The Population Moment Condition and Identification

In this section, we consider the conditions under which the population moment provides sufficient information to determine uniquely 0 0 from all other elements in the parameter space 0 C W. If this is the case then 0O is said to be identified. To begin, it is necessary to define formally the information contained in the popula­tion moment condition. From the heuristic discussion above, it is clear that the population moment condition is a statement about a q x 1 vector of functions, f (•,•), of the observable vector of random variables vt and the unknown (p x 1) parameter vector, 0O. Certain restrictions must be placed on these constituents as we proceed with the analysis, and we shall impose the most important as they become necessary. However, space constraints forbid a complete accounting of all the required conditions; these can be found in Hansen (1982) or Wooldridge (1994). Throughout the chapter we follow Hansen’s (1982) original framework and consider the following case.

Assumption 1. Strict Stationarity. The (r x 1) random vectors {vt; -<*> < t < ro} form a strictly stationary process with sample space V C ^r.

Recall that this assumption implies all expectations of functions of vt are inde­pendent of time.4

Assumption 2. Regularity Conditions for f (•,•). The functionf : V x 0 ^ satisfies: (i) it is continuous on 0 for each v Є V; (ii) E[ f(vt, 0)] exists and is finite for every 0 Є 0; (iii) E[ f (vt, 0)] is continuous on 0.

With these two assumptions in place, we can can now formally restate the popu­lation moment condition.

Assumption 3. Population Moment Condition. There exists 00 є 0 such that the following (q x 1) population moment condition holds: E[ f (vt, 00)] = 0.

Throughout this chapter, we focus on the properties of the GMM estimator of 00 based on this population moment condition. However, by itself, Assumption 3 does not provide sufficient information to identify 0 0. This is only the case if there are no other values of 0 at which the population moment condition is satisfied. This condition for parameter identification can be stated as follows.

Assumption 4. Global Identification. The parameter vector 00 is glo­bally identified by the population moment condition in Assumption 3 if and only if E[f(vt, 0)] Ф 0 for all 0 Є 0 such that 0 Ф 00.

The adjective "global" emphasizes that the population moment condition only holds at one value in the entire parameter space. While this condition is easily stated, it is often hard to verify a priori in nonlinear models. Identification can fail due to the properties of the data, vt, or due to the properties of f (•) as a func­tion of 0 or due to an interaction of the two. Fortunately, a more useful condition can be found if attention is limited to some suitably defined neighborhood of 00. The price of this approach is that we are now deriving conditions for identifica­tion only within this neighborhood and so these are referred to as conditions for local identification. As the names suggest, local identification is necessary but not sufficient for global identification. Therefore, a more transparent condition for local identification is useful because it provides insights into the circumstances in which identification can fail.

The condition for local identification is based on a first order Taylor series expansion, and so it is necessary to introduce the following definition and as­sumption. An г-neighborhood of 00 is defined to be the set Ne which satisfies Ne = {0; || 0 – 001| < г} where || a || = (a’a)1/2.

Assumption 5. Regularity Conditions on df (vt, 0)/Э0′. (i) The deriva­tive matrix df(v, 0)/Э0′ exists and is continuous on 0 for each v Є V; (ii) 0 0 is an interior point of 0; (iii) E [df(vt, 0 0)/Э0′] exists and is finite.

To derive the condition for local identification, we restrict attention to a suffi­ciently small г such that f() can be approximated by the following first order Taylor series expansion5 in Ne

where df (vt, 0О)/Э0′ is the q x p matrix with i – jth element 0o)/30;’. Taking

expectations on both sides of (11.6) and using Assumptions 3 and 5 yields

E[f(v, 0)] = {E[df(vt, 0o)/30′]} (0 – 0o). (11.7)

Equation (11.7) states that in the neighborhood of 0o, E[f(vt, 0)] is essentially a linear function of 0 – 0 0. This leads to the following condition for local identification.

Lemma 1. Local Identification. The parameter vector 0o is locally identified by the population moment condition in Assumption 3 if and only if E[df(vt, 0o)/30′] is of rank p.

While this condition needs to be verified on a case by case basis, it does provide some general insights into identification in nonlinear models. First, the rank con­dition immediately implies identification fails if there are fewer moment condi­tions than parameters, i. e. q < p. Second, notice that the dependence of the partial derivative matrix on 0 implies the population moment condition may provide enough information to identify the parameters at some values of 0o but not at others.

From this discussion, it is clear that the relationship between q and p is import­ant. This has led to the introduction of the following terminology. If p > q and hence the local identification condition is not satisfied then 0o is said to be ми – identified. If p = q and Assumption 4 is satisfied then 0o is said to just-identified. Finally, if q > p and Assumption 4 is satisfied then 0o is said to be over-identified.

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