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Asymptotics of regressions with stationary and nonstationary residuals

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  • Maller, R. A.

Abstract

A comprehensive description is given of the limiting behaviour of normalised pseudo-MLEs of the coefficients in a discrete-time autoregressive process, with nonstochastic regressors, for all cases: stationary, unit root and explosive situations. The residuals are assumed to be independent and identically distributed, with finite variance, and we allow a wide class of regressors: they need only be uniformly asymptotically negligible and not too regular, in a certain sense. Under these assumptions, the normalised estimator of the regression coefficient is shown to be asymptotically normal, regardless of the value of the autocorrelation coefficient, and asymptotically independent of the normalised estimator of the autocorrelation coefficient, which also has a proper, nondegenerate limiting distribution. The normalisation for the estimators can be based on the sample information matrix. Limiting distributions for likelihood ratio test statistics of hypotheses of interest are also given under the same assumptions.

Suggested Citation

  • Maller, R. A., 2003. "Asymptotics of regressions with stationary and nonstationary residuals," Stochastic Processes and their Applications, Elsevier, vol. 105(1), pages 33-67, May.
    • Handle: RePEc:eee:spapps:v:105:y:2003:i:1:p:33-67
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    References listed on IDEAS

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    1. Maller, R. A., 1993. "Quadratic Negligibility and the Asymptotic Normality of Operator Normed Sums," Journal of Multivariate Analysis, Elsevier, vol. 44(2), pages 191-219, February.
    2. Monsour, Michael J. & Mikulski, Piotr W., 1998. "On limiting distributions in explosive autoregressive processes," Statistics & Probability Letters, Elsevier, vol. 37(2), pages 141-147, February.
    3. Dickey, David A & Fuller, Wayne A, 1981. "Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root," Econometrica, Econometric Society, vol. 49(4), pages 1057-1072, June.
    4. Datta, Somnath, 1995. "Limit theory and bootstrap for explosive and partially explosive autoregression," Stochastic Processes and their Applications, Elsevier, vol. 57(2), pages 285-304, June.
    5. Davis, Richard A. & Mikosch, Thomas, 1998. "Gaussian likelihood-based inference for non-invertible MA(1) processes with SS noise," Stochastic Processes and their Applications, Elsevier, vol. 77(1), pages 99-122, September.
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    1. Ross Maller & Maryam Ghodsi, 2025. "A General Formulation for the Large-Sample Behaviour of a Class of Hypothesis Test Statistics," Sankhya A: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 87(1), pages 1-40, February.
    2. Hongchang Hu & Weifu Hu & Xinxin Yu, 2021. "Pseudo-maximum likelihood estimators in linear regression models with fractional time series," Statistical Papers, Springer, vol. 62(2), pages 639-659, April.

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