Durham, Garland

Geweke, John

English

DigitalCommons@CalPoly

Comment Garland DURHAM and John GEWEKE Department of Economics, University of Iowa, Iowa City, lA 52242-1 000 ({garland-durham, john-geweke} @uiowa.edu) This article provides a comprehensive overview and integra­Lion of state-of-the-art econometric methods for models that are natural! y stated in terms of latent variables but present signif­icant practical problems for inference from data. ln so doing it extends these methods in ignificant ways by incorporating the important concept of backfitt.ing. ft shows explicitly how this extension applies to GMM and ML estimators. The spe­cific problems inherent in estimating affine pricing models of the term structure motivate much of the art.icle and provi de its illustrative applications. Along the way. the article treats a wide variety of related problems . These comments focus on the main theme leading to the e mpirica l application, which is the devel­opment of IS-GMM and the exte nsion to backfitting in Sec­lions 3 and 4. This article addres es circumstances in which moment con­dition are most naturally form ulated in terms of latent variables Y~ rather than observable counterparts Y,. (As in the art.icle. the s uperscri pt ··Q"' indicate the value of the parameter or moment in the data-generating process. which in turn presumes correct specification of the moment conclitions and model.) Motivated by asset pricing models. it takes up the case in which the latent and observable variables are linked by a relation Yt=g(Yt,A0), Yt=g- 1(Y,A0). (2) This relation is completely determined by the unknown para­meter vector (}, so that A = A(9) and, in particular, A 0 = A(9o). Definine-¢[Yt. 0. A(9)J = 1/t{g-1[Yt. A(9)], 9}, ( I) becomes ~{¢[Yt.O,A{9)]) =0. (3) An attraction of the article is that it treats extremum estima­tors in general. Thi s includes the GMM estimator based on (3) set forth at the start of Section 3.2, with the criterion function Qr(9, A) I TT (4)=­ r - 1 L ¢(Y1, ¢,A) Wr 1 1 L ¢CYt . ¢,A) t=l 1=1 It also includes the MLE that works with the criterion function T Qr(O' A)= 1 1 L r{g- 1(Yt , A.)lg- 1( Yt-1· A): 0} t= l T +1 1 LlogiJyg- 1(Yt, A) I, (5) t=1 in which e• denotes the log-pdf of the latent r: and Jy is the Jacobian of transformation. Sections 3 and 4 contrast three clifferent ways of handling the identity ).. = >..(9) in (4) or (5) . The infeasible oracle estimator. introduced at the start of Sec­tion 3.2, is B:;. = arg max Qr(9. A0 ). Oe0 and the asymptotic variance of this estimator is presented in the same paragraph. The /S-GMM estimator is Oj.5 = argmaxQr[e. A(B)l . (6)oee introduced in the next paragraph. and its asymptotic variance is the semi parametric efficiency bound indicated in equation (3.3) in the article. ~ The /S-GMM backjilfing estimator is 8r. the limit (in p) of the sequence ().~'!> = arg max Qr[(). A(O,Y- 1 ')]. (7)He0 The formulation (7) is first presented in Section 3.2. after (3.7). If one defines Br(A} = argmax Qr(O, A).Oe0 which the article does in (3.7), then (7) can be expressed as (8) Computation of 8}5 entails evaluating derivatives of A(B)..... [see (6)] whereas computati on of Or requires only evaluation of the function A((:}) itself [see (7)]. Because A(B) is typically complicated, this is a significant practical advantage of back­fitting. For example, in ML the Jacobian term appearing in the last term of (5) involves only A. Whereas Bj-5 must contend di­rectly with the shape of this term in each iteration to maximum, ~ Br needs to evaluate it only once each step. The article defines -8r in (4.1) with reference to an iteration ­stopping rule p(D. but it is simpler to work with the limit the sequence in (8). and in any event conventional convergence cri­teria rather than a fixed function p(T) are used in the application in Section 6. The article develops conditions for the conver­gence of e?' in p to the estimator 8r that in tum guarantee weak consistency (Prop. 2). The most important of these, in our view, is the contraction mapping of Assumption 6. Proposition 3 pro­vides the asymptotic variance of -Br. The contrast between ej.S and Br is important in understand­ing the article's contribution, in interpreting either estimate in practice, and in choosing an estimator in an empirical appli­cation. The article States that. at least typically. ej.S and Br coincide so long as (8) has a fixed point rsec. 3.2. end of the paragraph inclucling (3. 7)]. If that were the case, then these esti­mators would have the same asymptotic variance. but the article demonstrates that these variances are different. We find that a geometric interpretation of the estimators ef-5 and Br is helpful in appreciating these differences. Fig­ure I portrays what is perhaps the simplest possible situation. The concentric circles in the figure represent level curves of Qr(e, A): in the case portrayed. there is no interaction between e and A in Qr(B. A). The positively sloped straight line rep­-resents the identity A= .\.(0) = a9. The function 9r(A) is the ti(>.) >.(8) • (IOr Figure 1. locus of tangencies of horizontal lies ro the concentric circle level curves, and therefore is the vertical (joe through the cen­ter of these circles portrayed in Figure I. The estimator {Jj.5 is the abscissa of the point of tangency between this li ne and the level curves. The sequence 8jl!1 in this example converges in exactly two s teps. From any initial value 8 (01 on the abscissa, the firs t step in (8) maps vertically to >.~1 = >- (8(01 ) and then horizontally to e~' l = Or(>.~' \ The second step maps verti­call y to >-i) = 1.(8}11) and then finds that 8?1 = Br(>.. W). Be­cause 8}2) = 8~1 ). the iterations have converged. and in fact 0; = 8}2' = ei'1• Thi s example illustrates a number of general points about G.f5and Br. Obvi ously the estimators are not the same, very special cases aside. If ).(B) were horizontal , [i.e ., ).(8) = >.*"18 E 9]. then efS =fir. Thi s. in tum . is a case in which there is no non­adaptivit y problem and illustrates the article's points about the importance of nonadaptivity in comparing the asymptotic dis­tributions of Bf5 and ffr. This very special case suggests that in any application in which A. is not very responsive to() and there is little interaction between 8 and >. in the criterion function (4) or (5), backfiLLing may achieve resu.lts close to [S-GMM , with the advantage of s ubstantial computational efficiency. On the other hand, given >. = >..(()) =a8 , a quadratic criterion function and a Gaussian data-generating process, it is straightforward to s how. in the context of Figure I. that var(Or)1var({Jj.5) = I + a2 . This raises general questions about efficiency loss due to back­fitting that might be investigated in more detail in future work. ff the model underlying the hypothetical situation in Fig­ure I is specified correctly. then the center of the concentric circles will move, stochas tically, toward the line >. (8 ) as sam­ple size T increases. Because both estimators are consistent. G.fS- 87 .!!. 0 , but i 112(1fj5 - 1fr) wilJ have a nondegenerate limiting distribution whose variance will depend on the rela­tive orientation of 1.(8) and the level cw·ves, as discussed in the previous paragraph. We would expect differences relative ro standard errors to persis t. (In the application in the article, it seems to us that this is the case. but the differences are not large, suggesting that the limiting case a = 0 might be an ide­alized. rough approximation in this application. ) On the other hand, if the model underlying the situation portrayed in Fig­ure I is misspecified, then the center of the circles will not, in general , converge to a point on 1.(8) . The estimators 8{.5 and O.r will converge to diffe rent pseudotrue values, and in the metric of the standard error of either one. differences between them will grow. Generali zing Figure I to the case in which level curves are ellipses rather than circl es is inf ormative in illustrating some of the other point~ in Section s 3 and 4 of the article. The function 8(1.) remains a straight line, but is no longer vertical. As long as A(t9) and (}()..) have different slopes, there will be exactly one fixed point. The sequence [ejfJ } either converges toward this point or diverges from it, depending on whether the contraction mapping conditions of Assumption 6 are violated. With linear A.(8) and{}(>.), these conditions reduce to simple inequalities involving the respective slopes of the two fw1ctions. Relevant app(jcations , including those in asset pricing. do not, of course, have the simplicity of either Figure I or this mild extension . The parameters 8 and >.. are vectors, not scalars. and so two-dimensional di agrams cannot represent the situation ad­equately. More important. in our view. is the fact that the func ­tions 1. (8) and 8(1.) are nonlinear in interesting application s, including those presented in Section 6 of the article. Thi s opens up possibilities like the very simplified one portrayed in Fig­ure 2. There are two fixed points. (e,. A.!) and ((h . J..2). The first point sati.sfies the contraction mapping condition. but the second point does not. Over the range portrayed in Figure 2. if e~O) > (h then limp-.co e!f' = () l , whereas if B~Ol < fh, then e~j!l rapidly diverges downward (to the left in Fig . 2). More imponant in Figure 2 is the fact that no level c<mtours of Q; have been included. In part, this was done to keep that il­lustration simple, but it is also to allow the reader lO s ketch level curves of his or her own. verifying that in this situation. dif­ferences between 8{.5 and ffr could be extremel y great, and er could be misleading. A reliable empirical application of back­fitting must rule out situations like the one portrayed in Fig­ure 2. l n the empirical work reported in Section 6 of the arti­cle, the contraction mapping conditions of Assumption 6 do not hold , and multiple stationary points of !8!(' l were found. The ~ selected Or (Tables 5 and 6) bear the imerpretation of not dif­fering drastically from {Jj.5 (Tables 3 and 4). Such compari sons will not always be possible, however, if this research progran1 realizes the goal of using backfitting for inferences in situations impracticable for IS-GMM . Identification of this critical area is one of the article ' s important contributi ons. ..\(0) Figure 2. 0 

Comment on  Iterative and Recursive Estimation in Structural Nonadaptive Models Iterative and Recursive Estimation in Structural Nonadaptive Models  by S. Pastorello, V. Patilea, and E. Renault

https://digitalcommons.calpoly.edu/cgi/viewcontent.cgi?article=1017&amp;context=fin_fac

Comment on Iterative and Recursive Estimation in Structural Nonadaptive Models Iterative and Recursive Estimation in Structural Nonadaptive Models by S. Pastorello, V. Patilea, and E. Renault

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