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Operational Algebra and Regression t-Tests

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Abstract

Data reduction involves a physical transition from sample data to econometric estimator and test statistic. This transition induces a mapping on the probability law of the sample, whose image is the distribution of the statistic of interest. At a general level, the mapping can often be captured by means of an operational algebra. Some methods than employ nonlinear functions of differential operators are suggested which can perform this task. The methods are related to pseudodifferential operator techniques that are used in abstract mathematics to solve systems of partial differential equations. They also generalize the fractional calculus methods developed by the author in earlier work (1984, 1985). Two examples are studied in detail. One of these deals with the feasible generalized least squares estimator and its regression t-statistic in the linear model with a non scalar error covariance matrix whose elements are functions of a finite dimensional vector of nuisance parameters. This includes a wide class of models such as general SUR systems and models with serially dependent or heterogeneous errors.

Suggested Citation

  • Peter C.B. Phillips, 1990. "Operational Algebra and Regression t-Tests," Cowles Foundation Discussion Papers 948, Cowles Foundation for Research in Economics, Yale University.
    • Handle: RePEc:cwl:cwldpp:948
    Note: CFP 830.
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    File URL: https://cowles.yale.edu/sites/default/files/files/pub/d09/d0948.pdf
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    References listed on IDEAS

    as
    1. Zinde-Walsh, Victoria, 1988. "Some Exact Formulae for Autoregressive Moving Average Processes," Econometric Theory, Cambridge University Press, vol. 4(3), pages 384-402, December.
    2. Rothenberg, Thomas J, 1984. "Approximate Normality of Generalized Least Squares Estimates," Econometrica, Econometric Society, vol. 52(4), pages 811-825, July.
    3. Phillips, P.C.B., 1984. "The exact distribution of the Stein-rule estimator," Journal of Econometrics, Elsevier, vol. 25(1-2), pages 123-131.
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    Cited by:

    1. Peter C.B. Phillips & Yixiao Sun & Sainan Jin, 2005. "Improved HAR Inference," Cowles Foundation Discussion Papers 1513, Cowles Foundation for Research in Economics, Yale University.

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