2. A MATRIX INVERSION RESULT

Consider the matrix

where A is a q × q positive (or negative) definite symmetric matrix of finite, real constants, B is a q × k matrix of finite real constants of rank k, and c_T is a scalar sequence of constants that satisfies c_T → ∞ as T → ∞.

LEMMA 1. M_T has the following properties.

For brevity, the proof is omitted but is available from the authors upon request. This result generalizes Theorem 1 of Hall (2000), which covers the case in which k = 1. Lemma 1 reveals an interesting relationship between the limiting behavior of M_T and that of its inverse. The term M_T is the sum of two matrices: A, which is nonsingular and O(1), and c_T BB′, which is singular with rank k and O(c_T). For finite T, M_T is nonsingular, but it is clear that as T → ∞ then c_T BB′ dominates in some sense. In the scalar case, the dominant role of the second term is readily apparent in the inverse because lim_T→∞(a + c_T b²)⁻¹ = 0. However, Lemma 1 indicates that the matrix case is far more subtle. As the rank of c_T BB′ increases then the rank of lim_T→∞(M_T)⁻¹ = N decreases. Notice that it is only in the case where k = q, and so c_T BB′ is nonsingular, that N equals the null matrix.

3. THE LIMITING BEHAVIOR OF THE OVERIDENTIFYING RESTRICTIONS TEST IN THE PRESENCE OF NEGLECTED STRUCTURAL INSTABILITY

In this section, we consider the limiting behavior of the overidentifying restrictions test in the presence of neglected structural instability when the long-run variance is estimated using a HAC matrix. To define this test statistic, we must briefly provide an outline of the GMM estimation framework.

Let

where

, and p ≤ q and let {v_t} be a sequence of d-dimensional random vectors in

. We consider the case in which the researcher (mistakenly) assumes that there exists an unknown, but unique, element θ₀ ∈ Θ such that

and proceeds to estimate θ₀ using GMM based on (2) using the sample {v_t; t = 1,2…,T}. Define the first step GMM estimator

by

where

and W_T⁽¹⁾ is a q × q positive semidefinite matrix that converges in probability to a positive definite matrix W⁽¹⁾. Because the first step estimator is typically asymptotically inefficient, it is assumed that the researcher undertakes a second step estimation. This second step GMM estimator is defined to be

where the minimand is

and W_T⁽²⁾ is an estimator that is believed to be consistent for the inverse of the long-run variance of the sample moment. In this paper, we consider the case in which W_T⁽²⁾ is either

where

for

and

for

. Subsequently we impose specific assumptions on the scalar function ω(.), known as the kernel, and the bandwidth, b_T. Both

fall into the generic class of HAC estimators. The only difference between them is in whether or not the sample autocovariance matrices are centered. For this reason, we refer to

as the uncentered HAC estimator and to

as the centered HAC estimator.

The overidentifying restrictions test statistic associated with second step estimation is given by

This statistic is the focus of our paper. However, we discuss the extension of our results to the overidentifying restrictions test associated with the iterated GMM estimator at the end of the section.

In view of the structure of (3) and (7), the analysis breaks down naturally into three parts. First we derive the limiting behavior of the inverse of a HAC matrix in the presence of neglected structural instability using Lemma 1. Second, we derive the probability limit of the second step GMM estimator. Third, we use aforementioned results to deduce the limiting behavior of the overidentifying restrictions test associated with the second step GMM estimation.

3.1. The Limiting Behavior of the Inverse of the Uncentered and Centered Full Sample HAC Estimators

Given the structure of

, such an analysis requires assumptions about the first step GMM estimator, the kernel and bandwidth used in the HACs, and certain aspects of the data generation process. We now present these assumptions in turn.

The first step estimator is assumed to satisfy the following condition.

Assumption 1. The first step GMM estimator

converges in probability to a well-defined limit denoted θ_*(W⁽¹⁾) where

W⁽¹⁾.

The kernel and bandwidth are restricted to satisfy the following conditions.

Assumption 2. (i) For all

, |ω(x)| ≤ 1, ω(−x) = ω(x), ω(0) = 1, ω(x) is continuous at zero, and for almost all

, ω(x) = 0 for |x| ≥ 1,

for all

where 0 < β < ∞; (ii) b_T is a sequence of nonnegative integers such that b_T = o(T^1/2) and b_T → ∞.

It can be verified that Assumption 2(i) restricts attention to truncated kernels such as the Bartlett and Parzen, which were proposed in this context, respectively, by Newey and West (1987) and Gallant (1987). Assumption 2(ii) specifies the rate of increase for b_T used by Andrews (1991) in his proof of the consistency of

in correctly specified models.

We now specify certain key aspects of the data generation process. We focus exclusively on the case in which there is instability at a single point in the sample, denoted [πT] , where π ∈ (0,1) and [.] denotes the integer part.

²

This scenario is also the primary focus of all the papers on structural stability based on GMM cited in note 1.

For the analysis, we need to impose certain restrictions on f (v_t,.) evaluated at both the first step GMM estimator and at the probability limit of the first step estimator. In the statement of these conditions, it is notationally more convenient to suppress the dependence of the estimator and its limit on the first step weighting matrix. Therefore, we set

and θ_* = θ_*(W⁽¹⁾).

Assumption 3. The sample {v_t} is a realization from a stochastic process that satisfies the following conditions.

(i) The first moment of f (v_t,θ_*) satisfies

where μ₁ ≠ μ₂.

(ii) The autocovariances of f (v_t,θ_*) satisfy

(iii) The long-run variances for each subsample are given by

, where V₁ and V₂ are positive definite matrices of constants.

(iv) The long-run variance of the full sample is given by

.

Notice that Assumption 4 is weaker than the requirement that the partial sums satisfy a central limit theorem.

The following theorem characterizes the large sample behavior of the two HAC estimators described previously.

THEOREM 1. Suppose that Assumptions 1–4 hold. Then the HAC estimators in (5) and (6) take the generic form

where A = V, {c_T} satisfies lim_T→∞ b_T /c_T = β, and (a) for

; (b) for

.

Theorem 1 indicates that neither estimator is consistent in the presence of neglected structural stability.³

However, the centered full sample estimator is consistent if μ₁ = μ₂. The latter condition implies that the misspecification is not characterized by structural instability but is constant over time, and so under these circumstances this result replicates Theorem 3(i) of Hall (2000).

Because both matrices are asymptotically equivalent to a matrix with the structure in (1), we obtain the following useful corollary to Lemma 1 and Theorem 1 via similar arguments to the proof of Theorem 1 of Hall (2000).

COROLLARY 1. Suppose that Assumptions 1–4 hold. Then (a)

where N_U has rank q − r_U where

and has a null space spanned by

; (b)

where N_C has rank q − 1 and a null space spanned by (μ₁ − μ₂).

In most cases r_U = 2; however, it is easily verified that r_U = 1 under the following circumstances.

Assumption 5. One of the following conditions holds: (i) μ₁ = 0, μ₂ ≠ 0; (ii) μ₁ ≠ 0, μ₂ = 0; (iii) μ₁ ∝ μ₂.

These three cases have an obvious interpretation. Parts (i) and (ii) imply that the population moment condition evaluated at θ_* is satisfied in one subsample but not in the other. Part (iii) implies that the instability is purely due to a scale effect, that is, μ_i = k_i μ for some nonzero vector μ and scalars k_i for which k₁ ≠ k₂. Notice that if Assumption 5 holds then the null spaces of both N_U and N_C are spanned by μ₁ − μ₂.

3.2. The Limiting Behavior of the Second Step GMM Estimator

Standard proofs of convergence in probability can be divided into two parts.⁴

For example, see Newey and McFadden (1994).

First, it is shown that the estimator minimizes the population analog to the minimand with probability one in the limit. Second, it is shown that the first property plus an identification condition implies that the estimator converges to some unique value in the parameter space. We follow this approach subsequently. So to begin, we must characterize the population analog to the two-step GMM minimand.

To characterize the population analog to Q_T(θ;W_T⁽²⁾), we introduce the following assumption.

Assumption 6. μ₁(θ) = E [f (v_t,θ)] for t = 1,2,…[πT] and μ₂(θ) = E [f (v_t,θ)] for t = [πT] + 1,…T for all θ ∈ Θ. Moreover, sup_θ∈Θ∥g_1,T(θ) − μ₁(θ)∥ = o_p(1) and sup_θ∈Θ∥g_2,T(θ) − μ₂(θ)∥ = o_p(1) where

.

Assumption 6 and Corollary 1 imply that⁵

This follows by the same argument as the proof of Theorem 2.6 in Newey and McFadden (1994).

Q_T(θ;W_T⁽²⁾) converges uniformly in probability to

where

denotes whichever HAC estimator is chosen to estimate the long-run variance. Therefore, we refer to Q₀(θ;W⁽²⁾) as the population analog to Q_T(θ;W_T⁽²⁾).

To derive the probability limit of

, it is necessary to impose an identification condition. It is most convenient to express this condition in the following generic form.

Assumption 7. Q₀(θ;W⁽²⁾) is minimized at θ = θ_**(W⁽²⁾), and Q₀(θ_**(W⁽²⁾);W⁽²⁾) < Q₀(θ;W⁽²⁾) for all θ ∈ Θ[setmn ]{θ_**(W⁽²⁾)}.

Subsequently we specialize this assumption to the choice of W⁽²⁾ implied by the particular choice of HAC estimator.⁶

Assumption 7 is similar to the “identifiable uniqueness” condition that underpins Gallant and White's analysis of misspecified models (see Gallant and White, 1988, p. 19).

The following theorem gives the probability limit of

.

THEOREM 2. Let Assumptions 1–4 and 6 hold.

(a) Let

and Assumption 7 hold for W⁽²⁾ = N_U; then

.

(b) Let

and Assumption 7 hold for W⁽²⁾ = N_C; then

. If, in addition, Assumption 5 holds then θ_**(W⁽²⁾) = θ_*(W¹).

Notice that if the uncentered full sample HAC estimator is used then the second step GMM estimator converges to the same probability limit as the first step estimator even though the model is misspecified.

3.3. The Limiting Behavior of the Overidentifying Restrictions Test

To facilitate the analysis, certain additional conditions are required. The first assumption imposes uniform convergence of derivative matrices in each subsample.

The second assumption imposes the uniform boundedness of the partial sums of the sample moment.

Assumption 9 is not commonly used but holds if

where the latter holds in a neighborhood of θ_**. For example, (9) and (10) are satisfied if the moment function and its gradient are bounded (in probability) uniformly in t.

The behavior of the overidentifying restrictions test depends crucially on

Note that it is possible for E [f (v_t,θ)] ≠ 0 for all t but μ(θ) = 0. In other words, there may be no single value of θ that satisfies the population moment condition for every observation, but there is a value that sets the average of these population moment conditions to zero.

The limiting behavior of the overidentifying restrictions test in the presence of neglected structural instability is given by the following theorem. To facilitate the presentation, we define μ_* = μ(θ_*), μ_** = μ(θ_**), θ_* = θ_*(W⁽¹⁾), and θ_** = θ_**(W⁽²⁾).

THEOREM 3. If Assumptions 1–4 and 6–9 hold then the large sample properties of the overidentifying restrictions test are as follows.

(a) Put

. If μ_* = 0 then J_T(W_T⁽²⁾) = O_p(1); otherwise J_T(W_T⁽²⁾) diverges at rate T/b_T.

(b) Put

. If μ_** = 0 then J_T(W_T⁽²⁾) = O_p(1). If μ_** ≠ 0 and μ_** is in the null space of N_C then J_T(W_T⁽²⁾) diverges at rate T/b_T. Otherwise J_T(W_T⁽²⁾) diverges at rate T.

Theorem 3 establishes certain important features about the limiting behavior of the overidentifying restrictions in the presence of neglected structural instability. First, the test need not be consistent. Notice that this failure is not specific to the case in which a HAC estimator is used but applies more generally. Second, even if it is consistent then the rate of divergence depends on the choice of covariance matrix. If the uncentered HAC is used and the test is consistent then the overidentifying restrictions test only diverges at rate T/b_T. Notice that this behavior conforms with the results presented in Hall (2000) for the case in which the model is misspecified but structurally constant. However, if a centered version of the HAC is used and the test is consistent then the situation is more complicated. If μ_** is in the null space of N_C then the statistic diverges at rate T/b_T and in others at rate T. The vector μ_** can lie in the null space either because Assumption 5 holds, and so μ_** = μ(θ_*), or because it so happens that μ_** = b(μ₁ − μ₂) ≠ 0 for some b.

Throughout the analysis, we have concentrated on the two-step estimator. It is straightforward to show that Theorem 2(a) extends to the iterated estimator and as a consequence that Theorem 3(a) holds for the overidentifying restrictions test based on the iterated estimator. The results in Theorems 2(b) and 3(b) hold if the second step estimator is replaced by the ith step estimator, the first step estimator is replaced by the i − 1th step estimator, and all other definitions are changed appropriately.

4. IMPLICATIONS FOR MOMENT SELECTION

Andrews (1999) proposes methods for moment selection that use either an information criterion or a sequential testing strategy based on the overidentifying restrictions test. He also establishes conditions for their consistency. One of these conditions states that the overidentifying restrictions test diverges at rate T if the moment condition is invalid. It follows from Theorem 3 that the overidentifying restrictions test does not behave in this way under certain conditions when the long-run variance is estimated using a HAC estimator and the moment conditions are invalid due to neglected structural instability. Therefore, in this section, we present an alternative set of conditions for the consistency of the methods of moment selection based on the information criterion and the downward testing strategy.⁷

To consider the problem of moment selection, it is necessary to introduce some additional notation. It is assumed that the candidate set of scalar functions that can form the basis for the population moment condition is finite. It is convenient to stack these scalar functions into a single vector f_max(.) whose dimension is denoted by q_max. Following Andrews (1999), we use a (q_max × 1) selection vector c to denote which elements of the candidate set are included in a particular moment condition. We therefore now index f (.) by c. If c_j = 1 this implies that the jth element of f_max(.) is included in f (.;c), and c_j = 0 implies that this element is excluded. Note that |c| = c′c equals the number of elements in f (.;c). The set of all possible selection vectors is denoted by C, that is,

Following Andrews (1999) we also define the sets

as follows:

These sets are used in the statement of an identification condition later.

To present the generic structure of the moment selection methods, we use J_T(c) to denote the overidentifying restrictions test associated with the selection vector c. The two moment selection procedures are as follows.

Information criterion.

The selected moment is given by

where

and the information criterion is given by

where −h(|c|)κ_T is a deterministic bonus term on whose components certain restrictions are placed subsequently.

Downward testing.

The selected moment condition is given by

, any choice of selection vector that satisfies the following conditions.

where γ_T,|c|−p is the 1 − α_T quantile of the χ_|c|−p² distribution. Conditions are placed on α_T later.

Given the results in Theorem 3, we consider the case in which the overidentifying restrictions test exhibits the following generic behavior.

Assumption 10. (i) If E [f (v_t,θ_*;c)] = 0 for some θ_* ∈ Θ then

. (ii) If E [f (v_t,θ_*;c)] ≠ 0 for all θ ∈ Θ then

where a(c) is a finite positive constant that depends on c and τ_T → ∞ as T → ∞.

The following lemma gives conditions for the consistency of the two methods of moment selection.⁸

Andrews (1999) also considers the case in which

is not a singleton, but for brevity we do not consider this situation here.

LEMMA 2. Let Assumption 10 hold and assume

. (i) If h(.) is strictly increasing and κ_T → ∞ as T → ∞ with κ_T = o(τ_T) then

. (ii) If α_T → 0 as T → ∞ with ln(α_T) = o(τ_T) then

.

This lemma is established using identical arguments to the proofs of Theorems 1 and 2 of Andrews (1999) and so is omitted for brevity. However, there are two important differences between Lemma 2 and this earlier work. Andrews (1999) only considers the case in which a centered covariance matrix estimator is used and τ_T = T. In contrast, Lemma 2 is valid for cases in which the overidentifying restrictions test diverges at some unspecified rate τ_T when the moment condition is misspecified. Therefore, it allows for cases in which a centered covariance matrix estimator is used, but the test diverges more slowly than T, and it also covers cases in which an uncentered HAC covariance matrix estimator is used.

We now consider the implications of this result for the case in which a HAC is used to estimate the long-run variance and moments are invalid because of neglected structural instability of the type considered in the previous section. Because Andrews (1999) recommends using the centered covariance matrix estimator for the purposes of model selection, we focus purely on that case for brevity. We split our discussion into two parts depending on whether or not the test is consistent. If the test is not consistent then it does not diverge in cases when E [f (v_t,θ_**;c)] ≠ 0 for all t, and this inevitably means that neither moment selection method is consistent. Now consider the case in which the test is consistent. Theorem 3 indicates that the rate of divergence can be either T/b_T or T. Because in practice it is impossible to know which rate is relevant, we consider the worst case scenario, that is, τ = T/b_T. For the information criterion approach, the standard choices for the bonus term are GMM-BIC, for which κ_T = ln(T), and GMM-HQIC, for which κ_T = ln[ln(T)] .⁹

The appropriate choices of h(.) are |c| − p for GMM-BIC and k(|c| − p) where k > 2 for GMM-HQIC. See Andrews (1999) for further discussion.

Both these choices satisfy κ_T = o(τ_T) and so satisfy Lemma 2(i). For the downward testing procedure, the researcher must pick α_T. If this value is chosen under the mistaken assumption that τ_T = T, then this may undermine the consistency of the method, as the following lemma shows.

LEMMA 3. Let Assumption 10 hold. If α_T → 0 as T → ∞ with ln(α_T) = o(T) but −ln(α_T)/τ_T → ∞ then

where c_max is a q_max × 1 vector of ones.

Therefore, even if the candidate set contains invalid moment conditions—for example,

and |c_o| < q_max—the downward testing sequence picks all the moment conditions with probability one. The lemma is deduced using Theorem 5.8 of Pötscher (1983), in which it is shown that γ_|c|−p,T = o(τ_T) if and only if ln(α_T) = o(τ_T). Therefore, it follows that if ln(α_T)/τ_T → ∞ then γ_|c|−p,T diverges faster than the test statistic.

5. CONCLUDING REMARKS

In this paper, we have analyzed the behavior of the overidentifying restrictions test in the presence of neglected structural instability at a single break point. It is shown that the test need not be consistent against this type of misspecification. However, if it is consistent then the rate of divergence of the statistic depends on the covariance matrix estimator employed. If an uncentered HAC estimator is used then the test diverges at rate T/b_T; if a centered HAC estimator is used then the test diverges at either rate T/b_T or T depending on the form of the instability. These results are used to explore the properties of certain moment selection procedures when certain elements of the candidate set are misspecified as a result of neglected structural instability. Specifically, conditions are provided for the consistency of both the moment selection criterion of Andrews (1999) and a downward testing strategy based on the overidentifying restrictions test. For the moment selection criterion of Andrews (1999), these conditions exclude certain forms of instability but show that the method is valid with the bonus terms recommended by Andrews (1999). These results apply regardless of whether a centered HAC estimator, as recommended by Andrews (1999), or an uncentered HAC estimator is used to estimate the long-run variance. For the downward testing strategy, these conditions again exclude certain forms of instability and also require that the size of the test be chosen in a way that reflects the true rate of divergence of the test.

We believe that these results have some important practical implications for the use of moment selection procedures. If it is suspected that the candidate set contains moments that are misspecified because of neglected structural instability, then our results suggest that both of the aforementioned moment selection procedures are likely to provide effective methods for excluding a broad class of moments that are invalid in this way, at least asymptotically. However, because the overidentifying restrictions test is not consistent against all possible misspecifications of this type, it is advisable to submit the chosen moment condition to conventional tests for structural instability.