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THE FORM OF THE OPTIMAL NONLINEAR INSTRUMENT FOR MULTIPERIOD CONDITIONAL MOMENT RESTRICTIONS

Published online by Cambridge University Press:  06 June 2003

Stanislav Anatolyev
Stanislav Anatolyev
Affiliation:
New Economic School
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Abstract

We present the form of the optimal instrument in a system of multiperiod conditional moment restrictions in the presence of conditional heteroskedasticity. Using Hansen's (1985, Journal of Econometrics 30, 203–228) and Hansen, Heaton, and Ogaki's (1988, Journal of the American Statistical Association 83, 863–871) work on efficiency bounds for generalized method of moments estimators, we show that this form is an autoregressive recurrence parametrized by a few auxiliary processes that are defined through a system of nonlinear stochastic restrictions, with a stability condition among them. In general, the system does not allow inversion and obtaining an explicit solution for the auxiliary parameters.This is a part of my Ph.D. dissertation (Anatolyev, 2000). I am grateful to Kenneth West for introducing me to the subject and providing advice and encouragement. I also thank Bruce Hansen, Yuichi Kitamura, seminar participants at Indiana University, Pennsylvania State University, Rutgers University, University of British Columbia, University of Virginia, and University of Wisconsin, the co-editor Richard Smith, and two referees for helpful comments and suggestions. Research for this paper was supported in part by a fellowship from the Economics Education and Research Consortium, with funds provided by the government of Sweden through the Eurasia Foundation. The opinions expressed in this paper are those of the author and do not necessarily reflect the views of the government of Sweden, the Eurasia Foundation, or any other member of the Economics Education and Research Consortium.

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Type
Research Article
Information
Econometric Theory , Volume 19 , Issue 4 , August 2003 , pp. 602 - 609
Copyright
© 2003 Cambridge University Press

1. INTRODUCTION

A variety of intertemporal macroeconomic and financial models give rise to multiperiod conditional moment restrictions, where the moment function is serially correlated. Such restrictions typically result in models with overlapping prediction horizons (e.g., Hansen and Hodrick, 1980), with temporal aggregation (e.g., Hall, 1988), or with complex decision rules (e.g., Eichenbaum, Hansen, and Singleton, 1988). An applied researcher typically estimates such models by converting them into an overidentified system of unconditional restrictions after having chosen a set of instruments, with successive application of the generalized method of moments (GMM) (Hansen, 1982). However, the set of possible instruments is infinite, which raises a question of their optimal choice. When the moment function has a martingale difference property, it is well known that the optimal instrument is an explicit function of certain conditional expectations as in Chamberlain (1987). In the context of multiperiod restrictions the conditions for an instrument to be optimal are significantly more complicated. For the special case of conditional homoskedasticity, Hansen (1985) derives the optimal instrument, which takes the form of a certain recurrence relation. This optimal instrument is put to the test in Hansen and Singleton (1996), West and Wilcox (1996), and a few other papers.

For models where both serial correlation and conditional heteroskedasticity are in effect, Hansen (1985) and Hansen, Heaton, and Ogaki (1988), using Gordin's (1969) martingale difference approximation and Hayashi and Sims's (1983) forward filtering idea, present a characterization of the efficiency bound for GMM estimators that correspond to a given system of conditional moment restrictions. In this paper, we obtain a more algorithmic description of Hansen et al.'s (1988) result and present the form taken by the optimal instrument. The process followed by the optimal instrument turns out to be a recurrence relation that extends Hansen's formula (Hansen, 1985, Lemma 5.7). The recurrence is parameterized by a few auxiliary processes whose law of motion is not explicit, but instead solves a system of nonlinear functional equations. In rare circumstances, it is possible to solve this system analytically, as in Heaton and Ogaki's (1991) example, but this is not typical.

The paper is organized as follows. Section 2 presents a formal setup of the problem and a useful result. Section 3 derives the process for the optimal instrument. Section 4 concludes. The proofs are in the Appendix.

2. FORMULATION OF THE PROBLEM

We consider a multiple equation model

where et is an s × 1 vector of errors, xt is a vector of observable variables, β is a k × 1 vector of parameters to be estimated, and f(β,xt) is a function known up to β that is possibly nonlinear in β. In addition, we are given a vector zt of observable basic instruments (as opposed to just instruments that may be generated from the basic ones). Some of the xt's may be among zt. Let us denote by

the information embedded in zt and all its history, i.e.,

and use notation

. The conditional moment restriction

1

Chances are that (2.2) is an implication of the decision rule rather than the decision rule itself, because the decision maker may use information beyond that in

.

implies that all measurable functions of the basic instrument and their lags are valid instruments. This condition does not preclude correlatedness of the error with the leads of the basic instrument, because the latter may be endogenous (Hayashi and Sims, 1983). As a result, one generally cannot hope that the Wold innovation in et has a martingale difference property. This precludes backward filtering of (2.1) with subsequent taking care of heteroskedasticity. Define the k × s matrix

We use the Euclidean norm

for vectors and the induced spectral norm

, where

, is the spectral radius, for matrices.

2

We use the spectral norm instead of the more usual

because it is desirable to keep |A| smaller.

We consider the class of instrumental variables (IV) estimators, indexed by

-measurable instruments. Let

be the space of all

-measurable k × s matrices Zt with finite E [|Zt|4]. We are interested in the best choice of the instrument from

in the sense of asymptotic theory.

DEFINITION 1. The instrument

is called optimal relative to

if the asymptotic covariance matrix of the corresponding IV estimator

does not exceed that of the IV estimator

corresponding to any other instrument

.

The instrument CZt has the same asymptotic properties as Zt for any nonsingular nonrandom k × k matrix C. Thus, if Ξt is optimal, then CΞt is too for any such C, so the object of investigation represents an equivalence class. We make the following set of assumptions on

.

Assumption 1 (Data-Generating Mechanism and Regularity Conditions).

(1) xt and zt are jointly strictly stationary ergodic processes, and E [|zt|4] is finite;

(2)

is a vector of parameters to be estimated, where

is compact;

(3) f(β,xt) is a Borel measurable function for all

and is continuously differentiable in the first argument for all

for all xt in its support;

(4) E [|Dt|2] is finite, and E [DtDt′] is of rank k;

(5) E [sup|b−β|<δ|∂f(b,xt)/∂b′ − ∂f(β,xt)/∂β′|2] < ∞ for some δ > 0.

Later we will add assumptions about the structure of the error et. Assumptions 1(1)–1(3) are self-explanatory. Assumption 1(4) guarantees local identification of β. From Assumption 1(5) it follows that Ztf(b,xt) is first-moment continuous for all Zt from

at all

, which is needed to apply Hansen's (1982) GMM theory.

We will need the following terminology and a result from the theory of random matrices and a so-called generalized autoregression for a k × m matrix process Ψt:

where At is m × m and Bt − k × m matrix processes. Denote max(a,0) by a+.

DEFINITION 2. The top (dominant) Lyapunov exponent associated with the sequence {At} is defined as

LEMMA 1. Suppose that a k × m matrix process Ψt satisfies the recurrence (2.4) and

(1) At and Bt are stationary and ergodic;

(2) E [(log|At|)+] and E [(log|Bt|)+] are finite;

(3) the top Lyapunov exponent associated with {At} satisfies λ(A) < 0.

Then there exists a unique stationary ergodic solution Ψt of (2.4).

3. MAIN RESULT

Let us define p + 1 fundamental quantities

to be used throughout. These strictly stationary and ergodic

-measurable processes may be viewed as s × s infinite-dimensional parameters that index conditional heteroskedasticity, Ωt being the conditional variance and each Γj,t being the conditional jth-order autocovariance of the errors. We assume that et satisfies the following conditions.

Assumption 2 (Error Term). The error et satisfies (2.2), has finite E [|et|4], and

(1) et is conditionally pth-order serially correlated: Et [et−jet′] = Os×s for j > p;

(2) essinf λmin(Γp,tΓp,t′) > 0, where λmin(A) denotes the smallest eigenvalue of A.

Assumption 2(1) indicates that et is a (p + 1)-period ahead forecast error. Assumption 2(2) means that the pth order of conditional serial correlation does not come too close to order p − 1 for any realizations of

. The following theorem obtains aparametrization of the process followed by one of the optimal instruments, given that there are such in

.

THEOREM 1. Let the model satisfy Assumptions 1 and 2. Consider the following recurrence relation:

where the stationary ergodic

-measurable processes, s × s matrix Φ1,t, Φ2,t, …, Φp,t, k × s matrix Δt, and symmetric almost surely positive definite s × s matrix Pt satisfy the following system:

where

, and for integers m and n, Im is an m × m identity matrix, and Om×n is an m × n zero matrix. If a solution to (3.3)–(3.5) satisfies λ(Φ) < 0, E [(log|Δt|)+] < ∞, then there exists a unique stationary solution Ξt of (3.2). If in addition E [|Ξt|4] < ∞, then Ξt is optimal relative to

.

Theorem 1 does not address the issue of existence of optimal instruments or solutions to the system and implicitly presumes existence of all needed conditional expectations. Making these conditions primitive is problematic because of a highly nonlinear environment and essentially requires knowledge of the solution to (3.3)–(3.5). The proof of Theorem 1 proceeds in two steps. We obtain equation (3.3) from Hansen et al.'s (1988) conditional forward moving-average representation for the error term. Then we “solve out” Hansen's (1985) optimality condition treating it as a stochastic difference equation of order 2p + 1 and obtain (3.2), (3.4), and (3.5). For details, see the Appendix.

The key relation is (3.2). It is an extension of Hansen's (1985, Lemma 5.7) formula for the process followed by the optimal instrument in a homoskedastic environment. Here, in contrast to Hansen (1985), Φt is time varying, and the product ΔtPt is a generalization of the projection of the discounted sum of future Dt-variables onto the space of instruments. The condition λ(Φ) < 0 rules out unstable solutions of the nonlinear equation (3.3). For example, if in the one-equation MA(1) case we have (using small plain font for scalar quantities) et = εt+1 − θεt, where θ ∈ (−1,1), {εt} is a martingale difference sequence relative to

with σt2 ≡ Et [εt2], then γ1,t = −θσt2, ωt = Et [σt+12] + θ2σt2, equation (3.3) can be rewritten as

and one notices immediately that φt = θ−1 for all t is an unstable solution. Unfortunately, the stable one is not (except under homoskedasticity or other special circumstances) simply φt = θ. Moreover, in general |φt| may exceed unity for a set of realizations of

of nonzero measure. One case where the condition λ(Φ) < 0 fails is conditional homoskedasticity and unit roots in et.

4. CONCLUSION

We have presented the form of the process followed by the optimal instrument in a system of multiperiod conditional moment restrictions in the presence of conditional heteroskedasticity. Using Hansen's (1985) and Hansen et al.'s (1988) work on efficiency bounds for GMM estimators, we have shown that this form is a recurrence parametrized by a few auxiliary processes that are defined through a system of nonlinear stochastic restrictions, with a stability condition among them. In general, the system does not allow inversion and obtaining an explicit solution for the auxiliary parameters.

Further research will show how to implement this result. The construction of a feasible instrument should take the derived form as a point of departure and attempt either to estimate the auxiliary processes directly from the system by designing a contractive iterative scheme or to approximate the system in such a way that an explicit though approximate solution can be obtained and estimated. Either way requires nonparametric estimation of conditional expectations that may depend on the entire history of the state vector.

APPENDIX

Proof of Lemma 1. See Brandt (1986) for a proof in the unidimensional case k = s = 1 and Bougerol and Picard (1992) for a generalization.

Proof of Theorem 1. Assumptions 1–3 of Hansen et al. (1988) are implied by Assumptions 1(1), 1(3), and 1(4). Let us first assume that Assumption 7 of Hansen et al. (1988) is also satisfied. Then, by Lemma 4.1 of Hansen et al. (1988) there exist

-measurable stationary ergodic s × s matrix processes Wt, V1,t,…,Vp,t such that E [|Wt|2] < ∞, E [|V1,t|2] < ∞,…,E [|Vp,t|2] < ∞, P{|Wt| = 0} = 0, and

where ηt is the normalized forward innovation with Et [ηt] = 0 and Et [ηtηt′] = Is. If Assumption 7 of Hansen et al. (1988) is not satisfied, from their proof of Lemma 4.1 it still follows that

, where ηt* is an unnormalized forward innovation such that Et [ηt*] = 0, but now P{|ηt*| = 0} > 0. However, this contradicts our Assumption 2(2) and hence can be ruled out.

Matching the 1st through pth conditional moments of et gives the following system:

Let us define Φj,t ≡ −Vj,tWt−1 for j = 1,…,p (Wt is nonsingular almost surely). Then each Φj,t is a stationary ergodic

-measurable stochastic process. By postmultiplying Et+p [(Γp,t+p … Γ1,t+1)] by Φt+p′,…,Φt+1′ and Φt′, and taking Et [·] of what emerges, it is straightforward to see that it boils down to −(Γ1,t′ … Γp,t′) − ΩtJΦt′; hence (3.3) follows.

Define Pt = −Γp,t−1Φp,t. It follows that Pt = (WtWt′)−1, an almost surely symmetric positive definite matrix. Note also that E [log|Pt|] ≤ 2E [log|Wt−1|] ≤ 2E [log|Γp,t−1|] + 2E [log|Vp,t|], and 2E [log|Γp,t−1|] ≤ 2 log esssup|Γp,t−1| = log esssup|λmax((Γp,tΓp,t′)−1)| = −log essinf|λmin(Γp,tΓp,t′)| < ∞, E [log|Vp,t|] ≤ E [|Vp,t|2] < ∞. Consider Δt defined by (3.5). Because E [(log|ΔtPt|)+] ≤ E [(log|Δt|)+] + E [(log|Pt|)+] < ∞, by Lemma 1 applied to the companion form of (3.2), the matrix (Ξt Ξt−1 … Ξt−p+1)′ is a uniquely defined stationary and ergodic process, and so is the matrix Ξt.

Consider the first-order condition (Hansen, 1985; Hansen et al., 1988) for optimality of Ξt:

Because this equality must hold for all

, it is equivalent to

We now show that (A.1) is satisfied for the constructed Ξt, Φ1,t, …, Φp,t, Pt, and Δt. Note that (3.2) implies (Ξt+p … Ξt+1) = (Ξt+p−1 … Ξt)Φt+p + Δt+pPt+pJ. Make repeated substitutions of this equation and that for Ξt into (A.1) until there are no Ξ's dated later than t − 1 on the right-hand side. It will turn out that the coefficient near (Ξt−1 … Ξt−p) is zero matrix as a result of (3.3), and everything that is left is exactly (3.5). █

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