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Inverse Test Confidence Intervals for Turning points: A

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  • J. N. Lye and J. G. Hirschberg

Abstract

In this paper we demonstrate the construction of inverse test confidence intervals for the turning points in estimated nonlinear relationships by the use of the marginal or first derivative function. First, we outline the inverse test confidence interval approach. Then we examine the relationship between the traditional confidence intervals based on the Wald test for the turning-points for a cubic, a quartic and fractional polynomials estimated via regression analysis and the inverse test intervals. We show that the confidence interval plots of the marginal function can be used to estimate confidence intervals for the turning points that are equivalent to the inverse test. We also provide a method for the interpretation of the confidence intervals for the second derivative function to draw inferences for the characteristics of the turning-point. This method is applied to the examination of the turning points found when estimating a quartic and a fractional polynomial from data used for the estimation of an Environmental Kuznets Curve. The Stata do files used to generate these examples are listed in the appendix along with the data.

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  • J. N. Lye and J. G. Hirschberg, 2012. "Inverse Test Confidence Intervals for Turning points: A," Department of Economics - Working Papers Series 1160, The University of Melbourne.
    • Handle: RePEc:mlb:wpaper:1160
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    2. 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.

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    Keywords

    Inverse Test Confidence Intervals; Likelihood Profile; Quartic;
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