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TESTING A PARAMETRIC TRANSFORMATION MODEL VERSUS A NONPARAMETRIC ALTERNATIVE

Published online by Cambridge University Press:  12 May 2020

Arkadiusz Szydłowski
Arkadiusz Szydłowski*
Affiliation:
University of Leicester
*
Address correspondence to Arkadiusz Szydłowski, Division of Economics, University of Leicester, University Road, Leicester LE1 7RH, UK; email: [email protected]
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Abstract

Despite an abundance of semiparametric estimators of the transformation model, no procedure has been proposed yet to test the hypothesis that the transformation function belongs to a finite-dimensional parametric family against a nonparametric alternative. In this article, we introduce a bootstrap test based on integrated squared distance between a nonparametric estimator and a parametric null. As a special case, our procedure can be used to test the parametric specification of the integrated baseline hazard in a semiparametric mixed proportional hazard model. We investigate the finite sample performance of our test in a Monte Carlo study. Finally, we apply the proposed test to Kennan’s strike durations data.

Type
ARTICLES
Information
Econometric Theory , Volume 36 , Issue 5 , October 2020 , pp. 871 - 906
Copyright
© Cambridge University Press 2020

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Footnotes

I am grateful to Joel Horowitz and Elie Tamer for their encouragement and valuable suggestions. I would also like to thank the co-editor, three anonymous referees, Yu Zhu, and participants at WIEM 2015 and ESEM 2016 conferences for their comments. This research used the ALICE High Performance Computing Facility at the University of Leicester and the Social Sciences Computing Cluster at Northwestern University.

References

REFERENCES

Abrevaya, J. (2003) Pairwise-difference rank estimation of the transformation model. Journal of Business & Economic Statistics 21, 437–447.CrossRefGoogle Scholar
Arcones, M.A. & Gine, E. (1993) Limit theorems for U-processes. The Annals of Probability 21, 1494–1542.CrossRefGoogle Scholar
Asparouhova, E., Golanski, R., Kasprzyk, K., Sherman, R.P., & Asparouhov, T. (2002) Rank estimators for a transformation model. Econometric Theory 18, 1099–1120.CrossRefGoogle Scholar
Bickel, P.J. & Doksum, K.A. (1981) An analysis of transformations revisited. Journal of the American Statistical Association 76, 296–311.CrossRefGoogle Scholar
Bickel, P.J. & Freedman, D.A. (1981) Some asymptotic theory for the bootstrap. The Annals of Statistics 9, 1196–1217.CrossRefGoogle Scholar
Blundell, R. & Horowitz, J.L. (2007) A non-parametric test of exogeneity. The Review of Economic Studies 74, 1035–1058.CrossRefGoogle Scholar
Cavanagh, C. & Sherman, R.P. (1998) Rank estimators for monotonic index models. Journal of Econometrics 84, 351–381.CrossRefGoogle Scholar
Chen, S. (2002) Rank estimation of transformation models. Econometrica 70, 1683–1697.CrossRefGoogle Scholar
Chen, S. (2012) Distribution-free estimation of the Box-Cox regression model with censoring. Econometric Theory 28, 680–695.CrossRefGoogle Scholar
Cheng, G. & Huang, J.Z. (2010) Bootstrap consistency for general semiparametric M-estimation. Annals of Statistics 38, 2884–2915.CrossRefGoogle Scholar
Durbin, J. & Knott, M. (1972) Components of Cramér-von Mises statistics. I. Journal of the Royal Statistical Society. Series B (Methodological) 34, 290–307.CrossRefGoogle Scholar
Durbin, J., Knott, M., & Taylor, C.C. (1975) Components of Cramér-von Mises statistics. II Journal of the Royal Statistical Society. Series B (Methodological) 37, 216–237.CrossRefGoogle Scholar
Ekeland, I., Heckman, J.J., & Nesheim, L. (2004) Identification and estimation of hedonic models. Journal of Political Economy 112, S60–S109.CrossRefGoogle Scholar
Foster, A.M., Tian, L., & Wei, L.J. (2001) Estimation for the Box-Cox transformation model without assuming parametric error distribution. Journal of the American Statistical Association 96, 1097–1101.CrossRefGoogle Scholar
Gine, E. & Zinn, J. (1990) Bootstrapping general empirical measures. The Annals of Probability 18, 851–869.CrossRefGoogle Scholar
Giné, E. & Zinn, J. (1992) On Hoffmann–Jørgensen’s Inequality for U-Processes. In Dudley, R.M., Hahn, M.G., Kuelbs, J. (eds.), Probability in Banach Spaces, 8: Proceedings of the Eighth International Conference, pp. 80–91. Birkhäuser.CrossRefGoogle Scholar
Gørgens, T. & Horowitz, J.L. (1999) Semiparametric estimation of a censored regression model with an unknown transformation of the dependent variable. Journal of Econometrics 90, 155–191.CrossRefGoogle Scholar
Hahn, J. (1994) The efficiency bound of the mixed proportional hazard model. The Review of Economic Studies 61, 607–629.CrossRefGoogle Scholar
Hainmueller, J., Hiscox, M.J., & Sequeira, S. (2015) Consumer demand for fair trade: Evidence from a multistore field experiment. The Review of Economics and Statistics 97, 242–256.CrossRefGoogle Scholar
Han, A.K. (1987) A non-parametric analysis of transformations. Journal of Econometrics 35, 191–209.CrossRefGoogle Scholar
Heckman, J. & Singer, B. (1984) A method for minimizing the impact of distributional assumptions in econometric models for duration data. Econometrica 52, 271–320.CrossRefGoogle Scholar
Honoré, B.E. (1990) Simple estimation of a duration model with unobserved heterogeneity. Econometrica 58, 453–473.CrossRefGoogle Scholar
Honoré, B., Khan, S., & Powell, J.L. (2002) Quantile regression under random censoring. Journal of Econometrics 109, 67–105.CrossRefGoogle Scholar
Horowitz, J. (2009) Semiparametric and Nonparametric Methods in Econometrics, Springer-Verlag New York.CrossRefGoogle Scholar
Horowitz, J.L. (1996) Semiparametric estimation of a regression model with an unknown transformation of the dependent variable. Econometrica 64, 103–137.CrossRefGoogle Scholar
Horowitz, J.L. (1999) Semiparametric estimation of a proportional hazard model with unobserved heterogeneity. Econometrica 67, 1001–1028.CrossRefGoogle Scholar
Horowitz, J.L. (2006) Testing a parametric model against a nonparametric alternative with identification through instrumental variables. Econometrica 74, 521–538.CrossRefGoogle Scholar
Horowitz, J.L. & Härdle, W. (1994) Testing a parametric model against a semiparametric alternative. Econometric Theory 10, 821–848.CrossRefGoogle Scholar
Horowitz, J.L. & Neumann, G.R. (1992) A generalized moments specification test of the proportional hazards model. Journal of the American Statistical Association 87, 234–240.CrossRefGoogle Scholar
Härdle, W. & Mammen, E. (1993) Comparing nonparametric versus parametric regression fits. Annals of Statistics 21, 1926–1947.CrossRefGoogle Scholar
Härdle, W., Spokoiny, V., & Sperlich, S. (1997) Semiparametric single index versus fixed link function modelling. Annals of Statistics 25, 212–243.CrossRefGoogle Scholar
Ichimura, H. (1993) Semiparametric least squares (SLS) and weighted least squares estimation of single index models. Journal of Econometrics 58, 71–120.CrossRefGoogle Scholar
Ishwaran, H. (1996) Identifiability and rates of estimation for scale parameters in location mixture models. Annals of Statistics 24, 1560–1571.CrossRefGoogle Scholar
Jessoe, K. & Rapson, D. (2014) Knowledge is (less) power: Experimental evidence from residential energy use. American Economic Review 104, 1417–1438.CrossRefGoogle Scholar
Jochmans, K. (2012) The variance of a rank estimator of transformation models. Economics Letters 117, 168–169.CrossRefGoogle Scholar
Karlan, D. & Zinman, J. (2019) Long-run price elasticities of demand for credit: Evidence from a countrywide field experiment in Mexico, The Review of Economic Studies 86, 1704–1746.CrossRefGoogle Scholar
Kennan, J. (1985) The duration of contract strikes in U.S. manufacturing. Journal of Econometrics 28, 5–28.CrossRefGoogle Scholar
Klein, R.W. & Sherman, R.P. (2002) Shift restrictions and semiparametric estimation in ordered response models. Econometrica 70, 663–691.CrossRefGoogle Scholar
Lin, J., Zhang, D., & Davidian, M. (2006) Smoothing spline-based score tests for proportional hazards models. Biometrics 62, 803–812.CrossRefGoogle ScholarPubMed
Linton, O., Sperlich, S., & Van Keilegom, I. (2008) Estimation of a semiparametric transformation model. Annals of Statistics 36, 686–718.CrossRefGoogle Scholar
Lo, S.-H. & Singh, K. (1986) The product-limit estimator and the bootstrap: Some asymptotic representations. Probability Theory and Related Fields 71, 455–465.CrossRefGoogle Scholar
McCall, B.P. (1994) Testing the proportional hazards assumption in the presence of unmeasured heterogeneity. Journal of Applied Econometrics 9, 321–334.CrossRefGoogle Scholar
Meyer, B.D. (1990) Unemployment insurance and unemployment spells. Econometrica 58, 757–782.CrossRefGoogle Scholar
Mu, Y. & He, X. (2007) Power transformation toward a linear regression quantile. Journal of the American Statistical Association 102, 269–279.CrossRefGoogle Scholar
Neumeyer, N. (2009) Smooth residual bootstrap for empirical processes of non-parametric regression residuals. Scandinavian Journal of Statistics 36, 204–228.CrossRefGoogle Scholar
Neumeyer, N., Noh, H., & Van Keilegom, I. (2016) Heteroscedastic semiparametric transformation models: Estimation and testing for validity. Statistica Sinica 26, 925–954.Google Scholar
Pakes, A. & Pollard, D. (1989) Simulation and the asymptotics of optimization estimators. Econometrica 57, 1027–1057.CrossRefGoogle Scholar
Romano, J.P. & Wolf, M. (2005) Stepwise multiple testing as formalized data snooping. Econometrica 73, 1237–1282.CrossRefGoogle Scholar
Serfling, R.J. (1980) Approximation Theorems of Mathematical Statistics. Wiley.CrossRefGoogle Scholar
Sherman, R.P. (1993) The limiting distribution of the maximum rank correlation estimator. Econometrica 61, 123–137.CrossRefGoogle Scholar
Sherman, R.P. (1994) Maximal inequalities for degenerate U-processes with applications to optimization estimators. Annals of Statistics 22, 439–459.CrossRefGoogle Scholar
Subbotin, V. (2007) Asymptotic and bootstrap properties of rank regressions. MPRA Working Paper 9030.CrossRefGoogle Scholar
Van der Vaart, A.W. & Wellner, J.A. (1996) Weak Convergence and Empirical Processes: With Applications to Statistics. Springer-Verlag New York.CrossRefGoogle Scholar
Ye, J. & Duan, N. (1997) Nonparametric $\sqrt {n}$-consistent estimation for the general transformation models. Annals of Statistics 25, 2682–2717.CrossRefGoogle Scholar
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