14. the multiple-equation gmm -...

14. The Multiple-Equation GMM

Hayashi p. 258-295

Advanced Econometrics I, Autumn 2010,Multiple-Equation GMM 1

Multiple-Equation GMM (MEGMM)

Allows handling a system of mulitple equations

Can be expressed as a single-equation GMM estimator by suitably spe-cifying the matrices and vectors comprising the single-equation GMMformula

We could develop its large-sample properties in the same way as forsingle-equation GMM

Special cases of MEGMM correspond to some well known estimators:

– Under conditional homoskedasticity, MEGMM reduces to the full-information iv efficient(FIVE) estimator, which in turn reduces to3SLS if the set of IVs is common to all equations.


Multiple-Equation GMM (cont’d)

– If we assume all the regressors are predetermined, 3SLS reduces toseemingly unrelated regressions (SUR), which in turn reduces tomultivariate regression when all the equations have the same regressors.

– The multiple-equation system can be written as an equation systemwith its coefficients constrained to be the same across equations. TheGMM estimator for this system is a special case of single-equationGMM.

– The GMM estimator with all the regressors predetermined and errorsconditionally homoskedastic is the random-effects(RE) estimator

– Thus, SUR and RE are equivalent estimators


The Multiple-Equation Model (cont’d)

Assumption 4.1 (Linearity): There are M linear equations

yim = z′imδm + εim (m = 1, 2, ...,M ; i = 1, 2, ..., n)

where:n = sample size,

zim = Lm-dimensional vector of regressors,

δm = the conformable coefficient vector, and

εim = an unobservable error term in the m-th equation.

Oryim = z

′im

(1×Lm)

δm(Lm×1)

+ εim (m = 1, 2, ...,M ; i = 1, 2, ..., n)

Note: (i) No a priori assumption about cross equation error correlation, (ii)No cross equation parameter restriction



The matrix representation of the linear multiple equation regression can begiven by1

y1n×1

...yMn×1

=

Z1

n×L1. . .

ZMn×LM

δ1

L1×1...δM

LM×1

+

ε1n×1

...εMn×1

or

ynM×1

= ZnM×L

δL×1

+ εnM×1

,

where

L =

M∑m=1

LM

1Refer to Hayashi, p. 670, for a useful appendix material on partitioned matrices and kronecker product



Assumption 4.2 (Ergodic stationarity): Let wi be unique and nonconstantelements of (yi1, ..., yiM , zi1, ..., ziM ,xi1, ...,xiM). {wi} is jointly stationaryand ergodic.

Assumption 4.3 (Orthogonality conditions): For each equation m, theKm variables in xim (i.e. the instruments) are predetermined.

E(xim · εim) = 0 (m = 1, 2, ...,M)

There are thus∑

mKm orthogonality contitions in total. Defining

gi

(∑M

m=1 Km×1)≡

xi1 · εi1...

xiM · εiM

the orthogonality conditions can be written compactly as E(gi) = 0.Note that we are not assuming cross-equation orthogonalities


GMM Moment Conditions and Model identification

Given the orthogonality condition, identification could be established inmuch the same way as in the single-equation GMM case

g(wi; δ) ≡

xi1 · (yi1 − z′i1δ1)

...

xiM · (yiM − z′iMδM)

,

The orthogonality condition can be written as E[g(wi; δ)] = 0. Thecoefficient vector is identified if δ = δ is the only solution to the system ofequations

E[g(wi; δ)] = 0


GMM Moment Conditions and Model identification (cont’d)

The expanded orthogonality conditions can be given as follows

E[g(wi; δ)] ≡

E[xi1 · (yi1 − z′i1δ1)]

...

E[xiM · (yiM − z′iM δM)]

=

E(xi1 · yi1)...

E(xiM · yiM)

− E(xi1z

′i1)δ1

...

E(xiMz′iM)δM

=

E(xi1 · yi1)...

E(xiM · yiM)

−E(xi1z

′i1) . . . 0

... . . . ...

0 . . . E(xiMz′iM)

δ1...δM

≡ σxy

(k×1)− Σxz

(K×L)δ

(L×1)



The moment conditions for each equation in the MEGMM are the same asthe moment condition we derived for the single-equation GMM model

The system of equations determining δ can also be arrived at in the sameway as

Σxzδ = σxy

The MEGMM Σxz is block diagonal. Recall that a necessary and sufficientcondition for identification of the single-equation GMM is that Σxz be offull column rank.



If Km = Lm for m = 1, . . . ,M , then δ is exactly identified and δ1...δM

=

Σ−1x1z1σx1y1...

Σ−1xMzMσxMyM

If Km > Lm for some m, then solving

E[gi(δ)] = σxy(k×1)

− Σxz(K×L)

δ(L×1)

= 0

requires the rank condition

Assumption 4.4 (rank condition for identification): For each m(=

1, 2, . . . ,M), E(ximz′im)(Km × Lm) is of full column rank.



That is, all coefficient vectors (δ1, . . . , δM) can be uniquely determined iffeach coefficient vector δM is uniquely determined.

This is the case if the orthogonality assumption holds for each equation.

Assumption 4.5 (gi is a martingale difference sequence with finitesecond moments):{gi} is a joint martingale difference sequence. E(gig

′i)

is nonsingular.

Note, as is also the case for ergodic stationarity, that this assumption isstronger than requiring the same for each equation.

We use S for Avar(g), the asymptotic variance of (g) (i.e. the variance ofthe limiting distribution of

√ng)



By the CLT for stationary and ergodic m.d.s., S equals E(gig′i), with the

following partitioned structure:

S = E(gig′i)

(∑M

m=1 Km×∑M

m=1 Km)

=

E(εi1εi1xi1x′i1) . . . E(εi1εiMxi1x

′iM)

... ...

E(εiMεi1xiMx′i1) . . . E(εiMεiMxiMx

′iM)

That is, the (m,h) block of S is E(εimεihximx

′ih) (m,h = 1, 2, . . . ,M).

The ME Model, thus, is a system of equations where the assumptions wemade for SE model apply to each equation, with the added requirement ofjointness (eg joint stationarity) where applicable.


MEGMM Sample Moment

As in the SEGMM case, let δ be a hypothetical value of the true parametervector δ and define gn(δ) to be its sample analogue, which is given by:

gn(δ)(∑M

m=1 Km×1)=

1n

∑ni=1 xi1 · (yi1 − z

′i1δ1)

...1n

∑ni=1 xiM · (yiM − z

′iM δM)

=

1n

∑ni=1 xi1 · yi1

...1n

∑ni=1 xiM · yiM

− 1

n

∑ni=1 xi1z

′i1δ1

...1n

∑ni=1 xiMz

′iM δM

=

1n

∑ni=1 xi1 · yi1

...1n

∑ni=1 xiM · yiM

−1

n

∑ni=1 xi1z

′i1

. . .1n

∑ni=1 xiMz

′iM

δ1...δM


MEGMM Sample Moment (cont’d)

Thus,

gn(δ)(∑M

m=1 Km×1)≡ sxy − Sxzδ,

where

sxy ≡

1n

∑ni=1 xi1 · yi1

...1n

∑ni=1 xiM · yiM

,Sxz ≡

[1n

∑ni=1 xi1z

′i1

. . . 1n

∑ni=1 xiMz

′iM

]

Orgn(δ)

(∑M

m=1 Km×1)= sxy

(K×1)− Sxz

(K×L)

δL×1)


MEGMM,identification

If Km = Lm for m = 1, . . . ,M then X′mZm is a square matrix and so

δm = S−1xmzmSxmym, m = 1, . . . ,M

Therefore,δ = (δ1, . . . , δM)

′

SolvesSxy

(K×1)− Sxz

(K×L)

δ(L×1)

= 0

If each equation is identified and Km > Lm for some m, then it is notpossible to find some δ that solves the moment conditions.


MEGMM,identification (cont’d)For the overidentified model, let W be a K×K positive definite symmetricmatrix given by

W =

W11 W12 . . . W1M

W′12 W22 . . . W2M

... ... . . . ...

W′1M W

′2M . . . WMM

Such that W→

pW. The GMM estimator then solves

δ(W) = argminδ

J(δ,W) = ngn(δ)′Wgn(δ)

= argminδ

n(sxy − Sxzδ

)′W(sxy − Sxzδ

)Advanced Econometrics I, Autumn 2010,Multiple-Equation GMM 16

MEGMM,identification (cont’d)

Straightforward but tedious algebra should give

δ(W) =(S′xzWSxz

)−1S′xzWsxy =

S′x1z1

W11Sx1z1 S′x1z1

W12Sx2z2 . . . S′x1z1

W1MSxMzM

S′x2z2

W′12Sx1z1 S

′x2z2

W21Sx2z2 . . . S′x2z2

W2MSxMzM... ... . . . ...

S′xMzM

W′1MSx1z1 S

′xMzM

W′2MSx2z2 . . . S

′xMzM

WMMSxMzM

−1

×

S′x1z1

∑Mm=1 W1msxmym

S′x2z2

∑Mm=1 W2msxmym

...

S′xMzM

∑Mm=1 WMmsxmym


Multiple-Equation GMM Defined

The definition of the MEGMM estimator is the same as in the SEGMM,provided that the weighting matrix W is now

∑mKm ×

∑mKm.

Multiple-equation GMM estimator: δ(W) = (S′xzWSxz)

−1S′xzWsxy,

Its sampluing error: δ(W)− δ = (S′xzWSxz)

−1S′xzWg.

Features specific to the MEGMM are:

(i) sxy is a stacked vector,

(ii) Sxz is a block diagonal matrix,

(iii) accordingly, the size of the weighting matrix is∑

mKm×∑

mKm,

(iv) g, the sample mean of gi, is stacked vector, given by


Multiple-Equation GMM Defined (cont’d)

g ≡ 1

n

n∑i=1

gi =

1n

∑ni=1 xi1 · εi1

...1n

∑ni=1 xiM · εiM

= gn(δ)

given these features, the MEGMM estimator

δ(W) = (S′xzWSxz)

−1S′xzWsxy

can be written out in full.

Required: the Wmh(Km ×Kh) has to be the (m,h) block of W(m,h =1, 2, . . . ,M)

[See Hayashi (2000), p. 267]


MEGMM Asymptotics

The mechanics of asymptotics for multiple-equation GMM is similar to thatfor single-equation GMM so long as:

(i) δ,Σxz,S,gi, sxy, and Sxz are as defined for the multiple-equationGMM, and

(ii) the multiple-equaion GMM assumptions are used.

δ in the case of multiple equations is a stacked vector of parameterscomposed of coefficients from different equation.

We could conduct hypothesis testing involving cross-equation restrictions.


MEGMM Asymptotics

Example: Two-equation system of wage equations for two years

LW69i = φ1 + β1Si + γ1IQi + π1EXPRi + εi1,

LW80i = φ2 + β2Si + γ2IQi + π2EXPRi + εi2,

Here

δ = (φ1, β1, γ1, π1, φ2, β2, γ2, π2)′

If all coefficients are thought not to change over time then

φ1 = φ2, β1 = β2, γ1 = γ2, π1 = π2 (also : εim = αi + ηim)


We could test this as:

H0 : φ1 = φ2, β1 = β2, γ1 = γ2, and, π1 = π2

The Hypothesis can be written as Rδ = r, where.

R =

1 0 0 0 −1 0 0 00 1 0 0 0 −1 0 00 0 1 0 0 0 −1 00 0 0 1 0 0 0 −1

, r =

0000

.


MEGMM Asymptotics

(Test of overidentifying restriction) The number of orthogonality conditionsis K =

∑mKm and the number of coefficients is L =

∑mLm. The degrees

of freedom for the J statistic is thus K − L =∑

mKm −∑

mLm.

Proposition 4.1 (consistent estimation of contemporaneous errorcross-equation moments): Let δm be a consistent estimator of δm, and

let εim ≡ yim − z′imδm be the implied residual for m = 1, 2, . . . ,M . Under

Assumptions 4.1 and 4.2, plus the assumption that E(zimz′ih) exists and is

finite for all m,h(= 1, 2, . . . ,M),

σmh →pσmh,

whereσmh ≡

1

n

n∑i=1

εimε′ih and σmh ≡ E(εimεih)

Provided that E(εimεih) exists and is finite.


MEGMM Asymptotics

Suppose that wi denote the unique and nonconstant elements ofyi1, . . . , yiM ; zi1, . . . , ziM , andxi1, . . . , xiM .

Assume that {wi} is jointly ergodic and stationary such that

1

n

n∑i=1

wi → E [wi]

Assume also that {gi} is an ergodic-stationary m.d.s. satisfying

1

n

n∑i=1

gi(δ)→dN(0,S)


where, as before

S = E[gi(δ)gi(δ′)]

=

E[xi1x

′i1ε

2i1] E[xi1x

′i2εi1εi2] . . . E[xi1x

′iMεi1εiM ]

E[xi2x′i1εi2εi1] E[xi2x

′i2ε

2i2] . . . E[xi2x

′iMεi2εiM ]

... ... . . . ...

E[xiMx′i1εiMεi1] E[xiMx

′i2εiMεi2] . . . E[xiMx

′iMε

2iM ]

Assumption 4.6(finite fourth moments): E[(ximk · · · zihj)2] exists andis finite for all k(= 1, 2, . . . ,Km), j(= 1, 2, . . . , Lh),m, h(= 1, 2, . . . ,M),where ximk is the k-th element of xim and zihj is the j-th element of zih.


MEGMM Asymptotics

Estimation of S: the multiple-equation version formula for consistentlyestimating S is given by

S =

S11 S12 . . . S1M

S′12 S22 . . . S2M... ... . . . ...

S′1M S

′2M . . . SMM

where

Smh =1

n

n∑i=1

ximx′ihεimεih

Proposition 4.2 (consistent estimation of S, the asymptotic varianceof g): Let εm be a consistent estimator of εm, and let εim ≡ yim − z

′imδm

be the implied residual for m = 1, 2, . . . ,M .


MEGMM Asymptotics

Under Assumptions 4.1, 4.2, and 4.6, S is consistent for S.

Potential initial consistent estimators of δ:

1. δ(IK) = (δ1(IK1)′, . . . , δM(IKM

)′)′

2. Single equation efficient GMM estimators:

δm(S−1mm), m = 1, . . . ,M


MEGMM Asymptotics

Since S is consistent for S, the MEGMM estimator that uses S−1 asthe weighting matrix, δ(S−1), is an efficient MEGMM estimator withminimum asymptotic variance.

Thus,

Avar(δ(S−1)) = (Σ′xzS−1Σxz),

Avar(δ(S−1)) = (S′xzS−1Sxz)

−1.


Single- vs. Multiple-Equation GMM

Single-equation GMM is a special case of multiple-equation GMM

In SEGMM, apply GMM on each equation separately with equationspecific weight matrix Wmm; so that

δm(Wmm) = (S′xmzmWmmSxmzm)

−1S′xmzmWmmSxmym

m = 1, . . . ,M

This is MEGMM with a block diagonal weight matrix

W = diag(W11, . . . ,WMM)

Question: When is multiple-equation efficient GMM equivalent to singleequation efficient GMM?


Single- vs. Multiple-Equation GMM

Two cases:

1. Each equation is just identified ⇒ the weight matrix does not matter(obvious case)

2. At least one equation is overidentified but S = E[gi(δ)gi(δ)′] is block

diagonal (not so obvious case)

S = diag(S11, . . . ,SMM)

= diag(E[xi1x′i1ε

2i1], . . . , E[xiMx

′iMε

2iM ])

That is, if the equations are unrelated as in

E[ximx′ihεimεih] = 0 for all m 6= h


15. Special Cases of MEGMM: FIVE, 3SLSand SUR

Hayashi, pp. 274-294


Assumption 4.7 (conditional homoskedasticity): Under conditionalhomoskedasticity, a number of prominent estimators can be derived asspecial cases of MEGMM.

E(εimεih|xim,xih) = σmh

for all m,h = 1, 2, . . . ,M.

This means:

Smh = E[ximx′ihεimεih] = σmhE[ximx

′ih]

Thus

S =

σ11E(xi1x′i1) . . . σ1ME(xi1x

′iM)

... ...

σM1E(xiMx′i1) . . . σMME(xiMx

′iM)


Full-Information Instrumental Variables Efficient (FIVE)

An estimator of S in this case is

S =

σ11 · (1n∑n

i=1 xi1x′i1) . . . σ1M · (1n

∑ni=1 xi1x

′iM)

... ...

σM1 · (1n∑n

i=1 xiMx′i1) . . . σMM · (1n

∑ni=1 xiMx

′iM)

,where, for some consistent estimator δm of δ,

σmh ≡1

n

n∑i=1

εimεih; εim ≡ yim − z′imδm (m,h = 1, ,M)


FIVE

- σmh →p σmh provided that E(zimz′ih) (and assumptions 4.1 and 4.2)

- 1n

∑i ximx

′ih →p E(ximx

′ih), which exists and is finite (by ergodic

stationarity)

Therefore, S is consistent for S.

The FIVE estimator of δ, denoted δFIV E, is thus

δFIV E ≡ δ(S−1)


FIVE

Large-sample properties of FIVE:

Under the conditions of

a. Assumptions 4.1-4.5 and 4.7,b. E(zimz

′ih) exists and finite for all m,h(= 1, 2, . . . ,M), and

c. S and S given above:

We have large-sample properties of FIVE in that

i. S→p S;

ii. δFIV E ≡ δ(S−1) is consistent, asymptotically normal, and efficient with

Avar(δ(S−1)) = (Σ′xzS−1Σxz)

−1;

iii.

Avar(δ(S−1)) = (S′xz

ˆS−1Sxz)−1 is consistent for Avar(δFIVE)


FIVE

iv. Sargan’s Statistic:

J(δFIVE, S−1) ≡ n · gn(δFIVE)

′S−1gn(δFIVE)→

dχ2

(∑m

(Km − Lm)

),

wheregn(·) = gn(δ)

(∑M

m=1 Km×1)≡ sxy − Sxzδ

The initial estimator δm needed to calculate S is usually the 2SLSestimator.


Three-Stage Least Squares (3SLS)

Assumptions:

(a) Conditional homoskedasticity

E[εimεih|xim,xih] = σmh

for all m,h = 1, 2, . . . ,M.

⇒ Smh = E[ximx′ihεimεih] = σmhE[ximx

′ih]

(b) Common set of instruments across all equations

xi1 = xi2 = . . . = xim = xik×1


3SLS

The 3SLS moment conditions can be given by

gi(Mk×1)

=

xiεi1...

xiεiM

= εi ⊗ xi, with εi(M×1)

≡

εi1...εiM

The 3SLS Efficient Weight matrix is

S3SLS(Mk×Mk)

=

σ11E[xix

′i] σ12E[xix

′i] . . . σ1ME[xix

′i]

σ12E[xix′i] σ22E[xix

′i] . . . σ2ME[xix

′i]

... ... . . . ...

σ1ME[xix′i] σ2ME[xix

′i] . . . σMME[xix

′i]


3SLS

Or compactly as

S3SLS(Mk×Mk)

= Σ(M×M)

⊗ E[xix′i]

(k×k),

where

Σ(M×M)

= E[εiε′i] =

σ11 σ12 . . . σ1M

σ12 σ22 . . . σ2M... ... . . . ...

σ1M σ2M . . . σMM

Then

S−13SLS = Σ−1(M×M)

⊗ E[xix′i]−1

(k×k)


3SLS

Estimation of S3SLS:

S3SLS = Σ⊗ Sxx,

where

Sxx =1

nX′X,

and

σmh =1

n(ym − Zmδm,2SLS)

′(yh − Zhδh,2SLS)

alsoδm,2SLS = (Z

′mPXZm)−1Z

′mPXym


3SLS

Large-sample properties of 3SLS: Under the conditions of

a. Assumptions 4.1-4.5 and 4.7,b. Common set of instruments (xim = xi),

c. E(zimz′ih) exists and finite for all m,h(= 1, 2, . . . ,M), and

d. Σ is the M ×M matrix of estimated error cross moments

Σ ≡

σ11 . . . σ1M... ...

σM1 . . . σMM

=1

n

n∑i=1

εiε′

i,

estimated using the 2SLS residuals.

Then (See Hayashi, p. 279):

i. δ3SLS is consistent, asymptotically normal, and efficient withAvar(δ3SLS);


Large-Sample properties of 3SLS

ii. The estimated asymptotic variance Avar(δ3SLS) is consistent forAvar(δ3SLS);

iii. Sargan’s statistic

J(δ3SLS, S−1) ≡ n · gn(δ3SLS)

′S−1gn(δ3SLS)→

dχ2

(MK −

∑m

Lm)

),

where

S = Σ⊗ (1

n

∑i

xix′i),Kis the number of common instruments, and

gn(·) = gn(δ)(∑M



Seemingly Unrelated Regression (SUR)

Assumptions:

(a) Conditional homoskedasticity

E[εimεih|xim,xih] = σmh for all m,h = 1, 2, . . . ,M

⇒ Smh = E[ximx′ihεimεih] = σmhE[ximx

′ih]

(b) Common set of instruments across all equations

xi1 = xi2 = . . . = xim = xik×1

(c) xi = union of(zi1, . . . , ziM) = zi ⇒ zi is not endogenous

E[zimεih] = 0, m, h = 1, . . . ,M


SUR

SUR Efficient Weight Matrix

SSUR(Mk×Mk)

=

σ11E[ziz

′i] σ12E[ziz

′i] . . . σ1ME[ziz

′i]

σ12E[ziz′i] σ22E[ziz

′i] . . . σ2ME[ziz

′i]

... ... . . . ...

σ1ME[ziz′i] σ2ME[ziz

′i] . . . σMME[ziz

′i]

Or compactly as

SSUR(Mk×Mk)

= Σ(M×M)

⊗ E[ziz′i]

(k×k),


SUR

Estimating SSUR

SSUR = ΣSUR ⊗ Szz,

where

Szz =1

nZ′Z,

and

σmh =1

n(ym − Zmδm,OLS)

′(yh − Zhδh,OLS)

alsoδm,OLS = (Z

′mZm)−1Z

′mym


Large-Sample properties of SUR

Under the conditions of

a. Assumptions 4.1-4.5 and 4.7,b. Common set of instruments (xim = xi),c. xi = union of(zi1, . . . , ziM)

d. Σ is the M ×M matrix of estimated error cross moments

Σ ≡

σ11 . . . σ1M... ...

σM1 . . . σMM

=1

n

n∑i=1

εiε′

i,

estimated using the OLS residuals.

Then (See Hayashi, p. 281):

i. δSUR is consistent, asymptotically normal, and efficient with Avar(δSUR);


Large-Sample properties of SUR

ii. The estimated asymptotic variance Avar(δSUR) is consistent forAvar(δSUR);

iii. Sargan’s statistic

J(δSUR, S−1) ≡ n · gn(δSUR)

′S−1gn(δSUR)→

dχ2

(MK −

∑m

Lm)

),

where

S = Σ⊗ (1

n

∑i

xix′i),Kis the number of common instruments, and

gn(·) = gn(δ)(∑M



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