07_lent_Topic 2 - Generalized Method of Moments, Part II - The Linear Model_mw217

8/19/2019 07_lent_Topic 2 - Generalized Method of Moments, Part II - The Linear Model_mw217

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M320

Cross Section and Panel Data Econometrics

Topic 2: Generalized Method of Moments

Part II: The Linear Model

Dr. Melvyn Weeks

Faculty of Economics and Clare College

University of Cambridge

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Outline

Outline I

1 IV and OLS Estimators: RevisionSmall Sample Issues

2 IV and 2SLS EstimatorsJust Identified

Over Identified: The Generalised IV Estimator

3 GMM Estimators for the Linear ModelMoment-Based EstimationTypes of GMM Estimators

4 SummaryGMM Estimators in the Linear Model

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Outline

Readings

Cameron, A. C., and P. K. Trivedi (2005). Microeconometric Methods and Applications. Cambridge University Press.Chapter 6.

A. Colin Cameron and P.K. Trivedi. Microeconometrics Using Stata. Stata Press, 2009. URL

http://www.stata.com/bookstore/mus.html

Hayashi, F. (2000) Econometrics. Princeton University Press,Princeton.

Wooldridge, J. M. (2001). Applications of Generalized Method of Moments Estimation, Journal of Economic Perspectives,15(4), 87.100. 1

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Outline

Part II: GMM and Linear Models

1 The Linear Model

IV: Finite and Large Sample Properties (Review)

IV as MOM2SLS [Generalised IV (GIVE)] as GMM

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Notes

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Outline

Road Map

In Generalized Method of Moments - Part II: The Linear Model,we consider the gmm as a canonical estimator.

In doing this we show how ols , iv and 2sls are special cases of the more general gmm estimator.

We show how just identified and over-identified models may berepresented as moment estimators. In the just-identified case wealso explicitly show that the gmm criterion function may be set tozero, whereas in the over-identified case we minimise the distancefrom zero.

We also provide some background material on the small-sampleproperties of the iv and 2sls estimators.

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IV and OLS Estimators: Revision

We first provide a brief overview of ols and iv estimators.

We consider the unbiasedness of the ols estimator under certainconditions and the small sample bias of the iv estimator.

We show how it is difficult to obtain the sample mean of the ivestimator

This leads us to the potential small (and large) sample bias thatmight be induced by weak instruments.

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OLS

Proposition

If E [X ε] = 0 ols is an unbiased estimator of β .

Proof.

ˆ β = (X X )−1X y

E [ ˆ β] = β + E [(X X )−1X ε]

= β + (X X )−1E [X ε] (1)

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Proposition

If E [X ε] = 0 the ols estimator is biased and inconsistent estimator of β

Proof.

ˆ β = (X X )−1X y E [ ˆ β] = β + E [(X X )−1X ε]

= β + E [(X X )−1X τ (X )] = β

for E [ε|X ] = τ (X )Question: why E [(X X )−1X τ (X )] cannot be factored, as in (1)?

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The IV Estimator

βIV = (Z X )−1Z (X β + ε)= (Z X )−1Z X β + (Z X )−1Z ε

Proposition

The IV estimator is biased in small samples.

Can we utilise the same sort of proof as used for the unbiasednessof the ols estimator?

E [ βIV ] = E [(Z X )−1Z X β + (Z X )−1Z ε]= (Z X )−1Z X β +E X ,Z ,ε[(Z

X )−1Z ε]

= β + E X ,Z ,ε[(Z X )−1Z ε] (2)

= β + E X ,Z [(Z X )−1 × [E [ε|Z ,X ]] (3)

= β + (Z X )−1E [Z ε]

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Note that the unconditional expectation wrt E X ,Z ,ε[.] in (2) isobtained by first taking expectations wrt ε given Z ,X in (3)

What if we imposed E [ε|Z ,X ] = 0?

But this is no use since it implies E [ε|X ] = 0, thereby negating arequirement for an instrument in the first place.

What if we exploit

βIV = β + N −1Z X −1 N −1Z ε

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In what way might large sample arguments help us here?

Lets consider the following simple case, where we make theassumption that E [εi |X i ] = 0.

A single instrument Z i is available.

Y i = β1 + β2X i + εi

βIV 2 = ∑ N i =1(Z i − Z̄ )(Y i − Ȳ )∑ N i =1(Z i − Z̄ )(X i − X̄ )

Proposition βIV 2 is consistent provided that σ ZX is nonzero 11


βIV 2 = ∑ N i =1(Z i − Z̄ )(Y i − Ȳ )∑ N i =1(Z i − Z̄ )(X i − X̄ )

= ∑

N i =1(Z i − Z̄ )([ β1 + β2X i + εi ] − [ β1 + β2 X̄ + ε̄])

∑ N i =1(Z i − Z̄ )(X i − X̄ )

= ∑

N i =1( β2(Z i − Z̄ )(X i − X̄ ) + (Z i − Z̄ )(εi − ε̄))

∑

N

i =1(Z i − ¯

Z )(X i − ¯

X )

= β2 + ∑

N i =1(Z i − Z̄ )(ε i − ε̄)

∑ N i =1(Z i − Z̄ )(X i − X̄ )

Nothing much can be said about the distribution of βIV 2 in smallsamples

However, we observe that the iv estimator is equal to the truevalue plus an error, which under certain conditions will vanish as N becomes large

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Divide the numerator and denominator by N so that they bothhave limits, then we can take plims

plim

βIV 2 = β2 +

plim 1N ∑

N i =1(Z i − Z̄ )(εi − ε̄)

plim 1N ∑

N i =1(Z i − Z̄ )(X i − X̄ )

= β2 + σ Z

εσ ZX

Slutskys theorem allows us (as opposed to the situation whentaking expectations) to split the problem. We can the evaluate thelimit of numerator and denominator separately

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IV and OLS Estimators: Revision Small Sample Issues

Small Sample Issues

The iv estimator requires σ εZ = 0. In small samples ε and Z maybe weakly correlated.

In this instance the iv estimator can have large (asymptotic bias)even if the correlation is moderate.

To see this we write the probability limit of the iv estimator as

plim ˆ βIV ,2 = β2 + Cov (Z , ε)

Cov (Z ,X ) (4)

= β2 + Corr (Z , ε)

Corr (Z ,X ) ×

σ εσ X

(5)

= β2 + 0

Cov (Z ,X ) (6)

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Comments

1 In small samples it is not possible to say much about thedistribution of ˆ βIV ,2

2 From (6) we observe that if Z is distributed independent of εthen ˆ βIV ,2 is consistent for β2.

3 In small (or even large) samples ε and Z may be weaklycorrelated.

In this instance the iv estimator can have large (asymptoticbias) even if the correlation is moderate

This will obviously depend upon the correlation between Z and X ; this is the problem of weak instruments.

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Comments - cont.

4. Even in the context of large sample results, it is notunequivocally better to use iv rather than ols .

To see this compare (5) with the plim of the ols estimatorˆ βOLS ,2 which we write as

plim ˆ βOLS ,2 = β2 + Corr (X , ε) σ εσ X

(7)

By finding a relationship between plim ˆ βOLS ,2 and plim ˆ βIV ,2,under what circumstances is iv preferred to ols onasymptotic grounds?

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IV and 2SLS Estimators

Below we briefly review the iv approach to identification in thepresence of exact and overidentified models.

In the following section we show that the gmm estimator for thelinear model is canonical in that it nests the iv (mom) and 2slsestimators as special cases.

We also show that in the case of an exactly identified model, thegmm criterion function can be set exactly to zero, just as the sumof residuals for the OLS estimator is exactly zero.

In this instance the sample realisation of the criterion function isnot a random variable and it is not possible to test the exogeneityof instruments.

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IV and 2SLS Estimators

Consider a first stage regression based on a linear combination of instruments

X = Z δ + u (8)

δ = (Z Z )−1Z X

X = P Z X = Z (Z Z )−1Z X Using X as instruments.

βIV = (X P Z X )−1X P Z y = (X Z (Z Z )−1Z X )−1X Z (Z Z )−1Z y (9)

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IV and 2SLS Estimators Just Identified

Just-identified: M = k The estimator with M = k (M the of cols. of Z ) isoften called the Instrumental Variable Estimator .

X Z is a square matrix since M = k .

(X Z (Z Z )−1Z X )−1 can then be decomposed as

(Z X )−1(Z Z )(X Z )−1, (10)

such that the iv estimator may then be rewritten as

ˆ βIV = (Z X )−1(Z Z )(X Z )−1X Z (Z Z )−1Z y

= (Z X )−1Z y .

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IV and 2SLS Estimators Just Identified

In the just-identified case we observe that the weighting matrixP Z = Z (Z

Z )−1Z falls out.

In situations where parameters β are exactly identified we have justenough moment conditions to estimate β.

Consequence: the minimum of a generalised distance measure(GMM) is exactly zero - all sample moments can be set to zero byappropriate choice of β.

There is therefore no need to weight the individual moments inorder to minimise a weighted sum.

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IV and 2SLS Estimators Over Identified: The Generalised IV Estimator

Over-identified: M > k The estimator is called the 2sls or GeneralizedInstrumental Variable Estimator (give).

X Z is not a square matrix since for M > k

(X

Z (Z

Z )−1

Z

X )−1

cannot be decomposed.

The iv estimator is then given by

ˆ βIV = (X Z (Z Z )−1Z X )−1X Z (Z Z )−1Z y

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GMM Estimators for the Linear Model Moment-Based Estimation

IV as Moment Estimators

We have motivated the iv estimator based on a transformed linearregression model of the form Z y = Z X β + Z ε.

Alternative derivation: minimise a quadratic form of vectormoments: functions of parameters and data.

The gmm estimator is obtained by minimising a quadratic form inthe analogous sample moments: n−1[Z y − Z X β]

Ignoring n−1 then the gmm estimator is defined as

ˆ β∗ = arg min[(Z y −Z X β)C N (Z

y −Z X β)] (11)

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GMM Estimators for the Linear Model Moment-Based Estimation

Solving (11) exactly or by minimising a weighted quadraticfunction depends on whether the system of equations is over orexactly identified.

C N is an M ×M p d symmetric weighting matrix; it tells us howmuch weight to attach to which (linear combinations) of thesample moments.

In general C N will depend upon the sample size N , because it mayitself be an estimate.Again we note the use of the Generalised prefix in gmm.

Given that we cannot set individual sample moments equal topopulation counterparts, we utilise a weighting matrix C N , whichweights each moment such that the sample moments are as closeas possible to zero.

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GMM Estimators for the Linear Model Types of GMM Estimators

A class of GMM estimators will depend on what is assumed sboutthe distribution of ε i

IV: M = K , errors homoscedastic; Var(εi |z i ) = σ 2I N . βMOM = (Z X )−1(X Z )−1X ZZ y

= (Z X )−1Z y

2SLS: M > K , errors homoscedastic: Var(εi |z i ) = σ 2I N . β2SLS = [X Z (Z Z )−1Z X ]−1X Z (Z Z )−1Z y GMM M > K , errors not restricted

βGMM = X ZC N Z X −1 X ZC N Z y

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2SLS

If ε ∼ (0 , σ 2I ) the covariance matrix of the moment conditions is

Var(Z (y − X β)) = Var(Z ε) = σ 2Z Z .An optimal weighting matrix is then

C N = 1N N ∑ i =1

z i z i −1The resulting GMM estimator is

β2SLS = [X Z (Z Z )−1Z X ]−1X Z (Z Z )−1Z y This estimator is often referred to as theGeneralised Instrumental Variables Estimator (give).

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Below we consider the following two propositions:

Proposition

I: If Var (Z ε) = σ 2Z Z then

βGMM = β2SLS = (X Z (Z Z )−1Z X )−1X Z (Z Z )−1Z y .Proposition

II: If M = K then ˆ βGMM = ˆ βIV and Q C ( β) can be set to 0

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Proposition

βGMM = β2SLS = (X Z (Z Z )−1Z X )−1X Z (Z Z )−1Z y .Proof.

Q C ( β) = (y − X

β)P Z (y − X

β) (12)

= (y −

X β)Z (Z Z Z )−1Z (y −X β)= [Z (y −X β)](Z Z )−1[Z (y − X β)] (13)= [Z (y −X β)]C N [Z (y − X β)] (14)= y P Z y + βX P Z X β− 2 βX P Z y (15)

∂Q C ( β)

∂ β = ∂ βX P Z X β

∂ β − ∂2 βXP Z y

∂ β= 2X P Z X β− 2X P Z y = 0 (16)= 2X ZC N Z

X

β− 2X Z C N Z

y = 0. (17)27


(16) and (17) are derived, respectively, from (13) and (14)

Both (16) and (17) represent systems of M equations and k unknowns, where X Z is k ×M .

The solution to (16) is given by

ˆ βGMM = (X P Z X )−1X P Z y

=X Z (Z Z )−1Z X

−1X Z (Z Z )−1Z y

= ˆ β2SLS

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Just Identified If M = k we may solve (11) exactly

ˆ βGMM = ˆ βIV = ˆ βMOM = (Z X )−1Z y

Proposition

ˆ βGMM = ˆ βIV and Q C ( β) can be set to 0

Proof.

Substituting ˆ βIV into (15) then

Q C ( β) = y P Z y + ˆ β

IV XP Z X

ˆ βIV − 2 ˆ βIV X

P Z y

= y Z (Z Z )−1Z y

+y Z (X Z )−1X Z (Z Z )−1Z X (Z X )−1Z y

−2Z (X Z )−1X Z (Z Z )−1Z y

= y Z (Z Z )−1Z y + y Z (Z Z )−1Z y

−2y Z (Z Z )−1Z y = 0

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Two Step GMM

The most efficient (feasible) GMM estimator based upon

E (z i εi ) = 0 uses weight matrix, say V −1. V is constructed using fitted residuals, obtained from a first step

using a consistent estimator of β , which is often the 2SLSestimator:

V = 1

N

N

∑ i =1

z i (y i − x i β2sls )(y i − x i β2sls )z i where V is given by

V = plimN →∞1

N

N

∑ i =1

z i εi εi z i This gives the two-step GMM estimator β2sGMM = X ZC N Z X −1 X ZC N Z y where C N =

V

−1

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The efficiency gains of the GMM estimator relative to thetraditional IV/2SLS estimator derive from the overidentifyingrestrictions of the model, the use of the optimal weighting matrix,and the relaxation of the i.i.d. assumption.

For an exactly-identified model, the efficient GMM and traditional

IV/2SLS estimators coincide.

Under the assumptions of conditional homoskedasticity andindependence, the efficient GMM estimator is the traditionalIV/2SLS estimator.

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Summary GMM Estimators in the Linear Model

Summary: GMM Estimators in the Linear IV Model

GMM ˆ βGMM =X ZC N Z

X −1

X ZC N Z y

2SLS ,GIVE ˆ β2SLS = X Z (Z Z )−1Z X −1 X Z (Z Z )−1Z y IV ˆ βIV =

Z X ]−1Z y

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07_lent_Topic 2 - Generalized Method of Moments, Part II - The Linear Model_mw217