Interview With Lars Peter Hansen
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- Bu səhifədəki naviqasiya:
- Lars Peter Hansen (L.P.H.)
- 2002 American Statistical Association Journal of Business Economic Statistics October 2002, Vol. 20, No. 4
- 4. Editors
| Interview With Lars Peter Hansen 1. Eric Ghysels and Alastair Hall (Editors): How did you come to be interested in estimation based on moment conditions? Lars Peter Hansen (L.P.H.): As a graduate stu- dent at the University of Minnesota, I had the opportu- nity to take classes from Chris Sims and Tom Sargent. Both emphasized the idea that dynamic econometric models should be viewed as restrictions on stochas- tic processes. Sims’s classroom development of large- sample econometrics broke with the more conventional view of the time of focusing on models with exogenous regressors or instrumental variables (IVs). His work on causality had emphasized the testable restrictions on stochastic processes of the notion of strict exogeneity. The idea of embedding estimation and testing prob- lems with a stochastic process framework proved to be a very useful starting point. Although this is common in econometrics today, it was not at the time I was a graduate student. I was exposed to nonlinear estimation and testing problems in Sims’s graduate course. Sims loved to experiment with new material and proofs in class, and I learned much from lling in details and some holes in lecture notes. His nonconventional teaching style was valuable for me. While I was working on my dissertation on exhaustible resources, I became interested in central limit theory for temporally dependent processes. This interest predated the excellent book by Hall and Heyde (1980). Sims had made reference to work by Scott (1973), but the central paper for me was that of Gordin (1969). (Scott indeed cites Gordin, which is how I became familiar with that work.) Gordin showed how to construct a martingale approximation to a wide class of stationary ergodic processes. It thus extended the applicability of earlier martingale central limit the- ory by Billingsley (1961) and others to processes with a rich temporal dependence structure.
estimation with temporally dependent disturbances. The motivation for this paper was simple. As an editor of Econometrica, Sims was handling a paper by Brown and Maital (1981) on assessing the ef - ciency of multiperiod forecasts. The multiperiod nature induced temporal dependence in the distur- bance term. Sims urged me to proceed quickly to get something written up as a paper that could be cited and used. Hodrick (my colleague at Carnegie Mellon at the time) made me aware of similar problems in the literature on the study for forward exchange rates as predictors of future spot rates. When the forward contract interval exceeds the sampling interval, tem- poral dependence is induced. Given the serial correla- tion in the disturbance term, many applied researchers at the time thought the right thing to do was to use a standard generalized least squares (GLS)-type correc- tion. In fact, while least squares remains consistent, the lack of strict exogeneity of the regressors prevents GLS from being consistent. This tripped up several researchers and was the impetus for my original least squares paper. In addition to the Brown and Maital paper, which with Sims’s in uence proceeded cor- rectly, Hodrick quickly informed authors of the papers submitted to the Journal of Political Economy that the GLS style correction that they had used was invalid. My least squares paper was rejected at Economet- rica because it failed to be ambitious enough. This irritated me and made me restructure the arguments in much greater generality. I had seen recent lecture notes of Sims that described a family of generalized method of moments (GMM) estimators, indexed by a matrix that selected which moment conditions to use in estimation. Sims treated time series applications, but with a focus on martingale difference disturbances. I adopted this formulation to present a more general central limit approximation for estimators with Sims’s encouragement. I learned subsequently from a referee that the selection matrix idea had been used by Sargan (1958, 1959) in studies of linear and nonlinear instru- mental variables estimators. I was a bit embarrassed that I had not cited the very nice Sargan (1958) paper in my original submission. This paper provided the impetus for my analysis of the limiting distribution of sample moment conditions evaluated at parameter estimators. Since previous referees complained about my casual treatment of consistency, I added some speci city by using a quadratic form construction as in the statis- tics literature on minimum chi-squared estimation and in Amemiya’s (1974, 1977) treatments of nonlinear two- and three-stage least squares. While developing consistency results, I became interested in the role of compactness in consistency arguments and explored alternative ways of justifying consistency based on tail characterizations of the objective functions.
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Journal of Business & Economic Statistics October 2002, Vol. 20, No. 4 DOI 10.1198/073500102288618577 442 Interview With Lars Peter Hansen 443
L.P.H.: Bob Hodrick and I began our discussion of econometric issues when I described to him the pitfalls in applying GLS in econometric equations that come from multiperiod forecasting problems. He immediately showed me working papers in the liter- ature on forward exchange rates in which GLS was applied as a remedy for serial correlation. At the same time he was educating people in international nance, he and I began working on our own analysis of for- ward exchange rates by applying least squares and adjusting the standard errors. Bob Hodrick was at Carnegie Mellon when I arrived there. Ken Singleton came to Carnegie Mellon after me, and after I had written my GMM paper. Both of us knew Hall’s work on consumption (Hall 1978), and Ken came back from a conference after hearing an underappreciated paper by Grossman and Shiller. A stripped-down version of this paper was subsequently published in the American Economic Review (Gross- man and Shiller 1981). Singleton suggested that the consumption Euler equation could be fruitfully posed as a conditional moment restriction and our collabora- tion began. After I arrived at Chicago, Jim Heckman showed me MaCurdy’s microeconomic counterpart to Hall’s paper (MaCurdy 1978). Ken’s and my second (and empirically more interesting) paper published in the Journal of Political Economy (Hansen and Single- ton 1983) came about by our construction of a maxi- mum likelihood counterpart to the GMM estimation in our Econometrica paper. Given the more transparent nature of how predictability had implications measur- ing and assessing risk aversion, this project became more ambitious and took on a life of its own.
my recent contribution to the Encyclopedia of the Social and Behavioral Sciences (Hansen 2002). The quadratic-form criterion that I used to formulate con- sistency certainly has its origins in the minimum chi- squared estimators used to produce estimators that are statistically ef cient and computationally tractable. An interesting difference is that GMM estimators are often used to study models that are only partially spec- i ed, whereas the earlier statistics literature provided computationally attractive estimators for fully speci- ed models. Implicit in many GMM applications is a semiparametric notion of estimation and ef ciency, in contrast to minimum chi-squared estimation. There is a more recent related statistics literature on esti- mating equations, with time series contributions by Godambe and Heyde (1987) and others. This litera- ture was developed largely independently of the cor- responding literature in econometrics, but its focus has been primarily on martingale estimating equations. My aim was in part to use Gordin (1969)’s martingale approximation device to adopt a more general starting point.
applications of GMM estimation were motivated by conditional moment restrictions or had the following time series structure. Any time an IV existed, lags of the IVs were also valid instruments. As in my other work, I was particularly interested in time series prob- lems. Strict exogeneity of IVs also allows for leading values of IVs to be valid instruments, but this orthog- onality was not in my econometric formulation. Prior to my GMM paper, Robert Shiller (1972) had empha- sized that in rational expectations models, omission of conditioning information gave rise to orthogonal dis- turbance. In a time series problem, this Shiller notion of an error term often had the feature that this error term was orthogonal to arbitrary past values of vari- ables in a conditioning information set used by an econometrician. This same argument will not work for arbitrary leading values of the conditioning informa- tion, because economic agents will not have seen these values.
My initial work in this area (Hansen 1985) used martingale approximations to give lower bounds on the ef ciency of a class of feasible GMM problems. In formulating this problem, I had to extend the selec- tion approach of Sargan, Sims, and others to cases in which the number of underlying moment conditions is in nite. This is a natural framework within which to think about estimation problems in which nonlin- ear functions of IVs are valid IVs and particularly for time series problems in which lagged values of IVs remain valid instruments. This gave rise to an explic- itly in nite-dimensional family of GMM estimators. Attaining this ef ciency, especially in a time series problem, is more problematic. There is a need asymp- totically to use information arbitrarily far into the past. While I was working in this area, I became aware of closely related work in engineering aiming to con- struct online estimators that are easy to compute (see, e.g., Soderstrum and Stoica 1983). Some of their cal- culations were similar to mine. Also, Gary Chamber- lain wrote his very nice paper (Chamberlain 1987) on semiparametric ef ciency bounds based on conditional moment restrictions parameterized in terms of a nite- dimensional vector and posed in an i.i.d. context. After my initial characterizations, Singleton and I took a stab at implementation in linear models using Kalman ltering (see Hansen and Singleton 1991; 1996). In many of our examples, we often found only very modest gains in even asymptotic ef ciency to using the more fancy procedure over ad hoc choices of variables. While a co-editor at Econometrica, I han- dled one of Whitney Newey’s papers (Newey 1990) that, in an i.i.d. context, attained the ef ciency bound. In the sampling experiments that he studied, the ef - ciency gains were often small, and sometimes the 444 Journal of Business & Economic Statistics, October 2002 actual construction of a nonparametric estimator of an ef cient IV undermined the ef ciency gains in nite samples. Of course, characterization of the ef ciency bound was needed to reach this conclusion. It is interesting that subsequently, West and Wilcox (1996) reached a rather different conclusion and pro- duced interesting examples in the study of economic time series where the use of the ef ciency bounds were of practical value, and they devised some esti- mation methods that appear to work quite well. There has been a variety of other interesting and related research on choice of moment conditions to use in actual estimation. Complementary work of Andrews (1999) and Hall and Inoue (2001) consid- ered the selection of which moment conditions to use in parameter estimation. These papers addressed the important practical problem of how to limit the number of moment conditions used in estimation, by excluding irrelevant and invalid ones using adaptive methods.
6. Editors: Method of moments is commonly used now in the estimation of diffusions. Do you expect that this line of research will continue, or do you think that the various simulation-based maximum likelihood estimators will become more widely used? L.P.H.: My interest in GMM estimation has been primarily to achieve partial identi cation. That is, sup- pose the aim of the econometric exercise is to extract a piece of say a fully speci ed dynamic general equi- librium model. Thus the semiparametric language that Chamberlain emphasized in his work is appropriate. My work with Scheinkman on estimation of nonlin- ear continuous-time models (Hansen and Scheinkman 1995) and the subsequent work with Conley and Luttmer (Conley, Hansen, Luttmer, and Scheinkman 1997) is best viewed in this light. The idea is that the model is not designed to t everything, but that there are certain data features such as the long-run station- ary distribution that are the appropriate targets. Both of these papers also explore misspeci cation that takes the form of an exogenous subordination process in the spirit of Clark’s fundamental work (Clark 1973). Of course, partial identi cation while it may be more realistic, is not a fully ambitious research goal. For many purposes, more is needed. But once a dynamic model is fully speci ed in a parametric way, maximum likelihood methods and their Bayesian counterpart become attractive. It may be that for numerical reasons, as in Pearson’s original work and the subsequent extensive literature on minimum chi- squared estimation, method of moments methods are attractive alternatives to likelihood-type estimators. It has been interesting, however, to watch the develop- ment of numerical Bayesian methods, methods that make prior sensitivity analyses feasible. The Bayesian problem based on integrating or averaging is numeri- cally simpler than hill-climbing to nd a maximum in many circumstances.
tors in terms of quadratic form minimization involved two-step approaches or iterated approaches. An initial consistent estimator was produced and used to esti- mate an ef cient weighting matrix designed so that the objective has the minimum chi-squared property. In the rst-order asymptotic theory, weighting matrix estimation is placed in the background, because all that is needed is a consistent estimator. Stopping after one iteration or continuing are both options. It was interesting, though, that when Sargan sought to compare IVs estimators to limited information max- imum likelihood (LIML) estimators, he produced a depiction of the LIML estimator as an IV-type esti- mator in which the variance for the structural distur- bance term is estimated at the same time the under- lying parameters are estimated (Sargan 1958). He essentially concentrated out this variance in terms of the underlying parameters, and produced what can be thought of as a continuous-updated weight- ing matrix estimator. Under conditional homoscedas- ticity and a martingale difference structure for the dis- turbance term, the optimal weighting matrix is the product of the disturbance term variance and the sec- ond moment of the IVs. By using the sample second- moment matrix of the IV and the sample variance of the disturbance term expressed as a function of the unknown parameter vector, in a effect a continuous- updating weighting matrix is produced. A very similar approach is one of the ways that the minimum chi-squared estimator was implemented for multinomial models. The counterpart to the weight- ing matrix could be fully parameterized in terms of the multinomial probabilities. In this case the weight- ing matrix can be produced without reference to the data, but only a function of the unknown parameters. This is because the multinomial model is a full model of the data, while Sargan considered only a partially speci ed model. In the more general GMM context with a quadratic form–type objective, it turned out to be quite easy to implement Sargan’ approach, except that the sim- ple separation between the disturbance term and the vector of IVs can no longer be exploited. Instead, one constructs the long-run covariance matrices of the function of the data and parameter vector used to pro- duce the parameterized moment condition. I became interested in this estimator in part to sidestep the issue of whether to iterate on the weight- ing matrix or not and in part because the sensitivity of both the two-step and iterated estimator to what seemed like arbitrary normalizations. For instance, if one takes the original function of the data x
and a
Interview With Lars Peter Hansen 445
hypothetical parameter vector ‚ and scales it by some arbitrary function of ‚, the moment conditions are pre- served. The two-step estimator will be sensitive to this transformation, but the continuous-updating estimator typically will not be. Moreover, in GMM estimation problems that Marty Eichenbaum and I encountered (Eichenbaum and Hansen 1990), we found that in one depiction of the moment conditions, there might be degenerate solutions at the boundary of the parameter space. The continuous-updating estimator would not reward these parameter values, because the degener- acy would cause the weighting matrix to diverge. In contrast, a two-step estimator could easily end up at these often-uninteresting parameter con gurations. In our original paper on continuous-updated GMM estimation, Heaton, Yaron and I found in our Monte Carlo experiments that we tended to replicate what was known in the simultaneous equations literature. The continuous-updating estimator was close to being median unbiased, but it had fat tails. The weighting matrix adjustment often leads to problems in which the objective function itself can be at in the tails. We found that inference methods based on studying the shape and degradation of the objective function were more reliable than computing quadratic approx- imations. One formal reason for why this can be true is provided in the paper by Stock and Wright (2000), which studies moment conditions with weak informa- tion. I have found the recent work by Newey and Smith (2000) and Bonnal and Renault (2000) to be very interesting. These papers show how to nest the continuous-updated estimator into a class of estima- tors that includes empirical likelihood. Newey and Smith (2000), in particular, use second-order asymp- totic theory to study this class of estimators. They characterize the advantages in using empirical likeli- hood to produce parameter estimates when the data are i.i.d. My own interest in the continuous-updating GMM estimator is not so much as a method for pro- ducing point estimates, but more as a method of mak- ing approximate inference. 8. Editors: A lot of work has focused on estimation of the optimal weighting matrix, which lead to the so- called HAC (heteroskedastic and autocorrelation consis- tent) estimators. In joint work with Ravi Jagannathan, you also advocate the use of weighting matrices that are suboptimal from a statistical point of view, but have desirable properties in nancial applications. What are your thoughts on the issue of weighting matrices? L.P.H.: The optimal weighting matrix that you refer to is obtained by asking a statistical question. Given that a nite set of moment conditions are satis- ed, what is the most ef cient linear combination to use in estimating a parameter vector of interest. Ravi and I (Hansen and Jagannathan 1997) and also John Heaton, Erzo Luttmer, and I (Hansen, Heaton, and Luttmer 1995) were interested in a different question. Suppose that, strictly speaking, the model is misspec- i ed. How might you pick a good model among the parameterized family of models? How does the per- formance of a parameterized family compare to other models? This leads to a rather different view. Under misspeci cation, statistical ef ciency is put to the side. It becomes a “higher-order” issue. We explore the question of weighting matrix selection in an asset- pricing environment and pose the problem as one in which the aim is to keep pricing errors small. The tar- get of the estimation problems of necessity comes to the forefront in this exercise. Consider an idealized Bayesian decision problem, but suppose that the likelihood function is misspec- i ed. One cannot separate this problem into the following two problems. Find the correct posterior distribution for the parameter vector and then solve the decision problem by minimizing the loss function aver- aging over the posterior. In the presence of misspeci - cation, we cannot separate the statistician’s problem of nding the posterior from that of the decision maker who wishes to use the parameter vector in practice. Consider now the ction of an in nite sample. In a GMM context, the weighting matrix that you refer to dictates how important each of the pricing error equations is in pinning down the parameter estimate. However, under correct speci cation, this choice does not alter the parameter vector that, say, minimizes the population quadratic criterion. Under misspeci cation, the choice of weighting matrix changes parameter val- ues as you change the weighting matrix, but this also changes the performance criterion. In an asset-pricing context, different weighting matrices alter the impor- tance of pricing errors coming from alternative securi- ties. Parameter choices that lead to moment conditions because of a large covariance matrix in a central limit approximation are rewarded for reasons that may not be very appealing. In its most simple form, Ravi and I jus- ti ed an estimation exercise based on minimizing pric- ing errors that led to a xed choice of weighting matrix independent of the parameter value being considered. Since misspeci cation can destroy the simple sepa- ration between estimation and decision, the aim of the parameter choice comes to the forefront. Ravi and I can be criticized for our ad hoc choice of loss func- tion, but it is also sometimes unappealing to study moment relations in which the weighting matrix can change with the underlying parameter vector. Model misspeci cation is an example.
446 Journal of Business & Economic Statistics, October 2002 environment and takes us away from the “distribution- free” approach of GMM-based Euler equation estima- tion. What are your thoughts on this development? L.P.H.: GMM approaches based on Euler equa- tions were designed to deliver only part of an eco- nomic model. Their virtue and liability is that they are based on partial speci cation of an econometric model. They allow an applied researcher to study a piece of a full dynamic model, without getting hung up on the details of the remainder of the model. For many purposes, including policy analysis and the conduct of comparative dynamic exercises, the entire model has to be lled out. We could require this full speci cation of all econometric exercises in applied time series, but I think this would be counterproduc- tive. On the other hand, to solve decision problems by policy makers or to conduct comparative dynamic exercises a full decision problem, and a complete dynamic evolution must be speci ed. A GMM esti- mation of a portion of the model can be an input into this, but more is required. Rational expectations compels a researcher to think about model speci cation both from the standpoint of the applied researcher and from the standpoint of indi- vidual decision makers. We could presume that when we ll out the entire model including the dynamic evolution of state variables, this is done correctly. Rational expectations becomes an equilibrium con- struct that compels the researcher to check for correct speci cation from all angles, econometrician and eco- nomic agents. It is a powerful and tremendously pro- ductive equilibrium concept. The question is what do you do if you suspect misspeci cation. Interestingly, if you look at applied research in economic dynamics that imposes rational expectations, one of the reasons people give for not getting distracted by econometrics is that the model is obviously misspeci ed. Once you go down this road, however, you naturally ask from whose per- spective. The consistency requirement of the equi- librium concept forces this misspeci cation on the economic agents, and the description “rational expec- tations” becomes a bit of a misnomer. There is no obvi- ously simple x to this problem. We could remove misspeci cation from the table as rational expectations does in such a clever way, but this seems incompati- ble with much applied research and arguably individual decision making. The work you describe (with a variety of co- authors) is one stab at confronting model misspeci - cation. Since we are compelled to put a rich class of models in play, typically in the form of an approxi- mating or benchmark model and a family of pertur- bations, there are hard questions about which models should be put into contention and how they should be confronted by decision makers. We have been working on some tractable ways to approach model misspeci - cation, building from a rich literature in robust control theory, but this is far from a mature literature. Our work to date has been very controversial, and some of this controversy is well justi ed. We are still sorting out strengths and weakness of alternative approaches in our own minds. Our robustness approach is related more closely to Sargent’s and my work (Hansen and Sargent 1993) and related work by Sims (1993) on estimation of fully speci ed dynamic rational expectations models using a misspeci ed likelihood function than it is to GMM estimation of partially speci ed models. Since we are compelled to work with well-posed decision problems, we need the entire model. 10. Editors: Over the last 20 years, there has been much progress in developing a framework for inference based on the GMM estimator. What do you perceive to be the strengths and weaknesses of the framework as it stands today? L.P.H.: GMM estimation is often best suited for models that are partially speci ed. This is both a strength and a weakness. Partial speci cation is convenient for a variety of reasons. It allows an econometrician to learn about something without needing to learn about everything. An appeal to par- tial speci cation, however, limits the questions that can be answered by an empirical investigation. For instance, the analysis of hypothetical interventions or policy changes typically requires a fully speci- ed dynamic, stochastic general equilibrium model. Applied researchers in macroeconomics and nance use highly stylized, and hence misspeci ed, ver- sions of such models. 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