Handling the Rank in Practice

The question is how one should apply this theorem in practice. Since 00 is an unknown parameter vector it is not yet clear how one should compute the rank of J(00). Furthermore, a question can be raised as to the restrictiveness of the assumption of regularity of 00. It appears that these problems are related. That is, if 00 is a regular point of J(0), then the rank of J(00) can be computed even though 00 itself is unknown. On the other hand, the assumption of regularity of 00, as it is stated in Theorem 5, is unnecessarily restrictive.

The assumption of regularity of 00 makes it possible to compute the rank of J(00). In the sequel we assume that f(0) is an analytic function, in which case almost all points 0 Є S are regular points: the irregular points constitute a set in Rl of Lebesgue measure zero. The proof of this statement is as follows. Let

0 = argmax{rank{ J(0)}},

0ЄRl

and let

Подпись:Ji(0)

_ J2 (0)__
so that rank{/(0)} = rank{/2(0)} and /2(0) is of full row rank. Then, due to continu­ity, | /2(0) /2(0)r1 > 0 in an open neighborhood of 0. That is, the analytic function |/2(0)/2(0)’| is not equal to zero on a set of positive Lebesgue measure. Hence |/2(0)/2(0)’ | > 0 almost everywhere, since an analytic function is either identical to zero or equal to zero on a set of Lebesgue measure zero only. Thus /2(0) will have full row rank almost everywhere in S so that rank{/(0)} = maxee]R {rank{/(0)}} almost everywhere. Consequently, if 00 is a regular point of /(0), then rank{/(0)} = rank{/(00)} for points 0 in a neighborhood with positive Lebesgue measure. Hence, rank{/(00)} = maxeeRi {rank{/(0)}}, which can be computed without knowing 00. We thus have shown the following result, see also Johansen (1995, theorem 2).

Theorem 6. Let o() and p() be analytic functions. Then 00 is a regular point of /(0) if and only if

rank{ /(00)} = max{rank{/(Є)}},

e^R1

which holds for almost all 00 Є Rl.

Thus if we know nothing about 00 except that 00 Є S C Rl, where S is some open set in Rl, then it would make sense to assume that 00 is a regular point, since almost all points of S are regular points. However, when doing so we would ignore the prior information contained in p(00) = 0. The set H (cf. (7.5)) constitutes a manifold of dimension l – r in Rl, where we assume that Эр(0)/Э0′ has full row rank in 00. Let then rank{R(0°)} = r. The set H is locally homeomorphic to an open set in Rl-r: r elements of 0, collected in the vector 0:, say, are locally unique functions of the remaining elements collected in 0n: 0j = g(0n), say. Without loss of generality we assume that 0: constitutes the first r elements of 0. Now it does make sense to assume that 0° is a regular point of /(g(0n); 0n) since almost all points in Rl-r are regular points and we do not have any further relevant prior information with respect to 0 n. This assumption is less restrictive than the assumption in Theorem 5 since the constancy of rank{/(0)} in an open neighborhood of 00 implies the con­stancy of this rank for points close to 00 that satisfy 0: = g(0n). Moreover, the rank of /(00) should be computed by rank{/(00)} = max0eH{ rank{/(0)}}, which may be less than the maximum over 0 Є Rl. To formalize this, we generalize Definition 5 and generalize Theorem 5.

Definition 6. Let M(0) be a continuous matrix function and let 00 Є H C Rl. Then 00 is a regular point of M(0)| Hif the rank of M(0) is constant for points in H in an open neighborhood of 00.

Подпись: /(0) Подпись: 3o(0)/ae' . R(0) Подпись: (7.8)

Theorem 7. Let /(0) be the Jacobian matrix of order (n + r) x l formed by taking the partial derivatives of (o(0); p(0)) with respect to 0 Є S C Rl:

Let rank{R(00)} = r. If 00 is a regular point of /(0)| H, then 00 is locally identified in H if and only if max0eH(rank{ /(0)}} = l.

Proof. Since 00 is a regular point of /(0)| H the rank of /(00) is equal to max0eH(rank{ /(0)}}. So if the latter rank is equal to l, then /(00) has full column rank and 00 is identified. If rank{ /(00)} < l, then there is a partitioning as in (7.7), where the elements of p(0) are elements of f2(0) and /2(00) has full row rank and a deficient column rank. Hence we may apply the implicit function theorem to show that there exist points 0 = (g(0n); 0n) arbitrarily close to 00 so that f2(0) = f2(00). Since the elements of p(0) are also elements of f2(0), these points 0 satisfy p(0) = p(00), so that they are elements of H Since 00 is a regular point of /(0)| Hand the points 0 = (g(0n); 0n) are located in H 0n is a regular point of /(g(0II); 0ц), so that close to 0° the rows of /i(g(0n); 0n) are linearly dependent on the rows of /2(g(0n); 0n) and so, by the same argument as in the proof of Theorem 5, f1(g(0n); 0i) = f1(00). As a result, 00 will not be locally identified in H ■

It should be noted that, if o(0) and p(0) are linear functions, so that /(0) = / does not depend on 0 and f(0) – f(00) = / (0 – 00), then 00 is globally identified if and only if /(00) = / is of full column rank.

3 Partial Identification

Up till now we have been concerned with identification of 00 as a whole. How­ever, if 00 is locally not identified, it may still be the case that some separate elements of 00 are identified. If, for example, the ith element of 00 is identified, this means that for a point 0 observationally equivalent to 00 there holds 0; = 00 if 0 is close enough to 00.

Insight into such partial identification could be relevant for estimation pur­poses, if one is interested in estimating the single parameter 00. It could also be relevant for the purpose of model specification in the sense that it may suggest a further restriction of the parameter space such that 00 is identified as yet.

In order to describe conditions for the local identification of single parameters we denote by

A) – {010 є H 0i = 00}

the restricted parameter space whose elements satisfy the a priori restrictions p(0) = 0 and 0; = 00, cf. (7.5). Furthermore, let /0(0) consist of the columns of /(0)
except the zth one, so that derivatives have been taken with respect to all ele­ments of 0 apart from the zth one. The main results on the identification of single parameters are contained in the following two theorems.

Theorem 8. Let A,} = {0 | 0 Є H, 0; = 00}. If 00 is a regular point of /(0)| H and rank{/(f){00)} = rank{/(00)}, then 00 is locally not identified in H

Proof. Let the rows of /(0) be rearranged and partitioned as in (7.7) so that /2(00) is of full row rank equal to the rank of /(00). Furthermore let the columns be rearranged so that

/2(0) = (/21(0), /22(0)) = (d/2/d0I, Э/2/Э0п),

where /21(00) is nonsingular. Now, if rank{/(00)} = rank{/(2)(00)}, then 0II can be taken such that 0, is an element of 0II. Application of the implicit function theo­rem shows that there is a unique function 0I = g(0n) so that /2( g(0n); 0II) = 0 for all 0II in an open neighborhood of 0д. Since the elements of p(0) are elements of /2(0), the points (g(0n); 0II) are located in H Furthermore, since 00 is a regular point of /(0)| H it follows, by the same argument as in the proof of Theorem 5, that also /1(g(0II), 0II) = 0. Hence all elements of 0II, including 0;, are locally not identified in H П

We assume that rank{3p(0)/30’| 00} = rank{R(00)} = rank{R(f)(00)} = r so that 00 is locally not identified by the a priori restrictions p(00) = 0 alone. In other words, H = {0 | 0 Є S, p(0) = 0} constitutes a manifold of dimension l – r in Rl and A(,) = {0 Є H 0, = 00} constitutes a manifold of dimension l – r – 1 in Rl.

Theorem 9. Let A^ = {0 | 0 Є H, 0, = 00}. If 00 is a regular point of /й(0)| A) and rank{ /(,)(00)} < rank{/(00)}, then 00 is locally identified in H

Proof. Applying the same rearrangement and partitioning of /(0) as in the proof of Theorem 8, we find that 0, is an element of 0:. Again there is a unique function so that /2(g(0n), 0n) = 0 for 0n close to 0ц. Now, if we let 0, = 00 and differentiate /2 with respect to the remaining elements of 0, we get the Jacobian matrix /2Й(0). Since rank{П(00)} < rank{/(00)}, the matrix /2Й(00) is not of full row rank and the rank of {/2(,)(00)} equals the rank of {/2(,)(00)}. So if 00 is a regular point of /й(0)| Ад it is also a regular point of /20(0)| A,). So we may use the same argument as in the proof of Theorem 8, where we assume that rank{R(,-)(00)} = r, to verify that if 0, = 00, then there exist unique functions 0; = fy(0n), j Ф Ї, so that, if we write h;(0n) = 00, /2(h(0n), 0n) = 0 for 0n close to 0 Ц. However, we already verified that g(0n) is a unique function, so g;(0n) = 00. Consequently 00 is locally identified. П

Again the ranks that occur in these theorems can be evaluated by computing the maximum rank over the relevant parameter space (or manifold), which is H in Theorem 8 and A^ in Theorem 9.

The importance of these two theorems in practice is best brought out by the following reformulation.

Corollary 1. Let the regularity assumptions of Theorems 8 and 9 be satisfied (which holds for almost all 0° Є H). Let N(00) be a basis for the null-space of /(0°), i. e. /(0°)N(0°) = 0 and let et be the ith unit vector. Then 0 ° is locally identified if and only if e’N(0 °) = 0.

So a zero-row in a null-space indicates an identified parameter.

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