ECON 4160, Spring term 2014. Lecture 8 › studier › emner › sv › oekonomi › ECON4160 › h... · ECON 4160, Spring term 2014. Lecture 8 Identi cation and estimation of SEMs

Just identified IV GIVE and 2SLS GMM FIML estimation of systems

ECON 4160, Spring term 2014. Lecture 8Identification and estimation of SEMs (Part 2)

Ragnar Nymoen

Department of Economics

10 Oct 2014

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Notation for a structural equation (recap Lect 7) I

I Consider equation # 1 in a SEM:

y1= Z1β11+Y1β21+ε1 (1)

I y1 is n× 1, with observations of the variable that # 1 in theSEM is normalized on.

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

I Y1, n× k12 holds the included endogenous explanatoryvariables.

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Notation for a structural equation (recap Lect 7) II

The total number of explanatory variables in the first equation is

k11 + k12 = k1 (2)

For simplicity assume that the structural disturbance is Gaussianwhite-noise.

ε1 = IN(0,σ21 Inxn)

By defining the two partitioned matrices:

X1 =(

Z1 : Y1

)(note change) (3)

β1 = ( β11 : β21 )′ (4)

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Notation for a structural equation (recap Lect 7) III

X1 =(

Z1 : Y1

)(5)

β1 = ( β11 : β21 )′ (6)

(1) can be written compactly as

y1= X1β1+ε1 (7)

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The IV estimator (exact/just identification) I

Define W1 asW1 =

(Z1 : X01

)(8)

where X01 is n× k12 (number of included endogenous variablesminus one), with asymptotic properties:

plim(1

nW′

1X1) = SW ′1X1

(invertible) (9)

plim(1

nW′

1ε1) = 0 (independence) (10)

plim(1

nW′

1W1) = SW ′1W1

(positive definite) (11)

We call W1 the instrumental variable matrix and find the IVestimator β1,IV for the first structural equation by solving

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The IV estimator (exact/just identification) II

W′1

[y1 −X1β1,IV

]= 0 (12)

givingβ1,IV = (W′

1X1)−1W′

1y1 (13)

β1,IV is clearly a method-of-moments estimator. The onlydifference from OLS is that W′

1 takes the place of X′1 in the IV“normal equations”, or orthogonality conditions, (12).This means that, by construction, the IV-residuals

εIV,1 = y1 −X1β1,IV (14)

are uncorrelated with (all the instuments in) W1:

W′1εIV,1 = 0. (15)

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The IV residual maker I

I If we want, we can define the IV-residual maker as

MIV ,1=[I−X1(W

′1X1)

−1W′1

](Check that

εIV,1 = MIV ,1y1)

I If MIV ,1 is a proper residual maker, regression of W1 on W1

should result in zero-residuals.

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The IV residual maker III Check:

MIV ,1W1 =[I−X1(W

′1X1)

−1W′1

]W1

Use thatM′IV ,1 = MIV ,1

(show for k1 = 2 for example), then

MIV ,1W1 = M′IV ,1W1

=[I−W′

1

(W′

1X1)−1′

X′1

]W1

=[I−W′

1

(W′

1X1)′−1

X′1

]W1

=[I−W′

1

X′1W1

−1X′1

]W1

= 08 / 61


The IV residual maker III

so that the orthogonality condition (12) can be interpreted as:

W′1εIV ,1= (M

′

IV ,1W1)′y1 = 0 (16)

confirming (15), and showing that the “only” differencecompared to OLS is that the set of instruments used to formorthogonality conditions (normal equations) has been changedfrom “X” to “W”.

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Optimal instruments in the overidentified case I

I In the case of overidentification, W1 is n× l1 wherel1 > k1 = k11 + k12, W′

1X1 is no longer quadratic.

I There is more that one moment-matrix (based on W′1X1) that

are quadratic and invertible

I Each one defines a consistent IV-estimator of β1 under theassumptions (9)-(11). Hence we have over-identification

I To solve this “luxury problem” we can define another IVmatrix W1 that has dimension n× k1:

W1 =(

Z1 : Y1

), (17)

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Optimal instruments in the overidentified case II

where Y1 is n× k12 and is made up of the best linearpredictors of the k12 endogenous variables included in the firstequation:

Y1 =(

y2 y3 . . .)n×k12

(18)

I Where does the optimal predictors come from? Since we arelooking at a single equation in a system-of-equations, theymust come from the reduced form equations for theendogenous variables:

yj = W1πj , j = 2, ..., k12 + 1. (19)

where

πj = (W′1W1)

−1W′1yj , (20)

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Optimal instruments in the overidentified case IIIare the OLS estimators of the regression coefficients in theconditional expectation function for each included endogenousvariables in the first equation, conditional on the full set ofpredetermined variables in the system of equations. We canwrite Y1 as:

Y1 =(

W1(W′1W1)−1W′

1y2 . . . W1(W′1W1)−1W′

1y(k21+1)

),

and more compactly:

Y1 = W1(W′1W1)

−1W′1Y1 = PW1Y1

in terms of the prediction-maker:

PW1 = W1(W′1W1)

−1W′1 (21)

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The GIV-estimator I

I We define the Generalized IV estimator as

β1,GIV = (W′1X1)

−1W′1y1. (22)

with

W1 =(

Z1 : Y1

),

and

Y1 = PW1Y1

I β1,GIV is also known as the 2-stage least squares estimator ofβ1 in (7)

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GIVE and the 2SLS estimator I

I In s separate note, as a supplement, we show that by carryingthrough the partitioning of β1 and X1, we can write

β1,GIV = (23)(β11,GIV

β21,GIV

)=

(Z′1Z1 Z

′1Y1

Y′1Z1 Y

′1Y1

)−1 (Z′1y1

Y′1y1

)and, by use of prediction maker and residual maker matrices:(

β11,GIV

β21,GIV

)=

(Z′1Z1 Z′1Y1

Y′1Z1 Y′1Y1

)−1 (Y′1y1

Z′1y1

)(24)

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GIVE and the 2SLS estimator II

I Next, call the reduced form regression that gives rise to Y1

the first-stage regression.

I Then consider the consequences of using OLS on thestructural equation (1), but after substitution of Y1 with Y1 :

y1= Z1β11+Y1β21 + disturbance

=(

Z1 : Y1

) ( β11

β21

)+ disturbance

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GIVE and the 2SLS estimator III

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

β11,2SLS

β21,2SLS

)=[(

Z1 : Y1

)′ (Z1 : Y1

)]−1(

Z′1Y′1

)y1

=

(Z′1Z1 Z′1Y

Y′1Z1 Y′1Y1

)−1 (Y′1y1

Z′1y1

)(25)

which is identical to (24).

I This establishes that in the over-identified case, GIVE isidentical to the Two-Stage Least squares estimator (2SLS)

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Quick reference back to the just identified case I

I Since W1 is T × k1 (i.e. l1 = k1), we have:

β1,IV = β1,GIV = β1,2SLS

I Moreover, there is also a third estimator called indirect leastsquares (β1,IV ) which is the unique and consistent estimatorfor β1 that can be obtained from the reduced form estimators.

I Indirect least squares is cumbersome for all but simplesystems-of-equation. But Introductory books usually includean example, for example Bardsen and Nymoen ch 9.

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Another equivalent expression of the GIV-estimator I

I Let us again consider over-identification: l1 > k1 = k11 + k12.

I A different method to obtain unique moments is topost-multiply W1 by the matrix J1 so that W1J1 is anIV-matrix of dimension n× k1 and which satisfies (9)-(11).

I J1 must be l1 × k1. If we choose

W1J1 = PW1X1 (26)

as the IV matrix (remember that PW1 = W1(W′1W1)−1W′

1)we can determine J1 as

J1 = (W′1W1)

−1W′1X1 (27)

which has the correct dimension l1 × k1.

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Another equivalent expression of the GIV-estimator II

I Let us, for the time being denote the estimator that usesW1J1 as IV-matrix by β1,JGIV

β1,JGIV = ((W1J1)′X1)

−1(W1J1)′y1

= ((PW1X1)′X1)

−1X′1PW1y1

= (X′1PW1X1)

−1X′1PW1y1

which is the expression for the GIV estimator in equation(8.29) in DM (p 321).

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Another equivalent expression of the GIV-estimator III

I But we have already

β1,GIV = (W′1X1)

−1W′1y1. (28)

with

W1 =(

Z1 : Y1

),

Y1 = PW1Y1

from before. Is the an internal inconsistency? Canβ1,GIV 6= β1,JGIV ?

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Another equivalent expression of the GIV-estimator IV

I The answer is ”no”, since

W1 =(

Z1 : Y1

)=(

PW1Z1 : PW1Y1

)= PW1X1

we can replace W1 in β1,GIV by PW1X1 and obtain

β1,GIV = (W′1X1)

−1W′1y1

= (X′1PW1X1)−1X′1PW1y1

= β1,JGIV

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IV-criterion function I

I The projection matrix PW1 is also central in the IV-criterionfunction Q(β1, y1) which DM defines on page 321:

Q(β1, y1) = (y1 −X1β1)′PW1(y1 −X1β1). (29)

note the close relationship to the sum of squared residuals.

I Minimization of Q(β1, y1) wrt β1 gives the 1 oc:

X′1PW1(y1 −X1β1,GIV ) = 0 (30)

which gives β1,GIV as solution

β1,GIV = (X′1PW1X1)−1X′1PW1y1

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IV-criterion function II

I Note that from (26)

X′1PW1 = J

′1W′

1

we can write (30) as:

J′1W

′1(y1 −X1β1,GIV ) = 0 (31)

We can then represent the just identified case by setting J1 toa non-singular k1 × k1 matrix. Pre-multiplication in (31) by(J′1)

−1 gives

W′1(y1 −X1β1,GIV ) = 0

confirming that in the just identified case, β1,GIV = β1,IV .

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IV-criterion function III

I In the just identified case the minimized value of theIV-criterion function is zero:

Q(β1,IV , y1) = (y1 −X1β1,IV )′PW1(y1 −X1β1,IV )

= ε′IV,1

[W1(W

′1W1)

−1W′1

]εIV,1

= ε′IV,1W1(W′1W1)

−1[W′

1εIV,1

]= 0 (32)

from (12).

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Overidentification tests

Sargan test and J-test I

I When l1 − k1 ≥ 1 the validity of the over-identifyinginstruments can be tested.

I DM Ch 8.6

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

I Simplest case. Assume that there is one endogenousexplanatory variable in the first structural equation.

I The regression of εGIV,1 on the instruments and a constantproduces an R2 that we can call R2

GIVres .

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Sargan test and J-test II

I The test called Specification test when IV-estimation Pc-Givesingle equation is used is:

Specification test = nR2GIVres ∼a χ2(l1 − k1) (33)

under the H0 of valid instrumental variables (HN p 230)

I As noted by DM on p 338 this is rightly called a Sargan test ,because of the contribution of Denis Sargan who worked onIV-estimation theory during the 1950 and 1960.

I In terms of computation Sargan test can be computed fromthe IV criterion function:

I From (32) Q(β1,IV , y1) = 0, the IV-residuals are orthogonal toall instrumental variables

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Sargan test and J-test III

I In the over-identified case, Q(β1,GIV , y1) > 0 since theGIV-residuals are uncorrelated with the optimal instruments,but not each individual.

I Therefore, a test of the validity of the over-identification canbe based on Q(β1,GIV , y1)−Q(β1,IV , y1):

Specification test =Q(β1,GIV , y1)

σ21

∼a

χ2(l1 − k1) (34)

where σ21 is the usual consistent estimator for σ2

1 .

I The two ways of computing the Sargan Specification test arenumerically-identical.

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Sargan test and J-test IV

I Many (papers) and software-packages report “J-tests” or“Hansen test” for instrument validity. This test uses theF -statistic from the auxiliary regression instead of R2

GIVres :

J − test = l1F 2GIVres ∼a χ2(l1 − k1)

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Distribution and IV strength

Asymptotic properties of GIV I

I As noted at the end of Lecture 7, with reference to theasymptotic nature of the IV requirements (9)-(11) it is notsurprising that the known properties of the IV and GIVEestimators are asymptotic.

I In the case of overidentification, and maintaining,

ε1 = IN(0,σ21 Inxn) (35)

β1,GIV and β1,2SLS are consistent and asymptotically efficient.The use of optimal instruments based on the reduced form ofthe system-of-equations (and (35)) are the main driversbehind this result.

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Asymptotic properties of GIV II

I If (35) cannot be maintained, or if we want to take intoaccount correlation between ε1 and other disturbances in theSEM, other estimation methods are more efficient: GMM and3SLS for example.

I Lecture 6: asymptotic normality of “t-ratios” etc, and theform of the variance estimator.

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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 analysis

I Weak instruments: The instruments are poorly correlated withthe included endogenous variables in the structural equation.

I When instrumentst 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 l1instrumentalvariables.

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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IV estimation of simultaneous equation models and lookingahead I

I We have focused on the “first equation” in SEM

I Obviously: Can apply the same 2-SLS method to all theidentified equations of a SEM.

I However, if we are interested in more than one identifiedstructural equation, there is really no reason to use limitedinformation methods (based on uncorrelated disturbances).

I We will therefore turn to Full Information MaximumLikelihood (FIML) and 3SLS

I First, we need to mention another important development of“single equation IV” called Generalized Method ofMoments (GMM, Ch 8 in DM) and show examples of IVestimation and test of forward-looking models.

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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 Efficient 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)

I 2013 NOBEL PRICE TO GMM!GMM is an extension of the GIV estimator that accounts forheteroskedasticity and autocorrelation. A famous paper byLars Petter Hansen from 1982 (joint Nobel prize winner 2013)was the “.. the first to propose GMM estimation under thatname“ (Davidson and MacKinnon page 367).

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GMM for a structural equation, motivation I

I Consider again equation # 1 in a SEM as in DM page 522:

y1= X1β1+ε1 (36)

I Memo:

X1 =(

Z1 : Y1

)(37)

β1 = ( β11 : β21 )′ (38)

I y1 is n× 1, with observations of the variable that equation #1 in the SEM is normalized on.

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

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GMM for a structural equation, motivation III Y1, n× k12 holds the included endogenous explanatory

variables.

Change assumption about hite-noise disturbances toautocorrelated/heteroskedastic disturbances:

E (ε1ε′1) = Ω1. (39)

I Consider the just-identified or over-identified case (by the rankand order conditions)

I When E (ε1ε′1) = IN(0,σ2

1 Inxn) we know how to find optimalinstruments: Use the reduced form to find the linearcombination of exogenous and predetermined variables thatgive the best predictors of the variables in Y1.

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GMM for a structural equation, motivation III

I When we have the more general E (ε1ε′1) = Ω1, the best

predictors for Y1 should take the correlation structure betweenthe random variables in ε1 into account.

I Generalized (weighted) LS analogy: The regressors in themodel are weighted so as to whiten the residuals to get backto the “classical assumptions”

I In the case of IV estimation, the weights must include theinstruments as well—this leads to Generalized Method ofMoments

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GMM-criterion function and the GMM estimator I

I In the same way as for the GIV estimator, the GMM estimatorof β1can be found by minimization of the GMM-criterionfunction:

QGMM(β1, Ω1, y1) = (y1 −X1β1)′PgW1(y1 −X1β1). (40)

where

PgW1 = W1(W′1Ω1W1)

−1W′1. (41)

is the generalization of the GIV prediction maker matrix.

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GMM-criterion function and the GMM estimator II

I Assume that Ω1 is a matrix of known parameters. The firstorder conditions (i.e. Method of Moments!) are thencompactly written as

X′1PgW1(y1 −X1β1,GMM) = 0 (42)

which gives β1,GMM as the solution

β1,GMM = (X′1PWg1X1)−1X′1PW1y1

I If you “write it out”, you get expression (9.10) on page 355 inDM (with some small changes in notation)

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GMM-criterion function and the GMM estimator III

I The generalization of the J1 matrix from GIV estimation isfound from:

X′1PgW1 = J

′g1W′

1

and becomes (check)

Jg1 = (W′1Ω1W1)

−1W′1X1 (43)

which gives the optimal weights to the instruments in theGMM case.

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MM an IV summary

1. When E (ε1ε′1) = Ω1 6= I , β1,GMM is asymptotically more

efficient than β1,GIV .

2. When Ω1=σ21 I: β1,GMM = β1,GIV and Jg1 = σ−2

1 J1 (thereduced form gives the optimal iv weights).

3. Ω1=σ21 I and exact identification: β1,GMM = β1,GIV = ffiIV

4. If the structural equation is a regression model, as in arecursive system of equations, we can set Z1 = X1 and theoptimal matrix with instruments is W1 = X1. Soβ1,GMM = β1,OLS in this case.

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Feasible GMM

I Have seen that GMM is the generalization of GIVE, to thecase of non-white-noise error terms. Just as GLS generalizedOLS estimation to that case.

I In the same way as with GLS, to make GMM a feasible ofpractical method, Ω1 has to be estimated by a consistentestimator Ω1. Use GIV residuals to get a first estimate.

I Can choose to iteration over β1,GMM and/or Ω1.

I The covariance matrix of the feasible GMM estimator is

Var(β1,GMM) = (X1W1(W′1Ω1W1)

−1W′1X1)

−1 (44)

and t-ratios are asymptotically N(0, 1) under their respectivenull hypotheses. So the inference procedure is as are before.

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Hansen-Sargan test for GMM I

I Since the GMM criterion function only depends on Ω1 troughthe square matrix W′

1Ω1W1 it is not surprising that the IVSpecfication test is also defined for GMM.

I It is usually interpreted as a test of the validity of theinstruments (orthogonality conditions), but as Davidson andMacKinnon (DM) discuss on

I p 338, for IVI p 367-368 for GMM

the second possibility is that the Specification test becomessignificant when an explanatory variable has incorrectly beenclassified as an instrument instead of (correctly) as anpredetermined of exogenous variable (i.e., would go into W1

via Z1 not via X01).

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Hansen-Sargan test for GMM II

I The general implication of a significant specification test istherefore re-specification.

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GMM for non-linear equations I

I Economic theory of intertemporal decisions leads toEuler-equations that are formulated as (say) l orthogonalityconditions that are similar to the moments conditions

I If the number of parameters to be estimated is less than orequal to l , we have identification.

I The moments conditions can be linear or non-linear inparameters

I The non-linear GMM can be obtained by minimization ofcertain quadratic forms. Chapter 9.5 is a relatively advancedchapter on non-linear GMM that you use as a reference if youapply this method.

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An example of IV and GMM estimation

New Keynesian Phillips curve (PCM) I

I DSGE macro models to a large extent are made up ofstructural equations that are

I first order conditions of agents intertemporal optimizationproblems

I price and wage equations that are based on Calvo-pricing: NewKeynesian Phillips curves (PCM)

I We will look at a famous example of IV estimation of a PCMfrom the euro area.

I Let pt be the log of a price level index. The (hybrid) PCMstates that inflation, defined as ∆pt ≡ pt − pt−1, is explainedby

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New Keynesian Phillips curve (PCM) II

I Et(∆pt+1), expected inflation one period ahead conditionalupon information available at time t, lagged inflation and avariable

I xt , often called a forcing variable, representing excess demandor marginal costs (e.g., output gap, the unemployment rate orthe wage share in logs):

∆pt = bfp1Et(∆pt+1) + bb

p1∆pt−1 + bp2xt + εpt , (45)

I εpt is assumed to be white noise

I Theory predicts (a little simplified) : 0 < bfp1 < 1 and

0 < bbp1 < 1 and bp2 > 0 if xt is measured by the logarithm

of the wage-share.

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The NPC system

I We cannot find Et(∆pt+1) from (45) alone.

I To make progress, we need a “completing” system.

I The term ’forcing variable’ suggests that xt is exogenous, sowe specify

∆pt = bfp1Et(∆pt+1) + bb

p1∆pt−1 + bp2xt + εpt (46)

xt = bx1xt−1 + εxt , − 1 < bx1 < 1 (47)

where εxt is white-noise, and uncorrelated with εpt

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The NPC in terms of observables

I A popular approach is to substitute the theoretical Et(∆pt+1)by the lead variable ∆pt+1.

I After substitution, the NPC is:

∆pt = bfp1∆pt+1 + bb

p1∆pt−1 + bp2xt + νpt (48)

the disturbance νpt contains the forecast error∆pt+1 − Et(∆pt+1) in addition to the white noise εpt .

I As usual with error-in-variables models, ∆pt+1is correlatedwith the νpt .

I OLS on (48) is inconsistent. Need IV estimation

I We postpone to E 5101 to show that νpt follow a first orderMoving-Average process

I In principle GMM should give more efficient estimation.

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Euro-area NPC I

I We replicate the results in European inflation dynamics, Gali,Gertler and Lopez-Solado, European Economic Review (2001).

I Reference: Econometric evaluation of the New KeynesianPhillips Curve, Bardsen, Jansen and Nymoen Oxford Bulletinof Economics and Statistics (2004)

I With the log of the wage-share wst as the forcing variable (xt):

∆pt = 0.60(0.06)

∆pt+1 + 0.35(0.06)

∆pt−1 + 0.03(0.03)

wst + 0.08(0.06)

(49)

GMM, T = 107 (1972 (4) to 1997 (4))

χ2J (8) = 6.74 [0.35]

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Euro-area NPC III Note that χ2

J (8) is the J−form of the Specification test

I Note that there are 8 overidentifying restrictions, indicatingthat implicitly GGL had a larger NPC-system in mind.

I The results in (49) are not very robust to the details abouthow we estimate Ω1. When we iterate over Ω1:

∆pt = 0.731(0.052)

∆pt+1 + 0.340(0.069)

∆pt−1 − 0.042(0.029)

wst

− 0.102(0.070)

(50)

GMM, T = 107 (1971 (3) to 1998 (1))

χ2J (8) = 7.34 [0.50]

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Euro-area NPC III

Lack of robustness with respect to such details need not e aproblem, but it is here.

I GIV estimation results

∆pt = 0.66(0.14)

∆pt+1 + 0.28(0.12)

∆pt−1 + 0.07(0.09)

wst + 0.10(0.12)

(51)

2SLS, T = 104 (1972 (2) to 1998 (1))χ2

Specification (6) = 11.88[0.06]

I Misspecification tests show that there is heavy residualautocorrelation in (51).

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Euro-area NPC IV

I Consistent with NPC but could also be a resultmisspecification: ∆pt+1 acting as a proxy for omitted currentand lagged variables.

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The VAR system and dynamic SEM I

I We now switch attention back to dynamic systems anddynamic models of the system such as a the bivariate VAR(

Y1t

Y2t

)︸︷︷︸

yt

=

(π11 π12

π21 π22

)︸︷︷︸

Π

(Y1t−1

Y2t−1

)︸︷︷︸

yt−1

+

(ε1t

ε2t

)︸︷︷︸

εt

, (52)

εt = IN(0, Σ2×2) (53)

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The VAR system and dynamic SEM II

I or the bivariate open-VAR (also called VARX)(Y1t

Y2t

)︸︷︷︸

yt

=

(π11 π12

π21 π22

)︸︷︷︸

Π1

(Yt−1

Yt−1

)︸︷︷︸

yt−1

+

(γ11 γ12

γ21 γ22

)︸︷︷︸

Γ1

(Z1t

Z2t

)︸︷︷︸

zt

+

(ε1t

ε2t

)︸︷︷︸

εt

(54)

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The VAR system and dynamic SEM IIIGeneralizations to higher dimensions (more variables) and longerlags are unproblematic, as long as yt and zt are covariancestationary:

yt =p

∑i=0

Πiyt−i +q

∑i=0

Γizt−i + εt (55)

εt = IN(0, Σ) (56)

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The VAR system and dynamic SEM IV

I As we have seen (!), dynamic linear structural models of(55)-(56) can be obtained by pre-multiplying (55) by anon-singular matrix B:

Byt =p

∑i=0

BΠiyt−i +q

∑i=0

BΓizt−i + Bεt (57)

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

′t) = Ω.

I Assume just identification, or overidentification of (57).

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ML estimation of the system and the SEM I

I We know already that ML estimation of the Gaussian VARsystem (55)-(56) is obtained by OLS on each reduced formequation.

I The maximum likelihood estimator of the linear SEM (57) 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 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 (55).

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ML estimation of the system and the SEM III In the over-identified case, the SEM restricts the maximized

log 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.

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 2SLS

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ML estimation of the system and the SEM III

I 3SLSI 1SLS (OLS equation by equation)

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3SLS estimation I

I 3SLS is a GMM type estimator which is efficient 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 coefficients

gives 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, Spring term 2014. Lecture 8 › studier › emner › sv › oekonomi › ECON4160 › h... · ECON 4160, Spring term 2014. Lecture 8 Identi cation and estimation of SEMs