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Distribution-free specification tests of conditional models. (English) Zbl 1418.62193
Summary: This article proposes a class of asymptotically distribution-free specification tests for parametric conditional distributions. These tests are based on a martingale transform of a proper sequential empirical process of conditionally transformed data. Standard continuous functionals of this martingale provide omnibus tests while linear combinations of the orthogonal components in its spectral representation form a basis for directional tests. Finally, Neyman-type smooth tests, a compromise between directional and omnibus tests, are discussed. As a special example we study in detail the construction of directional tests for the null hypothesis of conditional normality versus heteroskedastic contiguous alternatives. A small Monte Carlo study shows that our tests attain the nominal level already for small sample sizes.

##### MSC:
 62G10 Nonparametric hypothesis testing 62E20 Asymptotic distribution theory in statistics 62G20 Asymptotic properties of nonparametric inference 62P20 Applications of statistics to economics
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##### References:
 [1] Andrews, D.W.K., A conditional Kolmogorov test, Econometrica, 65, 1097-1128, (1997) · Zbl 0928.62019 [2] Bai, J., Weak convergence of sequential empirical processes of residuals in ARMA models, Annals of statistics, 22, 2051-2061, (1994) · Zbl 0826.60016 [3] Bai, J., Testing for parameter constancy in linear regressions: an empirical distribution function approach, Econometrica, 64, 597-622, (1996) · Zbl 0844.62032 [4] Bai, J., Testing parametric conditional distributions of dynamic models, The review of economics and statistics, 85, 531-549, (2003) [5] Behnen, K.; Neuhaus, G., A central limit theorem under contiguous alternatives, Annals of statistics, 3, 1349-1353, (1975) · Zbl 0322.62021 [6] Bickel, P.; Wichura, M., Convergence criteria for multiparameter stochastic processes, Annals of mathematical statistics, 42, 1656-1670, (1971) · Zbl 0265.60011 [7] Brown, R.L.; Durbin, J.; Evans, J.M., Techniques for testing the constancy of regression relationships over time, Journal of the royal statistical society series B, 37, 149-192, (1975) · Zbl 0321.62063 [8] Brownrigg, R.D., 2005. Tables of distribution functions of suprema of Brownian Motion on a line or in 2-space. At $$\langle$$http://www.mcs.vuw.ac.nz/ray/Brownian/⟩. [9] Cameron, A.C.; Trivedi, P.K., Regression analysis on count data, (1998), Cambridge University Press Cambridge · Zbl 0924.62004 [10] Durbin, J., Weak convergence of the sample distribution function when parameters are estimated, Annals of statistics, 1, 279-290, (1973) · Zbl 0256.62021 [11] Durbin, J.; Knott, M.; Taylor, C.C., Components of cramér – von Mises statistics II, Journal of the royal statistical society series B, 37, 216-237, (1975) · Zbl 0335.62032 [12] Eubank, R.L.; La Riccia, V.N., Asymptotic comparison of cramér – von Mises and nonparametric function techniques for testing goodness-of-fit, Annals of statistics, 20, 2071-2086, (1992) · Zbl 0769.62033 [13] Gikhman, I.I., Some remarks on A. Kolmogorov’s goodness of fit test, Dokladi akademii nauk, 91, 715-718, (1953), (in Russian) [14] Grenander, U., Stochastic processes and statistical inference, Arkiv för matematik, 1, 195-277, (1950) · Zbl 0058.35501 [15] Härdle, W.; Mammen, E., Comparing nonparametric versus parametric regression fits, Annals of statistics, 21, 1926-1947, (1993) · Zbl 0795.62036 [16] Kac, M.; Kiefer, J.; Wolfowitz, J., On tests of normality and other goodness of fit based on distance methods, Annals of mathematical statistics, 26, 189-211, (1955) · Zbl 0066.12301 [17] Kallenberg, C.M.; Ledwina, T., Data-driven rank tests for independence, Journal of the American statistical association, 94, 285-301, (1999) · Zbl 1072.62574 [18] Khmaladze, E.V., Martingale approach to the goodness of fit tests, Theory of probabilities and applications, 26, 246-265, (1981) · Zbl 0454.60049 [19] Khmaladze, E.V., An innovation approach in goodness-of-fit tests in $$\mathbb{R}^m$$, Annals of statistics, 16, 1503-1516, (1988) · Zbl 0671.62048 [20] Khmaladze, E.V., Goodness of fit problem and scanning innovation martingales, Annals of statistics, 21, 798-829, (1993) · Zbl 0801.62043 [21] Khmaladze, E.V.; Koul, H.L., Martingale transforms goodness-of-fit tests in regression models, Annals of statistics, 32, 995-1034, (2004) · Zbl 1092.62052 [22] Koul, H., Weighted empirical processes in dynamic nonlinear models, (2002), Springer New York · Zbl 1007.62047 [23] Koul, H.; Stute, W., Nonparametric model checks for time series, Annals of statistics, 27, 204-236, (1999) · Zbl 0955.62089 [24] Kuelbs, J., The invariance principle for a lattice of random variables, Annals of mathematical statistics, 39, 382-389, (1968) · Zbl 0164.46401 [25] Lancaster, T., The econometric analysis of transition data, (1990), Cambridge University Press Cambridge · Zbl 0717.62106 [26] Maddala, G.S., Limited-dependent and qualitative variables in econometrics, (1983), Cambridge University Press Cambridge · Zbl 0527.62098 [27] Neuhaus, G., On weak convergence of stochastic processes with multi-dimensional time parameter, Annals of mathematical statistics, 42, 1285-1295, (1971) · Zbl 0222.60013 [28] Neuhaus, G., Asymptotic properties of the cramér – von Mises statistic when parameters are estimated, (), 257-297 [29] Neuhaus, G., 1976. Weak convergence under contiguous alternatives of the empirical process when parameters are estimated: the $$D_k$$ approach. Lecture Notes in Mathematics, vol. 566. Springer, Berlin, pp. 68-82. · Zbl 0356.62039 [30] Neyman, J., Smooth tests for goodness of fit, Skandinavan aktuarietidskrijt, 20, 149-199, (1937) · JFM 63.1092.02 [31] Nikabadze, A., 1997. Scanning innovations and goodness of fit tests for vector random variables against the general alternative. A. Razmadze Mathematical Institute, Tbilisi, Preprint. [32] Nikabadze, A.; Stute, W., Model checks under random censorship, Statistics and probability letters, 32, 249-259, (1997) · Zbl 1003.62540 [33] Pollard, D., Convergence of stochastic processes, (1984), Springer New York, Berlin · Zbl 0544.60045 [34] Rosenblatt, M., Remarks on a multivariate transformation, Annals of mathematical statistics, 23, 470-472, (1952) · Zbl 0047.13104 [35] Schoenfeld, D.A., Asymptotic properties of tests based on linear combinations of the orthogonal components of the cramér – von Mises statistic, Annals of statistics, 5, 1017-1026, (1977) · Zbl 0369.62045 [36] Schoenfeld, D.A., Tests based on linear combinations of the orthogonal components of the cramér – von Mises statistic when parameters are estimated, Annals of statistics, 8, 1017-1022, (1980) · Zbl 0485.62039 [37] Shorack, G.R.; Wellner, J.A., Empirical processes with applications to statistics, (1986), Wiley New York · Zbl 1170.62365 [38] Straf, M.L., Weak convergence of stochastic processes with several parameters, (), 187-221 [39] Stute, W., Nonparametric model checks for regression, Annals of statistics, 25, 613-641, (1997) · Zbl 0926.62035 [40] Stute, W.; Zhu, L.X., Model checks for generalized linear models, Scandinavan journal of statistics, 29, 535-545, (2002) · Zbl 1035.62073 [41] Stute, W.; Thies, S.; Zhu, L.X., Model checks for regression: an innovation process approach, Annals of statistics, 26, 1916-1934, (1998) · Zbl 0930.62044 [42] Zheng, J.X., A conditional test of conditional parametric distributions, Econometric theory, 16, 667-691, (2000) · Zbl 0967.62032
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