity.
30) Instrumental Variables regression uses instruments to
A) establish the Mozart Effect.
B) increase the regression R2.
C) eliminate serial correlation.
D) isolate movements in X that are uncorrelated with u.
31) Endogenous variables
A) are correlated with the error term.
B) always appear on the LHS of regression functions.
C) cannot be regressors.
D) are uncorrelated with the error term.
32) Consider the following two equations to describe labor markets in various sectors of the economy
Nd = β0 + β1 + u
Ns = γ0 + γ1 + v
Nd = Ns = N
A) W/P is exogenous, n is endogenous
B) Both n and W/P are endogenous
C) n is exogenous, W/P is endogenous
D) the parameters cannot be estimated because it would require two equations to be estimated at the
same time (simultaneously)
12.2 Essays and Longer Questions
1) Write a short essay about the Overidentifying Restrictions Test. What is meant exactly by
“overidentification?” State the null hypothesis. Describe how to calculate the J–statistic and what its
distribution is. Use an example of two instruments and one endogenous variable to explain under what
situation the test will be likely to reject the null hypothesis. What does this example tell you about the
exactly identified case? If your variables pass the test, is this sufficient for these variables to be good
instruments?
2) Using some of the examples from your textbook, describe econometric studies which required
instrumental variable techniques. In each case emphasize why the need for instrumental variables arises
and how authors have approached the problem. Make sure to include a discussion of overidentification,
the validity of instruments, and testing procedures in your essay.
3) Describe the consequences of estimating an equation by OLS in the presence of an endogenous
regressor. How can you overcome these obstacles? Present an alternative estimator and state its
properties.
4) Write an essay about where valid instruments come from. Part of your explorations must deal with
checking the validity of instruments and what the consequences of weak instruments are.
5) You have estimated a government reaction function, i.e., a multiple regression equation, where a
government instrument, say the federal funds rate, depends on past government target variables, such as
inflation and unemployment rates. In addition, you added the previous period’s popularity deficit of the
government, e.g. the (approval rating of the president – 50%), as one of the regressors. Your idea is that the
Federal Reserve, although formally independent, will try to expand the economy if the president is
unpopular. One of your peers, a political science student, points out that approval ratings depend on the
state of the economy and thereby indirectly on government instruments. It is therefore endogenous and
should be estimated along with the reaction function. Initially you want to reply by using a phrase that
includes the words “money neutrality” but are worried about a lengthy debate. Instead you state that as
an economist, you are not concerned about government approval ratings, and that government approval
ratings are determined outside your (the economic) model. Does your whim make the regressor
exogenous? Why or why not?
6) You have been hired as a consultant to estimate the demand for various brands of coffee in the market.
You are provided with annual price data for two years by U.S. state and the quantities sold. You want to
estimate a demand function for coffee using this data. What problems do you think you will encounter if
you estimated the demand equation by OLS?