Generalized Method of Moments EstimatorLecture XXXI
Basic Derivation of the Linear Estimator•Starting with the basic linear model
where yt is the dependent variable, xt is the vector of independent variables, 0 is the parameter vector, and is the residual. In addition to these variables we will introduce the notion of a vector of instrumental variables denoted zt.
0t t ty x u
▫Reworking the original formulation slightly, we can express the residual as a function of the parameter vector
▫Based on this expression, estimation follows from the population moment condition
0 0t t tu y x
0 0tE z u
▫Or more specifically, we select the vector of parameters so that the residuals are orthogonal to the set of instruments.
▫Note the similarity between these conditions and the orthogonality conditions implied by the linear projection space:
1' 'cP X X X X
▫Further developing the orthogonality condition, note that if a single 0 solves the orthogonality conditions, or that 0 is unique that
Alternatively,
00 if and only if t tE z u
00 if t tE z u
Going back to the original formulation
Taking the first-order Taylor series expansion
t t t t tE z u E z y x
0 0t t t t t t t t
t t t
E z y x E z y x E z x
y x x
Given that
this expression implies
0 0 0t t t t tE z y x E z u
0t t t t tE z y x E z x
•B. Given this background, the most general form of the minimand (objective function) of the GMM model can be expressed as
▫T is the number of observations, ▫u(0) is a column vector of residuals, ▫Z is a matrix of instrumental variables, and▫WT is a weighting matrix.
1 1T TQ u Z W Z u
T T
▫Given that WT is a type of variance matrix, it is positive definite guaranteeing that
▫Building on the initial model
In the linear case
0Tz W z
1t tE z u Z u
T
1t tE z u Z y X
T
▫Given that WT is positive definite and the optimality condition when the residuals are orthogonal to the variances based on the parameters
0 0 01 0 0t t TE z u Z u QT
▫Working the minimization problem out for the linear case
2
2
2
1
1
1
T T
T T
T T T T
Q y X Z W Z y XT
y ZW X ZW Z y Z XT
y ZW Z y X ZW Z y y ZW Z X X ZW Z XT
▫Note that since QT() is a scalar,
are scalars so
T TX ZW Z y y ZW Z X
2
1 2T T T TQ y ZW Z y X ZW Z X X ZW Z yT
▫Solving the first-order conditions
2
1
1 2 2 0
ˆ
T T T
T T
Q X ZW Z X X ZW Z yT
X ZW Z X X ZW Z y
▫An alternative approach is to solve the implicit first-order conditions above. Starting with
2
1 2 2 0
1 1 1 1 1 02
1 1 0
1 1 0
T T T
T T T
T
T
Q X ZW Z X X ZW Z yT
Q X Z W Z X X Z W Z yT T T T
X Z W Z X Z yT T
X Z W Z X yT T
1 1 1 02
1 1 0
T T
T
Q X Z W Z y XT T
X Z W ZuT T
The Limiting Distribution•By the Central Limit Theory
1
1 1 0,T
dt t
t
Z u z u N ST T
Therefore
0
1
2
1 1
1
1 ˆ 0,
1lim
dT
t t t t t t
T T
T t s t ss t
t t t t
N MSMT
M E x z WE z x E x z W
S E u u z z E u zzT
MSM E z x S E x z
•Under the classical instrumental variable assumptions
2
1
2
1ˆ ˆ
ˆˆ
T
T t t tt
TCIV
S u z zT
S Z ZT
Example: Differential Demand Model•Following Theil’s model for the derived
demand for inputs
•The model is typically estimated as
1
ln ln lnn
i i i ij j ij
f d q d O d p
1
n
it it i t ij jt itj
f D q D O D p
12 , 1it it i tf f f 1ln lnt t tD x x x
•Applying this to capital in agricultural from Jorgenson’s database, the output is an index of all outputs and the inputs are capital, labor, energy, and materials.
1
2 2 2 2 2
1
Tt ct lt et mt t
T
t ct lt et mt t ct lt et mt t
X O p p p p
Z O p p p p O p p p p
▫Rewriting the demand model
▫Thus, the objective function for the Generalized Method of Moments estimator is
y X
T TQ y X ZW Z y X
▫Initially we let WT = I and minimize QT(). This yields a first approximation to the estimator
GMM 1 GMM 2 OLSOutput 0.01588 0.01592 0.01591
(0.00865) (0.00825) (0.00885)
Capital -0.00661 -0.00675 -0.00675
(0.00280) (0.00261) (0.00280)
Labor 0.00068 0.00058 0.00058
(0.03429) (0.00334) (0.00359)
Energy 0.00578 0.00572 0.00572
(0.00434) (0.00402) (0.00432)
Materials 0.02734 0.02813 0.02813
(0.01215) (0.01068) (0.01146)