×

zbMATH — the first resource for mathematics

Testing conditional moment restrictions. (English) Zbl 1044.62049
Summary: Let \((x,z)\) be a pair of observable random vectors. We construct a new “smoothed” empirical likelihood-based test for the hypothesis \(\mathbb{E}\{g(z, \theta)\mid x\}=0\) w.p.1, where \(g\) is a vector of known functions and \(\theta\) an unknown finite-dimensional parameter. We show that the test statistic is asymptotically normal under the null hypothesis and derive its asymptotic distribution under a sequence of local alternatives. Furthermore, the test is shown to possess an optimality property in large samples. Simulation evidence suggests that it also behaves well in small samples.

MSC:
62G10 Nonparametric hypothesis testing
62E20 Asymptotic distribution theory in statistics
62G20 Asymptotic properties of nonparametric inference
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Aït-Sahalia, Y., Bickel, P. and Stoker, T. (2001). Goodness-of-fit tests for kernel regression with an application to option implied volatilities. J. Econometrics 105 363–412. · Zbl 1004.62042 · doi:10.1016/S0304-4076(01)00091-4
[2] Andrews, D. (1997). A conditional Kolmogorov test. Econometrica 65 1097–1128. · Zbl 0928.62019 · doi:10.2307/2171880
[3] Andrews, D. W. and Ploberger, W. (1994). Optimal tests when a nuisance parameter is present only under the alternative. Econometrica 62 1383–1414. · Zbl 0815.62033 · doi:10.2307/2951753
[4] Andrews, D. W. and Ploberger, W. (1996). Testing for serial correlation against an ARMA\((1,1)\) process. J. Amer. Statist. Assoc. 91 1331–1342. · Zbl 0895.62089 · doi:10.2307/2291751
[5] Bierens, H. J. (1990). A consistent conditional moment test of functional form. Econometrica 58 1443–1458. · Zbl 0737.62058 · doi:10.2307/2938323
[6] Bierens, H. J. and Ploberger, W. (1997). Asymptotic theory of integrated conditional moment tests. Econometrica 65 1129–1151. · Zbl 0927.62085 · doi:10.2307/2171881
[7] Brillinger, D. (1977). Discussion of “Consistent nonparametric regression” by C. J. Stone. Ann. Statist. 5 622–623. JSTOR: · links.jstor.org
[8] Brown, B. W. and Newey, W. K. (1998). Efficient bootstrapping for semiparametric models. Unpublished manuscript. · Zbl 1008.62571
[9] Chamberlain, G. (1987). Asymptotic efficiency in estimation with conditional moment restrictions. J. Econometrics 34 305–334. · Zbl 0618.62040 · doi:10.1016/0304-4076(87)90015-7
[10] Chamberlain, G. (1992). Efficiency bounds for semiparametric regression. Econometrica 60 567–596. · Zbl 0774.62038 · doi:10.2307/2951584
[11] Chen, S., Härdle, W. and Kleinow, T. (2001). An empirical likelihood goodness-of-fit test for time series. Discussion Paper 1, Sonderforschungsbereich 373, Humboldt-Univ. Berlin.
[12] Cosslett, S. (1981a). Efficient estimation of discrete choice models. In Structural Analysis of Discrete Data with Econometric Applications (C. F. Manski and D. McFadden, eds.) 51–111. MIT Press. · Zbl 0551.62076
[13] Cosslett, S. (1981b). Maximum likelihood estimator for choice-based samples. Econometrica 49 1289–1316. · Zbl 0494.62097 · doi:10.2307/1912755
[14] de Jong, P. (1987). A central limit theorem for generalized quadratic forms. Probab. Theory Related Fields 75 261–277. · Zbl 0596.60022 · doi:10.1007/BF00354037
[15] de Jong, R. M. and Bierens, H. J. (1994). On the limit behavior of a chi-square type test if the number of conditional moments tested approaches infinity. Econometric Theory 10 70–90.
[16] Devroye, L. P. and Wagner, T. J. (1980). Distribution-free consistency results in nonparametric discrimination and regression function estimation. Ann. Statist. 8 231–239. JSTOR: · Zbl 0431.62025 · doi:10.1214/aos/1176344949 · links.jstor.org
[17] Ellison, G. and Ellison, S. F. (2000). A simple framework for nonparametric specification testing. J. Econometrics 96 1–23. · Zbl 0968.62046 · doi:10.1016/S0304-4076(99)00048-2
[18] Eubank, R. and Spiegelman, C. (1990). Testing the goodness of fit of a linear model via nonparametric regression techniques. J. Amer. Statist. Assoc. 85 387–392. · Zbl 0702.62037 · doi:10.2307/2289774
[19] Fan, Y. and Li, Q. (1996). Consistent model specification tests: Omitted variables and semiparametric functional forms. Econometrica 64 865–890. · Zbl 0854.62038 · doi:10.2307/2171848
[20] Fan, J., Zhang, C. and Zhang, J. (2001). Generalized likelihood ratio statistics and Wilks phenomenon. Ann. Statist. 29 153–193. · Zbl 1029.62042 · doi:10.1214/aos/996986505
[21] Fisher, N. I., Hall, P., Jing, B.-Y. and Wood, A. T. A. (1996). Improved pivotal methods for constructing confidence regions with directional data. J. Amer. Statist. Assoc. 91 1062–1070. · Zbl 0882.62048 · doi:10.2307/2291725
[22] Hannan, E. (1970). Time Series Analysis . Wiley, New York. · Zbl 0211.49804
[23] Härdle, W. (1989). Asymptotic maximal deviation of \(M\)-smoothers. J. Multivariate Anal. 29 163–179. · Zbl 0667.62028 · doi:10.1016/0047-259X(89)90022-5
[24] Härdle, W. (1990). Applied Nonparametric Regression . Cambridge Univ. Press. · Zbl 0714.62030
[25] Härdle, W. and Mammen, E. (1993). Comparing nonparametric versus parametric regression fits. Ann. Statist. 21 1926–1947. JSTOR: · Zbl 0795.62036 · doi:10.1214/aos/1176349403 · links.jstor.org
[26] Härdle, W. and Marron, J. (1990). Semiparametric comparison of regression curves. Ann. Statist. 18 63–89. JSTOR: · Zbl 0703.62053 · doi:10.1214/aos/1176347493 · links.jstor.org
[27] Hart, J. D. (1997). Nonparametric Smoothing and Lack-of-Fit Tests . Springer, New York. · Zbl 0886.62043
[28] Hastie, T. and Tibshirani, R. (1986). Generalized additive models (with discussion). Statist. Sci. 1 297–318. JSTOR: · Zbl 0645.62068 · doi:10.1214/ss/1177013604 · links.jstor.org
[29] Hong, Y. and White, H. (1995). Consistent specification testing via nonparametric series regression. Econometrica 63 1133–1159. · Zbl 0941.62125 · doi:10.2307/2171724
[30] Horowitz, J. L. and Spokoiny, V. G. (2001). An adaptive rate-optimal test of a parametric mean-regression model against a nonparametric alternative. Econometrica 69 599–631. · Zbl 1017.62012 · doi:10.1111/1468-0262.00207
[31] Imbens, G. W. (1997). One-step estimators for over-identified generalized method of moments models. Rev. Econom. Stud. 64 359–383. · Zbl 0889.90039 · doi:10.2307/2971718
[32] Johnston, G. (1982). Probabilities of maximal deviations for nonparametric regression function estimates. J. Multivariate Anal. 12 402–414. · Zbl 0497.62038 · doi:10.1016/0047-259X(82)90074-4
[33] Kitamura, Y. (1997). Empirical likelihood methods with weakly dependent processes. Ann. Statist. 25 2084–2102. · Zbl 0881.62095 · doi:10.1214/aos/1069362388
[34] Kitamura, Y. (2001). Asymptotic optimality of empirical likelihood for testing moment restrictions. Econometrica 69 1661–1672. · Zbl 0999.62012 · doi:10.1111/1468-0262.00261
[35] Kitamura, Y., Tripathi, G. and Ahn, H. (2002). Empirical likelihood based inference in conditional moment restriction models. Manuscript, Dept. Economics, Univ. Wisconsin, Madison. · Zbl 1142.62331
[36] LeBlanc, M. and Crowley, J. (1995). Semiparametric regression functionals. J. Amer. Statist. Assoc. 90 95–105. · Zbl 0818.62040 · doi:10.2307/2291133
[37] Liero, H. (1982). On the maximal deviation of the kernel regression function estimate. Math. Operationsforsch. Statist. Ser. Statist. 13 171–182. · Zbl 0494.62044 · doi:10.1080/02331888208801638
[38] Newey, W. K. (1985). Maximum likelihood specification testing and conditional moment tests. Econometrica 53 1047–1070. · Zbl 0629.62107 · doi:10.2307/1911011
[39] Newey, W. K. (1994). Kernel estimation of partial means and a general variance estimator. Econometric Theory 10 233–253.
[40] Newey, W. K. and McFadden, D. (1994). Large sample estimation and hypothesis testing. In Handbook of Econometrics (R. Engle and D. McFadden, eds.) 4 2111–2245. North-Holland, Amsterdam.
[41] Owen, A. (1984). The estimation of smooth curves. Stanford Linear Accelerator Center Publication 3394. · Zbl 0529.76094
[42] Owen, A. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika 75 237–249. · Zbl 0641.62032 · doi:10.1093/biomet/75.2.237
[43] Owen, A. (1990). Empirical likelihood ratio confidence regions. Ann. Statist. 18 90–120. JSTOR: · Zbl 0712.62040 · doi:10.1214/aos/1176347494 · links.jstor.org
[44] Owen, A. (1991). Empirical likelihood for linear models. Ann. Statist. 19 1725–1747. JSTOR: · Zbl 0799.62048 · doi:10.1214/aos/1176348368 · links.jstor.org
[45] Priestley, M. (1981). Spectral Analysis and Time Series . Academic Press, New York. · Zbl 0537.62075
[46] Qin, J. and Lawless, J. (1994). Empirical likelihood and general estimating equations. Ann. Statist. 22 300–325. JSTOR: · Zbl 0799.62049 · doi:10.1214/aos/1176325370 · links.jstor.org
[47] Qin, J. and Lawless, J. (1995). Estimating equations, empirical likelihood and constraints on parameters. Canad. J. Statist. 23 145–159. · Zbl 0839.62059 · doi:10.2307/3315441
[48] Staniswalis, J. G. (1987). A weighted likelihood motivation for kernel estimators of a regression function with biomedical applications. Technical report, Virginia Commonwealth Univ.
[49] Staniswalis, J. G. and Severini, T. A. (1991). Diagnostics for assessing regression models. J. Amer. Statist. Assoc. 86 684–692. · Zbl 0736.62063 · doi:10.2307/2290398
[50] Wald, A. (1943). Tests of statistical hypotheses concerning several parameters when the number of observations is large. Trans. Amer. Math. Soc. 54 426–482. · Zbl 0063.08120 · doi:10.2307/1990256
[51] Whang, Y. and Andrews, D. (1993). Tests of specification for parametric and semiparametric models. J. Econometrics 57 277–318. · Zbl 0786.62029 · doi:10.1016/0304-4076(93)90068-G
[52] Wooldridge, J. (1992). A test for functional form against nonparametric alternatives. Econometric Theory 8 452–475.
[53] Yatchew, A. (1992). Nonparametric regression tests based on least squares. Econometric Theory 8 435–451.
[54] Zheng, J. (1996). A consistent test of functional form via nonparametric estimation techniques. J. Econometrics 75 263–289. · Zbl 0865.62030 · doi:10.1016/0304-4076(95)01760-7
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.