Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T17:57:15.477Z Has data issue: false hasContentIssue false
  • English
  • Français

SMOOTHED QUANTILE REGRESSION PROCESSES FOR BINARY RESPONSE MODELS

Published online by Cambridge University Press:  20 May 2019

Stanislav Volgushev
Stanislav Volgushev*
Affiliation:
University of Toronto
*
*Address correspondence to Stanislav Volgushev, Department of Statistical Sciences, University of Toronto, Toronto, ON M5S, Canada; e-mail: [email protected].
Get access Rights & Permissions [Opens in a new window]

Abstract

In this article, we consider binary response models with linear quantile restrictions. Considerably generalizing previous research on this topic, our analysis focuses on an infinite collection of quantile estimators. We derive a uniform linearization for the properly standardized empirical quantile process and discover some surprising differences with the setting of continuously observed responses. Moreover, we show that considering quantile processes provides an effective way of estimating binary choice probabilities without restrictive assumptions on the form of the link function, heteroskedasticity, or the need for high dimensional nonparametric smoothing necessary for approaches available so far. A uniform linear representation and results on asymptotic normality are provided, and the connection to rearrangements is discussed.

Type
ARTICLES
Information
Econometric Theory , Volume 36 , Issue 2 , April 2020 , pp. 292 - 330
Copyright
Copyright © Cambridge University Press 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

The idea of considering binary response quantile processes originated from discussions with Prof. Roger Koenker. I am thankful to him for the encouragement and many insightful discussions on this topic. My thanks also go to Prof. Jiaying Gu for many helpful discussions. Any remaining mistakes are my sole responsibility. I also thank the Editor Prof. Peter C.B. Phillips, the co-Editor Prof. Yoon-Jae Whang and three anonymous Referees for constructive and insightful comments on previous versions of this manuscript that helped to considerably improve the presentation and content of this article. Part of this research was conducted while I was a visiting scholar at UIUC. I am very grateful to the Statistics and Economics departments for their hospitality. Financial support from the DFG (grant VO1799/1-1) and from a discovery grant from NSERC of Canada is gratefully acknowledged.

References

REFERENCES

Chaudhuri, P. (1991) Nonparametric estimates of regression quantiles and their local bahadur representation. The Annals of Statistics 19(2), 760–777.10.1214/aos/1176348119CrossRefGoogle Scholar
Chernozhukov, V., Fernández-Val, I., & Galichon, A. (2010) Quantile and probability curves without crossing. Econometrica 78(3), 1093–1125.Google Scholar
Coppejans, M. (2001) Estimation of the binary response model using a mixture of distributions estimator (mod). Journal of Econometrics 102(2), 231–269.CrossRefGoogle Scholar
Cosslett, S. (1983) Distribution-free maximum likelihood estimator of the binary choice model. Econometrica 51(3), 765–782.CrossRefGoogle Scholar
Dette, H., Neumeyer, N., & Pilz, K. (2006) A simple nonparametric estimator of a strictly monotone regression function. Bernoulli 12(3), 469–490.CrossRefGoogle Scholar
Dette, H. & Volgushev, S. (2008) Noncrossing nonparametric estimates of quantile curves. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 70(3), 609–627.CrossRefGoogle Scholar
Florios, K. & Skouras, S. (2008) Exact computation of max weighted score estimators. Journal of Econometrics 146(1), 86–91.10.1016/j.jeconom.2008.05.018CrossRefGoogle Scholar
Friedman, J., Hastie, T., & Tibshirani, R. (2001) The Elements of Statistical Learning. Springer series in Statistics, vol. 1. Springer.Google Scholar
Goffe, W.L., Ferrier, G.D., & Rogers, J. (1994) Global optimization of statistical functions with simulated annealing. Journal of Econometrics 60(1), 65–99.CrossRefGoogle Scholar
Hardy, G., Littlewood, J., & Polya, G. (1988) Inequalities. Cambridge University Press.Google Scholar
Horowitz, J. (1992). A smoothed maximum score estimator for the binary response model. Econometrica 60(3), 505–531.10.2307/2951582CrossRefGoogle Scholar
Horowitz, J.L. (2009) Semiparametric and Nonparametric Methods in Econometrics. Springer Series in Statistics. Springer.CrossRefGoogle Scholar
Ichimura, H. (1993) Semiparametric least squares (SLS) and weighted SLS estimation of single-index models. Journal of Econometrics 58(1), 71–120.CrossRefGoogle Scholar
Khan, S. (2013) Distribution free estimation of heteroskedastic binary response models using probit/logit criterion functions. Journal of Econometrics 172(1), 168–182.CrossRefGoogle Scholar
Kim, J. & Pollard, D. (1990) Cube root asymptotics. The Annals of Statistics 18(1), 191–219.CrossRefGoogle Scholar
Klein, R. & Spady, R. (1993) An efficient semiparametric estimator for binary response models. Econometrica 61(2), 387–421.CrossRefGoogle Scholar
Koenker, R. & Bassett, G. (1978) Regression quantiles. Econometrica 46(1), 33–50.CrossRefGoogle Scholar
Kordas, G. (2006) Smoothed binary regression quantiles. Journal of Applied Econometrics 21(3), 387–407.CrossRefGoogle Scholar
Manski, C. (1975) Maximum score estimation of the stochastic utility model of choice. Journal of Econometrics 3(3), 205–228.10.1016/0304-4076(75)90032-9CrossRefGoogle Scholar
Manski, C. (1985) Semiparametric analysis of discrete response: Asymptotic properties of the maximum score estimator. Journal of Econometrics 27(3), 313–333.10.1016/0304-4076(85)90009-0CrossRefGoogle Scholar
Manski, C. (1988) Identification of binary response models. Journal of the American Statistical Association 83(403), 729–738.CrossRefGoogle Scholar
Neumeyer, N. (2007) A note on uniform consistency of monotone function estimators. Statistics & Probability Letters 77(7), 693–703.10.1016/j.spl.2006.11.004CrossRefGoogle Scholar
Portnoy, S. (1998) Convergence rates for maximal score estimators in binary response regressions. In Szyszkowicz, B. (ed.), Asymptotic Methods in Probability and Statistics, pp. 775–783, Elsevier.CrossRefGoogle Scholar
Powell, J., Stock, J., & Stoker, T. (1989) Semiparametric estimation of index coefficients. Econometrica 57(6), 1403–1430.10.2307/1913713CrossRefGoogle Scholar
van der Vaart, A.W. & Wellner, J.A. (1996) Weak Convergence and Empirical Processes. Springer Series in Statistics. Springer.CrossRefGoogle Scholar
Volgushev, S., Birke, M., Dette, H., & Neumeyer, N. (2013) Significance testing in quantile regression. Electronic Journal of Statistics 7, 105–145.CrossRefGoogle Scholar