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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.

We take Cumulative Prospect Theory (CPT) seriously by rigorously estimating structural models using the full set of CPT parameters. Much of the literature only estimates a subset of CPT parameters, or more simply assumes CPT parameter values from prior studies. Our data are from laboratory experiments with undergraduate students and MBA students facing substantial real incentives and losses. We also estimate structural models from Expected Utility Theory (EUT), Dual Theory (DT), Rank-Dependent Utility (RDU), and Disappointment Aversion (DA) for comparison. Our major finding is that a majority of individuals in our sample locally asset integrate. That is, they see a loss frame for what it is, a frame, and behave as if they evaluate the net payment rather than the gross loss when one is presented to them. This finding is devastating to the direct application of CPT to these data for those subjects. Support for CPT is greater when losses are covered out of an earned endowment rather than house money, but RDU is still the best single characterization of individual and pooled choices. Defenders of the CPT model claim, correctly, that the CPT model exists “because the data says it should.” In other words, the CPT model was borne from a wide range of stylized facts culled from parts of the cognitive psychology literature. If one is to take the CPT model seriously and rigorously then it needs to do a much better job of explaining the data than we see here.

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%.