econ 4160, autumn term 2015. lecture 8 - universitetet i oslo · econ 4160, autumn term 2015....

GIVE and 2SLS Overidentification test GMM FIML and 3SLS estimation

ECON 4160, Autumn term 2015. Lecture 8Identification and estimation of SEMs (Cont’d from Lecture 7).

Ragnar Nymoen

Department of Economics

15 Oct 2015

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The GIV-estimator (GIVE) I

I With the notation from Lecture 7, we write the first equationin a simultaneous equations model (SEM) as

y1= Z1β11+Y1β21+ε1 (1)

I y1 is n× 1, with observations of the Y -variable that the firstequation in the SEM is normalized on.

I Z1 is n× k11 with observations of the k11 includedpredetermined or exogenous variables.

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The GIV-estimator (GIVE) III Y1, n× k12 holds the included endogenous explanatoryvariables.By defining:

X1 =(Z1 Y1

)(2)

β1 = ( β11 β21 )′ (3)

equation (1) can be written more compactly as:

y1= X1β1+ε1 (4)

The Generalized IV estimator is

β1,GIV = (W′1X1)

−1W′1y1. (5)

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The GIV-estimator (GIVE) III

withW1 =

(Z1 Y1

),

andY1 = PW1Y1

I PW1 is the prediction maker for Y1 conditional on thevariables in W1 =

(Z1 X01

).

I β1,GIV is also known as the 2-stage least squares estimatorof β1 in (4)

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GIVE and the 2SLS estimator I

I In a separate note (Lecture note 6), you can see, as asupplement,that β1,GIV can be expressed as

β1,GIV =

(β11,GIVβ21,GIV

)(6)

=

(Z′1Z1 Z′1Y1Y′1Z1 Y

′1Y1

)−1 (Z′1y1Y′1y1

)I Call the reduced form regression that gives rise to theoredicted Y -values in Y1 the first-stage regression.

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GIVE and the 2SLS estimator IIThen consider the consequences of using OLS on the structuralequation (1), but after substitution of Y1 by Y1 :

y1= Z1β11+Y1β21 + disturbance

=(Z1 Y1

) ( β11β21

)+ disturbance

Call the OLS estimator from this second least-square estimation,the Two-Stage Least Square estimator: 2SLS:(

β11,2SLSβ21,2SLS

)=[(Z1 Y1

)′ (Z1 Y1

)]−1 ( Z′1Y′1

)y1

=

(Z′1Z1 Z′1YY′1Z1 Y

′1Y1

)−1 (Z′1y1Y′1y1

)(7)

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GIVE and the 2SLS estimator IIIwhich is identical to (6).

I This establishes that in the over-identified case,(β11,GIVβ21,GIV

)≡(

β11,2SLSβ21,2SLS

)

GIVE is identical to the Two-Stage Least squares estimator(2SLS).

I The same is true for exact identification (since that is aspecial case), hence

β1,IV = β1,GIV = β1,2SLS

when the equation is exactly identified.

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Indirect least squares

I In the exactly identified case, there is a fourth estimatorconsistent estimator called indirect least squares (β1,ILS ).

I It is obtained by estimating the reduced form and solving theequations for the reduced form estimators.

I Exactly as we did in Lecture 7 when we discussedidentification by regarding RF-parameters as known, and thestructural from parameters as unknowns

I However, as long as there is no restrictions on E (ε1ε′1) = Ω1,

β1,ILS ≡ β1,2SLS

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Specification test (IV validity test) IWhen l1 − k1 ≥ 1, the validity of the over-identifying instrumentscan be tested.

I Reference: DM Ch 8.6I Intuition: If the instruments are valid, they should have nosignificant explanatory power in an auxiliary regression thathas the GIV-residuals as regressand.

I Call the R2 from that regression R2GIVres .I The test called Specification test (that we get) when wechoose IV-estimation in Pc-Give is:

nR2GIVres ∼a χ2 (l1 − k1︸︷︷︸degree of over-ID

)

under the H0 of valid instrumental variables.9 / 22


Specification test (IV validity test) II

I There is short reference to it on HN page p 230 and a longerin DM p 338. BN(2011) has it in Kap. 9.4.4. and BN(2014)in Kap. 3.1.7.

I The Specification-test can also be computed from theIV-criterion function introduced in Lecture note 6.

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Finite sample properties and weak instruments I

I Y1is asymptotically uncorrelated with ε1, but in a finitesample there is always some correlation so thatE (β1,GIV ) 6= β1.

I See Ch 8.4 in DM for discussion and analysisI Weak instruments: The instruments are poorly correlatedwith the included endogenous variables in the structuralequation.

I When instruments are weak, the asymptotic distributionreflect the true finite sample distribution poorly. Theasymptotic inference theory loses its relevance.

I The practical issue is then how poor the instruments can bebefore we must conclude that the IV estimation and inferencebecomes unreliable.

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Finite sample properties and weak instruments II

I For k12 = 1 (only one endogenous explanatory variable) thereis a simple test:

I Regress the endogenous variables on the l1 instrumentalvariables.

I A rule-of-thumb test due to Stock and Watson is that if theF − statistic of the test of joint significance > 10 , weakinstruments is not a problem

I In general, the danger of weak instruments means that thesystem, the reduced form, the VAR should be evaluated in itsown merit (do we have a well specified statistical system thathas predictive power for the variables) before moving toestimation of the structural equation.

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General method of moments GMM II GMM is an extension of the GIV estimator that accounts forheteroskedasticity and autocorrelation.

I A famous paper by Lars Petter Hansen from 1982 (joint Nobelprize winner 2013) was the “.. the first to propose GMMestimation under that name“ (Davidson and MacKinnon page367).

I Consider again equation # 1 in a SEM

y1= X1β1+ε1

Change the assumption about white-noise disturbances toautocorrelated/heteroskedastic disturbances:

E (ε1ε′1) = Ω1. (8)

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General method of moments GMM II

I For an identified equation, we can improve on the largesample effi ciency of the GIVE estimator by also making use ofa consistent estimator of Ω1.

I We call this the Generalized Method of Moments, GMM.I Lecture note 6 contains more details if you are interested

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GMM references I

References to Davidson and MacKinnon,

I Ch 9.1-9.4 (GMM linear single equations), 9.5 (Non-linearequations, just the motivation)

I Ch 12.4 (the section Effi cient GMM for systems, and the textabout 3SLS )

I Ch 12.5 (the line of argument in the Full-InformationMaximum Likelihood section) and the last part of p. 536 (testof overidentifying restrictions)

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The VAR system and dynamic SEM I

I We now switch attention back to complete systems, andmodels of such systems.

I For example,a bivariate VAR with two exogenous variables

(Y1tY2t

)︸︷︷︸

yt

=

(π11 π12π21 π22

)︸︷︷︸

Π1

(Yt−1Yt−1

)︸︷︷︸

yt−1

+

(γ11 γ12γ21 γ22

)︸︷︷︸

Γ1

(Z1tZ2t

)︸︷︷︸

zt

+

(ε1tε2t

)︸︷︷︸

εt

(9)

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The VAR system and dynamic SEM II

I Generalizations to higher dimensions (more variables) andlonger lags are unproblematic (as long as we have covariancestationarity):

yt =p

∑i=0

Πiyt−i +q

∑i=0

Γizt−i + εt (10)

εt = IN(0,Σ) (11)

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The VAR system and dynamic SEM III

I Structural models of (10)-(11) can be obtained bypre-multiplying (10) by a non-singular matrix B:

Byt =p

∑i=0BΠiyt−i +

q

∑i=0BΓizt−i +Bεt (12)

so that the structural coeffi cients are in B, BΠi , BΓi and thevector of structural disturbances are εt = Bεt withE (εtε

′t ) = Ω.

I Assume just identification, or overidentification of (12).

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ML estimation of the system and the SEM I

I We know already that ML estimation of the Gaussian VARsystem (10)-(11) is obtained by OLS on each reduced formequation.

I The maximum likelihood estimator of the linear SEM (12) iscalled the full information maximum likelihood estimator orFIML.

I Intuitively, FIML estimators of the structural parameters B,BΠi , BΓi are obtained by “solving back” from the MLestimates of the reduced form parameters.

I This can be done algebraically in the exactly identified case.But generally requires numerical optimization algorithms.PcGive has very good one.

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ML estimation of the system and the SEM II

I In the just identified case, the maximized SEM log-likelihoodis exactly the same as the unrestricted reduced form loglikelihood value LURF from the OLS estimation of the (10).

I In the over-identified case, the SEM restricts the maximizedlog likelihood value LRRF through the over-identifyingrestrictions.

I The over-identifying can be tested by the use of the LRtest-statistic

−2(LRRF − LURF )

which is Chi squared distributed with d.f equal to the degreeof overidentification,

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ML estimation of the system and the SEM III

I PcGive reports this LR statistic as the LR test ofover-identifying restrictions when a SEM is estimated byFIML, or any one of the other estimation methods in theMultiple-Equation Dynamic Modelling part of the program:

I 2SLSI 3SLSI 1SLS (OLS equation by equation)

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3SLS estimation I

I 3SLS is a GMM type estimator which is effi cient whenE (εtε

′t ) = Ω is not-diagonal.

I We first estimate the SEM, equation by equation, with 2SLSI From the 2SLS residuals, we construct the consistent Ω.I Using Ω in a GMM estimator of the structural coeffi cientsgives the 3SLS estimator

I See DM p 531-532 (3SLS) and p 522-524 (GMM forcontemporaneously correlated residuals)

I But since PcGive does an excellent FIML, little practical needfor 3SLS.

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econ 4160, autumn term 2015. lecture 8 - universitetet i oslo · econ 4160, autumn term 2015....

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