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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

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  • 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

    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 [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

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