Asymptotic properties of OLS estimates in autoregressions with bounded or slowly growing deterministic trends

Mynbaev, Kairat (2003): Asymptotic properties of OLS estimates in autoregressions with bounded or slowly growing deterministic trends. Published in: Communications in Statistics—Theory and Methods , Vol. 35, (2006): pp. 499-520.

Abstract

We propose a general method of modeling deterministic trends for autoregressions. The method relies on the notion of $L_2$-approximable regressors previously developed by the author. Some facts from the theory of functions play an important role in the proof. In its present form, the method encompasses slowly growing regressors, such as logarithmic trends, and leaves open the case of polynomial trends.

• Item Type: MPRA Paper
• Language: English
• Keywords: autoregression; deterministic trend; OLS estimator asymptotics
• Subjects: C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes; C - Mathematical and Quantitative Methods > C0 - General > C01 - Econometrics

References

Anderson, T. W. (1971). "The Statistical Analysis of Time Series". Wiley, New York.

Anderson, T. W., and Kunitomo, N. (1992). Asymptotic distribution of regression and autoregression coefficients with martingale difference disturbances. {\it J. Multivariate Anal.}, 40, 221-243.

Andrews, D. W. K. (1988). Laws of large numbers for dependent non-identically distributed random variables. {\it Econometric Theory}, 4, 458-467.

Andrews, D. W. K., and McDermott, C. J. (1995). Nonlinear econometric models with deterministically trending variables. {\it Rev. Econom. Stud.}, 62, no. 3, 343-360.

Burkholder, D. L. (1973). Distribution function inequalities for martingales. {\it Ann. Probab.}, 1, no. 1, 14-42.

Chow, Y. S. (1971). On the $L_p$-convergence for $n^{-1/p}S_n$, $0<p<2$. {\it Ann. Math. Statist.}, 36, 393-394.

Davidson, J. (1994). "Stochastic Limit Theory: An Introduction for Econometricians}. Oxford Univ. Press, Oxford.

Hamilton, J. D. (1994). "Time Series Analysis". Princeton Univ. Press, Princeton, NJ.

Leonenko, N. N., and \v Silac-Ben\v si\'c, M. (1997). On the asymptotic distributions of least square estimations in a regression model with singular errors. {\it Dopov. Nats. Akad. Nauk Ukr. Mat. Prirodozn. Tekh. Nauki}, no. 7, 26-31.

McLeish, D. L. (1974). Dependent central limit theorems and invariance principles. {\it Ann. Probab.}, 2, 620-628.

Moussatat, M. W. (1976). On the asymptotic theory of statistical experiments and some of its applications. Ph.D. dissertation, Univ. California, Berkeley.

Mynbaev, K. T. (1997). Linear models with regressors generated by square-integrable functions. In: { \it 7 Escola de S\'eries Temporais e Econometria. Sociedade Brasileira de Econometria.} August 6-8, 80--82.

Mynbaev, K. T. (2001). $L_p$-approximable sequences of vectors and limit distribution of quadratic forms of random variables. {\it Adv. Appl. Math.}, 26, 302--329.

Mynbaev, K. T. (2003). Modeling deterministic regressors. In: {\it Advances in Statistical Inferential Methods: Theory and Applications.} Almaty, Kazakhstan, June 9-12, 133--148.

Mynbaev, K. T., and Castelar, I. (2001). {\it The Strengths and Weaknesses of $L_2$-approximable Regressors. Two Essays on Econometrics.} Fortaleza: Express\~ao Gr\'afica, v.1.

Nabeya, S. (2000). Asymptotic distributions for unit root test statistics in nearly integrated seasonal autoregressive models. {\it Econometric Theory}, 16, no. 2, 200--230.

Ng, S., and Vogelsang, T. J. (2002). Forecasting autoregressive time series in the presence of deterministic components. {\it Econom. J.}, 5, no. 1, 196--224.

Rahbek, A.; Kongsted, H. C., and Jorgensen, C. (1999). Trend stationarity in the $I(2)$ cointegration model. {\it J. Econometrics}, 90, no. 2, 265--289.

Sibbertsen, Ph. (2001). $S$-estimation in the linear regression model with long-memory error terms under trend. {\it J. Time Ser. Anal.}, 22, no. 3, 353--363.

Sims, C. A.; Stock, J. H., and Watson, M. W. (1990). Inference in linear time series models with some unit roots. {\it Econometrica}, 58, no. 1, 113--144.

Tam, Wing-kuen, and Reinsel, G. C. (1998). Seasonal moving-average unit root tests in the presence of a linear trend. {\it J. Time Ser. Anal.}, 19, no. 5, 609--625.