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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.

“Economics is a Serious Subject.” Edwin Cannan.

Research on strategic choices available to the firm are often modeled as a limited number of possible decision outcomes and leads to a discrete limited dependent variable. A limited dependent variable can also arise when values of a continuous dependent variable are partially or wholly unobserved. This chapter discusses the methodological issues associated with such phenomena and the appropriate statistical methods developed to allow for consistent and efficient estimation of models that involve a limited dependent variable. The chapter also provides a road map for selecting the appropriate statistical technique and it offers guidelines for consistent interpretation and reporting of the statistical results.

This chapter reviews recent developments in the density discontinuity approach. It is well known that agents having perfect control of the forcing variable will invalidate the popular regression discontinuity designs (RDDs). To detect the manipulation of the forcing variable, McCrary (2008) developed a test based on the discontinuity in the density around the threshold. Recent papers have noted that the sorting patterns around the threshold are often either the researcher’s object of interest or may relate to structural parameters such as tax elasticities through known functions. This, in turn, implies that the behavior of the distribution around the threshold is not only informative of the validity of a standard RDD; it can also be used to recover policy-relevant parameters and perform counterfactual exercises.

Statistical methods of spatial analysis are often successful at either prediction or explanation, but not necessarily both. In a recent paper, Dearmon and Smith (2016) showed that by combining Gaussian Process Regression (GPR) with Bayesian Model Averaging (BMA), a modeling framework could be developed in which both needs are addressed. In particular, the smoothness properties of GPR together with the robustness of BMA allow local spatial analyses of individual variable effects that yield remarkably stable results. However, this GPR-BMA approach is not without its limitations. In particular, the standard (isotropic) covariance kernel of GPR treats all explanatory variables in a symmetric way that limits the analysis of their individual effects. Here we extend this approach by introducing a mixture of kernels (both isotropic and anisotropic) which allow different length scales for each variable. To do so in a computationally efficient manner, we also explore a number of Bayes-factor approximations that avoid the need for costly reversible-jump Monte Carlo methods.

To demonstrate the effectiveness of this Variable Length Scale (VLS) model in terms of both predictions and local marginal analyses, we employ selected simulations to compare VLS with Geographically Weighted Regression (GWR), which is currently the most popular method for such spatial modeling. In addition, we employ the classical Boston Housing data to compare VLS not only with GWR but also with other well-known spatial regression models that have been applied to this same data. Our main results are to show that VLS not only compares favorably with spatial regression at the aggregate level but is also far more accurate than GWR at the local level.