Asymptotic. Distribution Theory

JMie asymptotic distribution of Ml - parameter estimates is given by the fol-wing result:

w ierc 1 given by:

1(0) = lim -E

1 32£(0) w llfJar?7

(Л.4.7)

(A.4.8)

and is known as the information matrix. I can he estimated by the sample counterpart:

3030

(Л.4.9)

T ic information matrix gives ns a measure of the sensitivity of the value of с likelihood is to the values of the parameters in the neighborhood of tl с maximum. If small changes in the parameters produce large changes in likelihood near the maximum, then the parameters can be precisely es-tirnated.- Since the likelihood function is Hat at the maximum, the local sensitivity of the likelihood to the parameters is measured by the local curvature (the second derivative) of likelihood with respect to the parameters, evaluated at the maximum.

Information-Matrix Equality

An alternative estimator of the information matrix, %, uses the average outer product or sample variance of the score vectors:

(Л. 4.1

То see why % converges to the same limit as X , differentiate the third equality of equation (A.4.5) with respect to 0 to get

дЧ,(х1+ив) Э0Э0

Aif f-v. . 0)

L, dx,.

J mo + J

3£,(x(+i,fl) 3L,(x,+ i,0) Эв Э0

Щх,+ 0) 3£,(x,+ 0)

= E,

30 30

94,(x,+ i,0) 9c,(x,+ l,0) i)Mx,+ i.0)

9090

4- V.,

a2 ,(x,+l,0) ,. i)t,(x,+l,0) э ,(х,+ 0)

= E----:--r E -

a0a0

dx,+1 1.1 dxn i

(Л.4.11)

V*,. = (Dr(0r)Sy40y) Dv(0 r*))

ln this case

while

since the score vector is serially uncorrelated so S can he estimated from its sample variance. Therefore, the distribution of the CMM estimator can be expressed as:

/Г(0-0 ) ~ м(л1ЛЫ0и)1;,(вл)1Лв )ГУ (Л.4.14)

where I and I/, are the limits of X and X as T increases without bound, evaluated at the true parameter vector 0().

When the model is correctly specified, I and. If, both converge to the information matrix I, hence (A.4.14) simplifies in the limit to (XX-1!)-1 = X-1 which reduces to the conventional expression for the asymptotic variance in (Л.4.7). Therefore, either X or! (or both) can be used to estimate I in this case.

However, when the model is inisspecificd, I and X/, converge lo different limits in general; this has been used as the basis for a specification test by White (1982). But MI, estimates of the inisspecificd model are still consistent provided that the orthogonality conditions hold, and one can use the general variance formula (Л.4.14) provided that the score vector is serially uncorrelated. White (1982) suggests this approach, which is known as quasi-maximum likelihood estimation.

Hypothesis Testing

The asymptotic variances in (Л.4.7) and (Л.4.14) can be used in a straightforward manner lo construct Wald tests of restrictions on (he parameters.

This is known as the information-matrix equality, and implies that the expectations lo wliich the sample averages I and 7/, converge are equal. The information matrix equality holds under the assumption that the data are generated by (Л.4.1).

CMM analysis gives an alternative formula for the distribution of Ml. parameter estimates. Recall from (Л.Л! 1) that the CMM estimator is asymptotically normal with asymptotic variance estimator

.tt\l

Appendix

The idea ofsue h lesis is lo sec whether i lit* unrestricted parameter estimates are significantly different Irom tlieir restricted values, where the variance of the unrestricted estimates is calculated without imposing the restrictions.

Alternatively, one may want to test restrictions using estimates only of the restricted model. Once restrictions are imposed, the minimized GMM objective function is no longer identically zero. Instead, the 1 lansen (1982) result is that ! times the minimized objective function has a x distribution with degrees of freedom equal to the number of restrictions. In ibis case T limes the minimized objective function is just

Vgr(0)i gr(e), (A.4.15)

which is the 1 .agrange multiplier test statistic for a restricted model estimated by maximum likelihood.

The Delia Method

More complicated inferences for arbitrary nonlinear functions of tbe estimator 0 may be performed via Taylors Theorem or tbe delta method. If v/T(f) - On) ~ AftO, Vn), then a nonlinear function f(0) has the following asymptotic distribution:

sflfiO) - f(0)) ~ ЛЛО. V,), Vr = ~Vo~z (Л.4.К5)

\ / do da

which follows from a first-order Taylor series approximation forf(0) around 0n. Higher-order terms converge to zero faster than \/~/Т hence only the first term of the expansion matters for the asymptotic distribution of f(0).

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