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The authors derive a risk lower bound in estimating the threshold parameter without knowing whether the threshold regression model is continuous or not. The bound goes to zero as the sample size n grows only at the cube-root rate. Motivated by this finding, the authors develop a continuity test for the threshold regression model and a bootstrap to compute its p-values. The validity of the bootstrap is established, and its finite-sample property is explored through Monte Carlo simulations.

We employ various local generalizations of the Solow growth model that model parameter heterogeneity using human development at the beginning of the period with adult literacy rates and life expectancy at birth as a proxy. The model takes the form of a semiparametric varying coefficient model along the lines of Hastie and Tibshirani (1992). The empirical results show substantial parameter heterogeneity in the cross-country growth process, a finding that is consistent with the presence of multiple steady-state equilibria and the emergence of convergence clubs.

I review the burgeoning literature on applications of Markov regime switching models in empirical finance. In particular, distinct attention is devoted to the ability of Markov Switching models to fit the data, filter unknown regimes and states on the basis of the data, to allow a powerful tool to test hypotheses formulated in light of financial theories, and to their forecasting performance with reference to both point and density predictions. The review covers papers concerning a multiplicity of sub-fields in financial economics, ranging from empirical analyses of stock returns, the term structure of default-free interest rates, the dynamics of exchange rates, as well as the joint process of stock and bond returns.

This survey focuses on two families of nonlinear vector time series models, the family of vector threshold regression (VTR) models and that of vector smooth transition regression (VSTR) models. These two model classes contain incomplete models in the sense that strongly exogeneous variables are allowed in the equations. The emphasis is on stationary models, but the considerations also include nonstationary VTR and VSTR models with cointegrated variables. Model specification, estimation and evaluation is considered, and the use of the models illustrated by macroeconomic examples from the literature.

Responses to minimum income and minimum spending questions are used to produce economic well-being thresholds. Thresholds are estimated using a regression framework. Regression coefficients are based on U.S. Survey of Income and Program Participation (SIPP) data and then applied to U.S. Consumer Expenditure Survey (CE) data. Three different resource measures are compared to the estimated thresholds. The first resource measure is total before-tax money income, and the other two are expenditure based. The first of these two refers to expenditure outlays and the second to outlays adjusted for the value of the service flow of owner-occupied housing (rental equivalence). The income comparison is based on SIPP data while the outlays comparisons are based on CE data. Results using official poverty thresholds are shown for comparison. This is among the earliest work in the U.S. in which expenditure outlays have been used for economic well-being determinations in combination with personal assessments, and the first time rental equivalence has been used in such an exercise. Comparisons of expenditures for various bundles of commodities are compared to the CE derived thresholds to provide insight concerning what might be considered minimum or basic.

Results reveal that CE and SIPP MIQ thresholds are higher than MSQ thresholds, and resulting poverty rates are also higher with the MIQ. CE-based MSQ thresholds are not statistically different from average expenditure outlays for food, apparel, and shelter and utilities for primary residences. When reported rental equivalences for primary residences that are owner occupied are substituted for out-of-pocket shelter expenditures, single elderly are less likely to be as badly off as they would be with a strict outlays approach in defining resources.

Regression discontinuity (RD) models are commonly used to nonparametrically identify and estimate a local average treatment effect. Dong and Lewbel (2015) show how a derivative of this effect, called treatment effect derivative (TED) can be estimated. We argue here that TED should be employed in most RD applications, as a way to assess the stability and hence external validity of RD estimates. Closely related to TED, we define the complier probability derivative (CPD). Just as TED measures stability of the treatment effect, the CPD measures stability of the complier population in fuzzy designs. TED and CPD are numerically trivial to estimate. We provide relevant Stata code, and apply it to some real datasets.

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.

The recent financial and sovereign debt crises around the world have sparked a growing literature on models and empirical estimates of defaultable debt. Frequently households and firms come under default threat, local governments can default, and recently sovereign default threats were eminent for Greece and Spain in 2012–2013. Moreover, Argentina experienced an actual default in 2001. What causes sovereign default risk, and what are the escape routes from default risk? Previous studies such as Arellano (2008), Roch and Uhlig (2013), and Arellano et al. (2014) have provided theoretical models to explore the main dynamics of sovereign defaults. These models can be characterized as threshold models in which there is a convergence toward a good no-default equilibrium below the threshold and a default equilibrium above the threshold. However, in these models aggregate output is exogenous, so that important macroeconomic feedback effects are not taken into account. In this chapter, we (1) propose alternative model variants suitable for certain types of countries in the EU where aggregate output is endogenously determined and where financial stress plays a key role, (2) show how these model variants can be solved through the Nonlinear Model Predictive Control numerical technique, and (3) present some empirical evidence on the nonlinear dynamics of output, sovereign debt, and financial stress in some euro areas and other industrialized countries.

For over three decades, vector autoregressions have played a central role in empirical macroeconomics. These models are general, can capture sophisticated dynamic behavior, and can be extended to include features such as structural instability, time-varying parameters, dynamic factors, threshold-crossing behavior, and discrete outcomes. Building upon growing evidence that the assumption of linearity may be undesirable in modeling certain macroeconomic relationships, this article seeks to add to recent advances in VAR modeling by proposing a nonparametric dynamic model for multivariate time series. In this model, the problems of modeling and estimation are approached from a hierarchical Bayesian perspective. The article considers the issues of identification, estimation, and model comparison, enabling nonparametric VAR (or NPVAR) models to be fit efficiently by Markov chain Monte Carlo (MCMC) algorithms and compared to parametric and semiparametric alternatives by marginal likelihoods and Bayes factors. Among other benefits, the methodology allows for a more careful study of structural instability while guarding against the possibility of unaccounted nonlinearity in otherwise stable economic relationships. Extensions of the proposed nonparametric model to settings with heteroskedasticity and other important modeling features are also considered. The techniques are employed to study the postwar U.S. economy, confirming the presence of distinct volatility regimes and supporting the contention that certain nonlinear relationships in the data can remain undetected by standard models.

The authors provide new evidence in favor of the expectation hypothesis (EH) as a long-run theory of the term structure of interest rates. Using nonparametric techniques first, the authors show that the results of conventional tests that reject EH are strongly affected by the presence of extreme observations – only a handful in the case of longer maturities. The authors then provide a new general methodology that determines the number of outliers causing any theory to fail, and their approach quantifies the extent of this failure.