×

Testing constancy in varying coefficient models. (English) Zbl 1471.62339

Summary: This article proposes a coefficients constancy test in semi-varying coefficient models that only needs to estimate the restricted coefficients under the null hypothesis. The test statistic resembles the union-intersection test after ordering the data according to the varying coefficients’ explanatory variable. This statistic depends on a trimming parameter that can be chosen by a data-driven calibration method we propose. A bootstrap test is justified under fairly general regularity conditions. Under more restrictive assumptions, the critical values can be tabulated, and trimming is unnecessary. The finite sample performance is studied by means of Monte Carlo experiments, and a real data application for modeling education returns.

MSC:

62G10 Nonparametric hypothesis testing
62E20 Asymptotic distribution theory in statistics
62P20 Applications of statistics to economics
PDFBibTeX XMLCite
Full Text: DOI Link

References:

[1] Andrews, D. W.K., Tests for parameter instability and structural change with unknown change point, Econometrica, 61, 821-856 (1993) · Zbl 0795.62012
[2] Andrews, D. W.K., A conditional Kolmogorov test, Econometrica, 65, 1097-1128 (1997) · Zbl 0928.62019
[3] Andrews, D. W.K.; Ploberger, W., Optimal tests when a nuisance parameter is present only under the alternative, Econometrica, 62, 1383-1414 (1994) · Zbl 0815.62033
[4] Bhattacharya, P. K., Convergence of sample paths of normalized sums of induced order statistics, Ann. Stat., 2, 1034-1039 (1974) · Zbl 0307.62036
[5] Bhattacharya, P. K., An invariance principle in regression analysis, Ann. Stat., 4, 621-624 (1976) · Zbl 0331.62016
[6] Billingsley, P., Convergence of Probability Measures (1968), John Wiley & Sons · Zbl 0172.21201
[7] Blackburn, M. L.; Neumark, D., Are OLS estimates of the return to schooling biased downward? Another look, Rev. Econ. Stat., 77, 217-230 (1995)
[8] Cai, Z.; Fan, J.; Lin, M.; Su, J., Inferences for a partially varying coefficient model with endogenous regressors, J. Bus. Econ. Stat., 37, 158-170 (2017)
[9] Cai, Z.; Fan, J.; Yao, Q., Functional-coefficient regression models for nonlinear time series, J. Am. Stat. Assoc., 95, 941-956 (2000) · Zbl 0996.62078
[10] Chibisov, D. M., Some theorems on the limiting behaviour of an empirical distribution function, Tr. Mat. Inst. Steklova, 71, 104-112 (1964) · Zbl 0163.40602
[11] Chou, S.-Y.; Liu, J.-T.; Huang, C. J., Health insurance and savings over the life cycle - a semiparametric smooth coefficient estimation, J. Appl. Econom., 19, 295-322 (2004)
[12] Csörgő, M., Quantile Processes with Statistical Applications, Vol. 42 (1983), SIAM: SIAM Philadephia · Zbl 0518.62043
[13] Csörgő, M.; Horváth, L., (Limit Theorems in Change-Point Analysis. Limit Theorems in Change-Point Analysis, Wiley Series in Probability and Statistics (1997), Wiley) · Zbl 0884.62023
[14] Darling, D. A.; Erdős, P., A limit theorem for the maximum of normalized sums of independent random variables, Duke Math. J., 23, 143-155 (1956) · Zbl 0070.13806
[15] Davydov, Y.; Egorov, V., Functional limit theorems for induced order statistics, Math. Methods Stat., 9, 297-313 (2000) · Zbl 1010.60031
[16] Delgado, M. A.; Hidalgo, J.; Velasco, C., Distribution free goodness-of-fit tests for linear processes, Ann. Stat., 33, 2568-2609 (2005) · Zbl 1084.62038
[17] Delgado, M. A.; Stute, W., Distribution-free specification tests of conditional models, J. Econom., 143, 37-55 (2008) · Zbl 1418.62193
[18] Durbin, J.; Knott, M., Components of Cramér-von Mises statistics, J. R. Stat. Soc. Ser. B, 34, 290-307 (1972) · Zbl 0238.62052
[19] Fan, J.; Huang, T., Profile likelihood inferences on semiparametric varying-coefficient partially linear models, Bernoulli, 11, 1031-1057 (2005) · Zbl 1098.62077
[20] Fan, J.; Zhang, W., Simultaneous confidence bands and hypothesis testing in varying-coefficient models, Scand. J. Stat., 27, 715-731 (2000) · Zbl 0962.62032
[21] Fan, J.; Zhang, C.; Zhang, J., Generalized likelihood ratio statistics and wilks phenomenon, Ann. Stat., 29, 153-193 (2001) · Zbl 1029.62042
[22] Frölich, M., Parametric and non-parametric regression in the presence of endogenous control variables, Int. Stat. Rev., 76, 214-227 (2008)
[23] Gaenssler, P.; Stute, W., Empirical processes: A survey of results for independent and identically distributed random variables, Ann. Probab., 7, 193-243 (1979) · Zbl 0402.60031
[24] Grenander, U., Stochastic processes and statistical inference, Ark. Mat., 1, 195-277 (1950) · Zbl 0058.35501
[25] Hawkins, D. L., A U-I approach to retrospective testing for shifting, Comm. Statist. Theory Methods, 18, 3117-3134 (1989) · Zbl 0696.62287
[26] Horváth, L., The maximum likelihood method for testing changes in the parameters of normal observations, Ann. Stat., 21, 671-680 (1993) · Zbl 0778.62016
[27] Horváth, L.; Shao, Q.-M., Limit theorems for the union-intersection test, Stat. Plan. Infer., 44, 133-148 (1995) · Zbl 0816.62020
[28] James, B.; James, K. L.; Siegmund, D., Tests for a change-point, Biometrika, 74, 71-83 (1987) · Zbl 0632.62021
[29] Kauermann, G.; Tutz, G., On model diagnostics using varying coefficient models, Biometrika, 86, 119-128 (1999) · Zbl 0917.62063
[30] Kiefer, J., K-sample analogues of the Kolmogorov-Smirnov and Cramér-V. Mises tests, Ann. Math. Stat., 30, 420-447 (1959) · Zbl 0134.36707
[31] Lee, Y.; Stoyanov, A.; Zubanov, N., Olley and Pakes-style production function estimators with firm fixed effects, Oxf. Bull. Econ. Stat., 81, 79-97 (2019)
[32] Levinsohn, J.; Petrin, A., Estimating production functions using inputs to control for unobservables, Rev. Econ. Stud., 70, 317-341 (2003) · Zbl 1073.91048
[33] Li, Q.; Huang, C. J.; Li, D.; Fu, T.-T., Semiparametric smooth coefficient models, J. Bus. Econ. Stat., 20, 412-422 (2002)
[34] Olley, G. S.; Pakes, A., The dynamics of productivity in the telecommunications equipment industry, Econometrica, 64, 1263-1297 (1996) · Zbl 0862.90024
[35] Politis, D. N.; Romano, J. P.; Wolf, M., Subsampling (1999), Springer · Zbl 0931.62035
[36] Schoenfeld, D. A., Asymptotic properties of tests based on linear combinations of the orthogonal components of the Cramer-von Mises statistic, Ann. Stat., 5, 1017-1026 (1977) · Zbl 0369.62045
[37] Scholz, F. W.; Stephens, M. A., K-sample Anderson-Darling tests, J. Am. Stat. Assoc., 82, 918-924 (1987)
[38] Sen, P. K., A note on invariance principles for induced order statistics, Ann. Probab., 4, 474-479 (1976) · Zbl 0336.60022
[39] Shorack, G. R., The weighted empirical process of row independent random variables with arbitrary distribution functions, Stat. Neerl., 33, 169-189 (1979) · Zbl 0436.60009
[40] Sowell, F., Optimal tests for parameter instability in the generalized method of moments framework, Econometrica, 108, 5-1107 (1996) · Zbl 0859.62060
[41] Stute, W., U-functions of concomitants of order statistics, Probab. Math. Stat., 14, 143-155 (1993) · Zbl 0806.60020
[42] Stute, W., Nonparametric model checks for regression, Ann. Stat., 25, 613-641 (1997) · Zbl 0926.62035
[43] Stute, W.; Manteiga, W. G.; Quindimil, M. P., Bootstrap approximations in model checks for regression, J. Am. Stat. Assoc., 93, 141-149 (1998) · Zbl 0902.62027
[44] Wald, A., Tests of statistical hypotheses concerning several parameters when the number of observations is large, Trans. Am. Math. Soc., 54, 426-482 (1943) · Zbl 0063.08120
[45] Wang, H.; Xia, Y., Shrinkage estimation of the varying coefficient model, J. Am. Stat. Assoc., 104, 747-757 (2009) · Zbl 1388.62213
[46] Wellner, J. A., Limit theorems for the ratio of the empirical distribution function to the true distribution function, Z. Wahrscheinlichkeitstheor. Verwandte Geb., 45, 73-88 (1978) · Zbl 0382.60031
[47] Wooldridge, J. M., Introductory Econometrics: A Modern Approach (2009), Nelson Education
[48] Wooldridge, J. M., On estimating firm-level production functions using proxy variables to control for unobservables, Econom. Lett., 104, 112-114 (2009) · Zbl 1181.62196
[49] Xia, Y.; Li, W. K., On single-index coefficient regression models, J. Am. Stat. Assoc., 94, 1275-1285 (1999) · Zbl 1069.62548
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.