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Estimation and inference of threshold regression models with measurement errors

  • Terence Tai-Leung Chong EMAIL logo , Haiqiang Chen , Tsz-Nga Wong and Isabel Kit-Ming Yan


An important assumption underlying standard threshold regression models and their variants in the extant literature is that the threshold variable is perfectly measured. Such an assumption is crucial for consistent estimation of model parameters. This paper provides the first theoretical framework for the estimation and inference of threshold regression models with measurement errors. A new estimation method that reduces the bias of the coefficient estimates and a Hausman-type test to detect the presence of measurement errors are proposed. Monte Carlo evidence is provided and an empirical application is given.

JEL Classification: C12; C22


We would like to thank Leonard Stefanski, W.K. Li, George Kapetanios and seminar participants at City University of Hong Kong, Lingnan University and International Symposium on Econometric Theory and Applications (SETA) for helpful comments. We also thank Min Chen and Margaret Loo for their able research assistance. This work is partially supported by the China National Science Foundation grant #71571152 and the Fundamental Research Funds for the Central Universities #20720171002. The views expressed here are those of the authors and do not reflect those of the FRB Richmond or the Federal Reserve System. All remaining errors are ours.

Appendix: Mathematical proofs

Proof of Lemma 1

By plugging the true model


into Equation (8), and using Ψi0(γ0)=Ψi(γ0+ui), under Assumptions A1–A6, we have


Similarly, we can show that


Proof of Lemma 2

The proof is similar to that of Lemma 1. By plugging the true model


into Equation (10), under Assumptions A1–A6, we have


Similarly, we can show that


Proof of Theorem 1

We first prove that β~1(λ) is less biased than β^1(γ^). Based on Lemmas 1 and 2, we only need to prove the following inequality:


Note that both Ψi(γ0+ui)Ψi(max{γ0+ui,γ^}) and Ψi(γ0+ui)Ψi(max{γ0+ui,γλ_}) are non-negative. Thus, it suffices to show that


Using the definition of the indicator function Ψi(⋅), the left side of the inequality (19) can be written as


Given γ^>γλ_, we have i=1nxi2I(γ0+ui<ziγ^,γ0+uiγλ_)i=1nxi2I(γ0+ui<ziγλ_) and i=1nxi2I(γ0+ui<ziγ^,γ0+ui>γλ_)i=1nxi2I(γλ_<ziγ^).



Using the definition of the indicator function Ψi(⋅), the right side of the inequality (19) can be written as


Combining the inequality (20) and the equation (21), we have


which completes the proof.

Next, we prove that β~2(λ) is less biased than β^2(γ^). Using Lemmas 1 and 2, we only need to show that


Since both Ψi(γ^)Ψi(max{γ0+ui,γ^}) and Ψi(γλ¯)Ψi(max{γ0+ui,γλ¯}) are non-negative, we only need to show that


Given γ^<γλ¯, we have




Thus, we have


which completes the proof.

Proof of Theorem 2

Consider a general threshold regression with multiple regressors


where xi is a k × 1 vector of covariates. When k = 1, we have the univariate model given by the equation (1).

The model can be rewritten in matrix form as follows:




Ψi0(γ0) is an indicator function defined in the equation (2); Y = (y1, y2, …, yn)′, X = (x1, x2, …, xn) and ε = (ε1, ε2, …, εn)′. We observe






Note that Ψi(γ)=I(zi>γ)=I(zi0>γui)=Ψi0(γui) and Ψi(γ0+ui)=Ψi0(γ0), thus, I(γ0 + u) = I0(γ0).

Given any γ(γλ_, γλ¯), the conventional LS estimators for β are given by








Given any λ ∈ (0, 1/2), the new estimators β~1(λ) and β~2(λ) are




Under the null, we have u = 0 and thus I(γ0)=I0(γ0). Given the assumption that γ0(γλ_, γλ¯), the equation (24) can be written as


The equation (22) can be written as




Similarly, we have


Before proceeding further, for any γ, we define the following conditional moment functionals for xi as


For any γ1 and γ2, define the conditional moment matrix for xiεi as


The corresponding sample moment estimators are defined as




Under Assumptions A1–A6, the law of large number holds and thus M^1(γ)pM1(γ), M^2(γ)pM2(γ), Ω^ij(γ1,γ2)pΩij(γ1,γ2) for all i = 1, 2, j = 1, 2.

Next, we derive the covariance matrix of n(β~1(λ)β^1(γ0)). Note that


Using the convergence results of M^i and Ω^ij,


Similarly, we have


The covariance between n(β^1(γ0)β~1(λ)) and n(β^2(γ0)β~2(λ)) can be written as



Π(γλ_,γλ¯,γ0)=(Π11(γλ_,γ0), Π12(γλ_,γλ¯,γ0)Π12(γλ_,γλ¯,γ0),Π22(γλ¯,γ0)),Π^(γλ_,γλ¯,γ0)=(Π^11(γλ_,γ0), Π^12(γλ_,γλ¯,γ0)Π^12(γλ_,γλ¯,γ0),Π^22(γλ¯,γ0)).

We have


Applying the central limiting theorem for martingale processes, we have




Under the null, from the Lemma A.9 of Hansen (2000), we have γ^γ0=Op(1n) and thus the impact from the estimation is negligible. It follows that



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Published Online: 2017-9-26

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