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)