Inverse Test Confidence Intervals for Turning-Points: A Demonstration with Higher Order Polynomials

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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“…12 This method for inverse tests is more fully developed for the turning points in higher order polynomials in Lye and Hirschberg (2012).…”

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“…The implicit linear function or line plot method for the definition of the Fieller intervals is based on the interpretation of the Fieller 's 1954 approximation method as the inversion of a test for a linear function of normally distributed random variables. In Hirschberg and Lye (2010a) we demonstrate this method for the Fieller case and discuss the generalization of this method in Lye and Hirschberg (2012) for the related problem of function turning points in high order polynomials.…”

Ratios of Parameters: Some Econometric Examples

“…This method for inverse tests is more fully developed for the turning points in higher order polynomials in Lye and Hirschberg ().…”