Prior Information

The information on the parameters can come from two sources. One is from the observations, which are informative about о(0). But we may also have a priori information. Let this be of the following form: 00 satisfies p(00) = 0, where p() is an r-dimensional vector function. Therefore we study the submodel H C S, where

Подпись: (7.5)H = {0 | 0 Є S C Rl, p(0) = 0}.

Now the problem of local identification of the parameter vector 00 reduces to the problem of verifying whether the equations system

Подпись:Подпись: f (9)Подпись: = 0(7.6)

has a locally unique solution: if we can find parameter points 0 arbitrarily close to 00 for which f(0) = 0, then 00 is locally not identified. In order to find out whether 00 is locally identified we will apply the implicit function theorem. We will now first discuss this theorem.

Let C be the class of functions that are r times continuously differentiable in 00. Furthermore a C” function is analytic if the Taylor series expansion of each component converges to that function. In practice most functions are analytic. They share the important quality of being either equal to zero identically or equal to zero only on a set of Lebesgue measure zero. We can now formulate the implicit function theorem.

Theorem 4. Implicit function theorem. Let f(0) = f(0I; 0II) be a C function (p > 1), f: Rn+m ^ Rn; 0 Є Rn, 0 Є Rm such that f(0°; 0(°I) = 0 and the n x n-matrix df(0)/Э011 00 is nonsingular, then there exists an open neighborhood U C Rn of 0° and an open neighborhood V C Rm of 0° such that there exists a unique Cp function g : V ^ U with g(0n) = 0І and f( g(0n); 0II) = 0 for all 0II Є V.

If indeed df(0)/Э0] 100 is nonsingular, so thatf(g(0II); 0II) = 0 for all 0II Є V, then we also have for points (0:; 0n) = (g(0n); 0n) with 0П Є V:

f + f _dg_ = 0

Э0ц dg’ дв’п

Returning to the question of identification of 0° a first important result is easily derived if we assume that f(0), as defined in (7.6), is continuously differentiable in 0°. Let the Jacobian matrix J(0) be

Подпись: J(9) Подпись: Ш] = Гз^і(е)/зе'" [э/2(е)/эе'_' Подпись: (7.7)

Now assume that J(0°) has full column rank. Then, possibly after a rearrange­ment of elements of f(0) = (f1(0); f2(0)), we have

where J2(0°) is nonsingular. If we consider the function h(0; v) = f2(0) + 0 ■ v, we find, by using the implicit function theorem, that the function g(v) = 0°, for which h( g(v); v) = 0, is unique. Consequently, there is an open neighborhood of 0° where 0 = 0° is the only solution of f2(0) = 0, or f(0) = 0. Thus, if J(0°) has full column rank, then 0 ° is locally identified.

Now assume that J(0°) is not of full column rank. Does this imply that 0° is not locally identified? The answer is negative. For example, consider the special case wheref(0) = 01 + 02, and 0° = (0, 0). Here the Jacobian matrix J(0) is given by (201, 202) and J(0°) = (0, 0), which is clearly not of full column rank. However, 0° is the only point in R2 for which f(0) = 0, so that 0° is, in fact, globally identified.

This may seem a pathological case since the (row) rank of the Jacobian matrix simply was 0. What can we say if J(0 °) is not of full column rank while it is of full row rank? In that case 0° will not be identified as a direct consequence of the implicit function theorem. And so we are left with the general case where J(0 °) has both a deficient column rank and a deficient row rank. Without loss of generality we can rearrange the rows of J(0) as in (7.7) where rank{ J(0°)} = rank{ J2(0°)} while now J2(0°) has full row rank. According to the implicit function theorem we can rearrange the elements of 0 = (0I; 0II) so that locally there exists a unique function g(0,,) so that f2( g(0n); 0n) = 0. However, f1( g(0n); 0n) = 0 does not necessarily hold, and we have not yet established a lack of identification.

Let h(0n) = f( g(0n); 0n). Then we have locally

Щ%) = f + f _dg_ эе;, де;, dg aeji

Ji(g(en); en)’

dg/deii

_ШЄ„); e,,)_

_ ik _

Ji(g(e, i); e,,)

dg/de;i

ik J

_ 0

where k = l – rank{/(00)}. If it can be established that the rows of J1(g(0n); 0n) are linearly dependent on the rows of J2{g(0n); 0II) for all 0II in an open neighborhood of 0Ц Є Rk, then 3h(0n)/30n = 0 in that neighborhood, which can be taken to be convex, so that f(g(0II); 0n) = h(0II) = h(0Ц) = 0 in an open neighborhood of 0Ц. As a result 00 will not be locally identified.

Notice that 00 is a regular point of J(0) if J(00) has full row (or column) rank since in that case | J(0o)J(00)’| Ф 0 (or | J(0O)7(0°)| Ф 0) so that J(0) will have full row (or column) rank for points close enough to 00. If 00 is a regular point of J(0) and rank{ J(00)} = rank{ J2(00)}, where J(0) has been partitioned as in (7.7), and J2(00) is of full row rank, then 00 is a regular point of both J(0) and J2(0). So rank{ J(0)} = rank{ J2(0)} in an open neighborhood of 00. In other words, for all points in an open neighborhood of 00 the rows of J1(0) will depend linearly on the rows of J2(0).

Summarizing these results we may give the following theorem, which is also given by Fisher (1966, theorem 5.9.2).

Theorem 5. Let J(0) be the Jacobian matrix of order (n + r) x l formed by taking partial derivatives of (o(0); p(0)) with respect to 0,

Подпись: J(0)3a(0)/30′ dp(0)/30’_.

If 00 is a regular point of J(0), then a necessary and sufficient condition for 00 to be locally identified is that J(0) has rank l at 00.

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