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Penalized Sieve Estimation and Inference of Semi-nonparametric Dynamic Models

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PENALIZED SIEVE ESTIMATION AND INFERENCE OF SEMI-NONPARAMETRIC DYNAMIC MODELS: A SELECTIVE REVIEW By Xiaohong Chen May 2011 COWLES FOUNDATION DISCUSSION PAPER NO. 1804 COWLES FOUNDATION FOR RESEARCH IN ECONOMICS YALE UNIVERSITY Box 208281 New Haven, Connecticut 06520-8281 http://cowles.econ.yale.edu/Penalized Sieve Estimation and Inference of Semi-nonparametricDynamic Models: A Selective Review Xiaohong ChenyYale UniversityApril 2011AbstractIn this selective review, we …rst provide some empirical examples that motivate the usefulness ofsemi-nonparametric techniques in modelling economic and …nancial time series. We describe popularclasses of semi-nonparametric dynamic models and some temporal dependence properties. We thenpresent penalized sieve extremum (PSE) estimation as a general method for semi-nonparametricmodels with cross-sectional, panel, time series, or spatial data. The method is especially powerfulin estimating di¢ cult ill-posed inverse problems such as semi-nonparametric mixtures or conditionalmoment restrictions. We review recent advances on inference and large sample properties of thePSE estimators, which include (1) consistency and convergence rates of the PSE estimator of thenonparametric part; (2) limiting distributions of plug-in PSE estimators of functionals th at areeither smooth (i.e., root-n estimable) or non-smooth (i.e., slower than root-n estimable); (3) simplecriterion-based inference for plug-in PSE estimation of smooth or non-smooth functionals; and (4)root-n asymptotic normality of semiparametric two-step estimators and their consistent varianceestimators. Examples from dynamic asset pricing, nonlinear spatial VAR, semiparametric GARCH,and copula-based multivariate …nancial models are used to illustrate the general resu lts.Keywords: Nonlinear time series, Temporal dependence, Tail dependence, Penalized sieve M es-timation, Penalized sieve minimum distance, Semiparametric two-step, Nonlinear ill-posed inverse,Mixtures, Conditional moment restrictions, Nonparametric endogeneity, Dynamic asset pricing, Vary-ing coe¢ cient VAR, GARCH, Copulas, Value-at-risk.JEL: C13, C14, C20. This paper was pr esented as an invited le cture at the World Congress of the Econometric Socie ty, Shanghai, August20 10. It was subsequently presented as three invited graduate lectures at CEMFI, Madrid, March 201 1. I thank ManuelArellano, David Ch ilders, Tim Christensen, Michael Jansson, Oliver Linton, Demian Pouzo and Enrique Sentana for helpfuldiscussions and Kieran Walsh for excell ent research assistance. I am grateful to Manuel Arellano, Lars Hansen and PeterRobi nson for encouragement. I acknowledge …nancial support from t he National Science Fou ndation via g rant SES-0838161and the Cowles Foundation.yCowles Foundation for Research in Eco no mics, Yale University, 30 Hillhouse A ve, Box 208281, New Haven, CT 06520,USA . E-mail address : [email protected] Introduction 12 Vast Classes of Semi-nonparametric Dynamic Models 32.1 Motivating empirical applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Partial list of semi-nonparametric time series models . . . . . . . . . . . . . . . . . . . . 92.3 Digression: nonlinearity and temporal dependence . . . . . . . . . . . . . . . . . . . . . 133 Penalized Sieve Extremum (PSE) Estimation 183.1 Ill-posed versus well-posed problems and PSE estimation . . . . . . . . . . . . . . . . . . 193.2 Penalized sieve M estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3 Penalized sieve MD estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 Large Sample Properties of PSE Estimators 264.1 Consistency, c onvergence rates of PSE estimators . . . . . . . . . . . . . . . . . . . . . . 274.2 Limiting distributions and inference for PSE estimation of functionals . . . . . . . . . . 294.2.1 Simultaneous penalized sieve M estimators . . . . . . . . . . . . . . . . . . . . . 294.2.2 Simultaneous penalized sieve MD estimators . . . . . . . . . . . . . . . . . . . . 305 Semiparametric Two-step Estimation 335.1 Consistent sieve estimators of Avar(bn) . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.2 Semiparametric multi-step e stimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366 Concluding Remarks 381 IntroductionIn this paper we review some recent developments in large sample theory for estimation of and infer-ence on semi-nonparametric time series models via the method of penalized sieves. To avoid confusion,we u se the same terminology as that in Chen (2007). An econometric (or statistical) model is a familyof probability distributions indexed by unknown p arameters. We call a model “parametric” if all of itsunknown parameters belong to …nite dimensional Euclidean spaces. We call a model “nonparametric”if all of its unknown parameters belong to in…nite dimensional function spaces. A model is “semipara-metric” if its parameters of interest belong to …nite dimensional spaces but its nuisance parametersare in in…nite dimensional spaces. Finally, a model is “semi-nonparametric” if it contains both …nitedimensional and in…nite dimensional unknown parameters of interest.Semi-nonparametric models and methods have become popular in much theoretical and empiricalwork in e conomics . This is partly because it is often the case that economic theory suggests neitherparametric functional relationships amon g economic variables nor particular parametric forms f or er-ror distributions. Another reason f or the rising popularity of semi-nonparametric models is rapidlydeclining costs of collecting and analyzing large data sets. The semi-nonparametric approach is very‡exible in economic structural modelling and policy and welfare analysis. Compared to parametric andsemiparametric approaches, semi-nonparametrics are more robust to functional form misspeci…cationand are better able to discover nonlinear economic relations. Compared to fully nonparametric meth-ods, semi-nonparametrics su¤er less from the “curse of dimensionality” and allow for more accurateestimation of structural parameters of interest.Semi-nonparametric time series models and methods should be very useful for economic structu raltime series analysis. Many economic and …nancial time series (and panel time series) are nonlinearand non-Gaussian; see, e.g., Granger (2003). Examples include but are not


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