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Nonlinear time series models: consistency and asymptotic normality of nls under new conditions

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  • Mira, Santiago

Abstract

In this paper we study the consistency and asymptotic normality properties of nonlinear least squares (NLS) under a set of assumptions that are not difficult to verify. The statistical literature on estimation of nonlinear models by NLS rely on abstract theoretical conditions. See for example the books of Tong(1990), and Granger and Terasvirta(1993). There are alternative statistical frameworks but all of them depend on high level (very technical) assumptions that are difficult and tedious to verify, see for example Gallant and White(1988) and Wooldridge(1994). In this paper we show that for a general class of nonlinear dynamic regression models, there are explicit and easy to check conditions that satisfy the general framework of Gallant and White(1988). We show the usefulness of our assumptions with some examples from the class of Smooth Transition Autoregressive (STAR) models.

Suggested Citation

  • Mira, Santiago, 1995. "Nonlinear time series models: consistency and asymptotic normality of nls under new conditions," DES - Working Papers. Statistics and Econometrics. WS 6202, Universidad Carlos III de Madrid. Departamento de Estadística.
    • Handle: RePEc:cte:wsrepe:6202
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    References listed on IDEAS

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    1. Álvaro Escribano & Oscar Jordá, 2001. "Testing nonlinearity: Decision rules for selecting between logistic and exponential STAR models," Spanish Economic Review, Springer;Spanish Economic Association, vol. 3(3), pages 193-209.
    2. H.J. Bierens, 1981. "Robust Methods and Asymptotic Theory in Nonlinear Econometrics," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 35(3), pages 173-173, September.
    3. Terasvirta, T & Anderson, H M, 1992. "Characterizing Nonlinearities in Business Cycles Using Smooth Transition Autoregressive Models," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 7(S), pages 119-136, Suppl. De.
    4. Rothman, Philip, 1991. "Further evidence on the asymmetric behavior of unemployment rates over the business cycle," Journal of Macroeconomics, Elsevier, vol. 13(2), pages 291-298.
    5. M. B. Priestley, 1980. "State‐Dependent Models: A General Approach To Non‐Linear Time Series Analysis," Journal of Time Series Analysis, Wiley Blackwell, vol. 1(1), pages 47-71, January.
    6. Donald W. K. Andrews & C. John McDermott, 1995. "Nonlinear Econometric Models with Deterministically Trending Variables," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 62(3), pages 343-360.
    7. Burgess, Simon M. & Pfann, Gerard A., 1993. "Asymmetric and time-varying error-correction: an application to labour demand in the UK," DES - Working Papers. Statistics and Econometrics. WS 3681, Universidad Carlos III de Madrid. Departamento de Estadística.
    8. Granger, Clive W. J. & Terasvirta, Timo, 1993. "Modelling Non-Linear Economic Relationships," OUP Catalogue, Oxford University Press, number 9780198773207.
    9. Neftci, Salih N, 1984. "Are Economic Time Series Asymmetric over the Business Cycle?," Journal of Political Economy, University of Chicago Press, vol. 92(2), pages 307-328, April.
    10. Granger, Clive W J, 1995. "Modelling Nonlinear Relationships between Extended-Memory Variables," Econometrica, Econometric Society, vol. 63(2), pages 265-279, March.
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    Cited by:

    1. Marcelo C. Medeiros & Alvaro Veiga, 2003. "Diagnostic Checking in a Flexible Nonlinear Time Series Model," Journal of Time Series Analysis, Wiley Blackwell, vol. 24(4), pages 461-482, July.
    2. Mira, Santiago, 1996. "Nonlinear cointegration and nonlinear error correction," DES - Working Papers. Statistics and Econometrics. WS 4546, Universidad Carlos III de Madrid. Departamento de Estadística.
    3. Hillebrand Eric & Medeiros Marcelo C. & Xu Junyue, 2013. "Asymptotic Theory for Regressions with Smoothly Changing Parameters," Journal of Time Series Econometrics, De Gruyter, vol. 5(2), pages 133-162, April.

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    Keywords

    Nonlinear Dynamic Regression Models;

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