Efficient detection of random coefficients in autoregressive models. (English) Zbl 1039.62081

Summary: The problem of detecting randomness in the coefficients of an \(\text{AR}(p)\) model, that is, the problem of testing ordinary \(\text{AR}(p)\) dependence against the alternative of a random coefficient autoregressive \(\text{[RCAR}(p)]\) model is considered. A nonstandard LAN property is established for \(\text{RCAR}(p)\) models in the vicinity of \(\text{AR}(p)\) ones. Two main problems arise in this context. The first problem is related to the statistical model itself: Gaussian assumptions are highly unrealistic in a nonlinear context, and innovation densities should be treated as nuisance parameters. The resulting semiparametric model however appears to be severely nonadaptive. In contrast with the linear ARMA case, pseudo-Gaussian likelihood methods here are invalid under non-Gaussian densities; even the innovation variance cannot be estimated without a strict loss of efficiency.
This problem is solved using a general result by M. Hallin and B. J. M. Werker [Bernoulli 9, 137–165 (2003; Zbl 1020.62042)], which provides semiparametrically efficient central sequences without going through explicit tangent space calculations. The second problem is related to the fact that the testing problem under study is intrinsically one-sided, while the case of multiparameter one-sided alternatives is not covered by classical asymptotic theory under LAN. A concept of locally asymptotically most stringent somewhere efficient test is proposed in order to cope with this one-sided nature of the problem.


62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference


Zbl 1020.62042
Full Text: DOI


[1] ABELSON, R. P. and TUKEY, J. W. (1963). Efficient utilization of non-numerical information in quatitative analysis: General theory and the case of simple order. Ann. Math. Statist. 34 1347-1369. · Zbl 0121.13907
[2] AND EL, J. (1976). Autoregressive series with random parameters. Math. Oper. Statist. 7 735-741. · Zbl 0346.62066
[3] BARTHOLOMEW, D. J. (1961). A test of homogeneity of means under restricted alternatives. J. Roy. Statist. Soc. Ser. B 23 239-281. JSTOR: · Zbl 0209.50303
[4] BENGHABRIT, Y. and HALLIN, M. (1996). Locally asy mptotically optimal tests for autoregressive against bilinear serial dependence. Statist. Sinica 6 147-169. · Zbl 0848.62045
[5] BENGHABRIT, Y. and HALLIN, M. (1998). Locally asy mptotically optimal tests for AR(p) against diagonal bilinear dependence. J. Statist. Plann. Inference 68 47-63. · Zbl 0942.62099
[6] BERGER, R. L. (1989). Uniformly more powerful tests for hy potheses concerning linear inequalities and normal means. J. Amer. Statist. Assoc. 84 192-199. JSTOR: · Zbl 0683.62035
[7] BICKEL, P. J., KLAASSEN, C. A. J., RITOV, Y. and WELLNER, J. A. (1993). Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins Univ. Press, Baltimore, MD. · Zbl 0786.62001
[8] CHACKO, V. J. (1963). Testing homogeneity against ordered alternatives. Ann. Math. Statist. 34 945-956. · Zbl 0116.11104
[9] CHOI, S., HALL, W. J. and SCHICK, A. (1996). Asy mptotically uniformly most powerful tests in parametric and semiparametric models. Ann. Statist. 24 841-861. · Zbl 0860.62020
[10] CONLISK, J. (1976). A further note on stability in a random coefficient model. Internat. Econom. Rev. 17 759-764. · Zbl 0361.62092
[11] DROST, F. C., KLAASSEN, C. A. J. and WERKER, B. J. M. (1997). Adaptive estimation in timeseries models. Ann. Statist. 25 786-817. · Zbl 0941.62093
[12] EHM, W., MAMMEN, E. and MÜLLER, D. W. (1995). Power robustification of approximately linear tests. J. Amer. Statist. Assoc. 90 1025-1033. JSTOR: · Zbl 0843.62053
[13] FEIGIN, P. D. and TWEEDIE, R. L. (1985). Random coefficient autoregressive processes: A Markov chain analysis of stationarity and finiteness of moments. J. Time Ser. Anal. 6 1-14. · Zbl 0572.62069
[14] GAREL, B. and HALLIN, M. (1995). Local asy mptotic normality of multivariate ARMA processes with a linear trend. Ann. Inst. Statist. Math. 47 551-579. · Zbl 0841.62076
[15] GRANGER, C. W. J. and ANDERSEN, A. P. (1978). An Introduction to Bilinear Time Series Models. Vandenhoeck and Ruprecht, G ottingen. · Zbl 0379.62074
[16] GUÉGAN, D. (1994). Séries chronologiques non linéaires à temps discret. Economica, Paris.
[17] GUTMANN, S. (1987). Tests uniformly more powerful than uniformly most powerful monotone tests. J. Statist. Plann. Inference 17 279-292. · Zbl 0635.62021
[18] GUy TON, D. A., ZHANG, N. F. and FOUTZ, R. V. (1986). A random parameter process for modeling and forecasting time series. J. Time Ser. Anal. 7 105-115. · Zbl 0588.62170
[19] HÁJEK, J. (1972). Local asy mptotic minimax and admissibility in estimation. Proc. Sixth Berkeley Sy mp. Math. Statist. Probab. 1 175-194. Univ. California Press, Berkeley.
[20] HÁJEK, J. and SIDÁK, Z. (1967). Theory of Rank Tests. Academic Press, New York.
[21] HÁJEK, J., SIDÁK, Z. and SEN, P. K. (1999). Theory of Rank Tests, 2nd ed. Academic Press, New York.
[22] HALLIN, M. and PURI, M. L. (1994). Aligned rank tests for linear models with autocorrelated error terms. J. Multivariate Anal. 50 175-237. · Zbl 0805.62050
[23] HALLIN, M. and WERKER, B. (1999). Optimal testing for semiparametric AR models: From Gaussian Lagrange multipliers to autoregression rank scores and adaptive tests. In Asy mptotics, Nonparametrics and Time Series (S. Ghosh, ed.) 295-350. Dekker, New York. · Zbl 1069.62541
[24] HALLIN, M. and WERKER, B. (2003). Semi-parametric efficiency, distribution-freeness, and invariance. Bernoulli 9 137-165. · Zbl 1020.62042
[25] HWANG, S. Y. and BASAWA, I. V. (1993a). Asy mptotic optimal inference for a class of nonlinear time series models. Stochastic Process. Appl. 46 91-113. · Zbl 0776.62068
[26] HWANG, S. Y. and BASAWA, I. V. (1993b). Parameter estimation in a regression model with random coefficient autoregressive errors. J. Statist. Plann. Inference 36 57-67. · Zbl 0771.62069
[27] HWANG, S. Y., BASAWA, I. V. and REEVES, J. (1994). The asy mptotic distribution of residual autocorrelations and related tests of fit for a class of nonlinear time series models. Statist. Sinica 4 107-125. · Zbl 0823.62018
[28] HWANG, S. Y. and BASAWA, I. V. (1998). Parameter estimation for generalized random coefficient autoregressive processes. J. Statist. Plann. Inference 68 323-337. · Zbl 0942.62102
[29] KOUL, H. L. and SCHICK, A. (1996). Adaptive estimation in a random coefficient autoregressive model. Ann. Statist. 24 1025-1052. · Zbl 0906.62087
[30] KOUL, H. L. and SCHICK, A. (1997). Efficient estimation in nonlinear autoregressive time series models. Bernoulli 3 247-277. · Zbl 1066.62537
[31] KREISS, J. P. (1987). On adaptive estimation in stationary ARMA processes. Ann. Statist. 15 112- 133. · Zbl 0616.62042
[32] KUDÔ, A. (1963). A multivariate analogue of the one-sided test. Biometrika 50 403-418. · Zbl 0121.13906
[33] LE CAM, L. (1986). Asy mptotic Methods in Statistical Decision Theory. Springer, New York. · Zbl 0605.62002
[34] LE CAM, L. and YANG, G. L. (1990). Asy mptotics in Statistics: Some Basic Concepts. Springer, New York.
[35] LEE, S. (1998). Coefficient constancy test in a random coefficient autoregressive model. J. Statist. Plann. Inference 74 93-101. · Zbl 0924.62092
[36] MENÉNDEZ, J. A., RUEDA, C. and SALVADOR, B. (1992). Dominance of likelihood ratio tests under cone constraints. Ann. Statist. 20 2087-2099. · Zbl 0774.62057
[37] MENÉNDEZ, J. A. and SALVADOR, B. (1991). Anomalies of the likelihood ratio test for testing restricted hy potheses. Ann. Statist. 19 889-898. · Zbl 0734.62058
[38] NICHOLLS, D. F. and QUINN, B. G. (1980). The estimation of random coefficient autoregressive models. I. J. Time Ser. Anal. 1 37-46. · Zbl 0495.62083
[39] NICHOLLS, D. F. and QUINN, B. G. (1981). The estimation of random coefficient autoregressive models. II. J. Time Ser. Anal. 2 185-203. · Zbl 0498.62079
[40] NICHOLLS, D. F. and QUINN, B. G. (1982). Random Coefficient Autoregressive Models: An Introduction. Lecture Notes in Statist. 11. Springer, New York. · Zbl 0497.62081
[41] NÜESCH, P. E. (1966). On the problem of testing location in multivariate populations for restricted alternatives. Ann. Math. Statist. 37 113-119. · Zbl 0136.40002
[42] PAGAN, A. R. (1980). Some identification and estimation results for regression models with stochastically varying coefficients. J. Econometrics 13 341-363. · Zbl 0457.62056
[43] PERLMAN, M. D. (1969). One-sided testing problems in multivariate analysis. Ann. Math. Statist. 40 549-567. [Correction (1971) 42 1777.] · Zbl 0179.24001
[44] PRIESTLEY, M. B. (1988). Nonlinear and Nonstationary Time Series Analy sis. Academic Press, London.
[45] RAMANATHAN, T. V. and RAJARSHI, M. B. (1994). Rank tests for testing the randomness of autoregressive coefficients. Statist. Probab. Lett. 21 115-120. · Zbl 0818.62046
[46] RIEDER, H. (2000). One-sided confidence about functionals over tangent cones. Dept. Mathematics, Univ. Bay reuth. Available at www.uni-bay reuth.de/departments/math/org/mathe7/ rieder/publications.html. URL:
[47] ROBINSON, P. M. (1978). Statistical inference for a random coefficient autoregressive model. Scand. J. Statist. 5 163-168. · Zbl 0392.62072
[48] SCHAAFSMA, W. and SMID, L. J. (1966). Most stringent somewhere most powerful tests against alternatives restricted by a number of linear alternatives. Ann. Math. Statist. 37 1161- 1172. · Zbl 0146.40103
[49] SCHICK, A. (1996). n-consistent estimation in a random coefficient autoregressive model. Austral. J. Statist. 38 155-160. · Zbl 0884.62099
[50] SHORACK, G. R. (1967). Testing against ordered alternatives in model I analysis of variance: Normal theory and nonparametric. Ann. Math. Statist. 38 1740-1752. · Zbl 0157.48304
[51] SWENSEN, A. R. (1985). The asy mptotic distribution of the likelihood ratio for autoregressive time series with a regression trend. J. Multivariate Anal. 16 54-70. · Zbl 0563.62065
[52] TANIGUCHI, M. and KAKIZAWA, Y. (2000). Asy mptotic Theory of Statistical Inference for Time Series. Springer, New York. · Zbl 0955.62088
[53] TONG, H. (1990). Nonlinear Time Series: A Dy namical Approach. Oxford Univ. Press.
[54] VAN DER VAART, A. (1988). Statistical Estimation in Large Parameter Spaces. CWI, Amsterdam. · Zbl 0629.62035
[55] VAN DER VAART, A. (1989). On the asy mptotic information bound. Ann. Statist. 17 1487-1500. · Zbl 0698.62033
[56] VERVAAT, W. (1979). On a stochastic difference equation and a representation of non-negative infinitely divisible random variables. Adv. in Appl. Probab. 11 750-783. JSTOR: · Zbl 0417.60073
[57] WEISS, A. A. (1985). The stability of the AR(1) process with an AR(1) coefficient. J. Time Ser. Anal. 6 181-186.
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