Estimation and Testing Stationarity for Double-Autoregressive Models

Shiqing Ling1
1Hong Kong University of Science and Technology , People's Republic of China

Tóm tắt

SummaryThe paper considers the double-autoregressive model yt = φyt−1+ɛt with ɛt =ηt√(ω+αyt−12). Consistency and asymptotic normality of the estimated parameters are proved under the condition E ln |φ +√αηt|<0, which includes the cases with |φ|=1 or |φ|>1 as well as E(εt2)=∞. It is well known that all kinds of estimators of φ in these cases are not normal when ɛt are independent and identically distributed. Our result is novel and surprising. Two tests are proposed for testing stationarity of the model and their asymptotic distributions are shown to be a function of bivariate Brownian motions. Critical values of the tests are tabulated and some simulation results are reported. An application to the US 90-day treasury bill rate series is given.

Từ khóa


Tài liệu tham khảo

Amemiya, 1985, Advanced Econometrics

Billingsley, 1968, Convergence of Probability Measures

Borkovec, 2001, Asymptotic behavior of the sample autocovariance and autocorrelation function of the AR(1) process with ARCH(1), Bernoulli, 6, 847, 10.2307/3318623

Borkovec, 1998, The tail of the stationary distribution of an autoregressive process with ARCH(1) errors, Ann. Appl. Probab., 11, 1220

Bougerol, 1992, Strict stationarity of generalized autoregressive processes, Ann. Probab., 20, 1714, 10.1214/aop/1176989526

Chan, 1989, On the first-order autoregressive process with infinite variance, Econometr. Theory, 5, 354, 10.1017/S0266466600012561

Davis, 1992, M-estimation for autoregressions with infinite variance, Stoch. Processes Appl., 40, 145, 10.1016/0304-4149(92)90142-D

Dickey, 1979, Distribution of the estimators for autoregressive time series with a unit root, J. Am. Statist. Ass., 74, 427

Engle, 1982, Autoregressive conditional heteroskedasticity with estimates of variance of U. K. inflation, Econometrica, 50, 987, 10.2307/1912773

Feigin, 1985, Random coefficient autoregressive processes: a Markov chain analysis of stationarity and finiteness of moments, J. Time Ser. Anal., 6, 1, 10.1111/j.1467-9892.1985.tb00394.x

Francq, 1998, Estimating linear representations of nonlinear processes, J. Statist. Planng Inf., 68, 145, 10.1016/S0378-3758(97)00139-0

Francq, 2000, Covariance matrix estimation for estimators of mixing weak ARMA models, J. Statist. Planng Inf., 83, 369, 10.1016/S0378-3758(99)00109-3

Guégan, 1994, Probabilistic properties of the β-ARCH-model, Statist. Sin., 4, 71

Hall, 1980, Martingale Limit Theory and Its Applications

Hamilton, 1994, Time Series Analysis, 10.1515/9780691218632

Jeganathan, 1988, On the strong approximation of the distributions of estimators in linear stochastic models: I, II, Stationary and explosive AR models, Ann. Statist., 16, 1283, 10.1214/aos/1176350962

Kluppelberg, 2002, Testing for reduction to random walk in autoregressive conditional heteroscedasticity models, Econometr. J., 5, 387, 10.1111/1368-423X.t01-1-00090

Koul, 1990, Weakly adaptive estimators in explosive autoregression, Ann. Statist., 18, 939, 10.1214/aos/1176347634

Kurtz, 1991, Weak limit theorems to stochastic integrals and stochastic differential equations, Ann. Probab., 19, 1035, 10.1214/aop/1176990334

Lee, 1994, Asymptotic theory for GARCH(1,1) quasi-maximum likelihood estimator, Econometr. Theory, 10, 29, 10.1017/S0266466600008215

Li, 1992, On the asymptotic standard errors of residual autocorrelations in nonlinear time series modeling, Biometrika, 79, 435, 10.1093/biomet/79.2.435

Li, 2001, Estimation for partially nonstationary multivariate autoregressive models with conditional heteroscedasticity, Biometrika, 88, 1135, 10.1093/biomet/88.4.1135

Li, 1994, On the squared residual autocorrelations in non-linear time series with conditional heteroskedasticity, J. Time Ser. Anal., 15, 627, 10.1111/j.1467-9892.1994.tb00217.x

Ling, 1999, On probability properties of a double threshold ARMA conditional heteroskedasticity model, J. Appl. Probab., 36, 688, 10.1239/jap/1032374627

Ling, 2003, A double AR(p) model: structure and estimation, Working Paper

Ling, 1997, On fractionally integrated autoregressive moving-average time series models with conditional heteroskedasticity, J. Am. Statist. Ass., 92, 1184, 10.1080/01621459.1997.10474076

Ling, 1998, Limiting distributions of maximum likelihood estimators for unstable ARMA models with GARCH errors, Ann. Statist., 26, 84, 10.1214/aos/1030563979

Ling, 2003, Asymptotic inference for unit root processes with GARCH (1, 1) errors, Econometr. Theory, 19, 541, 10.1017/S0266466603194029

Ling, 2003, Asymptotic theory for a new vector ARMA-GARCH model, Econometr. Theory, 19, 280, 10.1017/S0266466603192092

Lumsdaine, 1996, Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH(1,1) and covariance stationary GARCH(1,1) models, Econometrica, 3, 575, 10.2307/2171862

Mikosch, 1995, Parameter estimation for ARMA models with infinite variance innovations, Ann. Statist., 23, 305, 10.1214/aos/1176324469

Nicholls, 1982, Random coefficient autoregressive models: an introduction, Lect. Notes Statist., 11, 10.1007/978-1-4684-6273-9

Phillips, 1987, Time series regression with a unit root, Econometrica, 55, 277, 10.2307/1913237

Phillips, 1989, Partially identified econometric models, Econometr. Theory, 5, 181, 10.1017/S0266466600012408

Robinson, 1994, Efficient tests of nonstationary hypotheses, J. Am. Statist. Ass., 89, 1420, 10.1080/01621459.1994.10476881

Tjøstheim, 1986, Estimation in nonlinear time series models, Stoch. Processes Appl., 21, 251, 10.1016/0304-4149(86)90099-2

Tjøstheim, 1990, Nonlinear time series and Markov chains, Adv. Appl. Probab., 22, 587, 10.2307/1427459

Tong, 1990, Nonlinear Time Series: a Dynamical System Approach, 10.1093/oso/9780198522249.001.0001

Tsay, 1987, Conditional heteroscedastic time series models, J. Am. Statist. Ass., 82, 590, 10.1080/01621459.1987.10478471

Weiss, 1986, Asymptotic theory for ARCH models: estimation and testing, Econometr. Theory, 2, 107, 10.1017/S0266466600011397

Wong, 1997, On a multivariate conditional heteroscedastic model, Biometrika, 84, 111, 10.1093/biomet/84.1.111