Single Equation Linear GMM
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1 Single Equation Linear GMM Eric Zivot Winter 2013
2 Single Equation Linear GMM Consider the linear regression model Engodeneity = z 0 δ 0 + =1 z = 1 vector of explanatory variables δ 0 = 1 vector of unknown coefficients = random error term The model allows for the possibility that some or all of the elements of may be correlated with the error term (i.e., [ ] 6= 0for some ). If [ ] 6= 0 then is called an endogenous variable. If z contains endogenous variables, then the least squares estimator of δ 0 in is biased and inconsistent.
3 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.
4 Moment Conditions for General Model Define g (w δ 0 )=x = x ( z 0 δ 0 ) It is assumed that the instrumental variables x satisfy the set of orthogonality conditions [g (w δ 0 )] = [x ]= [x ( z 0 δ 0 )] = 0 Expanding gives the relation [x ] [x z 0 ]δ 0 = 0 Σ ( 1) δ 0 ( )( 1) = Σ where Σ = [x ] and Σ = [x z 0 ].
5 Example: Demand-Supply model with supply shifter demand: supply: = = temp + 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 [ ]
6 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
7 Example: Wage Equation Assume ln = = wages = years of schooling = years of experience = score on IQ test 2 = rate of return to schooling [ ]= [ ]=0 Endogeneity: is a proxy for unobserved ability but is measured with error = + [ ] 6= 0
8 Instruments: = age in years = years of mother s education [ ] = [ ]=0
9 Wage Equation in Hayashi notation Remarks = z 0 δ + = ln z = (1 ) 0 x = (1 ) 0 w = (ln ) 0 = 4 = 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.
10 Identification Identification means that δ 0 is the unique solution to the moment equations [g (w δ 0 )] = 0 That is, δ 0 is identified provided [g (w δ 0 )] = 0 and [g (w δ)] 6= 0 for δ 6= δ 0 For the linear GMM model, this is equivalent to Σ ( 1) Σ ( 1) δ 0 ( )( 1) = Σ 6= Σ ( ) δ for δ 6= δ 0 ( 1)
11 Rank Condition For identification of δ 0, it is required that the matrix [x z 0 ]=Σ be of full rank. Thisrank condition is usually stated as rank(σ )= This condition ensures that δ 0 is the unique solution to [x ( z 0 δ 0)] = 0. Note: If =, thenσ is invertible and δ 0 may be determined using δ 0 = Σ 1 Σ
12 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 )=2if det(σ ) 6= 0 Further, det(σ ) = [temp ] [temp ] [ ] = cov(temp ) Hence, rank(σ )=2provided cov(temp ) 6= 0 That is, δ 0 is identified provided the instrument is correlated with the endogenous variable.
13 Order Condition A necessary condition for the identification of δ 0 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 δ 0 is said to be (apparently) just identified; if then δ 0 is said to be (apparently) overidentified; if then δ 0 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.
14 Conditional Heteroskedasticity In the regression model the error terms are allowed to be conditionally heteroskedastic 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 = 2 Σ
15 Remark: The matrix S is the asymptotic variance-covariance matrix of the sample moments ḡ = 1 P =1 g (w δ 0 ). 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)
16 Definition of the GMM Estimator The GMM estimator of δ 0 is constructed by exploiting the orthogonality conditions [x ( z 0 δ 0)] = 0. The idea is to create a set of estimating equations for δ 0 by making sample moments match the population moments. The sample moments for an arbitrary value δ are g (δ) = 1 = X X (w δ) = 1 x ( z 0 =1 δ) =1 P =1 1 ( z 0 δ) 1 1. P =1 ( z 0 δ) These moment conditions are a set of linear equations in unknowns.
17 Equating these sample moments to the population moment [x ]=0 gives the estimating equations where 1 X =1 are the sample moments. x ( z 0 δ) =S S δ = 0 S = 1 X =1 S = 1 X =1 x x z 0
18 Analytical Solution: Just Identified Model If = (δ 0 is just identified) and S is invertible, then the GMM estimator of δ 0 is ˆδ = S 1 S which is also known as the indirect least squares (ILS) or instrumental variables (IV) estimator.
19 Remark: It is straightforward to establish the consistency of ˆδ Since = z 0 δ 0 + = 1 X x = 1 =1 = S δ 0 + S X =1 x (z 0 δ 0 + ) Then ˆδ δ 0 = S 1 S
20 By the ergodic theorem and Slutsky s theorem S = 1 S 1 X =1 x z 0 [x z 0 ]=Σ Σ 1 provided rank(σ )= S = 1 X =1 x ε [x ]=0 ˆδ δ 0 = S 1 S 0
21 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 S δ as close to zero as possible. This is achieved by minimizing a weighted average of the moment equations.
22 GMM Objective Function Let Ŵ denote a symmetric and positive definite (p.d.) weight matrix, possibly dependent on the data, such that Ŵ W as with W symmetric and p.d. The GMM objective function is defined as (δ Ŵ) = g (δ) 0 Ŵg (δ) = (S S δ) 0 Ŵ(S S δ)
23 Remarks 1. The scaling by will be explained later. 2. The elements of Ŵ determine the weights put on the moment equations. For example, let =2 =1so that Ã! Ã! 1 ( ) ˆ 11 ˆ g ( ) = Ŵ = 12 2 ( ) ˆ 12 ˆ 22 Then (δ Ŵ) = g (δ) 0 Ŵg (δ) = ³ 1 ( ) 2 ( ) 0 Ã ˆ 11 ˆ 12 ˆ 12 ˆ 22!Ã 1 ( ) 2 ( ) (ˆ 11 1 ( ) 2 +2ˆ 12 1 ( ) 2 ( )+ ˆ 22 2 ( ) 2 ) 3. We will discuss how to determine the best Ŵ later!
24 Definition of GMM estimator The GMM estimator of δ 0, denoted ˆδ(Ŵ), isdefined as ˆδ(Ŵ) = argmin (δ Ŵ) δ arg min (S S δ) 0 Ŵ(S S δ) δ Since (δ Ŵ) is a simple quadratic form in δ, straightforward calculus may be used to determine the analytic solution for ˆδ(Ŵ): ˆδ(Ŵ) =(S 0 ŴS ) 1 S 0 ŴS
25 Derivation of GMM estimator First, note that Then (δ Ŵ) = (S S δ) 0 Ŵ(S S δ) = h S 0 ŴS 2S 0 ŴS δ + δ 0 S 0 ŴS δ i (δ Ŵ) δ Solving for ˆδ such that (ˆδ Ŵ) δ = 2S 0 ŴS +2S 0 ŴS δ = 0 gives ˆδ(Ŵ) =(S 0 ŴS ) 1 S 0 ŴS provided S 0 ŴS is invertible (which requires rank(s )= ).
26 Recall the results from matrix algebra (a 0 β) β (β 0 Aβ) β = a = 2Aβ for A symmetric
27 Asymptotic Properties Under standard regularity conditions (see Hayashi, 2000, C hap. 3), it can be shown that where ˆδ(Ŵ) =(S 0 ŴS ) 1 S 0 ŴS δ 0 ³ˆδ(Ŵ) δ 0 (0 avar(ˆδ(ŵ))) avar(ˆδ(ŵ)) = (Σ 0 WΣ ) 1 Σ 0 WSWΣ (Σ 0 WΣ ) 1
28 A consistent estimate of avar(ˆδ(ŵ)), denoted avar(ˆδ(ŵ)), d may be computed using avar(ˆδ(ŵ)) d = (S 0 ŴS ) 1 S 0 ŴŜŴS (S 0 ŴS ) 1 where Ŝ is a consistent estimate for S = avar(ḡ). Note: S can be consistently estimated using the White or heteroskedasticity consistent (HC) estimator because ˆ (Ŵ) δ 0 Ŝ = 1 X =1 x x 0 ˆ 2 ˆ = z ˆ (Ŵ)
29 Sketch of Proof For consistency, note that ˆδ(Ŵ) = (S 0 ŴS ) 1 S 0 ŴS = (S 0 ŴS ) 1 S 0 Ŵ(S δ 0 + S ) = δ 0 +(S 0 ŴS ) 1 S 0 ŴS
30 By assumption Ŵ W p.d. By the ergodic theorem and Slutsky s theorem S = 1 X =1 x z 0 [x z 0 ]=Σ S 0 Ŵ Σ 0 W ³ S 0 ŴS 1 ³ Σ 0 WΣ 1 provided rank(σ )= S = 1 X =1 x ε [x ]=0 ˆδ(Ŵ) δ 0 0
31 For asymptotic normality, write ³ ³ˆδ(Ŵ) δ 0 = S 0 ŴS 1 S 0 Ŵ S By the ergodic theorem and Slutsky s theorem S [x z 0 ]=Σ ³ S 0 ³ ŴS 1 Σ 0 WΣ 1 Since { } = {x } is a stationary and ergodic martingale difference sequence (MDS) satisfying [g g ] 0 = [x x 0 2 ]=S S = nonsingular matrix thecltforstationary-ergodicmdsgives S = 1 X =1 x (0 S)
32 Furthermore, Slutsky s theorem gives Therefore S 0 Ŵ S Σ 0 W (0 S) ³ ³ˆδ(Ŵ) δ 0 Σ 0 WΣ 1 Σ 0 W (0 S) µ0 ³ Σ 0 ³ WΣ 1 Σ 0 WSWΣ Σ 0 WΣ 1 Note: A (0 S) (0 ASA 0 )
33 The Efficient GMM Estimator For a given set of instruments x,thegmmestimatorˆδ(ŵ) is defined for an arbitrary p.d. and symmetric weight matrix Ŵ. The asymptotic variance of ˆδ(Ŵ) depends on the chosen weight matrix Ŵ avar(ˆδ(ŵ)) = (Σ 0 WΣ ) 1 Σ 0 WSWΣ (Σ 0 WΣ ) 1 A natural question to ask is: What weight matrix W produces the smallest value of avar(ˆδ(ŵ))? The GMM estimator constructed with this weight matrix is called the efficient GMM estimator.
34 Result: Hansen(1982)showed that efficient the GMM estimator results from setting Ŵ = Ŝ 1 such that Ŝ S = [x x 0 2 ] = avar ( g (δ 0 )). For this choice of Ŵ, the asymptotic variance formula reduces to of which a consistent estimate is avar(ˆδ(ŝ 1 )) = (Σ 0 S 1 Σ ) 1 avar(ˆδ(ŝ d 1 )) = (S 0 Ŝ 1 S ) 1
35 The efficient GMM estimator is defined as ˆδ(Ŝ 1 ) = argmin δ (δ Ŝ 1 ) = argmin δ g (δ) 0 Ŝ 1 g (δ) which requires a consistent estimate of S = [g g 0]= [x x 0 2 ]. However, a consistent estimation of S, in turn, requires a consistent estimate of δ 0. White (1982) showed that a consistent estimate of S has the form such that ˆδ δ 0. Ŝ = 1 X =1 x x 0 ˆ 2 = 1 X =1 x x 0 ( z 0 ˆδ) 2
36 Two-Step Efficient GMM The two-step efficient GMM estimator utilizes the result that a consistent estimate of δ 0 may be computed by GMM with an arbitrary positive definite and symmetric weight matrix Ŵ such that Ŵ W. Letˆδ(Ŵ) denote such an estimate. Common choices for Ŵ are Ŵ = I and Ŵ = S 1 =( 1 X 0 X) 1,where X is an matrix with the th row equal to x 0. Then, a firststepconsistentestimateofs is given by Ŝ(Ŵ) = 1 X =1 x x 0 ( z 0 ˆδ(Ŵ)) 2
37 The two-step efficient GMM estimator is then defined as ˆδ(Ŝ 1 (Ŵ)) = arg min δ g (δ) 0 Ŝ 1 (Ŵ)g (δ) ˆδ(Ŝ 1 (Ŵ)) = (S 0 Ŝ 1 (Ŵ)S ) 1 S 0 Ŝ 1 (Ŵ)S Drawback: The numerical value of ˆδ(Ŝ 1 (Ŵ)) depends on Ŵ
38 Iterated Efficient GMM The iterated efficient GMM estimator uses the two-step efficient GMM estimator ˆδ(Ŝ 1 (Ŵ)) to update the estimation of S and then recomputes the estimator. The process is repeated (iterated) until the estimates of δ 0 do not change significantly from one iteration to the next. Typically, only a few iterations are required. The resulting estimator is denoted ˆδ(Ŝ 1 iter ). The iterated efficient GMM estimator has the same asymptotic distribution as the two-step efficient estimator. However, in finite samples, the two estimators may differ. The iterated GMM estimator has a practical advantage over the two-step estimator in that the resulting estimates are invariant with respect to the scale of the data and to the initial weighting matrix Ŵ.
39 Continuous Updating Efficient GMM This estimator simultaneously estimates S, as a function of δ, and δ. It is defined as ˆδ(Ŝ 1 CU ) = argmin δ = argmin Ŝ(δ) = 1 δ X =1 (δ Ŝ 1 (δ)) g (δ) 0 Ŝ 1 (δ)g (δ) x x 0 ( z 0 δ) 2 Hansen, Heaton, and Yaron (1996) call ˆδ(Ŝ 1 CU ) the continuous updating (CU) efficient GMM estimator.
40 Remarks: 1. The CU estimator is asymptotically equivalent to the two-step and iterated estimators, but may differ in finite samples. 2. The CU estimator does not depend on an initial weight matrix W, and like the iterated efficient GMM estimator, the numerical value of CU estimator is invariant to the scale of the data. 3. The CU estimator is a non-linear function of δ and does not have a closed form expression like the two-step or iterated estimators. It is computationally more burdensome than the iterated estimator, especially for large nonlinear models, and is more prone to numerical instability. 4. Hansen, Heaton, and Yaron found that the finite sample performance of the CU estimator, and test statistics based on it, is often superior to the other estimators.
41 Remarks continued: 5. Under homoskedastic and normally distributed errors the CU estimator is equivalent to the (limited information) maximum likelihood estimator. 6. More generally, the CU estimator is in the class of generalized empirical likelihood (GEL) estimators, which are related to non-parametric maximum likelihood estimators, and have certain optimality properties. 7. The CU estimator is more robust to the presence of weak instruments than the two-step or interated GMM estimators. Here, robust loosely means that the asymptotic distribution of the CU estimator does not get as screwed up as the asymptotic distributions of the two-step and iterated GMM estimators when instruments are characterized as weak. We will more formally characterize weak instruments later on in the course.
42 Two-Stage Least Squares as an Efficient GMM Estimator If the errors are conditionally homoskedastic, then [x x 0 2 ]= 2 Σ = S A consistent estimate of S has the form Ŝ =ˆ 2 S where ˆ 2 2. Typically, ˆ 2 = 1 X ( z 0 ˆδ) 2 =1 ˆδ δ 0 The efficient GMM estimator becomes ˆδ(ˆ 2 S 1 ) = (S 0 ˆ 2 S 1 S ) 1 S 0 ˆ 2 S 1 S = (S 0 S 1 S ) 1 S 0 S 1 S = ˆδ(S 1 ) which does not depend on ˆ 2.
43 The estimator ˆδ(S 1 ) is, in fact, identical to the two-stage least squares (TSLS) estimator of δ 0 : where ˆδ(S 1 ) = (S 0 S 1 S ) 1 S 0 S 1 S = (Z 0 P Z) 1 Z 0 P y = (Ẑ 0 Ẑ) 1 Ẑ 0 y = ˆδ TSLS Z = matrix of observations with tth row z 0 X = matrix of observations with tth row x 0 P = X(X 0 X) 1 X 0 Ẑ = P Z
44 Theasymptoticvarianceofˆδ(S 1 )=ˆδ TSLS is avar(ˆδ TSLS )=(Σ 0 S 1 Σ ) 1 = 2 (Σ 0 Σ 1 Σ ) 1 Although ˆδ(S 1 ) does not depend on ˆ 2, a consistent estimate of the asymptotic variance does: avar(ˆδ d TSLS ) = ˆ 2 TSLS (S0 S 1 S ) 1 ˆ 2 TSLS = X 1 ( z 0 ˆδ TSLS ) 2 =1
45 Continuous Updating Efficient GMM under Homoskedasticity: LIML Under conditional homoskedasticity, S = 2 Σ and the CU efficient GMM estimator becomes ˆδ(Ŝ 1 CU ) = argmin δ A little algebra shows that Ŝ(δ) = ˆ 2 (δ)s = 1 g (δ) 0 Ŝ 1 (δ)g (δ) X =1 ( z 0 δ) 2 S ˆδ(Ŝ 1 CU )=ˆδ LIML = limited information maximum likelihood estima- Result: tor. g (δ) 0 Ŝ 1 (δ)g (δ) = (y Zδ)0 P X (y Zδ) (y Zδ) 0 (y Zδ)
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