Single Equation Linear GMM
Eric Zivot
Winter 2013
Single Equation Linear GMM
Consider the linear regression model
= z
0
δ0 + = 1
z = × 1 vector of explanatory variables
δ0 = × 1 vector of unknown coefficients
= random error term
Engodeneity
The model allows for the possibility that some or all of the elements of may
be correlated with the error term (i.e., [] 6= 0 for some ).
If [] 6= 0 then is called an endogenous variable.
If z contains endogenous variables, then the least squares estimator of δ0 in
is biased and inconsistent.
Instruments
It is assumed that there exists a ×1 vector of instrumental variables x that
may contain some or all of the elements of z.
Let w represent the vector of unique and nonconstant elements of { zx}.
It is assumed that {w} is a stationary and ergodic stochastic process.
Moment Conditions for General Model
Define
g(w δ0) = x = x( − z0δ0)
It is assumed that the instrumental variables x satisfy the set of orthogo-
nality conditions
[g(w δ0)] = [x] = [x( − z0δ0)] = 0
Expanding gives the relation
[x]−[xz0]δ0 = 0
⇒ Σ
(×1)
= Σ
(×)
δ0
(×1)
where Σ = [x] and Σ = [xz0].
Example: Demand-Supply model with supply shifter
demand: = 0 + 1 +
supply: = 0 + 1 + 2temp +
equilibrium: =
=
[temp] = [temp] = 0
Goal: Estimate demand equation; 1 = demand elasticity if data are in logs
Remark
Simultaneity causes to be endogenous in demand equation: [] 6= 0
Why? Use equilibrium condition, solve for and compute []
Demand/Supply model in Hayashi notation
= z
0
δ +
=
z = (1 )
0 δ = (0 1)
0
x = (1 temp)
0 w = ( temp)
0
= 2 = 2
Note: 1 is common to both z and x
Example: Wage Equation
ln = 1 + 2 + 3 + 4 +
= wages
= years of schooling
= years of experience
= score on IQ test
2 = rate of return to schooling
Assume
[] = [] = 0
Endogeneity: is a proxy for unobserved ability but is measured with error
= +
⇒ [] 6= 0
Instruments:
= age in years
= years of mother’s education
[] = [] = 0
Wage Equation in Hayashi notation
= z
0
δ +
= ln
z = (1 )
0
x = (1 )
0
w = (ln )
0
= 4 = 5
Remarks
1. 1 are often called included exogenous variables. That is, they
are included in the behavioral wage equation.
2. are often called excluded exogenous variables. That is, they
are excluded from the behavioral wage equation.
Identification
Identification means that δ0 is the unique solution to the moment equations
[g(w δ0)] = 0
That is, δ0 is identified provided
[g(w δ0)] = 0 and [g(w δ)] 6= 0
for δ 6= δ0
For the linear GMM model, this is equivalent to
Σ
(×1)
= Σ
(×)
δ0
(×1)
Σ
(×1)
6= Σ
(×)
δ
(×1)
for δ 6= δ0
Rank Condition
For identification of δ0, it is required that the × matrix [xz0] = Σ
be of full rank . This rank condition is usually stated as
rank(Σ) =
This condition ensures that δ0 is the unique solution to [x(−z0δ0)] = 0.
Note: If = , then Σ is invertible and δ0 may be determined using
δ0 = Σ
−1
Σ
Example: Demand/Supply Equation
x = (1 temp)
0 z = (1 )
0
= 2 = 2
Σ = [xz
0
] =
Ã
1 []
[temp] [temp]
!
For the rank condition, we need rank(Σ) = = 2 Now, rank(Σ) = 2 if
det(Σ) 6= 0 Further,
det(Σ) = [temp]−[temp][]
= cov(temp )
Hence, rank(Σ) = 2 provided cov(temp ) 6= 0 That is, δ0 is identified
provided the instrument is correlated with the endogenous variable.
Order Condition
A necessary condition for the identification of δ0 is the order condition
≥
which simply states that the number of instrumental variables must be greater
than or equal to the number of explanatory variables.
If = then δ0 is said to be (apparently) just identified;
if then δ0 is said to be (apparently) overidentified;
if then δ0 is not identified.
Note: the word “apparently” in parentheses is used to remind the reader that
the rank condition rank(Σ) = must also be satisfied for identification.
Conditional Heteroskedasticity
In the regression model the error terms are allowed to be conditionally het-
eroskedastic and/or serially correlated. Serially correlated errors will be treated
later on.
For the case in which is conditionally heteroskedastic, it is assumed that
{g} = {x} is a stationary and ergodic martingale difference sequence
(MDS) satisfying
[gg
0
] = [xx
0
2
] = S
S = nonsingular × matrix
Note: if var(|x) = 2 then S = 2Σ
Remark:
The matrix S is the asymptotic variance-covariance matrix of the sample mo-
ments ḡ = −1
P
=1 g(w δ0). This follows from the central limit theorem
for ergodic stationary martingale difference sequences (see Hayashi, 2000, p.
106)
√
ḡ =
1
√
X
=1
x
→ (0S)
S = [gg
0
] = avar(
√
ḡ)
Definition of the GMM Estimator
The GMM estimator of δ0 is constructed by exploiting the orthogonality condi-
tions [x(−z0δ0)] = 0. The idea is to create a set of estimating equations
for δ0 by making sample moments match the population moments.
The sample moments for an arbitrary value δ are
g(δ) =
1
X
=1
(w δ) =
1
X
=1
x( − z0δ)
=
⎛⎜⎜⎝
1
P
=1 1( − z0δ)...
1
P
=1 ( − z0δ)
⎞⎟⎟⎠
These moment conditions are a set of linear equations in unknowns.
Equating these sample moments to the population moment [x] = 0 gives
the estimating equations
1
X
=1
x( − z0δ) = S − Sδ = 0
where
S =
−1
X
=1
x
S =
−1
X
=1
xz
0
are the sample moments.
Analytical Solution: Just Identified Model
If = (δ0 is just identified) and S is invertible, then the GMM estimator
of δ0 is
δ̂ = S−1 S
which is also known as the indirect least squares (ILS) or instrumental variables
(IV) estimator.
Remark: It is straightforward to establish the consistency of δ̂
Since = z0δ0 +
=
−1
X
=1
x =
−1
X
=1
x(z
0
δ0 + )
= Sδ0 + S
Then
δ̂ − δ0 = S−1 S
By the ergodic theorem and Slutsky’s theorem
S =
−1
X
=1
xz
0
→ [xz0] = Σ
S−1
→ Σ−1 provided rank(Σ) =
S =
−1
X
=1
xε
→ [x] = 0
δ̂ − δ0 = S−1 S
→ 0
Analytic Solution: Overidentified Model
If then are equations in unknowns. In general, there is no solution
to the estimating equations
S − Sδ = 0
GMM Solution: Use the method of minium chi-square. The idea is to try to
find a δ that makes S
