Instrumental Variable Estimation of a Threshold Model

Abstract

Threshold models~sample splitting models! have wide application in economics+ Existing estimation methods are confined to regression models, which require that all right-hand-side variables are exogenous+ This paper considers a model with endogenous variables but an exogenous threshold variable+ We develop a twostage least squares estimator of the threshold parameter and a generalized method of moments estimator of the slope parameters+ We show that these estimators are consistent, and we derive the asymptotic distribution of the estimators+ The threshold estimate has the same distribution as for the regression case~Hansen, 2000, Econometrica 68, 575-603!, with a different scale+ The slope parameter estimates are asymptotically normal with conventional covariance matrices+ We investigate our distribution theory with a Monte Carlo simulation that indicates the applicability of the methods+ We thank the two referees and co-editor for constructive comments+ Hansen thanks the National Science Foundation for financial support+ Caner thanks

Is the Relationship Between Financial Development and Economic Growth Monotonic? Evidence from a Sample of Middle-Income Countries

“…considered by Law et al (2013), but they attempt to identify the institutional quality thresholds that may affect the finance-growth relationship. Using data for 85 countries for the 1980-2008 period, and using both Hansen (2000)-type threshold regressions and Caner and Hansen (2004)-type IV threshold regressions, they find that institutions shape the finance-growth relationship: financial development promotes growth after institutions exceed a certain threshold level. Owen and Temesvary (2014) contribute to the finance-growth literature by showing that the effect of bank finance on growth is heterogeneous across countries and across types of bank lending (domestic and foreign).…”

“…However, some macroeconomic variables such as GDP growth are highly persistent, meaning that a dynamic panel framework is more appropriate. To consider nonlinearity in dynamic panel data, Bick (2010) and Kremer et al (2013) proposed a dynamic panel threshold estimator which is an extension of the threshold models by Hansen (1999), Hansen (2000, and Caner and Hansen (2004). This estimator allows us to investigate the potential existence of a discrete shift in a dynamic framework.…”

Nonlinear mechanisms of the exchange rate pass-through: a Phillips curve model with threshold for Brazil

“…4 In the case of models with endogenous variables, in turn, the theory is still working in progress. Caner and Hansen (2004) develop an estimator and an inference theory for this kind of model, with the restriction that the threshold variable must be exogenous. We are interested in models with endogenous variables because we want to estimate a Phillips curve for inflation in Brazil with an expectations term.…”

“…Following the method proposed by Caner and Hansen (2004), in the first step of the estimation process, we estimate the endogenous variables as a function of instrumental variables, that is, we model the conditional mean of the endogenous variables as a function of exogenous variables. The fitted values for the endogenous variables are then plugged into the structural equation (4), and the threshold value is estimated by minimizing the sum of the squared residuals.…”

“…According to Franses and van Dijk (2000), a safe choice is at least 15% of the sample. Given the estimated threshold valueτ , the sample is divided into subsamples, and parameters of equation (10) can be estimated by 2SLS as: Caner and Hansen (2004) show that those estimators are consistent, although not necessarily efficient. Their applicability is conditioned on the exogeneity of the threshold variable.…”