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Inferences for the Extremum of Quadratic Regression Models

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  • Joseph G. Hirschberg
  • Jenny N. Lye

Abstract

Quadratic functions are often used in regression to infer the existence of an extremum in a relationship although tests of the location of the extremum are rarely performed. We investigate the construction of the following confidence intervals: Delta, Fieller, estimated first derivative, bootstrapping, Bayesian and likelihood ratio. We propose interpretations for the unbounded intervals that may be generated by some of these methods. The coverage of the confidence intervals is assessed by Monte Carlo; the Delta and studentized bootstrap can perform quite poorly. Of all the methods, the first derivative method is easiest to implement.

Suggested Citation

  • Joseph G. Hirschberg & Jenny N. Lye, 2004. "Inferences for the Extremum of Quadratic Regression Models," Department of Economics - Working Papers Series 906, The University of Melbourne.
    • Handle: RePEc:mlb:wpaper:906
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    File URL: http://www.economics.unimelb.edu.au/downloads/wpapers-04/906.pdf
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    References listed on IDEAS

    as
    1. Jenny Lye & Joe Hirschberg, 2004. "Alcohol consumption, smoking and wages," Applied Economics, Taylor & Francis Journals, vol. 36(16), pages 1807-1817.
    2. Herbert G. Thompson, Jr. & Lynda L. Wolf, 1993. "Regional Differences in Nuclear and Fossil-Fuel Generation of Electricity," Land Economics, University of Wisconsin Press, vol. 69(3), pages 234-248.
    3. Zellner, Arnold, 1978. "Estimation of functions of population means and regression coefficients including structural coefficients : A minimum expected loss (MELO) approach," Journal of Econometrics, Elsevier, vol. 8(2), pages 127-158, October.
    4. Hsing, Yu, 1996. "Estimating the laffer curve and policy implications," Journal of Behavioral and Experimental Economics (formerly The Journal of Socio-Economics), Elsevier, vol. 25(3), pages 395-401.
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    Citations

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    Cited by:

    1. Jo Thori Lind & Halvor Mehlum, 2010. "With or Without U? The Appropriate Test for a U‐Shaped Relationship," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 72(1), pages 109-118, February.
    2. J.G. Hirschberg & J. N. Lye, 2007. "Providing Intuition to the Fieller Method with Two Geometric Representations using STATA and Eviews," Department of Economics - Working Papers Series 992, The University of Melbourne.
    3. Richard F. J. Haans & Constant Pieters & Zi-Lin He, 2016. "Thinking about U: Theorizing and testing U- and inverted U-shaped relationships in strategy research," Strategic Management Journal, Wiley Blackwell, vol. 37(7), pages 1177-1195, July.
    4. Hirschberg, J.G. & Lye, J.N. & Slottje, D.J., 2008. "Inferential methods for elasticity estimates," Journal of Econometrics, Elsevier, vol. 147(2), pages 299-315, December.
    5. J. G. Hirschberg, J. N. Lye & D. J. Slottje, 2008. "Confidence Intervals for Estimates of Elasticities," Department of Economics - Working Papers Series 1053, The University of Melbourne.
    6. Joe Hirschberg & Jenny Lye, 2017. "Alternative Graphical Representations of the Confidence Intervals for the Structural Coefficient from Exactly Identified Two-Stage Least Squares," Department of Economics - Working Papers Series 2026, The University of Melbourne.
    7. Hirschberg, J.G. & Lye, J.N., 2010. "Two geometric representations of confidence intervals for ratios of linear combinations of regression parameters: An application to the NAIRU," Economics Letters, Elsevier, vol. 108(1), pages 73-76, July.

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    More about this item

    Keywords

    Inverted U-Shaped; turning point; Fieller method; Delta method; 1st derivative function; Bayesian; Likelihood ratio; Bootstrap.;
    All these keywords.

    JEL classification:

    • C21 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Cross-Sectional Models; Spatial Models; Treatment Effect Models
    • C40 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods: Special Topics - - - General
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection

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