Identification

Standard testsIdentification and OveridentificationTesting Overidentifying Restrictions

IV/GMM Inference. Identification andOveridentification

Walter Sosa-Escudero

Econ 507. Econometric Analysis. Spring 2009

March 8, 2009

Walter Sosa-Escudero IV/GMM Inference. Identification and Overidentification


Standard tests

Standard inference comes directly from the asymptotic normality result

n (g 0) d N(0,AV(g))

Then, it is easy to see that

Under H0 : k = k0

tk =ngk k0

AV(gk)

p N(0, 1)

where AV(gk) is any consistent estimator.

Under H0 : R0 r = 0

W = n(Rg r)[RAV(g)R]1(Rg r) p 2(q)with R, r and q defined as in our original large sample inferenceproblem.



Overidentification: Prelude

Suppose there are two instruments z1 and z2, for a modely = x0 + u where x is a single explanatory variable.Suppose invalid instrument means that for no E(zijui) = 0, j = 1, 2.

You use only one instrument, say z1 and get ui = yi xiwhere is the IV estimator using z1 as instrument.

What can you learn about instrument validity from thecorrelation between ui and z2i?.



Identification and Overidentification

Recall that our moment conditions for the IV case are

E[zi(yi xi0)] = 0,

and our problem was that the sample counterpart

1n

ni=1

zi(yi xib) = 0

implies a system of p linear equations with K unknowns, whichcannot produce a solution when p > K.

If all the moment conditions are valid we could discard pKmoment conditions in any arbitrary way and produce a MMestimator using the K moment conditions retained.



Another thing we can do is to combine the p moment conditionslinearly so as to produce K linearly independent momentconditions, that is, use:

BE[z(yi xi0)] = 0

where B is any K q matrix with (B) = KIn fact throwing away moment conditions implies a particularchoice of B! (which one?).

Note that the sample counterpart

1nB

ni=1

zi(yi xib) = 0

produces a single solution (why?).



GMM as a particular case of MM?

GMM is choosing a particular B. Let ui(b) yi xib andu(b) Y Xb. Recall that the GMM estimator minimizes:

J(b) = n{1nu(b)Z

}Wn

{1nZ u(b)

}so the FOCs are:

X Zn

WnZ u(b)n

= 0

which can be seen as the sample counterpart of:

E[xizi]W Kp

E[ziui(0)] p1

= 0

which are K moment conditions!



E[xizi]W Kp

E[ziui(0)] p1

= 0

Then

GMM is choosing a particular way of linearly combining the pmoment conditions so the system becomes exactly identified.



Identifying and Overidentifying Restrictions

It is very illuminating to explore how these linear combinations areproduced.

First note that W can be written as W =W 1/2W 1/2, where

W 1/2 is an invertible p p matrix.It will be more convenient to re-express our original p momentconditions as follows:

W 1/2E[ziui(0)] = 0

Note that this does not alter at all the informational structure ofthe problem (why?).



Now write the GMM version of the moment conditions as:

E[xizi]W E[ziui(0)] = 0

E[xizi]W1/2W 1/2 E[ziui(0)] = 0

F W 1/2E[ziui(0)] = 0

where F E[xizi]W 1/2 . Note F is K p with rank K.Let h W 1/2E[ziui(0)], so the moment conditions imply h = 0.



h =W 1/2E[ziui(0)] is a p vector, and it can be decomposedorthogonally as

h = Ph+Mh

where P and M are any pair of orthogonal projection matricesthat project h orthogonally onto some subspace and its orthogonalcomplement.

Note that the moment condition implies Ph = 0 and Mh = 0.

Also, note that the decomposition holds for any pair of projectionmatrices.



In particular for P = F (F F )1F Pf , whereF = E[xizi]W

1/2 .

Note F is K p with rank K, hence Pf is the projection matrixthat projects vectors of dimension p onto the span of the Kcolumn vectors in F , hence Pf has rank K andMf = I F (F F )1F has rank pK

(Remember the dimension theorem...).



ThenF (F F )1F h = 0,

forms a system of K linearly independent equations with Kunkowns.

Now note since F (F F )1F has rank K, F (F F )1F h = 0whenever

F h = 0

which are the MM conditions derived from the GMM procedure.

This leds us to a very important result.



What GMM is doing in when p > K is decomposing themoment conditions, h, in two orthogonal parts. One that isused to exactly identify the relevant parameters (Pfh = 0)and another part which is left unused (Mfh = 0).The moment conditions Pfh = 0 are called the identifyingconditions and the conditions Mfh = 0 are called theoveridentifying conditions.

In a certain sense, we can see GMM as using a particular setof K moment conditions and discarding the remaining pK.



Another interesting intuition is the following.

Let h W 1/2n n1n

i=1 ziui(g). Then the minimized GMMobjective function can be expressed as:

J(g) = n hh

Let Fn n1 X Z W 1/2

n , Pfn = Fn(F nFn)1F n andMfn = Ip Pfn.By orthogonal decomposition, h = (Pfn +Mfn)h. Replacingabove and using the properties of these matrices:

J(g) = n[h(Pfn +Mfn)(Pfn +Mfn)h

]= n

[h(Pfn +Mfn)h

]= n

[hPfnh+ hMfnh

]Walter Sosa-Escudero IV/GMM Inference. Identification and Overidentification


J(g) = n[hPfnh+ hMfnh

]The FOC of the GMM problem sets Pfnh = 0, which leds to avery useful intuituion about how GMM works and its usefulness fortesting:

The GMM estimator satisfies strictly the identifyingrestrictions and tries to make the overidentifying restrictionsas small as possible.

The minimized value of J(g) = n hMfnh measures how faris the sample from satisfying the overidentifying restrictions.



Testing overidentifying restrictions

Intuituion:

The fact that J(g) measures how far is the sample fromsatisfying the overidentifying restrictions can be exploited todesign a formal specification test.

If all the assumptions of the overidentified GMM model hold,then asymptotically the overidentifying conditions should besatisfied: the identifying conditions succeed in producing aconsistent estimator and hence force all moment conditions tohold.

If the J(g) is too large then some of the conditions thatguarantee consistency and asymptotic normality are likely tobe false (more on this later).



The J test: the test statistic for the overidentifying restrictionstest is:

J = Jn(g) = nu(n)Z

nS1n

Z u(i)n

and under H0 : E[ziui(0)] = 0 it converges in distribution to2(pK).

First note that the test uses the optimal GMM estimator, thatis, Wn = S1n .Intuition: the J test checks if the GMM is small. Accordingto our previous result, this means checking if theoveridentifying restrictions are small.

Under the null that the model is correctly specified (all GMMassumptions hold), GMM is consistent and hence theoveridentifying restrictions should be close to zero.



What happens under the alternative hypothesis?

Suppose the rank condition holds. The sample version of theidentifying conditions must be satisfied, so g

p +.Asymptotic normality of the moment conditions also holds,but centered at some other place that is not zero.

Rewrite the statistic as:

J = nu(n)Z

nS1n

Z u(i)n

Then when H0 does not hold, n and the remaining partconverges to something not centered at zero, then J .It is a global test for misspecification. It may mean that somemoment conditions are invalid and/or that the model ismisspecified.



Proof: By our previous result

Jn(g) = n[hMfnh

]with h W 1/2n Z

u(g)n , so

Jn =

{W 1/2n

Z u(g)n

}Mfn

{W 1/2n

Z u(g)n

}From the asymptotic normality proof of GMM we got

Z u(g)n

d N(0, S).

If Wn = S1n and W1/2n = S1/2

W 1/2nZ u(g)

n= S1/2

Z u(g)n

d N(O, Ip)



Hence, the J test is an quadratic form of a p vector of independentnormal variables, normed by an idempotent matrix with rankpK, then the result follows.

Recall that if y N(0, Ip) and A is an idempotent matrix with rank q, then yAy 2(q) (Hayashi (2000,p. 37)).



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