Econometrics II, Fall 2017
Department of Economics, University of Copenhagen
By Heino Bohn Nielsen (Edited by Morten Nyboe Tabor)
PS #5
Generalized Method of Moments
This problem sets deals with theoretical and empirical aspects of GMM estima-
tion. Exercise #5.1 focuses on the interpretation of GMM estimations and the
distinction between model variables and instruments. Exercise #5.2 illustrates
how to perform an empirical GMM estimation corresponding to the monetary
policy analysis in Lecture Note 8. Exercise #5.3 derives the relationship between
ordinary least squares (OLS) estimation, instrumental variables (IV) estimation,
generalized instrumental variables (GIV) estimation and generalized method of
moments (GMM) estimation for the case of a linear regression model.
#5.1 Interpretation of IV and GMM Estimations
Assume that a central bank sets the monetary policy instrument, the short term
interest rate rt , according to a simple policy rule of the form
rt = α0 + α1 · E[πt+12 − π ∗ | It ] + α2 · E[yt | It ], (5.1)
where E[· | It ] denotes the rational expectation conditional on the information
set available at time t; πt is the current inflation rate from the year before; π ∗
is the constant inflation target of the central bank; and yt is a measure of the
output gap of the economy so that an increase corresponds to an expansion of
the economy.
(1) Give an economic interpretation of the equation in (5.1).
What do you think could be reasonable values for the parameters?
(2) Explain why the parameters cannot be estimated by OLS.
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(3) Now let zt denote and R−dimensional vector of variables in the information
set at time t, It . Explain how the instruments in zt can be used to estimate
the model in (5.1) using GMM. In particular, replace the expected value
with actual observation and state the moment conditions for estimating the
parameters.
What is the requirement for zt to be valid instruments?
(4) Explain the minimal value of R necessary for identification of the param-
eters in (5.1). Explain in your own words how identification is related to
estimation.
(5) Now consider a specific choice of instruments:
zt = (1, rt−1 , rt−2 , πt−1 , πt−2 , yt−1 , yt−2 , bt−1 , bt−2 , xt−1 , xt−2 )0 ,
where the additional variables bt and xt denote, respectively, the long-term
bond yield and the unemployment rate.
Explain the difference between the role of the variables included in (5.1)
and the instrumental variables in zt . To explain the difference you can
think of two-stage least-squares (2SLS) as a special case of GMM.
Explain (with reference to 2SLS) what it means that a set of instruments
can be weak.
(6) How would the interpretation change if the unlagged variables bt and xt
are included as additional instruments in zt ?
How would the interpretation change if yt was also included?
Which of the cases do you think is the most reasonable in practice?
#5.2 Monetary Policy Rules for the US
This exercise illustrates how to perform a GMM estimation in practice, using the
simple GMM module for OxMetrics. The data and code can be obtained from
the software download page, and the GMM lecture note explains how to install
the GMM module.
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The data file MonetaryPolicyData.xls contains the following 11 variables:
ff Average effective Federal funds rate.
fftarget Target for the Federal funds rate.
bond Average 10 years bond yield.
inf Inflation from year to year.
infexcl Inflation from year to year excluding food and energy.
caputil Capacity utilization.
capgap Measure of output gap: Deviation of caputil from mean.
ip Industrial production.
ipgap Measure of output gap: Deviation of ip from a smooth HP trend.
unr Unemployment rate.
unrgap Measure of output gap: Deviation of unr from mean.
for the US, 1971 : 1 − 2005 : 8.
We want to estimate a monetary policy rule of the form
rt = α0 + α1 · E[πt+12 − π ∗ | It ] + α2 · E[yt | It ], (5.2)
and we focus on the period with Greenspan as a chairman for the Federal Reserve
Board: 1987 − 2005.
(1) Open the data set in OxMetrics, and construct time series graphs of the
data.
(a) Compare the three measures of the output gap.
(b) Can you see any apparent relation between inflation and the policy
instrument?
(c) Can you see any apparent relation between the output gap and the
policy instrument?
Do the graphs for the full period and for the period under Greenspan,
1987 : 1 − 2005 : 8, seem to tell the same story?
(2) We measure inflation by the variable inf and the output gap by the variable
capgap. As instruments we take lag one and two of the variables, i.e.
zt = (1, ff t−1 , ff t−2 , inf t−1 , inf t−2 , capgapt−1 , capgapt−2 )0 .
Set up the model using the GMM module and estimate the parameters
using a heteroskedasticity and autocorrelation consistent (HAC) estimator
of the weight matrix. Choose a Bartlett kernel HAC-estimator with a
bandwidth B = 12.
[Hint: Follow the steps taken in the GMM lecture note.]
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(3) Construct the predicted policy rate. To do this print the vector of residuals,
ufunc, from the [Other settings] in the GMM module. Copy this time
series to the data base and construct the prediction as predt = ff t − ufunct .
What is your impression of the model?
(4) Change the number of lags for the instruments and see whether the results
are robust.
(5) Change the settings for the estimator of the weight matrix and see whether
the results are robust.
#5.3 OLS, IV and Linear GMM Estimation
Consider the linear regression model
yt = x01t β1 + x02t β2 + t = x0t β + t , t = 1, 2, ..., T, (5.3)
where x1t is a K1 × 1 vector, x2t is a K2 × 1 vector, while xt = (x01t , x02t )0 and
β = (β10 , β20 )0 are both a K × 1 vectors with K = K1 + K2 . Assume that yt and xt
are stationary and weakly dependent, so that the usual statistical results hold.
(1) (OLS) State the minimal conditions for consistency of the OLS estimator
in the regression model (5.3). How is this related to the interpretation of
(5.3) as a conditional expectation: E[yt | xt ] = x0t β ?
Discuss how this assumption can be used to construct moment conditions,
g(β) = E[f (yt , xt , β)] = 0,
for estimating β. Write the corresponding sample moment conditions
T
1X
gT (β) = f (yt , xt , β) = 0,
T
t=1
and derive the OLS estimator, βbOLS .
Now assume that the K2 variables in x2t are endogenous, in the sense
E[x2t t ] 6= 0.
How does that effect the properties of OLS?
(2) (IV) Now we assume the existence of K2 new instrumental variables, z2t ,
with the property
E[z2t t ] = 0. (5.4)
Should the new instruments, z2t , fulfill other requirements besides (5.4) for
being valid and relevant instruments?
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Define the K × 1 vector of instruments, zt = (x01t , z2t
0 )0 . State the pop-
ulation moment conditions for the instrumental variables (IV) estimator,
βbIV , in this model. Write the corresponding sample moment conditions
and derive the IV estimator.
Discuss why the simple IV estimator does not work if the number of
instruments is larger than the number of parameters.
(3) (GMM) Now assume that the number of instruments in zt , R, is larger
than the number of parameters, K. Explain the intuition for the GMM
estimator by referring to the quadratic form:
QT (β) = gT (β)0 WT gT (β). (5.5)
What is the role of the weight matrix WT , and how should it be optimally
chosen?
State the sample moments, gT (β), for the case R > K. Insert the
moment conditions in (5.5) and derive the GMM estimator for a given
weight matrix, βbGM M (WT ), as the solution to
∂QT (β)
= 0.
∂β
Can you think of any difficulties in implementing GMM estimation in prac-
tice?
(4) (GIV) Discuss how the optimal weight matrix, WTopt , can be estimated if
t is identically and independently distributed, IID.
Insert the optimal weight matrix in the formula for the estimator to
obtain βbGM M (WTopt ) and show that it simplifies to the generalized IV esti-
mator. Show that the GIV estimator, βbGIV , can be derived as a two-stage
least squares estimator.
