Threshold Regression with Endogeneity

Abstract

This paper studies estimation and speci…cation testing in threshold regression with endogeneity. For estimation, there are three results that are di¤erent from those in regular models. First, the threshold point and the parameters of threshold e¤ects can be identi…ed without instruments. Second, in partially linear models, extra randomness in the parametric regressors above the nonparametric regressors is not needed to identify the parametric coe¢ cients. Third, besides identi…cation, instruments can play di¤erent roles to di¤erent parameters. For speci…cation testing, we test whether endogeneity exists and whether there are threshold e¤ects when instruments are absent. Especially, the second test extends the usual parametric structural change tests to the nonparametric case. Furthermore, a wild bootstrap procedure is suggested to obtain critical values for both tests in …nite samples.

Threshold regressions for the resource curse

“…Yu (2013), however, shows that even in the case where the threshold variable is exogenous the 2SLS estimator of Caner and Hansen (2004) may be inconsistent. Yu (2013) and Yu and Phillips (2018) give examples where the true relationship between the endogenous, right-hand-variables and their instrumental variables are non-linear but mistakenly considered as linear to support this result. To overcome this issue, Yu and Phillips (2018) suggest a non-parametric estimator of the threshold which is, unfortunately, data consuming.…”

“…Yu (2013) and Yu and Phillips (2018) give examples where the true relationship between the endogenous, right-hand-variables and their instrumental variables are non-linear but mistakenly considered as linear to support this result. To overcome this issue, Yu and Phillips (2018) suggest a non-parametric estimator of the threshold which is, unfortunately, data consuming. For these reasons, special attention should be paid to the instrumental equations to ensure that they are well specified.…”

Adaptive estimation of the threshold point in threshold regression

“…Based on this lemma, the above argument is not surprising. Actually, Yu (2013a) shows that even under the nonparametric setup of E [y|x, q] in the two regimes, γ can still be adaptively estimated.…”

Behavioral barriers to success in education

“…Caner and Hansen (2004) and Kourtellos et al (2016) further discuss identification and inference when some of the regressors, including the threshold variable, are endogenous. Yu and Phillips (2018) propose a nonparametric estimator, the integrated difference kernel estimator (IDKE), which presents the advantage that the discrepancy coefficients can be statistically identified using only the local information around each threshold points. Thus, in contrast to Caner and Hansen (2004) and Kourtellos et al (2016), the approach in Yu and Phillips (2018) does not require additional instruments for identifying either threshold points or discrepancy coefficients, although instruments can be used to increase efficiency -instruments remain, however, necessary for identifying slope parameter β j .…”