# Single Equation Linear GMM · PDF file The matrix S is the asymptotic variance-covariance matrix of the sample mo-ments ¯g = −1 P =1 g (w δ 0). This follows from the central limit

• View
0

0

Embed Size (px)

### Text of Single Equation Linear GMM · PDF file The matrix S is the asymptotic variance-covariance...

• 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 { zx}.

It is assumed that {w} is a stationary and ergodic stochastic process.

• Moment Conditions for General Model

Define

g(w δ0) = x = x( − z0δ0)

It is assumed that the instrumental variables x satisfy the set of  orthogo- nality conditions

[g(w δ0)] = [x] = [x( − z0δ0)] = 0

Expanding gives the relation

[x]−[xz0]δ0 = 0 ⇒ Σ

(×1) = Σ (×)

δ0 (×1)

where Σ = [x] and Σ = [xz0].

• 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

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 [xz0] = 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(−z0δ0)] = 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 = [xz 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

[gg 0 ] = [xx

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 → (0S)

S = [gg 0 ] = avar(

√ ḡ)

• Definition of the GMM Estimator

The GMM estimator of δ0 is constructed by exploiting the orthogonality condi- tions [x(−z0δ0)] = 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( − z0δ)

=

⎛⎜⎜⎝ 1 

P =1 1( − z0δ)...

1 

P =1 ( − z0δ)

⎞⎟⎟⎠ 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( − z0δ) = S − Sδ = 0

where

S =  −1

X =1

x

S =  −1

X =1

xz 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  = z0δ0 + 

 =  −1

X =1

x =  −1

X =1

x(z 0 δ0 + )

= Sδ0 + S

Then

δ̂ − δ0 = S−1 S

• By the ergodic theorem and Slutsky’s theorem

S =  −1

X =1

xz 0  → [xz0] = Σ

S−1 → Σ−1  provided rank(Σ) = 

S =  −1

X =1

xε → [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 − Sδ = 0

GMM Solution: Use the method of minium chi-square. The idea is to try to find a δ that makes S

Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Engineering
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents
Documents