Hardle, Wolfgang and LIang, Hua and Gao, Jiti (2000): Partially linear models. Published in: PhysicaVerlag (1 September 2000): pp. 1202.

PDF
MPRA_paper_39562.pdf Download (1MB)  Preview 
Abstract
In the last ten years, there has been increasing interest and activity in the general area of partially linear regression smoothing in statistics. Many methods and techniques have been proposed and studied. This monograph hopes to bring an uptodate presentation of the state of the art of partially linear regression techniques. The emphasis of this monograph is on methodologies rather than on the theory, with a particular focus on applications of partially linear regression techniques to various statistical problems. These problems include least squares regression, asymptotically efficient estimation, bootstrap resampling, censored data analysis, linear measurement error models, nonlinear measurement models, nonlinear and nonparametric time series models. We hope that this monograph will serve as a useful reference for theoretical and applied statisticians and to graduate students and others who are interested in the area of partially linear regression. While advanced mathematical ideas have been valuable in some of the theoretical development, the methodological power of partially linear regression can be demonstrated and discussed without advanced mathematics. This monograph can be divided into three parts: part one–Chapter 1 through Chapter 4; part two–Chapter 5; and part three–Chapter 6. In the first part, we discuss various estimators for partially linear regression models, establish theo retical results for the estimators, propose estimation procedures, and implement the proposed estimation procedures through real and simulated examples. The second part is of more theoretical interest. In this part, we construct several adaptive and efficient estimates for the parametric component. We show that the LS estimator of the parametric component can be modified to have both Bahadur asymptotic efficiency and second order asymptotic efficiency. In the third part, we consider partially linear time series models. First, we propose a test procedure to determine whether a partially linear model can be used to fit a given set of data. Asymptotic test criteria and power investigations are presented. Second, we propose a CrossValidation (CV) based criterion to select the optimum linear subset from a partially linear regression and estab lish a CV selection criterion for the bandwidth involved in the nonparametric kernel estimation. The CV selection criterion can be applied to the case where the observations fitted by the partially linear model (1.1.1) are independent and identically distributed (i.i.d.). Due to this reason, we have not provided a sepa rate chapter to discuss the selection problem for the i.i.d. case. Third, we provide recent developments in nonparametric and semiparametric time series regression. This work of the authors was supported partially by the Sonderforschungs bereich373“QuantifikationundSimulationO ̈konomischerProzesse”.Thesecond author was also supported by the National Natural Science Foundation of China and an Alexander von Humboldt Fellowship at the Humboldt University, while the third author was also supported by the Australian Research Council. The second and third authors would like to thank their teachers: Professors Raymond Car roll, Guijing Chen, Xiru Chen, Ping Cheng and Lincheng Zhao for their valuable inspiration on the two authors’ research efforts. We would like to express our sin cere thanks to our colleagues and collaborators for many helpful discussions and stimulating collaborations, in particular, Vo Anh, Shengyan Hong, Enno Mam men, Howell Tong, Axel Werwatz and Rodney Wolff. For various ways in which they helped us, we would like to thank Adrian Baddeley, Rong Chen, Anthony Pettitt, Maxwell King, Michael Schimek, George Seber, Alastair Scott, Naisyin Wang, Qiwei Yao, Lijian Yang and Lixing Zhu. The authors are grateful to everyone who has encouraged and supported us to finish this undertaking. Any remaining errors are ours.
Item Type:  MPRA Paper 

Original Title:  Partially linear models 
English Title:  Partially Linear Models 
Language:  English 
Keywords:  Partially linear model 
Subjects:  C  Mathematical and Quantitative Methods > C5  Econometric Modeling > C52  Model Evaluation, Validation, and Selection C  Mathematical and Quantitative Methods > C1  Econometric and Statistical Methods and Methodology: General > C14  Semiparametric and Nonparametric Methods: General C  Mathematical and Quantitative Methods > C2  Single Equation Models ; Single Variables > C22  TimeSeries Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes 
Item ID:  39562 
Depositing User:  Jiti Gao 
Date Deposited:  20 Jun 2012 22:03 
Last Modified:  25 Aug 2016 04:28 
References:  Akahira, M. & Takeuchi, K.(1981). Asymptotic Efficiency of Statistical Estima tors. Concepts and Higher Order Asymptotic Efficiency. Lecture Notes in Statistics 7, SpringerVerlag, New York. An, H. Z. & Huang, F. C. (1996). The geometrical ergodicity of nonlinear au toregressive models. Statistica Sinica, 6, 943–956. Andrews, D. W. K. (1991). Asymptotic normality of series estimates for nonpara metric and semiparametric regression models. Econometrica, 59, 307–345. Anglin, P.M. & Gencay, R. (1996). Semiparametric estimation of a hedonic price function. Journal of Applied Econometrics, 11, 633648. Bahadur, R.R. (1967). Rates of convergence of estimates and test statistics. Ann. Math. Stat., 38, 303324. Begun, J. M., Hall, W. J., Huang, W. M. & Wellner, J. A. (1983). Information and asymptotic efficiency in parametricnonparametric models. Annals of Statistics, 11, 432452. ALS, 14, 1295–1298. Bhattacharya, P.K. & Zhao, P. L.(1997). Semiparametric inference in a partial linear model. Annals of Statistics, 25, 244262. Bickel, P.J. (1978). Using residuals robustly I: Tests for heteroscedasticity, non linearity. Annals of Statistics, 6, 266291. Bickel, P. J. (1982). On adaptive estimation. Annals of Statistics, 10, 647671. Bickel, P. J., Klaasen, C. A. J., Ritov, Ya’acov & Wellner, J. A. (1993). Efficient and Adaptive Estimation for Semiparametric Models. The Johns Hopkins University Press. Blanchflower, D.G. & Oswald, A.J. (1994). The Wage Curve, MIT Press Cambridge, MA. Boente, G. & Fraiman, R. (1988). Consistency of a nonparametric estimator of a density function for dependent variables. Journal of Multivariate Analysis, 25, 90–99. Bowman, A. & Azzalini,A. (1997). Applied Smoothing Techniques: the Kernel Approach and Splus Illustrations. Oxford University Press, Oxford. Box, G. E. P. & Hill, W. J. (1974). Correcting inhomogeneity of variance with power transformation weighting. Technometrics, 16, 385389. Buckley, M. J. & Eagleson, G. K. (1988). An approximation to the distribution of quadratic forms in normal random variables. Australian Journal of Statistics, 30A, 150–159. Buckley, J. & James, I. (1979). Linear regression with censored data. Biometrika, 66, 429436. Carroll, R. J. (1982). Adapting for heteroscedasticity in linear models. Annals of Statistics, 10, 12241233. Carroll, R. J. & H ̈ardle, W. (1989). Second order effects in semiparametric weighted least squares regression. Statistics, 2, 179186. Carroll, R. J. & Ruppert, D. (1982). Robust estimation in heteroscedasticity linear models. Annals of Statistics, 10, 429441. Carroll, R. J., Ruppert, D. & Stefanski, L. A. (1995). Nonlinear Measurement Error Models. Vol. 63 of Monographs on Statistics and Applied Probability, Chapman and Hall, New York. Carroll, R. J., Fan, J., Gijbels, I. & Wand, M. P. (1997). Generalized partially singleindex models. Journal of the American Statistical Association, 92, 477489. Chai, G. X. & Li, Z. Y. (1993). Asymptotic theory for estimation of error distri butions in linear model. Science in China, Ser. A, 14, 408419. Chambers, J. M. & Hastie, T. J. (ed.) (1992). Statistical Models in S. Chapman and Hall, New York. Chen, H.(1988). Convergence rates for parametric components in a partly linear model. Annals of Statistics, 16, 136146. Chen, H. & Chen, K. W. (1991). Selection of the splined variables and conver gence rates in a partial linear model. The Canadian Journal of Statistics, 19, 323339. Chen, R. & Tsay, R. (1993). Nonlinear additive ARX models. Journal of the American Statistical Association, 88, 955–967. Chen, R., Liu, J. & Tsay, R. (1995). Additivity tests for nonlinear autoregression. Biometrika, 82, 369–383. Cheng, B. & Tong, H. (1992). On consistent nonparametric order determination and chaos. Journal of the Royal Statistical Society, Series B, 54, 427–449. Cheng, B. & Tong, H. (1993). Nonparametric function estimation in noisy chaos. Developments in time series analysis. (In honour of Maurice B. Priestley, ed. T. Subba Rao), 183–206. Chapman and Hall, London. Cheng, P. (1980). Bahadur asymptotic efficiency of MLE. Acta Mathematica Sinica, 23, 883900. Cheng, P., Chen, X., Chen, G.J & Wu, C.Y. (1985). Parameter Estimation. Science & Technology Press, Shanghai. Chow, Y. S. & Teicher, H. (1988). Probability Theory. 2nd Edition, Springer Verlag, New York. Cuzick, J. (1992a). Semiparametric additive regression. Journal of the Royal Statistical Society, Series B, 54, 831843. Cuzick, J. (1992b). Efficient estimates in semiparametric additive regression mod els with unknown error distribution. Annals of Statistics, 20, 11291136. Daniel, C. & Wood, F. S. (1971). Fitting Equations to Data. John Wiley, New York. Daniel, C. & Wood, F. S. (1980). Fitting Equations to Data: Computer Analysis of Multifactor Data for Scientists and Engineers. 2nd Editor. John Wiley, New York. Davison, A. C. & Hinkley, D. V. (1997). Bootstrap Methods and their Application. Cambridge University Press, Cambridge. Devroye, L. P. & Wagner, T.J. (1980). The strong uniform consistency of kernel estimates. Journal of Multivariate Analysis, 5, 5977. DeVore, R. A. & Lorentz, G. G. (1993). Constructive Approximation. New York: Springer. Dinse, G.E. & Lagakos, S.W. (1983). Regression analysis of tumour prevalence. Applied Statistics, 33, 236248. Doukhan, P. (1995). Mixing: Properties and Examples. Lecture Notes in Statis tics, 86, Springer, New York. Efron, B. & Tibshirani, R. J. (1993). An Introduction to the Bootstrap. Vol. 57 of Monographs on Statistics and Applied Probability, Chapman and Hall, New York. Engle, R. F., Granger, C. W. J., Rice, J. & Weiss, A. (1986). Semiparametric estimates of the relation between weather and electricity sales. Journal of the American Statistical Association, 81, 310320. Eubank, R.L. (1988). Spline Smoothing and Nonparametric Regression. New York, Marcel Dekker. Eubank, R.L., Kambour,E.L., Kim,J.T., Klipple, K., Reese C.S. and Schimek, M. (1998). Estimation in partially linear models. Computational Statistics & Data Analysis, 29, 2734. Eubank, R. L. & Spiegelman, C. H. (1990). Testing the goodnessoffit of a linear model via nonparametric regression techniques. Journal of the American Statistical Association, 85, 387–392. Eubank, R. L. & Whitney, P. (1989). Convergence rates for estimation in certain partially linear models. Journal of Statistical Planning & Inference, 23, 33 43. Fan, J. & Truong, Y. K. (1993). Nonparametric regression with errors in variables. Annals of Statistics, 21, 1900–1925. Fan, J. & Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Vol. 66 of Monographs on Statistics and Applied Probability, Chapman and Hall, New York. Fan, Y. & Li, Q. (1996). Consistent model specification tests: omitted variables and semiparametric functional forms. Econometrica, 64, 865–890. Fu, J. C. (1973). On a theorem of Bahadur on the rate of convergence of point estimations. Annals of Statistics, 1, 745749. Fuller, W. A. (1987). Measurement Error Models. John Wiley, New York. Fuller, W. A. & Rao, J. N. K. (1978). Estimation for a linear regression model with unknown diagonal covariance matrix. Annals of Statistics, 6, 11491158. Gallant, A. R. (1981). On the bias in flexible functional forms and an essentially unbiased form: the Fourier flexible form. J. Econometrics, 15, 211–245. Gao, J. T. (1992). Large Sample Theory in Semiparametric Regression Models. Ph.D. Thesis, Graduate School, University of Science & Technology of China, Hefei, P.R. China. Gao, J. T. (1995a). The laws of the iterated logarithm of some estimates in partly linear models. Statistics & Probability Letters, 25, 153162. Gao, J. T. (1995b). Asymptotic theory for partly linear models. Communications in Statistics, Theory & Methods, 24, 19852010. Gao, J. T. (1998). Semiparametric regression smoothing of nonlinear time series. Scandinavian Journal of Statistics, 25, 521539. Gao, J. T. & Anh, V. V. (1999). Semiparametric regression under longrange dependent errors. Journal of Statistical Planning & Inference, 80, 3757. Gao, J. T., Chen, X. R. & Zhao, L. C.(1994). Asymptotic normality of a class estimates in partly linear models. Acta Mathematica Sinica, 37, 256268. Gao, J. T., Hong, S.Y. & Liang, H. (1995). Convergence rates of a class of estimates in partly linear models. Acta Mathematica Sinica, 38, 658669. Gao, J. T., Hong, S. Y. & Liang, H. (1996). The BerryEsseen bounds of para metric component estimation in partly linear models. Chinese Annals of Mathematics, 17, 477490. Gao, J. T. & Liang, H. (1995). Asymptotic normality of pseudoLS estimator for partially linear autoregressive models. Statistics & Probability Letters, 23, 27–34. Gao, J. T. & Liang, H. (1997). Statistical inference in singleindex and partially nonlinear regression models. Annals of the Institute of Statistical Mathematics, 49, 493–517. Gao, J. T. & Shi, P. D. (1997). Mtype smoothing splines in nonparametric and semiparametric regression models. Statistica Sinica, 7, 1155–1169. Gao, J. T., Tong, H. & Wolff, R. (1998a). Adaptive series estimation in additive stochastic regression models. Technical report No. 9801, School of Mathe matical Sciences, Queensland University of Technology, Brisbane, Australia. Gao, J. T., Tong, H. & Wolff, R. (1998b). Adaptive testing for additivity in addi tive stochastic regression models. Technical report No. 9802, School of Math ematical Sciences, Queensland University of Technology, Brisbane, Australia. Gao, J. T. & Zhao, L. C. (1993). Adaptive estimation in partly linear models. Sciences in China Ser. A, 14, 1427. Glass, L. & Mackey, M. (1988). From Clocks to Chaos: the Rhythms of Life. Princeton University Press, Princeton. Gleser, L. J. (1992). The importance of assessing measurement reliability in multivariate regression. Journal of the American Statistical Association, 87, 696707. Golubev, G. & H ̈ardle, W. (1997). On adaptive in partial linear models. Dis cussion paper no. 371, WeierstrsseInstitut fu ̈r Angewandte Analysis und Stochastik zu Berlin. Green, P., Jennison, C. & Seheult, A. (1985). Analysis of field experiments by least squares smoothing. Journal of the Royal Statistical Society, Series B, 47, 299315. Green, P. & Silverman, B. W. (1994). Nonparametric Regression and Generalized Linear Models: a Roughness Penalty Approach. Vol. 58 of Monographs on Statistics and Applied Probability, Chapman and Hall, New York. Green, P. & Yandell, B. (1985). Semiparametric generalized linear models. Generalized Linear Models (R. Gilchrist, B. J. Francis and J. Whittaker, eds), Lecture Notes in Statistics, 30, 4455. Springer, Berlin. GSOEP (1991). Das Sozioo ̈konomische Panel (SOEP) im Jahre 1990/91, Projektgruppe “Das Sozio ̈okonomische Panel”, Deutsches Institut fu ̈r Wirtschaftsforschung. Vierteljahreshefte zur Wirtschaftsforschung, pp. 146– 155. Gu, M. G. & Lai, T. L. (1990 ). Functional laws of the iterated logarithm for the productlimit estimator of a distribution function under random censorship or truncated. Annals of Probability, 18, 160189. Gy ̈orfi, L., H ̈ardle, W., Sarda, P. & Vieu, P. (1989). Nonparametric curve esti mation for time series. Lecture Notes in Statistics, 60, Springer, New York. Hall, P. & Carroll, R. J. (1989). Variance function estimation in regression: the effect of estimating the mean. Journal of the Royal Statistical Society, Series B, 51, 314. Hall, P. & Heyde, C. C. (1980). Martingale Limit Theory and Its Applications. Academic Press, New York. Hamilton, S. A. & Truong, Y. K. (1997). Local linear estimation in partly linear models. Journal of Multivariate Analysis, 60, 119. H ̈ardle, W. (1990). Applied Nonparametric Regression. Cambridge University Press, New York. H ̈ardle, W. (1991). Smoothing Techniques: with Implementation in S. Springer, Berlin. H ̈ardle, W., Klinke, S. & Mu ̈ller, M. (1999). XploRe Learning Guide. Springer Verlag. H ̈ardle, W., Lu ̈tkepohl, H. & Chen, R. (1997). A review of nonparametric time series analysis. International Statistical Review, 21, 49–72. H ̈ardle, W. & Mammen, E. (1993). Testing parametric versus nonparametric regression. Annals of Statistics, 21, 1926–1947. H ̈ardle, W., Mammen, E. & Mu ̈ller, M. (1998). Testing parametric versus semiparametric modeling in generalized linear models. Journal of the American Statistical Association, 93, 14611474. H ̈ardle, W. & Vieu, P. (1992). Kernel regression smoothing of time series. Journal of Time Series Analysis, 13, 209–232. Heckman, N.E. (1986). Spline smoothing in partly linear models. Journal of the Royal Statistical Society, Series B, 48, 244248. Hjellvik, V. & Tjøstheim, D. (1995). Nonparametric tests of linearity for time series. Biometrika, 82, 351–368. Hong, S. Y. (1991). Estimation theory of a class of semiparametric regression models. Sciences in China Ser. A, 12, 12581272. Hong, S. Y. & Cheng, P.(1992a). Convergence rates of parametric in semiparametric regression models. Technical report, Institute of Systems Science, Chinese Academy of Sciences. Hong, S. Y. & Cheng, P.(1992b). The BerryEsseen bounds of some estimates in semiparametric regression models. Technical report, Institute of Systems Science, Chinese Academy of Sciences. Hong, S. Y. & Cheng, P. (1993). Bootstrap approximation of estimation for parameter in a semiparametric regression model. Sciences in China Ser. A, 14, 239251. Hong, Y. & White, H. (1995). Consistent specification testing via nonparametric series regression. Econometrica, 63, 1133–1159. Jayasuriva, B. R. (1996). Testing for polynomial regression using nonparametric regression techniques. Journal of the American Statistical Association, 91, 1626–1631. Jobson, J. D. & Fuller, W. A. (1980). Least squares estimation when covari ance matrix and parameter vector are functionally related. Journal of the American Statistical Association, 75, 176181. Kashin, B. S. & Saakyan, A. A. (1989). Orthogonal Series. Translations of Math ematical Monographs, 75. Koul, H., Susarla, V. & Ryzin, J. (1981). Regression analysis with randomly rightcensored data. Annals of Statistics, 9, 12761288. Kreiss, J. P., Neumann, M. H. & Yao, Q. W. (1997). Bootstrap tests for simple structures in nonparametric time series regression. Private communication. Lai, T. L. & Ying, Z. L. (1991). Rank regression methods for lefttruncated and rightcensored data. Annals of Statistics, 19, 531556. Lai, T. L. & Ying, Z. L. (1992). Asymptotically efficient estimation in censored and truncated regression models. Statistica Sinica, 2, 1746. Liang, H. (1992). Asymptotic Efficiency in Semiparametric Models and Related Topics. Thesis, Institute of Systems Science, Chinese Academy of Sciences, Beijing, P.R. China. Liang, H. (1994a). On the smallest possible asymptotically efficient variance in semiparametric models. System Sciences & Matematical Sciences, 7, 2933. Liang, H. (1994b). The BerryEsseen bounds of error variance estimation in a semiparametric regression model. Communications in Statistics, Theory & Methods, 23, 34393452. Liang, H. (1995a). On Bahadur asymptotic efficiency of maximum likelihood estimator for a generalized semiparametric model. Statistica Sinica, 5, 363 371. Liang, H. (1995b). Second order asymptotic efficiency of PMLE in generalized linear models. Statistics & Probability Letters, 24, 273279. Liang, H. (1995c). A note on asymptotic normality for nonparametric multiple regression: the fixed design case. SooChow Journal of Mathematics, 395399. Liang, H. (1996). Asymptotically efficient estimators in a partly linear autore gressive model. System Sciences & Matematical Sciences, 9, 164170. Liang, H. (1999). An application of Bernstein’s inequality. Econometric Theory, 15, 152. Liang, H. & Cheng, P. (1993). Second order asymptotic efficiency in a partial linear model. Statistics & Probability Letters, 18, 7384. Liang, H. & Cheng, P. (1994). On Bahadur asymptotic efficiency in a semiparametric regression model. System Sciences & Matematical Sciences, 7, 229240. Liang, H. & H ̈ardle, W. (1997). Asymptotic properties of parametric estimation in partially linear heteroscedastic models. Technical report no 33, Sonderforschungsbereich 373, HumboldtUniversit ̈at zu Berlin. Liang, H. & H ̈ardle, W. (1999). Large sample theory of the estimation of the error distribution for semiparametric models. Communications in Statistics, Theory & Methods, in press. Liang, H., H ̈ardle, W. & Carroll, R.J. (1999). Estimation in a semiparametric partially linear errorsinvariables model. Annals of Statistics, in press. Liang, H., H ̈ardle, W. & Sommerfeld, V. (1999). Bootstrap approximation of the estimates for parameters in a semiparametric regression model. Journal of Statistical Planning & Inference, in press. Liang, H., H ̈ardle, W. & Werwatz, A. (1999). Asymptotic properties of nonpara metric regression estimation in partly linear models. Econometric Theory, 15, 258. Liang, H. & Huang, S.M. (1996). Some applications of semiparametric partially linear models in economics. Science Decision, 10, 616. Liang, H. & Zhou, Y. (1998). Asymptotic normality in a semiparametric partial linear model with rightcensored data. Communications in Statistics, Theory & Methods, 27, 28952907. Linton, O.B. (1995). Second order approximation in the partially linear regression model. Econometrica, 63, 10791112. Lu, K. L. (1983). On Bahadur Asymptotically Efficiency. Dissertation of Master Degree. Institute of Systems Science, Chinese Academy of Sciences. Mak, T. K. (1992). Estimation of parameters in heteroscedastic linear models. Journal of the Royal Statistical Society, Series B, 54, 648655. Mammen, E. & van de Geer, S. (1997). Penalized estimation in partial linear models. Annals of Statistics, 25, 10141035. Masry, E. & Tjøstheim, D. (1995). Nonparametric estimation and identification of nonlinear ARCH time series. Econometric Theory, 11, 258–289. Masry, E. & Tjøstheim, D. (1997). Additive nonlinear ARX time series and projection estimates. Econometric Theory, 13, 214–252. Mu ̈ller, H. G. & Stadtmu ̈ller, U. (1987). Estimation of heteroscedasticity in regression analysis. Annals of Statistics, 15, 610625. Mu ̈ller, M. & R ̈onz, B. (2000). Credit scoring using semiparametric methods, in Springer LNS (eds. J. Franks, W. H ̈ardle and G. Stahl). Nychka, D., Elliner, S., Gallant, A. & McCaffrey, D. (1992). Finding chaos in noisy systems. Journal of the Royal Statistical Society, Series B, 54, 399426. Pollard, D. (1984). Convergence of Stochastic Processes. Springer, New York. Rendtel, U. & Schwarze, J. (1995). Zum Zusammenhang zwischen Lohnhoehe und Arbeitslosigkeit: Neue Befunde auf Basis semiparametrischer Schaet zungen und eines verallgemeinerten VarianzKomponenten Modells. German Institute for Economic Research (DIW) Discussion Paper 118, Berlin, 1995. Rice, J.(1986). Convergence rates for partially splined models. Statistics & Prob ability Letters, 4, 203208. Robinson, P.M.(1983). Nonparametric estimation for time series models. Journal of Time Series Analysis, 4, 185208. Robinson, P.M.(1988). Rootnconsistent semiparametric regression. Econometrica, 56, 931954. Schick, A. (1986). On asymptotically efficient estimation in semiparametric model. Annals of Statistics, 14, 11391151. Schick, A. (1993). On efficient estimation in regression models. Annals of Statis tics, 21, 14861521. (Correction and Addendum, 23, 18621863). Schick, A. (1996a). Weighted least squares estimates in partly linear regression models. Statistics & Probability Letters, 27, 281287. Schick, A. (1996b). Rootn consistent estimation in partly linear regression mod els. Statistics & Probability Letters, 28, 353358. Schimek, M. (1997). Non and semiparametric alternatives to generalized linear models. Computational Statistics. 12, 173191. Schimek, M. (1999). Estimation and inference in partially linear models with smoothing splines. Journal of Statistical Planning & Inference, to appear. Schmalensee, R. & Stoker, T.M. (1999). Household gasoline demand in the United States. Econometrica, 67, 645662. Severini, T. A. & Staniswalis, J. G. (1994). Quasilikelihood estimation in semi parametric models. Journal of the American Statistical Association, 89, 501 511. Speckman, P. (1988). Kernel smoothing in partial linear models. Journal of the Royal Statistical Society, Series B, 50, 413436. Stefanski, L. A. & Carroll, R. (1990). Deconvoluting kernel density estimators. Statistics, 21, 169184. Stone, C. J. (1975). Adaptive maximum likelihood estimation of a location pa rameter. Annals of Statistics, 3, 267284. Stone, C. J. (1982). Optimal global rates of convergence for nonparametric estimators. Annals of Statistics, 10, 10401053. Stone, C. J. (1985). Additive regression and other nonparametric models. Annals of Statistics, 13, 689705. Stout, W. (1974). Almost Sure Convergence. Academic Press, New York. Ter ̈asvirta, T., Tjøstheim, D. & Granger, C. W. J. (1994). Aspects of modelling nonlinear time series, in R. F. Engle & D. L. McFadden (eds.). Handbook of Econometrics, 4, 2919–2957. Tjøstheim, D. (1994). Nonlinear time series: a selective review. Scandinavian Journal of Statistics, 21, 97–130. Tjøstheim, D. & Auestad, B. (1994a). Nonparametric identification of nonlinear time series: projections. Journal of the American Statistical Association, 89, 13981409. Tjøstheim, D. & Auestad, B. (1994b). Nonparametric identification of nonlinear time series: selecting significant lags. Journal of the American Statistical Association, 89, 1410–1419. Tong, H. (1977). Some comments on the Canadian lynx data (with discussion). Journal of the Royal Statistical Society, Series A, 140, 432–436. Tong, H. (1990). Nonlinear Time Series. Oxford University Press. Tong, H. (1995). A personal overview of nonlinear time series analysis from a chaos perspective (with discussions). Scandinavian Journal of Statistics, 22, 399–445. Tripathi, G. (1997). Semiparametric efficiency bounds under shape restric tions. Unpublished manuscript, Department of Economics, University of WisconsinMadison. Vieu, P. (1994). Choice of regression in nonparametric estimation. Computational Statistics & Data Analysis, 17, 575594. Wand, M. & Jones, M.C. (1994). Kernel Smoothing. Vol. 60 of Monographs on Statistics and Applied Probability, Chapman and Hall, London. Whaba, G. (1990). Spline Models for Observational Data, CBMSNSF Regional Conference Series in Applied Mathematics. 59, Philadelphia, PA: SIAM, XII. Willis, R. J. (1986). Wage Determinants: A Survey and Reinterpretation of Hu man Capital Earnings Functions in: Ashenfelter, O. and Layard, R. The Handbook of Labor Economics, Vol.1 North HollandElsevier Science Pub lishers Amsterdam, 1986, pp 525602. Wong, C. M. & Kohn, R. (1996). A Bayesian approach to estimating and forecast ing additive nonparametric autoregressive models. Journal of Time Series Analysis, 17, 203–220. Wu, C. F. J. (1986). Jackknife, Bootstrap and other resampling methods in regression analysis (with discussion). Annals of Statistics, 14, 12611295. Yao, Q. W. & Tong, H. (1994). On subset selection in nonparametric stochastic regression. Statistica Sinica, 4, 51–70. Zhao, L. C. (1984). Convergence rates of the distributions of the error variance estimates in linear models. Acta Mathematica Sinica, 3, 381392. Zhao, L. C. & Bai, Z. D. (1985). Asymptotic expansions of the distribution of sum of the independent random variables. Sciences in China Ser. A, 8, 677697. Zhou, M. (1992). Asymptotic normality of the synthetic data regression estimator for censored survival data. Annals of Statistics, 20, 10021021. 
URI:  https://mpra.ub.unimuenchen.de/id/eprint/39562 