Annals of Statistics

Asymptotic inference in some heteroscedastic regression models with long memory design and errors

Hongwen Guo and Hira L. Koul

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This paper discusses asymptotic distributions of various estimators of the underlying parameters in some regression models with long memory (LM) Gaussian design and nonparametric heteroscedastic LM moving average errors. In the simple linear regression model, the first-order asymptotic distribution of the least square estimator of the slope parameter is observed to be degenerate. However, in the second order, this estimator is n1/2-consistent and asymptotically normal for h+H<3/2; nonnormal otherwise, where h and H are LM parameters of design and error processes, respectively. The finite-dimensional asymptotic distributions of a class of kernel type estimators of the conditional variance function σ2(x) in a more general heteroscedastic regression model are found to be normal whenever H<(1+h)/2, and non-normal otherwise. In addition, in this general model, log(n)-consistency of the local Whittle estimator of H based on pseudo residuals and consistency of a cross validation type estimator of σ2(x) are established. All of these findings are then used to propose a lack-of-fit test of a parametric regression model, with an application to some currency exchange rate data which exhibit LM.

Article information

Ann. Statist., Volume 36, Number 1 (2008), 458-487.

First available in Project Euclid: 1 February 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M09: Non-Markovian processes: estimation
Secondary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84] 62M99: None of the above, but in this section

Local Whittle estimator moving average errors model diagnostics OLS


Guo, Hongwen; Koul, Hira L. Asymptotic inference in some heteroscedastic regression models with long memory design and errors. Ann. Statist. 36 (2008), no. 1, 458--487. doi:10.1214/009053607000000686.

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