Hostname: page-component-74656f9884-npj9w Total loading time: 0 Render date: 2026-11-10T05:00:00.609Z Has data issue: false hasContentIssue false
  • English
  • Français

k-NEAREST NEIGHBOR ESTIMATION OFINVERSE-DENSITY-WEIGHTED EXPECTATIONS WITH DEPENDENT DATA

Published online by Cambridge University Press:  21 February 2012

Ba Chu  and
David T. Jacho-Chávez
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the 'Save PDF' action button.

This paper considers the problem of estimating expected values of functionsthat are inversely weighted by an unknown density using the k-nearest neighbor (k-NN) method. Itestablishes the $\root \of T $-consistency and the asymptotic normality of anestimator that allows for strictly stationary time-series data. Theconsistency of the Bartlett estimator of the derived asymptotic variance isalso established. The proposed estimator is also shown to be asymptoticallysemiparametric efficient in the independent random sampling scheme. MonteCarlo experiments show that the proposed estimator performs well in finitesample applications.

Information

Type
Research Article
Information
Econometric Theory , Volume 28 , Issue 4 , August 2012 , pp. 769 - 803
Copyright
Copyright © Cambridge University Press 2012

Footnotes

The authors would like to thank the co-editor and three referees for their valuable comments that led to corrections and various improvements in the paper. Francesco Bravo and Juan Carlos Escanciano also provided many helpful comments, suggestions, and corrections. The authors acknowledge funding from the Social Science and Humanities Research Council of Canada (MBF Grant 410-2011-1700). The usual disclaimer applies.

References

REFERENCES

Aaronson, J., Burton, R., Dehling, H., Gilat, D., Hill, T., & Weiss, B. (1996) Strong laws for L- and U- statistics. Transactions of the American Mathematical Society 348, 2845–2865.10.1090/S0002-9947-96-01681-9CrossRefGoogle Scholar
Abadie, A. & Imbens, G. (2006) Large sample properties of matching estimators for average treatment effects. Econometrica 74(1), 235–267.10.1111/j.1468-0262.2006.00655.xCrossRefGoogle Scholar
Abadie, A. & Imbens, G. (2011) Bias corrected matching estimators for average treatment effects. Journal of Business Economics and Statistics 29(1), 1–11.10.1198/jbes.2009.07333CrossRefGoogle Scholar
Andrews, D.W.K. (1991) Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59(3), 817–858.10.2307/2938229CrossRefGoogle Scholar
Bhattacharya, P.K. & Mack, Y.P. (1987) Weak convergence of k-nn density and regression estimators with varying k and applications. Annals of Statistics 15(3), 976–994.CrossRefGoogle Scholar
Boente, G. & Fraiman, R. (1988) Consistency of a nonparametric estimate of a density function for dependent variables. Journal of Multivariate Analysis 25, 90–99.10.1016/0047-259X(88)90154-6CrossRefGoogle Scholar
Boente, G. & Fraiman, R. (1990) Asymptotic distribution of robust estimators for nonparametric models from mixing processes. Annals of Statistics 18, 891–906.CrossRefGoogle Scholar
Boente, G. & Fraiman, R. (1991) Strong order of convergence and asymptotic distribution of nearest neighbor density estimates from dependent observations. Sankhyā: Series A 53(2), 194–205.Google Scholar
Boente, G. & Fraiman, R. (1995) Asymptotic distribution of smoothers based on local means and local medians under dependence. Journal of Multivariate Analysis 54(1), 77–90.CrossRefGoogle Scholar
Bradley, R.C. (2007) Introduction to Strong Mixing Conditions, vol. 1. Kendrick Press.Google Scholar
Clauser, J.F. & Horne, M.A. (1974) Experimental consequences of objective local theories. Physical Review D 10, 526–535.10.1103/PhysRevD.10.526CrossRefGoogle Scholar
Cox, D. & Kim, T.Y. (1995) Moment bounds for mixing random variables useful in nonparametric function estimation. Stochastic Processes and Their Applications 56, 151–159.CrossRefGoogle Scholar
Delgado, M.A. (1992) Semiparametric generalized least squares in the multivariate nonlinear regression model. Econometric Theory 8, 203–222.10.1017/S0266466600012767CrossRefGoogle Scholar
Delgado, M.A. & Stengos, T. (1994) Semiparametric testing in non-nested econometric models. Review of Economic Studies 61(2), 291–303.CrossRefGoogle Scholar
Escanciano, J.C. & Jacho-Chávez, D.T. (2010) Approximating the critical values of Cramér-von Mises tests in general parametric conditional specifications. Computational Statistics & Data Analysis 54(3), 625–636.10.1016/j.csda.2008.05.016CrossRefGoogle Scholar
Hall, P. & Heyde, C.C. (1980) Martingale Limit Theory and Its Application. Academic Press.Google Scholar
Hall, P. & Yatchew, A. (2005) Unified approach to testing functional hypotheses in semiparametric contexts. Journal of Econometrics 127(2), 225–252.10.1016/j.jeconom.2004.08.005CrossRefGoogle Scholar
Hansen, B.E. (1992) Consistent covariance matrix estimation for dependent heterogeneous processes. Econometrica 60(4), 967–972.CrossRefGoogle Scholar
Härdle, W. & Stoker, T.M. (1989) Investigating smooth multiple regression by the method of average derivatives. Journal of the American Statistical Association 84, 986–995.Google Scholar
Hausman, J.A. & Newey, W.K. (1995) Nonparametric estimation of exact consumers surplus and deadweight loss. Econometrica 63(6), 1445–1476.10.2307/2171777CrossRefGoogle Scholar
Heckman, J.J., Ichimura, H., & Todd, P. (1998) Matching as an econometric evaluation estimator. Review of Economic Studies 65(2), 261–94.10.1111/1467-937X.00044CrossRefGoogle Scholar
Hong, Y. & White, H. (2005) Asymptotic distribution theory for nonparametric entropy measures of serial dependence. Econometrica 73(3), 837–901.CrossRefGoogle Scholar
Honoré, B.E. & Lewbel, A. (2002) Semiparametric binary choice panel data models without strictly exogenous regressors. Econometrica 70(5), 2053–2063.10.1111/1468-0262.00363CrossRefGoogle Scholar
Ibragimov, I.A. & Linnik, Y.V. (1971) Independent and stationary sequences of random variables. Wolters-Noordhoff.Google Scholar
Jacho-Chávez, D.T. (2008) k nearest-neighbor estimation of inverse density weighted expectations. Economics Bulletin 3(48), 1–6.Google Scholar
Jacho-Chávez, D.T. (2010) Optimal bandwidth choice for estimation of inverse conditional density – weighted expectations. Econometric Theory 26(01), 94–118.CrossRefGoogle Scholar
Jun, S.J. & Pinkse, J. (2009a) Efficient semiparametric seemingly unrelated quantile regression estimation. Econometric Theory 25(05), 1392–1414.CrossRefGoogle Scholar
Jun, S.J. & Pinkse, J. (2009b) Semiparametric tests of conditional moment restrictions under weak or partial identification. Journal of Econometrics 152(1), 3–18.10.1016/j.jeconom.2009.02.005CrossRefGoogle Scholar
Jun, S.J. & Pinkse, J. (2009c) Testing Under Weak Identification with Conditional Moment Conditions. Unpublished manuscript.Google Scholar
Khan, S. & Lewbel, A. (2007) Weighted and two-stage least squares estimation of semiparametric truncated regression models. Econometric Theory 23(2), 309–347.10.1017/S0266466607070132CrossRefGoogle Scholar
Khan, S. & Tamer, E. (2011) Irregular identification, support conditions, and inverse weight estimation. Econometrica 78(6), 2021–2042.Google Scholar
Lewbel, A. (1997) Semiparametric estimation of location and other discrete choice moments. Econometric Theory 13(01), 32–51.10.1017/S0266466600005636CrossRefGoogle Scholar
Lewbel, A. (1998) Semiparametric latent variable model estimation with endogenous or mismeasured regressors. Econometrica 66(1), 105–121.10.2307/2998542CrossRefGoogle Scholar
Lewbel, A. (2000) Semiparametric qualitative response model estimation with unknown heteroscedasticity or instrumental variables. Journal of Econometrics 97(1), 145–177.CrossRefGoogle Scholar
Lewbel, A. (2007) Endogenous selection or treatment model estimation. Journal of Econometrics 141(2), 777–806.10.1016/j.jeconom.2006.11.004CrossRefGoogle Scholar
Lewbel, A., Linton, O.B., & McFadden, D. (2011) Estimating features of a distribution from binomial data. Journal of Econometrics 162(2), 170–188.10.1016/j.jeconom.2010.11.006CrossRefGoogle Scholar
Lewbel, A. & Schennach, S.M. (2007) A simple ordered data estimator for inverse density weighted expectations. Journal of Econometrics 127(1), 189–211.CrossRefGoogle Scholar
Li, Q. (2006) Testing Parametric Regression Functional Forms Using k-Nearest Neighbor Method. Unpublished manuscript.Google Scholar
Liu, Z. & Lu, X. (1997) Root- n-consistent semiparametric estimation of partially linear models based on k-NN method. Econometric Reviews 16(4), 411–420.10.1080/07474939708800396CrossRefGoogle Scholar
Loftsgaarden, D.O. & Quesenberry, C.P. (1965) A nonparametric estimate of a multivariate density function. Annals of Mathematical Statistics 36(3), 1049–1051.CrossRefGoogle Scholar
Lu, Z. & Cheng, P. (1998) Strong consistency of nearest neighbor kernel regression estimation for stationary dependent samples. Science in China (Series A) 41(9), 918–926.10.1007/BF02880000CrossRefGoogle Scholar
Mack, Y.P. & Rosenblatt, M. (1979) Multivariate k-nearest neighbor density estimates. Journal of Multivariate Analysis 9(1), 1–15.10.1016/0047-259X(79)90065-4CrossRefGoogle Scholar
McFadden, D. (1999) Computing willingness to pay in random utility models. In Moore, J.C., Reizman, R., & Melvin, J.R. (eds.), Trade, Theory and Econometrics, chap. 15, pp. 253–274. Routledge.CrossRefGoogle Scholar
McLeish, D.L. (1975) Invariance principles for dependent variables. Z. Wahrscheinlichkeitstheorie verw. Gebiete 32, 165–178.CrossRefGoogle Scholar
Newey, W.K. (1990) Efficient instrumental variables estimation of nonlinear models. Econometrica 58(4), 809–37.10.2307/2938351CrossRefGoogle Scholar
Newey, W.K. (1997) Convergence rates and asymptotic normality for series estimators. Journal of Econometrics 79(1), 147–168.CrossRefGoogle Scholar
Newey, W.K. & McFadden, D. (1994) Large sample estimation and hypothesis testing. In McFadden, D. & Engle, R.F. (eds.), Handbook of Econometrics, vol. IV, pp. 2111–2245. Elsevier, North-Holland.Google Scholar
Newey, W.K. & Ruud, P.A. (2005) Density weighted linear least squares. In Andrews, D.W.K. & Stock, J.H. (eds.), Identification and Inference for Econometric Models: Essays in Honor of Thomas Rothenberg, chap. 23, p. 554–573. Cambridge University Press.10.1017/CBO9780511614491.024CrossRefGoogle Scholar
Newey, W.K. & West, K.D. (1987) A simple positive definite heteroscedasticity and autocorrelation consistent covariance matrix. Econometrica 55(3), 703–708.10.2307/1913610CrossRefGoogle Scholar
Newey, W.K. & West, K.D. (1994) Automatic lag selection in covariance matrix estimation. The Review of Economic Studies 61(4), 631–653.CrossRefGoogle Scholar
Phillips, P.C.B. (1987) Time series regression with a unit root. Econometrica 55(2), 277–301.10.2307/1913237CrossRefGoogle Scholar
Pinkse, J. (2006) Heteroskedasticity Correction and Dimension Reduction. Unpublished manuscript.Google Scholar
Rao, B.L.S.P. (1999) Semimartingales and their Statistical Inference. Chapman & Hall/CRC.Google Scholar
Rao, B.L.S.P. (2009) Conditional independence, conditional mixing and conditional association. Annals of the Institute of Statistical Mathematics 61, 441–460.Google Scholar
Robinson, P.M. (1987) Asymptotically efficient estimation in the presence of heteroskedasticity of unknown form. Econometrica 55(4), 875–891.10.2307/1911033CrossRefGoogle Scholar
Robinson, P.M. (1995) Nearest-neighbour estimation of semiparametric regression models. Journal of Nonparametric Statistics 5(1), 33–41.10.1080/10485259508832632CrossRefGoogle Scholar
Rosenblatt, M. (1956a) A central limit theorem and a strong mixing condition. Procedure of the National Academy of Science USA 42, 43–47.10.1073/pnas.42.1.43CrossRefGoogle Scholar
Rosenblatt, M. (1956b) Remarks on some non-parametric estimates of a density function. The Annals of Mathematical Statistics, 27, 832–837.CrossRefGoogle Scholar
Roussas, G.G. & Ioannides, D. (1987) Moment inequalities for mixing sequences of random variables. Stochastic Analysis and Applications 5(1), 61–120.10.1080/07362998708809108CrossRefGoogle Scholar
Severini, T.A. & Tripathi, G. (2001) A simplified approach to computing efficiency bounds in semiparametric models. Journal of Econometrics 102, 23–66.10.1016/S0304-4076(00)00090-7CrossRefGoogle Scholar
Smith, R.J. (2005) Automatic positive semidefinate HAC covariance matrix and GMM estimation. Econometric Theory 21(1), 158–170.CrossRefGoogle Scholar
Tran, L.T. (1993) Nonparametric function estimation for time-series by local average estimators. Annals of Statistics 21(2), 1040–1057.CrossRefGoogle Scholar
Tran, L.T. & Yakowitz, S. (1993) Nearest neighbor estimators for random fields. Journal of Multivariate Analysis 44, 23–46.CrossRefGoogle Scholar
Truong, Y.K. & Stone, C.J. (1992) Nonparametric function estimation involving time series. Annals of Statistics 20(1), 77–97.10.1214/aos/1176348513CrossRefGoogle Scholar
Wheeden, R.L. & Zygmund, A. (1977) Measure and Integral. Dekker.10.1201/b15702CrossRefGoogle Scholar
White, H. (2001) Asymptotic Theory for Econometricians, revised ed. Academic Press.Google Scholar
Yakowitz, S. (1987) Nearest-neighbor methods for time series analysis. Journal of Time Series Analysis 8(2), 235–247.CrossRefGoogle Scholar