Generalized Instrumental Variables Estimation of Nonlinear Rational Expectations Models Author(s): Hansen and . : Econometrica, Vol. 50, No. 5 (Sep., 1982), pp. 1269-1286
Published by: The Econometric Society
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Econometrica, Vol. 50, No. 5 (September, 1982)
GENERALIZED INSTRUMENTAL VARIABLES ESTIMATION OF NONLINEAR RATIONAL EXPECTATIONS MODELS’
BY LARS PETER HANSEN AND KENNETH J. SINGLETON
This paper describes a method for estimating and testing nonlinear rational expectations models directly from stochastic Euler equations. The estimation procedure makes sample counterparts to the population orthogonality conditions implied by the economic model close to zero. An attractive feature of this method is that the parameters of the dynamic objective functions of economic agents can be estimated without explicitly solving for the stochastic equilibrium.
1. INTRODUCTION
THE ECONOMETRIC IMPLICATIONS of dynamic rational expectations models in which economic agents are assumed to solve quadratic optimization problems, subject to linear constraints, have been analyzed extensively in [11, 12, 25, 29, and 30]. Linear-quadratic models lead to restrictions on systems of constant coefficient linear difference equations, which provide complete characterizations of the equilibrium time paths of the variables being studied. Hence, the parame- ters of these models can be estimated using the rich body of time series econometric tools developed for the estimation of restricted vector difference equations [e.g., 11, 20, and 32]. Once the linear-quadratic framework is aban- doned in favor of alternative nonquadratic objective functions, dynamic rational expectations models typically do not yield representations for the variables that are as convenient from the standpoint of econometric analysis. Indeed, in many models, closed-form solutions for the equilibrium time paths of the variables of interest have been obtained only after imposing strong assumptions on the stochastic properties of the “forcing variables,” the nature of preferences, or the production technology. See, for example, the models in Merton [24], Brock [4], and Cox, Ingersoll, and Ross [5].
The purpose of this paper is to propose and implement an econometric estimation strategy that circumvents the theoretical requirement of an explicit representation of the stochastic equilibrium, yet permits identification and esti- mation of parameters of economic agents’ dynamic (nonquadratic) objective functions, as well as tests of the over-identifying restrictions implied by the theoretical model. The procedures we propose do not require a complete, explicit representation of the economic environment and, in particular, do not require strong a priori assumptions about the nature of the forcing variables. Conse- quently, estimation and inference can be conducted when only a subset of the economic environment is specified a priori. While our strategy involves specifying the objective functions of a subset of agents, it is distinct from specifying the
‘This research was supported in part by NSF Grant SES-8007016. We wish to thank , , , and two anonymous referees for helpful comments on earlier drafts of this paper, and for research assistance.
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1270 L. P. HANSEN AND K. J. SINGLETON
decision rules (e.g., dynamic demand or supply schedules) of a subset of agents without specifying the entire economic environment.2 The latter approach is characteristic of many applications of limited information methods to conven- tional simultaneous equations models. Lucas [21] has criticized this approach by noting that, under the assumption of rational expectations, the dynamic decision rules of economic agents depend explicitly on the stochastic specification of the forcing variables and possibly the structural specification of the entire economic environment.
The basic idea underlying our estimation strategy is as follows. The dynamic optimization problems of economic agents typically imply a set of stochastic
Euler equations that must be satisfied in equilibrium. These Euler equations in
turn imply a set of population orthogonality conditions that depend in a nonlinear way on variables observed by an econometrician and on unknown parameters characterizing preferences, profit functions, etc. We construct nonlin ear instrumental variables estimators for these parameters in the manner sug-
gested by Amemiya [1, 2], Jorgenson and Laffont [18], and Hansen [10] by making sample versions of the orthogonality conditions close to zero according to a certain metric. An important feature of these estimators is that they are consistent and have a limiting normal distribution under fairly weak assumptions about the stochastic processes generating the observable time series. Also, more orthogonality conditions are typically available for use in estimation than there are parameters to be estimated and, in this sense, the models are “over- identified.” The overidentifying restrictions can be tested using a procedure, justified in Hansen [10], that examines how close sample versions of population orthogonality conditions are to zero.
Other authors have proposed using stochastic Euler equations to estimate parameters (but1not test restrictions) in the context of models containing linear quadratic optimization problems (e.g., Hayashi [16] and Kennan [19]). By focus-
ing on the Euler equations in these linear models, some of the restrictions implied
by the model are ignored at the gain of computational simplicity (see Hansen
and Sargent [13]). In addition to computational simplicity, there is perhaps a
more compelling reason for using instrumental variables procedures in the nonlinear environments considered here. Namely, there is the added difficulty of obtaining a complete characterization of the stochastic equilibrium under weak assumptions about the forcing variables.3 Fair and Taylor [7] proposed an alternative, approximate maximum likelihood procedure that can be applied to
2By the economic environment we mean a specification of preferences, technology, and the stochastic process underlying the forcing variables. By a decision rule we mean a rule used by economic agents to determine the current period “decision” as a function of the current “state” of the economy.
3The procedures discussed in this paper are also of use in quadratic optimization environments
when it is important to allow conditional variances to be dependent on variables in the information
set. Allowing for these dependencies complicates decision rule derivation and does not permit use of conventional asymptotic distribution theory results for instrumental variables estimators. See Section 3 for further discussion of these issues.
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INSTRUMENTAL VARIABLES ESTIMATION 1271
an equation system that includes a set of stochastic Euler
certainty equivalence on the nonlinear rational expectati vent some of the difficulties in obtaining a complete characterization of the stochastic equilibrium. The instrumental variables procedure proposed here
avoids their approximations for an important class of models.
The remainder of this paper is organized as follows. In Section 2 the class of stochastic Euler equations that can be used in estimation is discussed, and then
an example from the literature on multi-period asset pricing is presented. In
Section 3 the generalized instrumental variables estimator is formulated, and its large sample properties are discussed. In Section 4 this estimator is compared to
the maximum likelihood estimator in the context of a nonlinear model of stock
market returns. This discussion contrasts the orthogonality conditions exploited
by the instrumental variables estimator with those exploited by the maximum likelihood estimator when a specific distributional assumption is made. In Section 5 results from applying the generalized instrumental variables estimator to this stock return model are presented. Finally, some concluding remarks are
made in Section 6.
2. THE IMPLICATIONS OF RATIONAL EXPECTATIONS MODELS USED IN CONSTRUCTING ESTIMATORS
Discrete-time models of the optimizing behavior of economic agents o to first-order conditions of the form:
(2.1) Eth(xt+? bo) = 0,
where xt+n is a k dimensional vector of variables observed by agents an econometrician as of date t + n, bo is an 1 dimensional parameter vector unknown to the econometrician, h is a function mapping Rk X R’ into Rm, and
Et is the expectations operator conditioned on agents’ period t information set, It. Expectations are assumed to be formed rationally and, hence, Et denotes both
the mathematical conditional expectation and agents’ subjective expectations as
of date t. For the purposes of this paper, we shall think of equation (2.1) as emerging from the first-order conditions of a representative agent’s utility maxi- mization problem in an uncertain environment. Our procedures can also be
applied with some modification to models of panel data in which (2.1) represents the first-order conditions associated with the optimum problems of heteroge-
neous agents, so long as the heterogeneity is indexed by individual characteristics observed by the econometrician. More generally, our approach to estimation is appropriate for any model that yields implications of the form (2.1) with x observed. This latter qualification does rule out some models in which the
implied Euler equations involve unobservable forcing variables.
An example will be useful both for interpreting (2.1) and understanding the estimation procedure discussed in Section 3. Following Lucas [22], Brock [4], Breeden [3], and Prescott and Mehra [26], suppose that a representative con-
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1272 L. P. HANSEN AND K. J. SINGLETON
sumer chooses stochastic consumption and investment plans so as to maximize – 00
(2.2) Eo E i8tU(Ct) X t=O
where Ct is consumption in time period t, fi Ee (0, 1) is a di U(.) is a strictly concave function. Further, suppose that the consumer has the
choice of investing in a collection of N assets with maturities Mj, j= 1, . .. , N. Let Qjt denote the quantity of asset j held at the end of date t, Pj, the price of asset j at date t, Rjt the date t payoff from holding a unit of an M1-period asset purchased at date t -MA1j, and W, (real) labor income at date t.4 All prices are denominated in terms of the consumption good. The feasible consumption and investment plans must satisfy the sequence of budget constraints
(2.3) Ct + 2 < - M + Wt. j23 j=1
The maximization of (2.2) subject to (2.3) gives the first-order necessary conditions (Lucas [22], Brock [4], Prescott and Mehra [26]):
(2.4) Pjt U(t) A (j]= 1, . . ., N).
If, for example, the jth asset is a default-free, zero coupon bond with term to maturity Mj, then Rjt+M in (2.4) equals the real par value of the bond at date t + Mj. Alternatively, if ' t denotes the quantity of shares of stock of a firm held
at date t, Djt denotes the dividend per share of stock j at date t, and Mj = 1, then Rjt+I= (Pjt+I + Djt +) and (2.4) becomes
(2.5) Pjt U'(Ct ) = fEt[(Pjt+ I + Djt+ 1) U'(Ct+ 1)
with Pjt interpreted as the exdividend price per share. Note that (2.5) is a generalization of the model studied by Hall [9] in which preferences were
quadratic and real interest rates were assumed to be constant over time.
Estimation and testing using (2.4) or (2.5) requires that the function U be explicitly parameterized. For the moment, we assume only that preferences are described by a vector of parameters -y, U(-, y), in order to emphasize the generality of our estimation strategy. At this level of generality, the representative agent assumption plays a critical role in the derivation of (2.4) and (2.5).
4This income term W, could emerge under the assumption that labor is supplied inelastic this case W, can be thought of as not being controllable by the representative consumer. A tively, we can introduce a period t labor supply variable L4 into the specification of U and let
U(Ct, Lt)-=U, (Ct ) -U2(4t)
where L, is a choice variable of the consumer. For this case, W, = L4wt where wt is the real wa at period t.
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in (2.1) is given by
INSTRUMENTAL VARIABLES ESTIMATION 1273
Rubinstein [28] has shown, however, that (2.4) and (2.5) can implicitly accommo- date certain types of heterogeneity when agents' preference functions are mem- bers of the HARA class. In Sections 4 and 5, we consider in detail the special case of (2.5) with a constant relative risk averse preference function. Using a theorem in Rubinstein [28], this version of (2.5) can be derived from a model in which agents are allowed to have different stochastic endowment streams.
Relation (2.4) can be used to construct the h function specified in (2.1).
Suppose the econometrician has observations on Pi and Rj for a subset the assets (m < N) with maturities n , n2, . .. ., nm, and on consumption C.
C, and Pjt are known to agents at time t, (2.4) implies (2.6) Et[3 An/ J, ) Xjt+n(f j = 0,
where Xjt+n/ = nt + t, for j = 1, .. ., m. Let n = nm and x' = ... Xmt + n,, Ct*') where the n * constituents of Ct* are observable fu
of Ct and the distinct values of Ct + n , i = 1, . . . , n. For example, U'( U'(Ct) in (2.6) may involve the functio'n Ct + n / Ct. Then the function h (xt
U' (CtIn) X11tfl - 1
UA(n I t+n) Xlt+n -1
Notice that x has m + n* coordinates. We can interpret
Ut+n = h(xt+n?bo)
as the disturbance vector in our econometric estimation. The matrix Eutut' is assumed to have full rank. This assumption implicitly imposes some structure
on the link between u and the "forcing variables" not observed by the econometrician that enter, for example, through the production technology. The autocovariance structure of u depends on the nature of the assets being studied.
If the m assets are stocks and n1 through n equal unity, for example, then
h (xt + I, bo) is constructed from the Euler equations by setting Xjt + 1 = (Pjt + 1
Djlt +)/Pt. In this case, u is serially uncorrelated, since observations on xt s ? 0, are contained in It and Et[h(xt+ 1, bo)] = 0. On the other hand, if n > 1
somej, as in the model of the term structure of bond prices implied by (2.4), condition Ej[h(xt+n, bo)] = 0 does not preclude serial correlation in u. This be seen by noting that x + – is not necessarily included in It if nj > 1.
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1274 L. P. HANSEN AND K. J. SINGLETON
presence of serial correlation in u leads to a more complicated asymptotic covariance matrix for our proposed estimator, but it does not affect consistency (see Section 3).
Before describing our estimation strategy in detail, we briefly consider what is perhaps a more “natural” approach to estimation in order to help motivate our approach. A possible way of proceeding with estimation is: (i) to explicitly
specify the rest of the economic environment, including the production technol-
ogy and the stochastic properties of the forcing variables; (ii) to solve for an equilibrium representation for the endogenous variables in terms of past endoge- nous variables and current and past forcing variables; and (iii) to estimate the parameters of tastes, technology, and the stochastic process governing the forcing variables using a full information procedure such as maximum likelihood. This approach will yield an exact relationship among current and past endogenous variables and current and past forcing variables. To avoid an implication of a stochastic singularity among variables observed by the econometrician, it can be assumed that the econometrician does not have observations on some of the forcing variables. This approach is viable if both the form of (2.1) and the stochastic specification of the forcing variables are relatively simple. To allow for
a general representation of the forcing variables, explicitly solve for an equilib-
rium representation of the observables, and proceed to estimate the parameters
of tastes and technology together with the parameters of the forcing processes appears to be an overly ambitious task outside of linear environments.5 For this reason we adopt an alternative estimation strategy that can be viewed as an extension to nonlinear environments of the procedures of McCallum [23] and Cumby, Huizinga, and Obstfeld [6] for estimating linear rational expectations models.
3. ESTIMATION
In this section we describe how to estimate the vector bo usin instrumental variables procedure. The basic idea underlying our proposed esti-
mation strategy is to use the theoretical economic model to generate a family of orthogonality conditions. These orthogonality conditions are then used to con-
struct a criterion function whose minimizer is our estimate of bo. This criterion function is constructed in a manner that guarantees that our parameter estimator
is consistent, asymptotically normal, and has an asymptotic covariance matrix
that can be estimated consistently. The orthogonality conditions also can be used
to construct a test of the overidentifying restrictions implied by the theoretical model. We elaborate on each of the steps in the following discussion.
Let ut + = h (x, + n, bo) and consider again the first-order conditions (3.1) Ej[ut+n] = 0,
with the additional assumption that the m constituents of ut,n have finite second
5Even in linear environments this is a nontrivial econometric endeavor. See Hansen and Sargent [11, 12] and Sargent [30] for a discussion of these issues in linear environments.
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INSTRUMENTAL VARIABLES ESTIMATION 1275
moments. Also, let zt denote a q dimensional ve second moments that are in agents’ information set and observed by the econometrician; and define the function f by
(3.2) f(xt+W,tz b) = h(Xt+n,b) 0 Zt,
where f maps R k x R q X R ‘ into R r r = m – q, and 0 is the Kronecker product. Then an implication of (3.1), (3.2), and the accompanying assumptions is that
(3X3) ES f (Xt + n 9 Zt) bo) 0 ?
where E is the unconditional expectations operator.6 Equation (3.3) represents a
set of r population orthogonality conditions from which an estimator of bo can b constructed, provided that r is at least as large as the number of unknown parameters, 1.
We proceed by constructing an objective function that depends only on the
available sample information {(x +n, Z1), (X2+n, Z2)’ . . . (XT+n, ZT)} and the
unknown parameters. Let go(b) = E [f(x, +n,Z,, b)], where b E R’ and it is as- sumed that the left-hand side does no
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