Specification testing for regression models with dependent data. (English) Zbl 1418.62331
Summary: We examine a consistent test for the correct specification of a regression function with dependent data. The test is based on the supremum of the difference between the parametric and nonparametric estimates of the regression model. Rather surprisingly, the behaviour of the test depends on whether the regressors are deterministic or stochastic. In the former situation, the normalization constants necessary to obtain the limiting Gumbel distribution are data dependent and difficult to estimate, so it may be difficult to obtain valid critical values, whereas, in the latter, the asymptotic distribution may not be even known. Because of that, under very mild regularity conditions, we describe a bootstrap analogue for the test, showing its asymptotic validity and finite sample behaviour in a small Monte-Carlo experiment.
MSC:
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 62G10 | Nonparametric hypothesis testing |
| 62G07 | Density estimation |
| 62G09 | Nonparametric statistical resampling methods |
| 62E20 | Asymptotic distribution theory in statistics |
| 62G20 | Asymptotic properties of nonparametric inference |
| 62P20 | Applications of statistics to economics |
Keywords:
functional specification; variable selection; nonparametric kernel regression; frequency domain bootstrapReferences:
| [1] | Berk, K. N., Consistent autoregressive spectral estimates, The Annals of Statistics, 2, 489-502 (1974) · Zbl 0317.62064 |
| [2] | Bickel, P. J.; Rosenblatt, M., On some global measures of the deviations of density function estimates, The Annals of Statistics, 1, 1071-1095 (1973) · Zbl 0275.62033 |
| [3] | Bierens, H.; Ploberger, W., Asymptotic theory of integrated conditional moment test, Econometrica, 65, 1153-1174 (1997) |
| [4] | Brillinger, D. R., Time Series, Data Analysis and Theory (1981), Holden-Day: Holden-Day San Francisco, CA · Zbl 0486.62095 |
| [5] | Brockwell, P. J.; Davis, R. A., Time Series: Theory and Methods (1991), Springer: Springer New York · Zbl 0673.62085 |
| [6] | Bühlmann, P., Sieve-bootstrap for time series, Bernoulli, 3, 123-148 (1997) · Zbl 0874.62102 |
| [7] | Csörgő, S.; Mason, D. M., Bootstrapping empirical functions, The Annals of Statistics, 17, 1447-1471 (1989) · Zbl 0701.62057 |
| [8] | Csörgő, S.; Mielniczuk, J., Nonparametric regression under long-range dependent normal errors, The Annals of Statistics, 23, 1000-1014 (1995) · Zbl 0852.62035 |
| [9] | Dahlhaus, R.; Janas, D., A frequency domain bootstrap for ratio statistics in time series analysis, The Annals of Statistics, 24, 1934-1963 (1996) · Zbl 0867.62072 |
| [10] | Davies, R. B.; Harte, D. S., Tests for Hurst effect, Biometrika, 74, 95-101 (1987) · Zbl 0612.62123 |
| [11] | Delgado, M. A.; Gonzalez-Manteiga, W., Significance testing in nonparametric regression based on the bootstrap, The Annals of Statistics, 29, 1469-1507 (2001) · Zbl 1043.62032 |
| [12] | Delgado, M. A.; Hidalgo, J.; Velasco, C., Distribution free goodness-of-fit tests for linear processes, The Annals of Statistics, 33, 2568-2609 (2005) · Zbl 1084.62038 |
| [13] | Efron, B., Bootstrap methods: another look at the jackknife, The Annals of Statistics, 7, 1-26 (1979) · Zbl 0406.62024 |
| [14] | Eubank, R.; Spiegelman, S., Testing the goodness of fit of a linear model via nonparametric functional forms, Journal of the American Statistical Association, 85, 387-392 (1990) · Zbl 0702.62037 |
| [15] | Fan, Y.; Li, Q., Consistent model specification test: omitted variables and semiparametric functional forms, Econometrica, 64, 865-890 (1996) · Zbl 0854.62038 |
| [16] | Giné, E.; Zinn, J., Necessary conditions for the bootstrap of the mean, The Annals of Statistics, 17, 684-691 (1989) · Zbl 0672.62026 |
| [17] | Härdle, W.; Mammen, E., Comparing nonparametric versus parametric regression fits, The Annals of Statistics, 21, 1926-1947 (1993) · Zbl 0795.62036 |
| [18] | Hidalgo, J., Nonparametric estimation with strongly dependent multivariate time series, Journal of Time Series Analysis, 18, 95-122 (1997) · Zbl 0923.62045 |
| [19] | Hidalgo, J., Nonparametric tests for model selection with time series data, Test, 8, 365-398 (1999) · Zbl 0938.62052 |
| [20] | Hidalgo, J., Nonparametric test for causality with long-range dependence, Econometrica, 68, 1465-1490 (2000) · Zbl 1056.62518 |
| [21] | Hidalgo, J., An alternative bootstrap to moving blocks for time series regression models, Journal of Econometrics, 117, 369-399 (2003) · Zbl 1029.62041 |
| [22] | Hidalgo, J., 2007. Specification testing for regression models with dependent data. STICERD econometric series, London School of Economics.; Hidalgo, J., 2007. Specification testing for regression models with dependent data. STICERD econometric series, London School of Economics. |
| [23] | Holly, A., A remark on Hausman’s specification test, Econometrica, 50, 749-760 (1982) · Zbl 0523.62095 |
| [24] | Hong, Y.; White, H., Consistent specification testing via nonparametric series regressions, Econometrica, 63, 1133-1159 (1995) · Zbl 0941.62125 |
| [25] | Hurvich, C. M., Model selection for broadband semiparametric estimation of long-memory in time series, Journal of Time Series Analysis, 22, 679-709 (2001) · Zbl 0984.62068 |
| [26] | Hurvich, C.M., Zeger, S.L., 1987. Frequency domain bootstrap methods for time series. Technical Report, vol. 87-115, Graduate School of Business Administration, New York University.; Hurvich, C.M., Zeger, S.L., 1987. Frequency domain bootstrap methods for time series. Technical Report, vol. 87-115, Graduate School of Business Administration, New York University. |
| [27] | Ibragimov, I. A.; Rozanov, Y. A., Gaussian Random Processes (1978), Springer: Springer Berlin · Zbl 0392.60037 |
| [28] | Johnston, G. J., Probabilities of maximal deviation for nonparametric regression function estimates, Journal of Multivariate Analysis, 12, 402-414 (1982) · Zbl 0497.62038 |
| [29] | Koul, H. L.; Stute, W., Nonparametric model checks for time series, The Annals of Statistics, 27, 204-236 (1999) · Zbl 0955.62089 |
| [30] | Kreiss, J.-P., 1988. Asymptotic statistical inference for a class of stochastic processes. Habilitationsschrift, Fachbereich Mathematik, Univ. Hamburg.; Kreiss, J.-P., 1988. Asymptotic statistical inference for a class of stochastic processes. Habilitationsschrift, Fachbereich Mathematik, Univ. Hamburg. |
| [31] | Kreiss, J.-P.; Paparoditis, E., Autoregressive-aided periodogram bootstrap for time series, The Annals of Statistics, 31, 1923-1955 (2003) · Zbl 1042.62081 |
| [32] | Künsch, H. A., The jackknife and the bootstrap for general stationary observations, The Annals of Statistics, 17, 1217-1241 (1989) · Zbl 0684.62035 |
| [33] | Marinucci, D.; Robinson, P. M., Weak convergence of multivariate fractional processes, Stochastic Processes and its Applications, 80, 103-120 (2003) · Zbl 1028.60030 |
| [34] | Prichard, D., Theiler, J., 1994. Generating surrogate data for time series with several simultaneously measured variables. Working paper #94-04-023, Santa Fe Institute.; Prichard, D., Theiler, J., 1994. Generating surrogate data for time series with several simultaneously measured variables. Working paper #94-04-023, Santa Fe Institute. |
| [35] | Robinson, P. M., Automatic frequency domain inference on semiparametric and nonparametric models, Econometrica, 59, 1329-1363 (1991) · Zbl 0779.62037 |
| [36] | Robinson, P. M., Log-periodogram regression for time series with long range dependence, The Annals of Statistics, 23, 1048-1072 (1995) · Zbl 0838.62085 |
| [37] | Robinson, P. M., Gaussian semiparametric estimation of long-range dependence, The Annals of Statistics, 23, 1630-1661 (1995) · Zbl 0843.62092 |
| [38] | Robinson, P. M., Large-sample inference for nonparametric regression with dependent errors, The Annals of Statistics, 25, 2054-2083 (1997) · Zbl 0882.62039 |
| [39] | Stute, W., Nonparametric model checks for regression, The Annals of Statistics, 25, 613-641 (1997) · Zbl 0926.62035 |
| [40] | Theiler, J., Galdrikian, B., Longtin, A., Eubank, S., Farmer, J.D., 1992. Using surrogate data to detect nonlinearity in time series. In: Casdagli, M., Eubank, S. (Eds.), A proceedings volume in the Santa Fe Institute Studies in the Sciences of Complexity. Nonlinear Modeling and Forecasting. pp. 163-188.; Theiler, J., Galdrikian, B., Longtin, A., Eubank, S., Farmer, J.D., 1992. Using surrogate data to detect nonlinearity in time series. In: Casdagli, M., Eubank, S. (Eds.), A proceedings volume in the Santa Fe Institute Studies in the Sciences of Complexity. Nonlinear Modeling and Forecasting. pp. 163-188. · Zbl 1194.37144 |
| [41] | Wang, Q.; Lin, Y-X.; Gulati, C. M., Strong approximation for long memory processes with applications, Journal of Theoretical Probability, 16, 377-389 (2003) · Zbl 1020.60017 |
| [42] | Woodroofe, M., On the maximum deviation of the sample density, Annals of Mathematical Statistics, 38, 475-481 (1967) · Zbl 0157.48002 |
| [43] | Yong, C. H., Asymptotic behaviour of trigonometric series (1974), Chinese University of Hong Kong: Chinese University of Hong Kong Hong Kong · Zbl 0338.42004 |
| [44] | Zheng, J. X., A consistent test of functional form via nonparametric estimation techniques, Journal of Econometrics, 75, 263-289 (1996) · Zbl 0865.62030 |
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