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Efficiency Tradeoffs in Estimating the Trend and Error Structure of the Linear Model

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  • Falk, Barry
  • Roy, Anindya

Abstract

Assume that the observed time series has been generated by the model Yt=a + bt + yt, t=l,...,T (1) yt = pyt-i+YiAyt-i+-.--h'p-i^yt-p+i+£t, st~i.i.d.(0,c^) (2) where A denotes the first difference operator and p e (-1,1] is the largest autoregressive root in the autoregressive representation of yt implied by (2). Thus, yt can be an 1(1) or an 1(0) process according to whether p = 1 or p e (-1,1), respectively. If p e (-1,1), the Grenander and Rosenblatt (1957) result implies that the ordinaiy least squares (OLS) estimator of (a,b) in (1) is asymptotically equivalent to the generalized least squares (GLS) estimator of (a,b) using (1) and (2). If p = 1, the parameter a is not identified and although the OLS estimator of b is consistent, it is not asymptotically efficient. In this case, the sample mean of Ayt is an asymptotically efficient estimator of b, being equivalent to the GLS estimator. We will refer to the f sample mean of Ayt as the first-difference estimatorof b. Of course, in practice we do not know a priori whether p is equal to or less than one.

Suggested Citation

  • Falk, Barry & Roy, Anindya, 1999. "Efficiency Tradeoffs in Estimating the Trend and Error Structure of the Linear Model," ISU General Staff Papers 199908010700001327, Iowa State University, Department of Economics.
    • Handle: RePEc:isu:genstf:199908010700001327
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    References listed on IDEAS

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    1. Rudebusch, Glenn D, 1992. "Trends and Random Walks in Macroeconomic Time Series: A Re-examination," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 33(3), pages 661-680, August.
    2. MacKinnon, James G. & Smith Jr., Anthony A., 1998. "Approximate bias correction in econometrics," Journal of Econometrics, Elsevier, vol. 85(2), pages 205-230, August.
    3. Roy, Anindya & Falk, Barry & Fuller, Wayne A., 1999. "Estimation of the Trend Model with Autoregressive Errors," ISU General Staff Papers 199907010700001328, Iowa State University, Department of Economics.
    4. Roy, Anindya & Falk, Barry L. & Fuller, Wayne A., 2004. "Estimation of the Trend Model with Autoregressive Errors," Staff General Research Papers Archive 12005, Iowa State University, Department of Economics.
    5. Andrews, Donald W K, 1993. "Exactly Median-Unbiased Estimation of First Order Autoregressive/Unit Root Models," Econometrica, Econometric Society, vol. 61(1), pages 139-165, January.
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    Cited by:

    1. Roy, Anindya & Falk, Barry & Fuller, Wayne A., 1999. "Estimation of the Trend Model with Autoregressive Errors," ISU General Staff Papers 199907010700001328, Iowa State University, Department of Economics.
    2. Ernst, Matthew & Rodecker, Jared & Luvaga, Ebby & Alexander, Terence & Kliebenstein, James & MIRANOWSKI, JOHN A, 1999. "The Viability of Methane Production by Anaerobic Digestion on Iowa Swine Farms," ISU General Staff Papers 199910010700001329, Iowa State University, Department of Economics.

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