Estimation of Conditional Moment Restrictions. without Assuming Parameter Identifiability in the Implied Unconditional Moments

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1 Estimation of Conditional Moment Restrictions without Assuming Parameter Identifiability in the Implied Unconditional Moments Shih-Hsun Hsu Department of Economics National Chengchi University Chung-Ming Kuan Institute of Economics Academia Sinica his version: August 6, 2008 Author for correspondence: Chung-Ming Kuan, Institute of Economics, Academia Sinica, aipei 5, aiwan; his paper was originally entitled: Consistent Parameter Estimation for Conditional Moment Restrictions. We are indebted to Z. Cai, J. Chao, J.-C. Duan, J. Hamilton, J. Hahn, Y. Hong, Y.-Y. Lee, J. Park, M. H. Pesaran, P. C. B. Phillips, M. Stinchcombe, A. immermann, Y.-J. Whang, and H. White for their useful comments and suggestions on early drafts of this paper. We also thank the seminar participants at Boston College, Rice, oronto, UCSD, U. of Hong Kong, USC, U Austin, and the SEA 2007 meeting in Hong Kong for their comments. he research support from the National Science Council of the Republic of China (NSC H ) for C.-M. Kuan is gratefully acknowledged.

2 Abstract A well known difficulty in estimating conditional moment restrictions is that the parameters of interest need not be globally identified by the implied unconditional moments. In this paper, we propose an approach to constructing a continuum of unconditional moments that can ensure parameter identifiability. hese unconditional moments depend on the instruments generated from a generically comprehensively revealing function, and they are further projected along the exponential Fourier series. he objective function is based on the resulting Fourier coefficients, from which an estimator can be easily computed. A novel feature of our method is that the full continuum of unconditional moments is incorporated into each Fourier coefficient. We show that, when the number of Fourier coefficients in the objective function grows at a proper rate, the proposed estimator is consistent and asymptotically normally distributed. An efficient estimator is also readily obtained via the conventional two-step GMM method. Our simulations confirm that the proposed estimator compares favorably with that of Domínguez and Lobato (2004, Econometrica) in terms of bias, standard error, and mean squared error. JEL classification: C2, C22 Keywords: conditional moment restrictions, Fourier coefficients, generically comprehensive revealing function, global identifiability, GMM

3 Introduction Many economic and econometric models can be characterized in terms of conditional moment restrictions. Consistent and efficient estimation of the parameters in such restrictions is thus a crucial step in empirical studies. It is typical to find a finite set of unconditional moment restrictions implied by the original, conditional restrictions and apply a suitable estimation method, such as the generalized method of moment (GMM) of Hansen (982) and Hansen and Singleton (982), or the empirical likelihood method of Qin and Lawless (994) and Kitamura (997). his approach will be referred to as the unconditional moment approach; a leading example is the instrumental-variable estimation method for regression models. On the other hand, there are nonparametric methods that deal with the conditional moments directly, e.g., Ai and Chen (2003) and Kitamura, ripathi, and Ahn (2004). A critical assumption for the unconditional moment approach is that the parameters in the conditional restrictions can be globally identified by the implied, unconditional restrictions. With this assumption, estimator consistency is not really an issue and can be easily established under suitable regularity conditions. herefore, much research interest focuses on estimator efficiency, e.g., Chamberlain (987), Newey (990, 993), Carrasco and Florens (2000), and Donald, Imbens, and Newey (2003). Domínguez and Lobato (2004) challenge the assumption of global identifiability and show that the unconditional moments, when chosen arbitrarily, need not be equivalent to the original conditional restrictions. hey also demonstrate that the identification problem may arise even when the unconditional moments are based on the so-called optimal instruments. Without assuming the global identifiability of parameters, Domínguez and Lobato (2004) construct a continuum of unconditional moment restrictions that are equivalent to the original, conditional restrictions and obtain consistent estimate from these restrictions. In particular, their unconditional moments are determined by the instruments generated from an indicator function. here are some disadvantages of their method, however. First, the indicator function takes only the values one and zero and hence may not well present the information in the conditioning variables. Second, their estimation method does not utilize the full continuum of moment restrictions. his may result in further efficiency loss (Car-

4 rasco and Florens, 2000). hird, it is not easy to obtain an efficient estimate from their consistent estimate. In this paper, we propose a different approach to constructing a continuum of unconditional moments that can ensure parameter identifiability. hese unconditional moments depend on the instruments generated from the class of generically comprehensively revealing (GCR) functions (Stinchcombe and White, 998), and they are further projected along the exponential Fourier series. he objective function is then based on the resulting Fourier coefficients, from which an estimator can be easily computed. A novel feature of our method is that it in effect utilizes all possible information in the conditioning variables because all unconditional moments have been incorporated into each Fourier coefficient. Moreover, an efficient estimator is readily obtained via the conventional two-step GMM method. his efficient GMM estimator is computationally simpler than that of Carrasco and Florens (2000). We first show that the proposed estimator is consistent and asymptotically normally distributed when the number of Fourier coefficients in the objective function grows at a proper rate. We also specialize on the instruments generated from the exponential function, a special case in the class of GCR functions. For such instruments, the unconditional moments and Fourier coefficients have analytic forms, which greatly facilitate estimation in practice. Our simulations confirm that, under various settings, the proposed consistent and efficient estimators perform significantly better than that of Domínguez and Lobato (2004) in terms of bias, standard error, and mean squared error. he proposed estimators also outperform the estimator based on the optimal instruments. his paper is organized as follows. We introduce the new class of consistent estimators in Section 2 and establish its consistency and asymptotic normality in Section 3. he Efficient estimator based on the proposed consistent estimator is discussed in Section 4. he simulation results are reported in Section 5. Section 6 concludes this paper. All proofs are deferred to Appendix. 2

5 2 Consistent Estimation We are interested in estimating θ o, the q vector of unknown parameters, in the following conditional moment restriction: IE[h(Y, θ o ) X] = 0, with probability one (w.p.), () where h is a p vector of functions, Y is a r vector of data variables, and X is an m vector of conditioning variables. Without loss of generality, we shall work on the case that X is bounded with probability one; see e.g., Bierens (994, heorem 3.2.). It is well known that () is equivalent to the unconditional moment restriction: IE[h(Y, θ o )f(x)] = 0, (2) for all measurable functions f, where each f(x) may be interpreted as an instrument that helps to identify θ o. In practice, it is infeasible to consider all possible functions. hus, one typically forms an estimating function by subjectively choosing certain instruments, such as the square and cross product of the elements in X. his would not be a problem in a linear model if the resulting unconditional moments can exactly identify θ o. Yet, when h is nonlinear in θ o, Domínguez and Lobato (2004) show that θ o is not necessarily identified when unconditional moments are determined arbitrarily, and its identifiability may depend on the marginal distributions of the conditioning variables X. his concern is practically relevant because models with nonlinear restrictions are quite common in econometric applications; see e.g., Hansen and Singleton (982) and Hansen and West (2002). One way to ensure parameter identifiability is to employ a class of instruments that span a space of functions of X (Bierens, 982, 990; Stinchcombe and White, 998). Domínguez and Lobato (2004) set the instruments as (X τ ) = m j= (X j τ j ), where (B) is the indicator function of the event B. his leads to a continuum of unconditional moments indexed by τ that are equivalent to (): IE[h(Y, θ o )(X τ )] = 0, τ R m. (3) Hansen and West (2002) studied the papers published in 7 top economics journals in 990 and 2000 and found that, among 35 articles that employed the GMM technique, 4 of them deal with models with nonlinear restrictions. 3

6 hen, θ o can be globally identified by an L 2 -norm of these moments, i.e., θ o = argmin IE [ h(y, θ)(x τ ) ] 2 dp (τ ), (4) θ Θ R m with P (τ ) a distribution function of τ and denotes the Euclidean norm. Here, a natural choice of P (τ ) is P X (τ ), the distribution function of X. he L 2 -norm in (4) is thus an expectation with respect to P X (τ ) and can be well approximated by the sample average. Domínguez and Lobato (2004) thus propose the following estimator: 2 θ DL ( ) = argmin h(y θ Θ t, θ)(x t τ k ), (5) k= t= where y t and x t are the sample observations of Y and X, respectively, and τ k = x k, k =,...,. his is precisely a GMM estimator based on unconditional moments induced by the indicator function. By the analogy between the L 2 -norm in (4) and the objective function in (5), θ DL ( ) is consistent for θ o under regularity conditions. 2. A Class of Consistent Estimators he indicator function is not the only choice for the desired instruments; Stinchcombe and White (998) show that any GCR function will also do. In particular, for a real analytic function G that is not a polynomial, 2 G(A(X, τ )) can serve as an instrument in (2), where A is the affine transformation such that A(X, τ ) = τ 0 + m j= X jτ j. For example, G may be the exponential function (Bierens, 982, 990) or the logistic function (White, 989). A striking property of the instruments resulted from a GCR function is that (2) holds for the instruments with the index τ in an arbitrarily chosen index set in R m+ ; see Stinchcombe and White (998, p. 304). As such, the unconditional moment restrictions induced by a GCR function are IE [ h(y, θ o )G ( A(X, τ ) )] = 0, for almost all τ R m+, (6) where may be a small subset with a nonempty interior. Note that the indicator function is not GCR; hence (3) must hold for all τ in R m. Similar to (4), θ o now can be globally identified by the L 2 -norm of (6): θ o = argmin IE [ h(y, θ)g ( A(X, τ ) )] 2 dp (τ ). (7) θ Θ 2 A function G is said to be analytic if it locally equals its aylor expansion at every point of its domain. 4

7 In contrast with Domínguez and Lobato (2004), there is no natural choice of P (τ ), and it is not easy to find a proper sample counterpart of the L 2 -norm in (7). Although an objective function for estimating θ o can be constructed using randomized τ, the resulting estimate is arbitrary and may not be preferred. In this paper, we take a different approach to deriving a class of consistent estimators for θ o without assuming parameter identifiability in the implied unconditional moments. his approach finds a condition equivalent to the L 2 -norm in (7). o this end, we project the unconditional moments in (6) along the exponential Fourier series and obtain IE [ h(y, θ)g ( A(X, τ ) )] = (2π) m+ C G,k (θ) exp (ik τ ), k S where S := {k = [k 0, k,..., k m ] Z m+ } with k i = 0, ±, ±2,, ±, and C G,k (θ) is a Fourier coefficient: C G,k (θ) = IE [ h(y, θ)g ( A(X, τ ) )] exp ( ik τ ) dτ [ = IE h(y, θ) G ( A(X, τ ) ) ] exp ( ik τ ) dτ, k S. It can be seen that each C G,k (θ) incorporates the full continuum of the original instruments G(A(X, τ )) into a new instrument: ϕ G,k (X) = G ( A(X, τ ) ) exp ( ik τ ) dτ, (8) in which the index parameter τ has been integrated out. We shall use the following notations. Given a complex number f, let f denote its complex conjugate and Re(f) and Im(f) denote its real and imaginary parts, respectively. For a vector of complex numbers f, its complex conjugate, real part and imaginary part are defined elementwise. hen, f 2 = f f. Apart from a scaling factor, Parseval s heorem implies that the L 2 -norm in (7) is equivalent to C G,k (θ) 2 = IE [ h(y, θ)ϕ G,k (X) ] 2. k S k S It follows that θ o can be identified as θ o = argmin θ Θ IE [ h(y, θ)ϕ G,k (X) ] 2, (9) k S 5

8 where the right-hand side no longer involves τ, cf. (7). By replacing IE[h(Y, θ)ϕ G,k (X) ] in (9) with its sample counterpart, an objective function for estimating θ o is readily obtained. It is well known that C G,k (θ) 0 as k tends to infinity by Bessel s inequality. his suggests that the new instruments ϕ G,k (X), and hence IE[h(Y, θ)ϕ G,k (X)], contain little information for identifying θ o when k is large. As such, we may omit remote Fourier coefficients and compute an estimator of θ o as θ(g, K ) = argmin θ Θ k S(K ) h(y t, θ)ϕ G,k (x t ) where S(K ) is a subset of S with k i = 0, ±,..., ±K, such that K t= 2, (0) grows with but at a slower rate. he proposed estimator (0) depends on the function G, and it is also a GMM estimator based on (2K +) m+ unconditional moments with the identity weighting matrix. Hence, θ(g, K ) is not an efficient estimator. Note that the Domínguez-Lobato estimator (5) relies only on a finite number of unconditional moments determined by the sample observations. By contrast, the proposed estimator (0) utilizes all possible information in estimation because each ϕ G,k has included the full continuum of the instruments required for identifying θ o. Our estimator is also computationally simpler than that of Carrasco and Florens (2000), which requires preliminary estimation of a covariance operator and its eigenvalues and eigen-functions. Moreover, a regularization parameter must be determined in practice so as to ensure the invertibility of the estimated covariance operator. 2.2 A Specific Estimator o compute the proposed estimator, we follow Bierens (982, 990) and set G as the exponential function. his choice has some advantages relative to the indicator function. First, the indicator function takes only the values one and zero, whereas the exponential function is more flexible and hence may better presents the information in the conditioning variables. hat is, the exponential function may generate better instruments for identifying θ o. Second, the exponential function is smooth and hence is convenient in an optimization program. Further, exp(a(x, τ )) with τ R m+ and exp(x τ ) with τ R m only differ by a constant and hence play the same role in function approximation (Stinchcombe and 6

9 White, 998). By employing exp(x τ ) as a desired instrument, we are able to reduce the dimension of integration in (7) by one, i.e., R m, and the summation in (9) is over S = {k = [k,..., k m ] Z m }. More importantly, choosing exp(x τ ) results in an analytic form for the instrument ϕ exp,k which in turn facilitates estimation in practice. In particular, setting = [, π] m, the new instruments that integrate out τ are ϕ exp,k (X) = exp(x τ ) exp ( ik τ ) dτ where ϕ exp,kj (X j ) = = ϕ exp,k (X ) ϕ exp,km (X m ), k S, π exp(x j τ j ) exp( ik j τ j ) dτ j = ( )kj 2 sinh (πx j ), j =,..., m, (X j ik j ) and sinh(w) = (exp (w) exp ( w))/2. Based on ϕ exp,k (X), θ o can be identified as in (9). he proposed estimator thus reads θ(exp, K ) = argmin θ Θ where k is m. k S(K ) h(y t, θ)ϕ exp,k (x t ) t= 2, () 3 Asymptotic Properties We now establish the asymptotic properties of the proposed estimator θ(g, K ). o ease our illustration and proof, we begin our analysis with the case that m = ; the univariate X is denoted as X (no boldface). he asymptotic properties for the case with multivariate X are discussed in Section Consistency We impose the following conditions. [A] he observed data (y t, x t ), t =,...,, are independent realizations of (Y, X). 7

10 [A2] For each θ Θ, h(, θ) is measurable, and for each y R r, h(y, ) is continuous on Θ, where Θ is a compact subset in R q. Also, θ o in Θ is the unique solution to IE[h(Y, θ) X] = 0. [A3] IE[sup θ Θ h(y, θ) 2 ] <. [A4] G is real analytic but not a polynomial such that, w.p., sup τ G(A(X, τ )) <, sup τ G i (A(X, τ )) <, and sup τ G ij (A(X, τ )) <, where G i (A(X, τ )) = G(A(X, τ ))/ τ i and G ij (A(X, τ )) = 2 G(A(X, τ ))/( τ i τ j ), for i, j = {0, }. hese conditions are convenient and quite standard in the GMM literature. hey may be relaxed at the expense of more technicality. For example, it is possible to extend [A] to allow for weakly dependent and heterogeneously distributed data; see, e.g., Gallant and White (988) and Chen and White (996). Note that in [A2], θ o is assumed to be the unique solution to the original conditional restrictions; we do not require θ o to be the unique solution to some implied, unconditional moment restrictions. As in Stinchcombe and White (998), [A4] requires G to be real analytic but not a polynomial. [A4] also imposes additional restrictions on G and its derivatives, yet it still permits quite general G functions. Setting = [, π] 2, the instruments resulted from G are ϕ G,k (X) = G ( A(X, τ ) ) exp ( ik τ ) dτ. (2) [,π] 2 Here, k = (k 0, k ). Define c(k i ) = k i for k i 0 and c(k i ) = for k i = 0, i = 0,. he result below provides a bound on ϕ G,k (X). Lemma 3. Given [A4], ϕ G,k (X) /[c(k 0 )c(k )] w.p., where is a real number. Define the sample counterpart of C G,k (θ) as m G,k, (θ) = h(y t, θ)ϕ G,k (x t ). t= With Lemma 3., we are able to characterize the approximating capability of m G,k, (θ). 8

11 Lemma 3.2 Given [A] [A4], if K and K = o( /2 ), then where sup Θ K k 0,k = K m G,k, (θ) C G,k (θ) 2 IP 0, IP stands for convergence in probability. Lemma 3.2 implies K k 0,k = K m G,k, (θ) 2 IP k 0,k = C G,k (θ) 2, (3) uniformly for all θ in Θ. As θ o is the unique minimizer of the right-hand side of (3), the consistency result below follows from heorem 2. of Newey and McFadden (994). heorem 3.3 Given [A] [A4], if K and K = o( /2 ), then θ(g, K ). IP θ o as For the estimator θ(exp, K ) in (), note that exp(xτ) satisfies [A4] with τ a scalar. It is easy to deduce that Lemma 3. holds with ϕ exp,k (X) /k. Lemma 3.2, we also have K k= K m exp,k, (θ) C exp,k (θ) 2 In analogy with IP 0, (4) when K = o( ). he result below follows from (4) and is analogous to heorem 3.3. Corollary 3.4 Given [A] [A3], if K and K = o( ), then θ(exp, K ). IP θ o as 3.2 Asymptotic Normality Recall that the Fourier coefficient C G,k (θ) can be expressed as IE [ h(y, θ)ϕ G,k (X) ] = IE [ h(y, θ)g ( A(X, τ ) )] exp( ik τ ) dτ, [,π] 2 which is the integral of the product of two functions in τ, i.e., IE[h(Y, θ)g(a(x, ))] and exp( ik ). o establish asymptotic normality, we work on IE[h(Y, θ)g(a(x, ))] and its 9

12 sample counterpart directly. below. his requires some results in the function space, as given Consider functions in the Hilbert space L 2 [, π]. he inner product of two p vectors of functions f and g is f, g = π f(τ) ḡ(τ) dτ, and the norm induced by the inner product is f, f /2. A random element U has mean IE(U) if IE[ U, g ] = IE(U), g for any g in L 2 [, π]. he covariance operator K associated with U is, for any g in L 2 [, π], Kg = IE [ U IE(U), g ( U IE(U) )] such that (Kg)(τ) = IE [ U IE(U), g ( U(τ) IE(U(τ)) )] ( p ) π = κ ji (τ, s)g i (s) ds, i= j=,...,p with the kernel κ ji (τ, s) = IE[(U j (τ) IE U j (τ))(u i (s) IE U i (s))]. U is said to be Gaussian if for any g in L 2 [, π], U, g has a normal distribution on R with mean IE(U), g and variance Kg, g. Analogous results also hold in L 2 ([, π] m ). For more discussions on random elements in Hilbert space; see, e.g., Chen and White (998) and Carrasco and Florens (2000). In view of (0), θ(g, K ) must satisfy the first order condition: 0 = K k 0,k = K θ m G,k, (θ) m G,k, (θ) + θ m G,k, (θ) m G,k, (θ) K ( = 2 Re θ m G,k, (θ) m G,k, (θ) ), k 0,k = K where θ m G,k, (θ) is a p q matrix with θi m G,k, (θ) its i-th column. A mean-value expansion of m G,k, ( θ(g, K ) ) about θ o gives m G,k, ( θ(g, K ) ) = m G,k, (θ o ) + θ m G,k, ( θ )( θ(g, K ) θ o ), where θ is between θ(g, K ) and θ o, and its value may be different for each row in the matrix θ m G,k, ( θ ). hus, K k 0,k = K Re ( θ m G,k, ( θ(g, K ) ) [ mg,k, (θ o ) + θ m G,k, ( θ )( θ(g, K ) θ o )] ) = 0. (5) 0

13 o derive the limiting distribution of normalized θ(g, K ), we impose the following conditions. [A5] θ o is in the interior of Θ. [A6] For each y, h(y, ) is continuously differentiable in a neighborhood N of θ o such that IE [ sup θ N θ h(y, θ) 2] <, where is a matrix norm. [A7] he q q matrix M q, with the (i, j)-th element IE [ θi h(y, θ o )G(A(X, )) ] [ ], IE θj h(y, θ o )G(A(X, )), is symmetric and positive definite. [A8] /2 t= h(y t, θ o )G(A(x t, )) D Z, where D denotes convergence in distribution, and Z is a p-dimensional Gaussian random element that has mean zero and the covariance operator K with (Kg)(τ) = IE [ h(y, θ o )G(A(X, )), g ( h(y, θ o )G(A(X, τ)) )], for any p-dimensional function g. Here, [A5] is needed for mean-value expansion; [A6] is a standard smoothness condition in nonlinear models. By [A7], M q is invertible so that the normalized estimator has a unique representation, as given in (6) below. We directly assume functional convergence in [A8] for convenience; this condition is the same as Assumption in Carrasco and Florens (2000). o ensure such convergence, one may also impose primitive conditions on h, G and the data; see, e.g., Chen and White (998). o study the behavior of the normalized estimator via (5), we give two limiting results for the terms on the right-hand side of (5). Lemma 3.5 Given [A] [A6], if K and K = o( /4 ), then K k 0,k = K Re ( θ m G,k, ( θ(g, K ) ) θ m G,k, (θ ) ) IP k 0,k = θ C G,k (θ o ) θ C G,k (θ o ).

14 he limit in Lemma 3.5 is precisely the matrix M q defined in [A7], because its (i, j)-th element is k 0,k = θi C G,k (θ o ) θj C G,k (θ o ) = IE [ θi h(y, θ o )G ( A(X, ) )], IE [ θj h(y, θ o )G ( A(X, ) )], by the Multiplication theorem (e.g. Stuart, 96). Lemma 3.6 Given [A] [A6], if K and K = o( /4 ), then K k 0,k = K Re ( θ m G,k, ( θ(g, K ) ) ) m G,k, (θ o ) = k 0,k = θ C G,k (θ o ) m G,k, (θ o ) + o IP (). With Lemma 3.5 and Lemma 3.6, (5) can be expressed as ( θ(g, K ) θ o ) = M q k 0,k = θ C G,k (θ o ) m G,k, (θ o ) + o IP (). (6) he functional convergence condition [A8] now ensures that the term in the square bracket on the right-hand side of (6) has a limiting normal distribution, which in turn leads to the asymptotic normality of θ(g, K ). heorem 3.7 Given [A] [A8], if K and K = o( /4 ), then ) D ( θ(g, K ) θ o N (0, V), where V = M q Ω q M q IE [ θi h(y, θ o )G(A(X, )) ], K IE and Ω q is a q q matrix with the (i, j)-th element: [ ] θj h(y, θ o )G(A(X, )). For the estimator θ(exp, K ) with G(A(X, τ )) = exp(xτ), it can be verified that the results analogous to Lemma 3.5 and Lemma 3.6 hold when K is o( /2 ). In particular, K k= K Re ( θ m exp,k, ( θ(exp, K ) ) θ m exp,k, (θ ) ) IP k= θ C exp,k (θ o ) θ C exp,k (θ o ), (7) 2

15 which is the matrix M q with the (i, j)-th element: IE [ θi h(y, θ o ) exp(x ) ], IE [ θj h(y, θ o ) exp(x ) ], and K k= K Re ( θ m exp,k, ( θ(exp, K ) ) ) m exp,k, (θ o ) = k= In this case, (6) becomes ( θ(exp, K ) θ o ) = M q [ k= θ C exp,k (θ o ) m exp,k, (θ o ) + o IP (). (8) θ C exp,k (θ o ) m exp,k, (θ o ) ] + o IP (), (9) which also has a limiting normal distribution. he result below is analogous to heorem 3.7. Corollary 3.8 Given [A] [A3] and [A5] [A8], if K and K = o( /2 ), then ( θ(exp, K ) θ o ) where V = M q Ω q M q D N (0, V), IE [ θi h(y, θ o ) exp(x ) ], K IE and Ω q is a q q matrix with the (i, j)-th element: [ ] θj h(y, θ o ) exp(x ). For estimation of V in heorem 3.8, note from (7) that M q can be consistently estimated by K θ m exp,k, ( θ(exp, K ) ) θ m exp,k, ( θ ( exp, K ) ). k= K From [A8] and (8), Ω q can be consistently estimated by the real part of K K [ θ m exp,l, ( θ(exp, K ) ) ] k= K l= K [ h ( y t, θ(exp, K ) ) ϕ exp,l (x t )ϕ exp,k (x t )h ( y t, θ(exp, K ) ) ] t= [ θ m exp,k, ( θ(exp, K ) )]. A consistent estimator of V is readily computed from these two estimators. 3

16 3.3 he Results for Multivariate X We now extend the asymptotic properties above to the case with multivariate X. Recall that X is an m vector of conditioning variables Setting = [, π] m+, the proposed instruments based on G are ϕ G,k (X) = G ( A(X, τ ) ) exp ( ik τ ) dτ, [,π] m+ where k = (k 0, k,..., k m ). he required conditions for asymptotics are unchanged, except [A4] is changed to [A4 ]. [A4 ] G is real analytic but not a polynomial such that, w.p., j G(A(X, τ )) m i=0 ( τ i) l i <, sup τ where i = 0,,..., m, j =,..., m, and l i = 0,,..., j such that m i= l i = j. Again, let c(k i ) = k i for k i 0 and c(k i ) = for k i = 0, i = 0,,..., m. Similar to Lemma 3., we obtain the following bound on ϕ G,k (X) when X is multivariate. Lemma 3.9 Given [A4 ], ϕ G,k (X) /[ m i=0 c(k i)] w.p., where is a real number. With Lemma 3.9, the results below include heorem 3.3 and heorem 3.7 as special cases. Note that the growth rates of K depend on m, the dimension of X. 3 he results for the specific estimator θ(g, K ) can be obtained similarly. heorem 3.0 Given [A] [A3] and [A4 ], if K and K = o( /(m+) ), then θ(g, K ) IP θ o as. heorem 3. Given [A] [A3], [A4 ] and [A5] [A8], if K and K = o( /(2m+2) ), then ( θ(g, K ) θ o ) D N (0, V), where V = M q Ω q M q and Ω q is a q q matrix with the (i, j)-th element: IE [ θi h(y, θ o )G(A(X, )) ], K IE [ ] θj h(y, θ o )G(A(X, )). 3 he dimension m affects the growth rates of K only through the implication rule and the generalized Chebyshev inequality in the proofs. 4

17 4 Efficient Estimation It now remains to show how an efficient estimator can be computed; this is the topic to which we now turn. Following Newey (990, 993) and Domínguez and Lobato (2004), an efficient estimate may be obtained from the proposed consistent estimate via an additional Newton-Raphson step. hat is, an efficient estimator can be computed as: θ e = θ(g, [ K ) θθ Q ( θ(g, K ) )] θ Q ( θ(g, K ) ), where Q (θ) is an objective function for the efficient estimator that can locally identify θ o, and θ Q (θ) and θθ Q (θ) are its gradient vector and Hessian matrix, both evaluated at the consistent estimate θ(g, K ). In practice, identifying such objective function and estimating its gradient and Hessian matrix may not be as straightforward as one would like (Newey, 990, 993). Carrasco and Florens (2000) consider efficient estimation based on the the objective function that takes into account the covariance structure: θ o = argmin K /2 IE [ h(y, θ) exp(τx) ] 2 dp (τ), θ Θ where K is the covariance operator introduced in section 3.2, and the corresponding estimation method is based on projection along preliminary estimates of the eigenfunctions of K. here are some drawbacks of this approach. First, this estimator depends on various user-chosen parameters and hence is arbitrary to some extent. Second, the generalized inverse of the covariance operator exists only for a subset of Hilbert space, namely, the reproducing kernel Hilbert space. Moreover, it is difficult to generalize their results to allow for multivariate X. Alternatively, an efficient estimator is readily computed via the conventional two-step GMM method. As ϕ G,k (X) is complex, we now consider ϕ r G,k (X) and ϕi G,k (X), the real 5

18 part and imaginary part of ϕ G,k (X). 4 Equivalent to (9), θ o can also be identified as: θ o = argmin θ Θ IE [ h(y, θ)ϕ r G,k (X)] 2 + IE [ h(y, θ)ϕ i G,k (X)] 2, k S where the minimum of this objective function is zero. A new set of unconditional moment restrictions now consists of IE [ h(y, θ o )ϕ r G,k (X)] = 0 and IE [ h(y, θ o )ϕ i G,k (X)] = 0 with k S. Given ϕ r G,k (X) = ϕr G, k (X) and ϕi G,k (X) = ϕi G, k (X) for any k S, some of these unconditional moment restrictions are redundant and can be omitted. Let q t (θ, G, K ) = h(y t, θ) Z G,K (x t ), where Z G,K (x t ) is the (2K +) (4K +) m - dimensional vector that contains ϕ r G,k (x t) and ϕ i G,k (x t), where k = [k 0, k,..., k m ] with k 0 = 0,,..., K, and k i = 0, ±,..., ±K the asymptotic covariance matrix of q t (θ, G, K ) is for i =, 2,..., m. he sample counterpart of V (θ, G, K ) = q t (θ, G, K )q t (θ, G, K ). t= Evaluating the inverse of V at the consistent estimate θ(g, K ) and taking the resulting matrix as the weighting matrix, an efficient GMM estimator of θ o is θ e (G, K ) = argmin θ Θ ( ) q t (θ, G, K ) V ( θ(g, K ), G, K ) t= ( ) q t (θ, G, K ). (20) In the homoskedasticity case that IE[h(y t, θ o )h(y t, θ o ) X] is constant, V simplifies to [ ] [ ] V (θ, G, K ) = h(y t, θ)h(y t, θ) Z G,K (x t )Z G,K (x t ). t= he inverse of V is easy to compute in practice because the first term in V is positive definite and the inverse of the second term can be obtained by any generalized inverse method. 4 For example, when X is univariate and G is the exponential function, ϕ r exp,k(x) = ( ) k 2X X 2 + k sinh(πx), 2 ϕ i exp,k(x) = ( ) k 2k X 2 + k sinh(πx), 2 are the real and imaginary parts of ϕ exp,k (X). t= t= 6

19 By treating Z G,K (x t ) as a class of approximating functions, the results in Donald et al. (2003) may be employed to establish the asymptotic properties of the efficient estimator (20). 5 It should be emphasized that, with the proposed unconditional moments, the two-step GMM estimation method is not the only way to obtain an efficient estimator. Other methods, such as empirical likelihood estimation (e.g., Qin and Lawless, 994) and continuously updated estimation (e.g., Hansen, Heaton, and Yaron, 996) will also do. 5 Simulations In this section, we focus on the finite-sample performance of the proposed consistent and efficient estimators: θ(exp, K ) and θ e (exp, K ). We compare their performance with the nonlinear least squares (NLS) estimator: θ NLS = argmin θ Θ h(y t, θ) 2, t= and the DL estimator of Domínguez and Lobato (2004), θ DL in (5). Our comparison is based on the bias, standard error (SE), and mean squared error (MSE) of these estimators. he parameter estimates are computed using the GAUSS optimization procedure, OP- MUM, with the BFGS algorithm. In all experiments, the samples are = 50, 00, 200; the number of replications is In each replication, we randomly draw 0 initial values for all estimators, and for each estimator, the estimate that leads to the smallest value of the objective function is chosen. he data x t are transformed using a logistic mapping: exp(x t )/[ + exp(x t )], so that they are bounded between 0 and. Note that we set K = 5 for the proposed estimators; the effect of different K examined in Section 5.4. on the proposed estimator will be 5 Some stronger conditions are needed. For example, when G is the exponential function and X is univariate, heorem 5.3 and heorem 5.4 in Donald et al. (2003) require the growth rate of K being o( /2 ). his is more restrictive than the rate required for the consistent estimator: θ(exp, K ), cf. Corollary

20 5. he Experiments in Domínguez and Lobato (2004) Following Domínguez and Lobato (2004), we postulate the following nonlinear model with exogenous regressors: Y = θ 2 ox + θ o X 2 + ɛ, ɛ N (0, ), where θ o =.25 is the unique solution to the conditional moment restriction: IE(ɛ X) = 0. We consider two cases: X N (0, ) and X N (, ). When X N (0, ), θ o =.25 is the only real solution to the unconditional moment restriction resulted from the feasible optimal instrument (2θX + X 2 ); the other two solutions are complex: ±.0533i. When X N (, ), in addition to θ o =.25, θ =.25 and θ = 3 also satisfy the unconditional moment restriction with the feasible optimal instrument. In this case,.25 is the global minimum of the NLS objective function; the other two solutions are only local minima. For comparison, our simulations here also includes the optimal instrument variable (OPIV) estimator: ( 2 θ OPIV = argmin (y θ Θ t θ 2 x t θx 2 t )(2θx t + x 2 t )), t= which is different from the NLS estimator, cf. Domínguez and Lobato (2004, p. 608). he simulation results are summarized in able. In most cases, the NLS estimator outperforms the other estimators in terms of bias, SE and MSE, as it ought to be. On the other hand, θ OPIV has severe bias and large SE and is dominated by the other estimators. Note that when X N (, ), the existence of 3 possible solutions (.25,.25 and 3) suggests that the bias of the OPIV estimator should be close to his is confirmed in our simulation. 6 It is also clear that the proposed consistent and efficient estimators, θ(exp, K ) and θ e (exp, K ), both dominate the DL estimator in terms of bias, SE and MSE in all cases. When the sample size is not too small ( = 00, 200), the performance of the proposed efficient estimator is comparable with that of the NLS estimator. somewhat surprising to see that, compared with the proposed consistent estimator, our efficient estimator has not only smaller SE and MSE but also slightly smaller bias in many cases. he trade-off between SE and bias can be seen in the experiments in Section Yet, Domínguez and Lobato (2004) report a much smaller bias (about 0.4) under the same simulation design. It is 8

21 5.2 Model with an Endogenous Regressor We extend the previous experiment to the case that there is an endogenous regressor. he model specification is: Y = θoz 2 + θ o Z 2 + ɛ, and Z = X + ν, with ɛ N 0, ρ ν ρ, where θ o =.25, ρ = 0.0, 0., 0.3, 0.5, 0.7, 0.9, and X N (0, ) is independent of ɛ and ν. Given this specification, IE(ɛ X) = 0. he simulation results are collected in able 2. It is clear that the bias of these estimators all increases with ρ. In particular, the NLS estimator has very large biases, and such biases do not diminish when the sample size increases. his should not be surprising because the NLS estimator is inconsistent (due to the endogenous regressor). On the other hand, the proposed consistent and efficient estimators perform remarkably well. hey have much smaller bias than the NLS estimator, and they again outperform the DL estimator in terms of bias, SE, and MSE for any ρ and any sample size. Although the NLS estimator typically has a smaller SE, the proposed estimators may yield smaller MSE when the correlation between ɛ and ν is not too small (e.g., ρ 0.3). 5.3 Noisy Disturbances We now examine the effect of the disturbance variance on the performance of various estimators. he model is again Y = θox 2 + θ o X 2 + ɛ, ɛ N (0, σ 2 ), where θ o =.25, X is the uniform random variable on (, ) and independent of ɛ, and σ 2 =, 4 and 9. It can be verified that there are 3 solutions to the unconditional moment restriction resulted from the feasible optimal instrument (2θX + X 2 ): θ =.25 and ( 25 ± 45)/40, where.25 is the global minimum. 9

22 he results are summarized in able 3. We note first that, in contrast with the results in able, the NLS estimator is no longer the best estimator even when there is a unique global minimum and the regressor is exogenous. he proposed consistent estimator has smaller biases than all other estimators in all cases, except its bias is slightly larger than the NLS estimator when σ 2 = and = 200. In terms of MSE, the proposed consistent estimator dominates the DL estimator in all cases and outperforms the OPIV estimator when σ 2 is not too large. Although the proposed efficient estimator has larger bias than θ(exp, K ) in most cases, it still outperform the other estimators in terms of bias in all cases (except when = 50 and σ 2 = ). Moreover, the efficient estimator has the smallest MSE in all cases with = 00, 200, and its MSE is only slightly larger than the NLS estimator for = 50. As far as MSE is concerned, the proposed efficient estimator is to be preferred to the other estimators. 5.4 he Proposed Estimator with Various K We now examine the effect of K on the performance of the proposed estimator. he model specification is the same as that in Section 5.2, where the regressor is endogenous. We consider the cases that ρ equals 0., 0.5 and 0.9, and the sample = 50, 00 and 200. We simulate the DL estimator and θ(exp, K ) with K =, 2,..., 0, 5, 20. We do not consider the NLS estimator because its performance is too poor when regressor is endogenous. o ease our computation, we do not simulate the efficient estimator here. We report only the results for ρ = 0.5 and ρ = 0.9, each with = 00, 200 in ables 4 and 5. In addition to the bias, SE and MSE, we also report their percentage changes when K increases. For instance, for ρ = 0.9 and = 00, the bias decreases 0.46%, SE decreases.22%, and MSE decreases 2.40% when K increases from to 2. hese tables show that, when K increases, the proposed estimator becomes more efficient (with a smaller SE), while its bias typically decreases. 7 he percentage changes of bias and SE are small; in most cases, such changes are less than 0.% when K is greater than 5 or 6. hese results suggest that the first few Fourier coefficients indeed contain the most information for identifying θ o. Further increase of K can only result in marginal 7 his ill behavior may be due to the convergence criteria in our procedure. he tolerance for gradient of estimated coefficients is set to

23 improvements on the bias and SE. Note that the proposed estimator again dominates the DL estimator in terms of bias, SE and MSE in all cases. 6 Concluding Remarks his paper is concerned with consistent and efficient estimation of conditional moment restrictions when the parameters of interests are not assumed to be identified in the implied unconditional moments. o ensure proper identification of these parameters, we propose to construct a continuum of unconditional moments based on a generically comprehensively revealing function and obtain an objective function based on the projection of these unconditional moments. hen, consistent and efficient estimators can be easily computed using the conventional GMM method. Our simulations confirm that the proposed estimators perform very well in finite samples and compare favorably with the existing estimators, such as that proposed by Domínguez and Lobato (2004). It must be emphasized that we do not have to confine ourselves with GMM estimation. Based on the proposed moment conditions, other estimation methods, such as the empirical likelihood method, can also be employed to obtain consistent and/or efficient estimators. hese are some open questions about the proposed consistent estimator. First, it is practically useful to find a criterion to determine the optimal number of the required Fourier coefficients, K, in the objective function. Second, it is of great interest to know if a better estimator can be obtained when the unconditional moments are generated from a different generically comprehensively revealing function. hese topics are left to future research. 2

24 Appendix Proof of Lemma 3.: Let be a generic constant whose value varies in different cases. Recall that A(X, τ) = τ 0 + τ X and X is univariate. We have ϕ G,k (X) = = π π π G(τ 0 + τ X) exp( ik 0 τ 0 ) exp( ik τ )dτ 0 dτ [ π ] G(τ 0 + τ X) exp( ik 0 τ 0 )dτ 0 exp( ik τ ) dτ. By integration by parts, for k 0, k expressed as hen, so that π G(τ 0 + τ X) exp( ik 0 τ 0 ) dτ 0 = i { ( ) k 0 [ G(π + τ X) G( + τ k X) ] 0 }{{} ϕ G,k (X) = i k 0 π ϕ G,k (X) k 0 Q (τ ) 0, the term in the square brackets above can be π [ Q (τ ) Q 2 (τ ) ] exp( ik τ ) dτ, { π Q (τ ) exp( ik τ ) dτ + Again by integration by parts, π } G 0 (τ 0 + τ X) exp( ik 0 τ 0 ) dτ 0. } {{ } Q 2 (τ ) π Q 2 (τ ) exp( ik τ ) dτ }. and Q (τ ) exp( ik τ ) dτ = ( )k 0 i { ( ) k [ G(π + πx) G( + πx) G(π πx) + G( πx) ] k π π [ G (π + τ X) G ( + τ X) ] exp( ik τ ) dτ }, Q 2 (τ ) exp( ik τ ) dτ = i k { ( ) k π π [ G0 (τ 0 + πx) G 0 (τ 0 πx) ] exp( ik 0 τ 0 ) dτ 0 ) G 0 (τ 0 + τ X) exp( ik 0 τ 0 ) dτ 0 exp( ik τ ) dτ }. ( π π 22

25 Given [A4], we have π Q (τ ) exp( ik τ ) dτ k k, [ 4 sup G(τ 0 + τ X) π + 2 sup G (τ 0 + τ X) exp( ik τ ) ] dτ τ τ and π k k. Q 2 (τ ) exp( ik τ ) dτ [ π 2 + sup ( π τ π G0 (τ 0 + τ X) exp( ik 0 τ 0 ) dτ 0 π sup G 0 (τ 0 + τ X) exp( ik 0 τ 0 ) ) dτ 0 exp( ik τ ) } ] dτ τ It follows that ϕ G,k (X) /( k 0 k ) for k 0, k 0. Similarly, we can show that ϕ G,k (X) / k for k 0 = 0 and k 0 and that ϕ G,k (X) / k 0 for k 0 0 and k = 0. Also, it is clear that ϕ G,0 (X). he proof is thus complete. Proof of Lemma 3.2: Again let denote a generic constant whose value varies in different cases. Define η G,k,t = h(y t, θ)ϕ G,k (x t ) IE [ h(y, θ)ϕ G,k (X) ], for t =,..., and k = (k 0, k ). By Lemma 3., ϕ G,k (X) /[c(k 0 )c(k )]. With [A3], we have IE [ η G,k,t 2] IE [ h(y, θ) 2 ϕ G,k (X) 2] c(k 0 ) 2 c(k ) 2. 23

26 Under [A], these bounds lead to K 2 η G,k,t k 0,k = K IE t= = 2 K k 0,k = K t= IE [ η G,k,t 2] 4 K K k 2 k 0 = 0 k 2 k = + 2 K k 2 k 0 = K k 2 k = +, by the fact that n k= k 2 2 /n 2. It follows from the implication rule and the generalized Chebyshev inequality that K IP 2 η G,k,t ε k 0,k = K t= K IP 2 ε η G,k,t (2K k 0,k = K t= + ) 2 (2K + ) 2 K 2 η ε G,k,t (2K + ) 2 ε k 0,k = K IE, which holds uniformly in θ, because does not depend on θ. It is clear that this bound can be made arbitrarily small when K = o( /2 ). t= Proof of heorem 3.3: he proposed estimator, θ(g, K ), is the solution to the left-hand side of (3). Hence, it must converge to the unique minimizer, θ o, of the right-hand side of (3) by heorem 2. of Newey and McFadden (994). Proof of Corollary 3.4: Given G(A(X, τ )) = exp(xτ), we have from the text that (4) holds when K = o( ). Analogous to (3), we obtain K k= K m exp,k, (θ) 2 IP k= C exp,k (θ) 2, 24

27 uniformly in θ. he assertion again follows from heorem 2. of Newey and McFadden (994). Proof of Lemma 3.5: Given [A] [A4] and K = o( /4 ), θ(g, K ) θ θ o. With [A6], we can apply a standard argument to get IP θ o. Hence, θ m G,k, ( θ(g, K ) ) θ m G,k, (θ o ) IP 0, θ m G,k, (θ ) θm G,k, (θ o ) IP 0. Also note that θ C G,k (θ o ) θ C G,k (θ o ) is real and K k 0,k = K θ C G,k (θ o ) θ C G,k (θ o ) herefore, it suffices to show k 0,k = θ C G,k (θ o ) θ C G,k (θ o ). K ( θ m G,k, (θ o ) θ m G,k, (θ o ) θ C G,k (θ o ) θ C G,k (θ o ) ) IP 0. k 0,k = K We shall show this convergence holds elementwise. For notation simplicity, we drop the subscript G and the argument θ o and write η i,k = θi m k, IE[ θi m k, ]. he (i, j)-th element of the matrix above can be expressed as η i,k θ j m k, + θi C k η j,k. We need to show K k 0,k = K ( ) η i,k θ j m k, + θi C k η IP j,k 0. Again by the implication rule and the generalized Chebyshev inequality, we have K IP η i,k θ j m k, + θi C k η j,k ɛ k 0,k = K K k 0,k = K IP (2K + ) 2 ɛ (2K + ) 2 ɛ { η i,k θ j m k, + θi C k η j,k K k 0,k = K IE K } ɛ (2K + ) 2 [ ] η i,k m θj k, + C θi k j,k η k 0,k = K [IE η i,k 2 ] /2 [IE θj m k, 2 ] /2 + [IE θi C k 2 ] /2 [IE η j,k 2 ] /2. 25

28 By [A], [A6] and Lemma 3., IE θj m k, 2 = IE θ j h(y, θ)ϕ k (X) 2 Similarly, θi C k 2 /[c(k 0 ) 2 c(k ) 2 ], and c(k 0 ) 2 c(k ) 2. IE η i,k 2 = IE θi m k, 2 IE θi C k 2 IE θi m k, 2 c(k 0 ) 2 c(k ) 2. Putting these results together we have, similar to the proof of Lemma 3.2, K IP η i,k θ j m k, + θi C k η j,k ɛ k 0,k = K (2K + ) 2 ɛ (2K + ) 2 ɛ K k 0,k = K, ( ) c(k 0 ) 2 c(k ) 2 + c(k0 ) 2 c(k ) 2 which can be made arbitrarily small when K = o( /4 ). Proof of Lemma 3.6: Similar to the proof of Lemma 3.5, given [A] [A6] and K = o( /4 ), θ(g, K ) IP θ o, it is thus sufficient to show K [ θ m G,k, (θ o ) θ C G,k (θ o ) ] m G,k, (θ o ) IP 0, k 0,k = K since θ m G,k, ( θ(g, K ) ) θ m G,k, (θ o ) IP 0 and K θ C G,k (θ o ) mg,k, (θ o ) k 0,k = K where, by invoking the multiplication theorem, k 0,k = θ C G,k (θ o ) m G,k, (θ o ) = IE [ θ h(y, θ o )G ( A(X, ) )], k 0,k = t= θ C G,k (θ o ) m G,k, (θ o ), h(y t, θ o )G ( A(x t, ) ) 26

29 is real. Again let η i,k = θi m G,k, (θ o ) IE[ θi m G,k, (θ o )] and by the implication rule and the generalized Chebyshev inequality, we have K IP η i,k mg,k, (θ o ) ɛ k 0,k = K K k 0,k = K IP (2K + ) 2 ɛ (2K + ) 2 ɛ (2K + ) 2 ɛ { η i,k mg,k, (θ o ) K k 0,k = K IE } ɛ (2K + ) 2 [ η ] i,k mg,k, (θ o ) K [ [IE η i,k 2 ] /2 IE m G,k, (θ o ) 2] /2 k 0,k = K K k 0,k = K [IE η i,k 2 ] /2 [ IE h(y, θo )ϕ G,k (X) 2] /2, where the last inequality, given [A], is due to the fact that IE m G,k, (θ o ) 2 = IE From the proof of Lemma 3.5 we have and IE η i,k 2 c(k 0 ) 2 c(k ) 2, IE h(y, θ o )ϕ G,k (X) 2 It follows that IP K k 0,k = K t= c(k 0 ) 2 c(k ) 2. η i,k mg,k, (θ o ) ɛ h(y t, θ o )ϕ G,k (x t ) 2 (2K + ) 2 ɛ = IE h(y, θ o )ϕ G,k (X) 2.. he proof is complete because this bound can be made arbitrarily small when K = o( /4 ) and. Proof of heorem 3.7: From [A8], we know /2 t= h(y t, θ o )G(A(x t, )) D Z, where Z is a Gaussian random element in L 2 ([, π] 2 ) with the covariance operator K. By 27

30 invoking the multiplication theorem, we have K θ C G,k, (θ o ) mg,k, (θ o ) k 0,k = K = = = k 0,k = θ C G,k, (θ o ) m G,k, (θ o ) + o IP () ( θi IE [ h(y, θ o )G ( A(X, ) )], ( θi IE [ h(y, θ o )G ( A(X, ) )] ), Z D N (0, Ω q ). t= i=,...,p h(y t, θ o )G(A(x t, )) + o IP () ) i=,...,p + o IP () he assertion follows from (6). Proof of Corollary 3.8: In this case, [A8] ensures /2 t= h(y t, θ o ) exp(x t, ) Z, where Z is a Gaussian random element in L 2 [, π] with the covariance operator K. Analogous to the proof for heorem 3.7, the conclusion follows from (9). D 28

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33 able : Models in Domínguez and Lobato (2004) with exogenous regressors. Sample X N (0, ) X N (, ) Estimator Bias SE MSE Bias SE MSE 50 ˆθNLS ˆθ OPIV ˆθ DL ˆθ(exp, K ) θ e (exp, K ) ˆθNLS ˆθ OPIV ˆθ DL ˆθ(exp, K ) θ e (exp, K ) ˆθNLS ˆθ OPIV ˆθ DL ˆθ(exp, K ) θ e (exp, K )