DOCSLIB.ORG

A Semiparametric Statistical Approach to Model-Free Policy Evaluation

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Semiparametric Statistical Approach to Model-Free Policy Evaluation A Semiparametric Statistical Approach to Model-Free Policy Evaluation Tsuyoshi Ueno† [email protected] Motoaki Kawanabe‡ [email protected] Takeshi Mori† [email protected] Shin-ichi Maeda† [email protected] Shin Ishii† [email protected] †Graduate School of Informatics, Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan ‡Fraunhofer FIRST, IDA, Kekul´estr. 7, 12489 Berlin, Germany Abstract parametric models. Linear function approximation has mostly been used due to their simplicity and com- Reinforcement learning (RL) methods based putational convenience. on least-squares temporal difference (LSTD) have been developed recently and have shown To estimate the value function with a linear model, good practical performance. However, the an online procedure called temporal difference (TD) quality of their estimation has not been well learning (Sutton & Barto, 1998) and a batch proce- elucidated. In this article, we discuss LSTD- dure called least-squares temporal difference (LSTD) based policy evaluation from the new view- learning are widely used (Bradtke & Barto, 1996). point of semiparametric statistical inference. LSTD can achieve fast learning, because it uses en- In fact, the estimator can be obtained from a tire sample trajectories simultaneously. Recently, ef- particular estimating function which guaran- ficient procedures for policy improvement combined tees its convergence to the true value asymp- with policy evaluation by LSTD have been developed, totically, without specifying a model of the and have shown good performance in realistic prob- environment. Based on these observations, lems. For example, the least squares policy itera- we 1) analyze the asymptotic variance of an tion (LSPI) method maximizes the Q-function esti- LSTD-based estimator, 2) derive the opti- mated by LSTD (Lagoudakis & Parr, 2003), and the mal estimating function with the minimum natural actor-critic (NAC) algorithm uses the natu- asymptotic estimation variance, and 3) derive ral policy gradient obtained by LSTD (Peters et al., a suboptimal estimator to reduce the com- 2005). Although variance reduction techniques have putational burden in obtaining the optimal been proposed for other RL algorithms (Greensmith estimating function. et al., 2004; Mannor et al., 2007), the important issue of how to evaluate and reduce the estimation variance of LSTD learning remains unresolved. 1. Introduction In this article, we discuss LSTD-based policy evalua- Reinforcement learning (RL) is a machine learning tion in the framework of semiparmetric statistical in- framework based on reward-related interactions with ference, which is new to the RL field. Estimation of environments (Sutton & Barto, 1998). In many RL linearly-represented value functions can be formulated methods, policy evaluation, in which a value function as a semiparametric inference problem, where the sta- is estimated from sample trajectories, is an important tistical model includes not only the parameters of in- step for improving a current policy. Since RL problems terest but also additional nuisance parameters with in- often involve high-dimensional state spaces, the value numerable degrees of freedom (Godambe, 1991; Amari functions are often approximated by low-dimensional & Kawanabe, 1997; Bickel et al., 1998). We approach this problem by using estimating functions, which pro- th Appearing in Proceedings of the 25 International Confer- vide a well-established method for semiparametric es- ence on Machine Learning, Helsinki, Finland, 2008. Copy- timation (Godambe, 1991). We then show that the in- right 2008 by the author(s)/owner(s). strumental variable method, a technique used in LSTD A Semiparametric Statistical Approach to Model-Free Policy Evaluation learning, can be constructed from an estimating func- rewritten as tion which guarantees its consistency (asymptotic lack V π(s ) = p(s |s )¯r(s ,s ) − r¯ of bias) by definition. t t+1 t t t+1 stX+1∈S As the main results, we show the asymptotic esti- + p(s |s )V π(s ), (3) mation variance in a general instrumental variable t+1 t t+1 stX+1∈S method (Lemma 2) and the optimal estimating func- tion that yields the minimum asymptotic variance of where the estimation (Theorem 1). We also derive a sub- p(st+1|st) := π(st, at)p(st+1|st, at) and optimal instrumental variable, based on the idea of atP∈A π(st,at)p(st+1|st,at)r(st,at,st+1) a the c-estimator (Amari & Kawanabe, 1997), to reduce r¯(s ,s ) := t∈A . t t+1 P p(st+1|st) the computational difficulty of estimating the optimal Throughout this article, we assume that the linear instrumental variable (Theorem 2). As a proof of con- function approximation is faithful, and discuss only cept, we compare the mean squared error (MSE) of asymptotic estimation variance. (In general cases, our new estimators with that of LSTD on a simple bias becomes non-negligible and selection of basis example of the Markov decision process (MDP). functions is more important.) Assumption 2. The value function can be repre- 2. Background sented as a linear function of some features: 2.1. MDPs and Policy Evaluation π ⊤ ⊤ V (st) = φ(st) θ = φt θ, (4) RL is an approach to finding an optimal policy for where φ(s) : S → Rm is a feature vector and θ ∈ Rm sequential decision-making in an unknown environ- is a parameter vector. ment. We consider a finite MDP, which is defined as a quadruple (S, A,p,r): S is a finite set of states; A Here, the symbol ⊤ denotes a transpose and the di- mensionality of the feature vector m is smaller than the is a finite set of actions; p(st+1|st, at) is the transition number of states |S|. Substituting eq. (4) for eq. (3), probability to a next state st+1 when taking an action we obtain the following equation at at state st; and r(st, at,st+1) is a reward received ⊤ with the state transition. Let π(st, at) = p(at|st) be a stochastic policy that the agent follows. We introduce φt − p(st+1|st)φt+1 θ = the following assumption concerning the MDP. stX+1∈S Assumption 1. An MDP has a stationary state dis- π p(st+1|st)¯r(st,st+1) − r.¯ (5) tribution d (s) = p(s) under the policy π(st, at). stX+1∈S There are two major choices in definition of the state When the matrix ⊤ value function: discounted reward accumulation and π E φt− p(st+1|st)φt+1 φt− p(st+1|st)φt+1 average reward (Bertsekas & Tsitsiklis, 1996). With " st+1∈S ! st+1∈S ! # P P the former choice, the value function is defined as is non-singular and p(st+1|st) is known, we can easily ∞ obtain the parameter θ. However, since p(st+1|st) π π t V (s) := E γ rt+1|s0 = s , (1) is unknown in normal RL settings, we have to Xt=0 estimate this parameter from the sample trajectory {s , a , r , · · · ,s , a , r } alone, instead of Eπ 0 0 1 N−1 N−1 N where [·|s0 = s] is the expectation with respect using it directly. to the sample trajectory conditioned on s0 = s and rt+1 := r(st, at,st+1). γ ∈ [0, 1) is the discount factor. Eq. (5) can be rewritten as With the latter choice, on the other hand, the value ⊤ yt = x θ + ǫt, (6) function is defined as t ∞ where yt, xt and ǫt are defined as V π(s) := Eπ [r − r¯|s = s] , (2) t+1 0 yt := rt+1 − r,¯ xt := φt − φt+1 t=0 X ⊤ wherer ¯ := dπ(s)π(s, a)p(s′|s, a)r(s,a,s′) ′ ǫt := φt+1 − p(st+1|st)φt+1 θ sP∈S aP∈A sP∈S denotes the average reward over the stationary distri- stX+1∈S bution. +rt+1 − p(st+1|st)¯r(st,st+1). (7) According to the Bellman equation, eq. (2) can be stX+1∈S A Semiparametric Statistical Approach to Model-Free Policy Evaluation When we use the discounted reward accumulation for p(x) and the conditional distribution p(y|x) of output the value function, eq. (6) also holds with y given x, respectively. Then, the joint distribution becomes yt := rt+1, xt := φt − γφt+1 ⊤ p(x,y; θ, kx, kǫ) = p(y|x; θ, kǫ)p(x; kx). (11) ǫt := γ φt+1 − p(st+1|st)φt+1 θ We would like to estimate the parameter θ represent- stX+1∈S ing the value function in the presence of the extra un- knowns kx and kǫ, which can have innumerable de- + rt+1 − p(st+1|st)¯r(st,st+1). (8) grees of freedom. Statistical models which contain stX+1∈S such (possibly infinite-dimensional) nuisance param- π Because E [ǫt] = 0, eq. (6) can be seen as a linear eters in addition to parameters of interest are called regression problem, where x, y and ǫ are an input, an semiparametric (Bickel et al., 1998). In semiparamet- output and observation noise, respectively (Bradtke ric inference, one established way of estimating param- & Barto, 1996). Note that eters is to employ an estimating function (Godambe, 1991), which can give a consistent estimator of θ with- π E [ǫtg(st,st−1, · · · ,s0)] = 0 (9) out estimation of the nuisance parameters kx and kǫ. Now we begin with a short overview of the estimating holds for any function g(st,st−1, · · · ,s0) because of function in the simple i.i.d. case, and then discuss the the Markov property. The regression problem (6) has Markov chain case.
Load more

Recommended publications
  • Instrumental Regression in Partially Linear Models
    10-167 Research Group: Econometrics and Statistics September, 2009 Instrumental Regression in Partially Linear Models JEAN-PIERRE FLORENS, JAN JOHANNES AND SEBASTIEN VAN BELLEGEM INSTRUMENTAL REGRESSION IN PARTIALLY LINEAR MODELS∗ Jean-Pierre Florens1 Jan Johannes2 Sebastien´ Van Bellegem3 First version: January 24, 2008. This version: September 10, 2009 Abstract We consider the semiparametric regression Xtβ+φ(Z) where β and φ( ) are unknown · slope coefficient vector and function, and where the variables (X,Z) are endogeneous. We propose necessary and sufficient conditions for the identification of the parameters in the presence of instrumental variables. We also focus on the estimation of β. An incorrect parameterization of φ may generally lead to an inconsistent estimator of β, whereas even consistent nonparametric estimators for φ imply a slow rate of convergence of the estimator of β. An additional complication is that the solution of the equation necessitates the inversion of a compact operator that has to be estimated nonparametri- cally. In general this inversion is not stable, thus the estimation of β is ill-posed. In this paper, a √n-consistent estimator for β is derived under mild assumptions. One of these assumptions is given by the so-called source condition that is explicitly interprated in the paper. Finally we show that the estimator achieves the semiparametric efficiency bound, even if the model is heteroscedastic. Monte Carlo simulations demonstrate the reasonable performance of the estimation procedure on finite samples. Keywords: Partially linear model, semiparametric regression, instrumental variables, endo- geneity, ill-posed inverse problem, Tikhonov regularization, root-N consistent estimation, semiparametric efficiency bound JEL classifications: Primary C14; secondary C30 ∗We are grateful to R.
    [Show full text]
  • Arxiv:1609.06421V4 [Math.ST] 26 Sep 2019 Semiparametric Identification and Fisher Information∗
    Semiparametric Identification and Fisher Information∗ Juan Carlos Escanciano† Universidad Carlos III de Madrid September 25th, 2019 Abstract This paper provides a systematic approach to semiparametric identification that is based on statistical information as a measure of its “quality”. Identification can be regular or irregular, depending on whether the Fisher information for the parameter is positive or zero, respectively. I first characterize these cases in models with densities linear in a nonparametric parameter. I then introduce a novel “generalized Fisher information”. If positive, it implies (possibly irregular) identification when other conditions hold. If zero, it implies impossibility results on rates of estimation. Three examples illustrate the applicability of the general results. First, I find necessary conditions for semiparametric regular identification in a structural model for unemployment duration with two spells and nonparametric heterogeneity. Second, I show irregular identification of the median willingness to pay in contingent valuation studies. Finally, I study identification of the discount factor and average measures of risk aversion in a nonparametric Euler Equation with nonparametric measurement error in consumption. Keywords: Identification; Semiparametric Models; Fisher Information. arXiv:1609.06421v4 [math.ST] 26 Sep 2019 JEL classification: C14; C31; C33; C35 ∗First version: September 20th, 2016. †Department of Economics, Universidad Carlos III de Madrid, email: [email protected]. Research funded by the Spanish Programa de Generaci´on de Conocimiento, reference number PGC2018-096732-B-I00. I thank Michael Jansson, Ulrich M¨uller, Whitney Newey, Jack Porter, Pedro Sant’Anna, Ruli Xiao, and seminar par- ticipants at BC, Indiana, MIT, Texas A&M, UBC, Vanderbilt and participants of the 2018 Conference on Identification in Econometrics for useful comments.
    [Show full text]
  • Semiparametric Efficiency
    Faculteit Wetenschappen Vakgroep Toegepaste Wiskunde en Informatica Semiparametric Efficiency Karel Vermeulen Prof. Dr. S. Vansteelandt Proefschrift ingediend tot het behalen van de graad van Master in de Wiskunde, afstudeerrichting Toegepaste Wiskunde Academiejaar 2010-2011 To my parents, my brother Lukas To my best friend Sara A mathematical theory is not to be considered complete until you have made it so clear that you can explain it to the first man whom you meet on the street::: David Hilbert Preface My interest in semiparametric theory awoke several years ago, two and a half to be precise. That time, I was in my third year of mathematics. I had to choose a subject for my Bachelor thesis. My decision was: A geometrical approach to the asymptotic efficiency of estimators, based on the monograph by Anastasios Tsiatis, [35], under the supervision of Prof. Dr. Stijn Vansteelandt. In this manner, I entered the world of semiparametric efficiency. However, at that point I did not know it was just the beginning. Shortly after I wrote this Bachelor thesis, Prof. Dr. Stijn Vansteelandt asked me to be involved in research on semiparametric inference for so called probabilistic index models, in the context of a one month student job. Under his guidance, I applied semiparametric estimation theory to this setting and obtained the semiparametric efficiency bound to which efficiency of estimators in this model can be measured. Some results of this research are contained within this thesis. While short, this experience really convinced me I wanted to write a thesis in semiparametric efficiency. That feeling was only more encouraged after following the course Causality and Missing Data, taught by Prof.
    [Show full text]
  • Efficient Estimation in Single Index Models Through Smoothing Splines
    arXiv: 1612.00068 Efficient Estimation in Single Index Models through Smoothing splines Arun K. Kuchibhotla and Rohit K. Patra University of Pennsylvania and University of Florida e-mail: [email protected]; [email protected] Abstract: We consider estimation and inference in a single index regression model with an unknown but smooth link function. In contrast to the standard approach of using kernels or regression splines, we use smoothing splines to estimate the smooth link function. We develop a method to compute the penalized least squares estimators (PLSEs) of the para- metric and the nonparametric components given independent and identically distributed (i.i.d.) data. We prove the consistency and find the rates of convergence of the estimators. We establish asymptotic normality under under mild assumption and prove asymptotic effi- ciency of the parametric component under homoscedastic errors. A finite sample simulation corroborates our asymptotic theory. We also analyze a car mileage data set and a Ozone concentration data set. The identifiability and existence of the PLSEs are also investigated. Keywords and phrases: least favorable submodel, penalized least squares, semiparamet- ric model. 1. Introduction Consider a regression model where one observes i.i.d. copies of the predictor X ∈ Rd and the response Y ∈ R and is interested in estimating the regression function E(Y |X = ·). In nonpara- metric regression E(Y |X = ·) is generally assumed to satisfy some smoothness assumptions (e.g., twice continuously differentiable), but no assumptions are made on the form of dependence on X. While nonparametric models offer flexibility in modeling, the price for this flexibility can be high for two main reasons: the estimation precision decreases rapidly as d increases (“curse of dimensionality”) and the estimator can be hard to interpret when d > 1.
    [Show full text]
  • Introduction to Empirical Processes and Semiparametric Inference1
    This is page i Printer: Opaque this Introduction to Empirical Processes and Semiparametric Inference1 Michael R. Kosorok August 2006 1c 2006 SPRINGER SCIENCE+BUSINESS MEDIA, INC. All rights re- served. Permission is granted to print a copy of this preliminary version for non- commercial purposes but not to distribute it in printed or electronic form. The author would appreciate notification of typos and other suggestions for improve- ment. ii This is page iii Printer: Opaque this Preface The goal of this book is to introduce statisticians, and other researchers with a background in mathematical statistics, to empirical processes and semiparametric inference. These powerful research techniques are surpris- ingly useful for studying large sample properties of statistical estimates from realistically complex models as well as for developing new and im- proved approaches to statistical inference. This book is more a textbook than a research monograph, although some new results are presented in later chapters. The level of the book is more introductory than the seminal work of van der Vaart and Wellner (1996). In fact, another purpose of this work is to help readers prepare for the mathematically advanced van der Vaart and Wellner text, as well as for the semiparametric inference work of Bickel, Klaassen, Ritov and Wellner (1997). These two books, along with Pollard (1990) and chapters 19 and 25 of van der Vaart (1998), formulate a very complete and successful elucida- tion of modern empirical process methods. The present book owes much by the way of inspiration, concept, and notation to these previous works. What is perhaps new is the introductory, gradual and unified way this book introduces the reader to the field.
    [Show full text]
  • Variable Selection in Semiparametric Regression Modeling
    The Annals of Statistics 2008, Vol. 36, No. 1, 261–286 DOI: 10.1214/009053607000000604 c Institute of Mathematical Statistics, 2008 VARIABLE SELECTION IN SEMIPARAMETRIC REGRESSION MODELING By Runze Li1 and Hua Liang2 Pennsylvania State University and University of Rochester In this paper, we are concerned with how to select significant vari- ables in semiparametric modeling. Variable selection for semipara- metric regression models consists of two components: model selection for nonparametric components and selection of significant variables for the parametric portion. Thus, semiparametric variable selection is much more challenging than parametric variable selection (e.g., linear and generalized linear models) because traditional variable selection procedures including stepwise regression and the best subset selection now require separate model selection for the nonparametric compo- nents for each submodel. This leads to a very heavy computational burden. In this paper, we propose a class of variable selection proce- dures for semiparametric regression models using nonconcave penal- ized likelihood. We establish the rate of convergence of the resulting estimate. With proper choices of penalty functions and regulariza- tion parameters, we show the asymptotic normality of the resulting estimate and further demonstrate that the proposed procedures per- form as well as an oracle procedure. A semiparametric generalized likelihood ratio test is proposed to select significant variables in the nonparametric component. We investigate the asymptotic behavior of the proposed test and demonstrate that its limiting null distri- bution follows a chi-square distribution which is independent of the nuisance parameters. Extensive Monte Carlo simulation studies are conducted to examine the finite sample performance of the proposed variable selection procedures.
    [Show full text]
  • An Averaging Estimator for Two Step M Estimation in Semiparametric Models
    An Averaging Estimator for Two Step M Estimation in Semiparametric Models Ruoyao Shi∗ February, 2021 Abstract In a two step extremum estimation (M estimation) framework with a finite dimen- sional parameter of interest and a potentially infinite dimensional first step nuisance parameter, I propose an averaging estimator that combines a semiparametric esti- mator based on nonparametric first step and a parametric estimator which imposes parametric restrictions on the first step. The averaging weight is the sample analog of an infeasible optimal weight that minimizes the asymptotic quadratic risk. I show that under mild conditions, the asymptotic lower bound of the truncated quadratic risk dif- ference between the averaging estimator and the semiparametric estimator is strictly less than zero for a class of data generating processes (DGPs) that includes both cor- rect specification and varied degrees of misspecification of the parametric restrictions, and the asymptotic upper bound is weakly less than zero. Keywords: two step M estimation, semiparametric model, averaging estimator, uniform dominance, asymptotic quadratic risk JEL Codes: C13, C14, C51, C52 ∗Department of Economics, University of California Riverside, [email protected]. The author thanks Colin Cameron, Xu Cheng, Denis Chetverikov, Yanqin Fan, Jinyong Hahn, Bo Honoré, Toru Kitagawa, Zhipeng Liao, Hyungsik Roger Moon, Whitney Newey, Geert Ridder, Aman Ullah, Haiqing Xu and the participants at various seminars and conferences for helpful comments. This project is generously supported by UC Riverside Regents’ Faculty Fellowship 2019-2020. Zhuozhen Zhao provides great research assistance. All remaining errors are the author’s. 1 1 Introduction Semiparametric models, consisting of a parametric component and a nonparametric com- ponent, have gained popularity in economics.
    [Show full text]
  • Robust Estimates in Generalized Partially Linear Models (With Full Appendix)
    The Annals of Statistics 2006, Vol. 34, No. 6, 2856–2878 DOI: 10.1214/009053606000000858 c Institute of Mathematical Statistics, 2006 ROBUST ESTIMATES IN GENERALIZED PARTIALLY LINEAR MODELS1 By Graciela Boente, Xuming He and Jianhui Zhou Universidad de Buenos Aires and CONICET, University of Illinois at Urbana-Champaign and University of Virginia In this paper, we introduce a family of robust estimates for the parametric and nonparametric components under a generalized par- tially linear model, where the data are modeled by yi|(xi,ti) ∼ F (·, µi) T with µi = H(η(ti)+ xi β), for some known distribution function F and link function H. It is shown that the estimates of β are root-n consistent and asymptotically normal. Through a Monte Carlo study, the performance of these estimators is compared with that of the clas- sical ones. 1. Introduction. Semiparametric models contain both a parametric and a nonparametric component. Sometimes, the nonparametric component plays the role of a nuisance parameter. Much research has been done on estima- tors of the parametric component in a general framework, aiming to obtain asymptotically efficient estimators. The aim of this paper is to consider semi- parametric versions of the generalized linear models where the response y is to be predicted by covariates (x,t), where x Rp and t R. It will be assumed that the conditional distribution of y∈(x,t) belongs∈T to ⊂ the canonical exponential family exp[yθ(x,t) B(θ(x,t)) +|C(y)] for known functions B and C. Then µ(x,t)= E(y (x,t))− = B′(θ(x,t)), with B′ denoting the deriva- tive of B.
    [Show full text]
  • Semiparametric Theory, Using As a Running Example the Common Problem of Estimating an Average Causal Effect
    Semiparametric theory Edward H. Kennedy∗ Carnegie Mellon University, PA, USA September 20, 2017 Abstract In this paper we give a brief review of semiparametric theory, using as a running example the common problem of estimating an average causal effect. Semiparamet- ric models allow at least part of the data-generating process to be unspecified and unrestricted, and can often yield robust estimators that nonetheless behave similarly to those based on parametric likelihood assumptions, e.g., fast rates of convergence to normal limiting distributions. We discuss the basics of semiparametric theory, focusing on influence functions. Keywords: causal inference, double robustness, efficiency theory, functional estimation, influence functions, nonparametric theory, robust inference, tangent space. arXiv:1709.06418v1 [stat.ME] 15 Sep 2017 ∗Edward Kennedy ([email protected]) is Assistant Professor in the Department of Statistics, Carnegie Mellon University, Pittsburgh, PA 15213. In this paper we give a review of semiparametric theory, using as a running example the common problem of estimating an average causal effect. Our review draws heavily on foundational work in general semiparametrics by Begun et al. (1983), Bickel et al. (1993), Pfanzagl (1982), van der Vaart (2000), and van der Vaart (2002), among others (Newey 1994; Kosorok 2007), as well as many modern developments in missing data and causal inference problems by Robins & van der Laan (Robins 1986; Robins et al. 1994; Robins et al. 1995; Robins 2000; van der Laan & Robins 2003; van der Laan & Rose 2011; van der Vaart 2014; Robins et al. 2017), and colleagues (Hahn 1998; Tsiatis 2006). We refer to Tsiatis (2006) for a very readable review with more details.
    [Show full text]
  • Semiparametric Regression in Stata
    Introduction PLM Stata Semipar Heteroskedasticity Endogeneity Heterogeneity Mfx Single index Semiparametric regression in Stata Vincenzo Verardi 2013 UK Stata Users Group meeting London, UK September 2013 Vincenzo Verardi Semiparametric regression 12/09/2013 1/66 Introduction PLM Stata Semipar Heteroskedasticity Endogeneity Heterogeneity Mfx Single index Introduction Semiparametric regression models Semiparametric regression Deals with the introduction of some very general non-linear functional forms in regression analysis Generally used to …t a parametric model in which the functional form of a subset of the explanatory variables is not known and/or in which the distribution of the error term cannot be assumed to be of a speci…c type beforehand. Most popular semiparametric regression models are the partially linear models and single index models Remark: for an excellent description of semiparametric estimators, we advice to read the book by Ahamada and Flachaire (2010) from which many of the explanations available here come from. Vincenzo Verardi Semiparametric regression 12/09/2013 2/66 Introduction PLM Stata Semipar Heteroskedasticity Endogeneity Heterogeneity Mfx Single index Introduction Semiparametric regression models Partially linear models The partially linear model is de…ned as: y = X β + m(z) + ε Advantage 1: This model allows "any" form of the unknown function m Advantage 2: βˆ is pn-consistent Single index models The single index model is de…ned as: y = g(X β) + ε Advantage 1: generalizes the linear regression model (which assumes g( ) is linear) Advantage 2: the curse of dimensionality is avoided as there is only one nonparametric dimension Vincenzo Verardi Semiparametric regression 12/09/2013 3/66 Introduction PLM Stata Semipar Heteroskedasticity Endogeneity Heterogeneity Mfx Single index Introduction PLM example Hedonic pricing equation of houses Wooldridge (2000): What was the e¤ect of a local garbage incinerator on housing prices in North Andover in 1981? .
    [Show full text]
  • Semiparametric Regression Pursuit
    Statistica Sinica 22 (2012), 1403-1426 doi:http://dx.doi.org/10.5705/ss.2010.298 SEMIPARAMETRIC REGRESSION PURSUIT Jian Huang, Fengrong Wei and Shuangge Ma University of Iowa, University of West Georgia and Yale University Abstract: The semiparametric partially linear model allows flexible modeling of covariate effects on the response variable in regression. It combines the flexibility of nonparametric regression and the parsimony of linear regression. The most important assumption in the existing methods for the estimation in this model is that a priori it is known which covariates have a linear effect and which do not. However, in applied work, this is rarely known in advance. We consider the problem of estimation in the partially linear models without assuming a priori which covariates have linear effects. We propose a semiparametric regression pursuit method for identifying the covariates with a linear effect. Our proposed method is a penalized regression approach using a group minimax concave penalty. Under suitable conditions we show that the proposed approach is model-pursuit consistent, meaning that it can correctly determine which covariates have a linear effect and which do not with high probability. The performance of the proposed method is evaluated using simulation studies that support our theoretical results. A data example is used to illustrated the application of the proposed method. Key words and phrases: Group selection, minimax concave penalty, model-pursuit consistency, penalized regression, semiparametric models, structure estimation. 1. Introduction Suppose we have a random sample (yi; xi1; : : : ; xip); 1 ≤ i ≤ n, where yi is the response variable and (xi1; : : : ; xip) is a p-dimensional covariate vector.
    [Show full text]
  • Parametric Versus Semi and Nonparametric Regression Models
    Parametric versus Semi and Nonparametric Regression Models Hamdy F. F. Mahmoud1;2 1 Department of Statistics, Virginia Tech, Blacksburg VA, USA. 2 Department of Statistics, Mathematics and Insurance, Assiut University, Egypt. Abstract Three types of regression models researchers need to be familiar with and know the requirements of each: parametric, semiparametric and nonparametric regression models. The type of modeling used is based on how much information are available about the form of the relationship between response variable and explanatory variables, and the random error distribution. In this article, differences between models, common methods of estimation, robust estimation, and applications are introduced. The R code for all the graphs and analyses presented here, in this article, is available in the Appendix. Keywords: Parametric, semiparametric, nonparametric, single index model, robust estimation, kernel regression, spline smoothing. 1 Introduction The aim of this paper is to answer many questions researchers may have when they fit their data, such as what are the differences between parametric, semi/nonparamertic models? which one should be used to model a real data set? what estimation method should be used?, and which type of modeling is better? These questions and others are addressed by examples. arXiv:1906.10221v1 [stat.ME] 24 Jun 2019 Assume that a researcher collected data about Y, a response variable, and explanatory variables, X = (x1; x2; : : : ; xk). The model that describes the relationship between Y and X can be written as Y = f(X; β) + , (1) where β is a vector of k parameters, is an error term whose distribution may or may not be normal, f(·) is some function that describe the relationship between Y and X.
    [Show full text]