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In this chapter 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 Appendix A along with the data.

The collection of chapters in this 30th volume of Advances in Econometrics provides a well-deserved tribute to Thomas B. Fomby and R. Carter Hill, who have served as editors of the Advances in Econometrics series for 25 and 21 years, respectively. Volume 30 contains a more varied collection of chapters than previous volumes, in essence mirroring the wide variety of econometric topics covered by the series over 30 years. Volume 30 starts with a chapter discussing the history of this series over the last 30 years. The next five chapters can be broadly categorized as focusing on model specification and testing. Following this section are three contributions that examine instrumental variables models in quite different settings. The next four chapters focus on applied macroeconomics topics. The final chapter offers a practical guide to conducting Monte Carlo simulations.

We review the experimental evidence on risk aversion in controlled laboratory settings. We review the strengths and weaknesses of alternative elicitation procedures, the strengths and weaknesses of alternative estimation procedures, and finally the effect of controlling for risk attitudes on inferences in experiments.

Bayesian A/B inference (BABI) is a method that combines subjective prior information with data from A/B experiments to provide inference for lift – the difference in a measure of response in control and treatment, expressed as its ratio to the measure of response in control. The procedure is embedded in stable code that can be executed in a few seconds for an experiment, regardless of sample size, and caters to the objectives and technical background of the owners of experiments. BABI provides more powerful tests of the hypothesis of the impact of treatment on lift, and sharper conclusions about the value of lift, than do legacy conventional methods. In application to 21 large online experiments, the credible interval is 60% to 65% shorter than the conventional confidence interval in the median case, and by close to 100% in a significant proportion of cases; in rare cases, BABI credible intervals are longer than conventional confidence intervals and then by no more than about 10%.

Zero-failure reliability testing aims at demonstrating whether the product has achieved the desired reliability target with zero failure and high confidence level at a given time. Incorporating accelerated degradation testing in zero-failure reliability demonstration test (RDT) facilitates early failure in high reliability items developed within short period of time to be able to survive in fiercely competitive market. The paper aims to discuss these issues.

The triangular cyclic stress uses one test chamber thus saving experimental cost. The parameters in model are estimated using maximum likelihood methods. The optimum plan consists in finding out optimum number of cycles, optimum specimens, optimum stress change point(s) and optimum stress rates.

The optimum plan consists in finding out optimum number of cycles, optimum specimens, optimum stress change point(s) and optimum stress rates by minimizing asymptotic variance of estimate of quantile of the lifetime distribution at use condition subject to the constraint that total testing or experimental cost does not exceed a pre-specified budget. Confidence intervals of the design parameters have been obtained and sensitivity analysis carried out. The results of sensitivity analysis show that the plan is robust to small deviations from the true values of baseline parameters.

For some highly reliable products, even accelerated life testing yields little failure data of units in a feasible amount of time. In such cases accelerated degradation testing is carried out, wherein the failure termed as soft failure is defined in terms of performance characteristic of the product exceeding its critical (threshold) value.

Briefly reviews the standard Poisson distribution and then examines a set of derivative, modified Poisson distributions for testing hypotheses derived from positive deviation‐amplifying feedback models, which do not lend themselves to ordinary statistically based hypothesis testing. The “reinforcement” or “contagious” Poisson offers promise for a subset of such models, in particular those models with data in the form of rates (rather than magnitudes). The practical difficulty lies in distinguishing reinforcement effects from initial heterogeneity, since both can form negative binomial distributions, with look‐alike data. Illustrates these difficulties, and also opportunities, for various feedback models employing the self‐fulfilling prophecy, and especially for confidence loops, which incorporate particular self‐fulfilling prophecies as part of a larger dynamic process. Describes an actual methodology for testing hypotheses regarding confidence loops with the aid of a “reinforcement” Poisson distribution, as well as its place within sociocybernetics.

The study of the link between debt and growth has been full of debates, both in theory and empirics. However, there is a growing consensus that the relationship is sensitive to the level of debt. The purpose of this paper is to address the question of non-linearity in the long-term relationship between public debt and economic growth. Specifically, the author set out to test if there exists an established “laffer curve” type relationship, where debt contributes to economic growth up to a certain point (maximal threshold) and then starts to have a negative effect on growth afterwards.

To carry out the tests, the author has used a methodology that delivers a superior test of inverse U-shapes (Lind and Mehlum, 2010), in addition to the traditional test based on a regression with a quadratic specification.

The results in the paper present evidence of a bell-shaped relationship between economic growth and total public debt in a panel of low-income Sub-Saharan African economies. This supports the hypothesis that debt has some positive contribution to economic growth in low-income countries, albeit up to a point.

The overall result supports the claim that public debt may start to be a drag on economic growth if it goes on increasing beyond the level where it would be sustainable.

This paper leads the way by implementing a robust test of non-linearity (“inverse-U” test) to the analyses the debt-growth nexus and the laffer curve in Sub-Saharan Africa.