 # Estimation and inference of threshold regression models with measurement errors

## Abstract

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

## Acknowledgement

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

yi=β1xi+δxiΨi0(γ0)+εi,

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

β^1(γ^)=i=1nxiyi(1Ψi(γ^))(i=1nxi2(1Ψi(γ^)))1=β1+δi=1nxi2Ψi0(γ0)(1Ψi(γ^))i=1nxi2(1Ψi(γ^))+i=1nxiεi(1Ψi(γ^))i=1nxi2(1Ψi(γ^))=β1+δi=1nxi2Ψi(γ0+ui)(1Ψi(γ^))i=1nxi2(1Ψi(γ^))+Op(1n)=β1+δi=1nxi2[Ψi(γ0+ui)Ψi(max{γ0+ui,γ^})]i=1nxi2(1Ψi(γ^))+Op(1n).

Similarly, we can show that

β^2(γ^)β2=δi=1nxi2[Ψi(γ^)Ψi(max{γ0+ui,γ^})]i=1nxi2Ψi(γ^)+Op(1n).

### Proof of Lemma 2

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

yi=β1xi+δxiΨi0(γ0)+εi,

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

β~1(λ)=i=1nxiyi(1Ψi(γλ_))(i=1nxi2(1Ψi(γλ_)))1=β1+δi=1nxi2Ψi0(γ0)(1Ψi(γλ_))i=1nxi2(1Ψi(γλ_))+i=1nxiεi(1Ψi(γλ_))i=1nxi2(1Ψi(γλ_))=β1+δi=1nxi2Ψi(γ0+ui)(1Ψi(γλ_))i=1nxi2(1Ψi(γλ_))+Op(1n)=β1+δi=1nxi2[Ψi(γ0+ui)Ψi(max{γ0+ui,γλ_})]i=1nxi2(1Ψi(γλ_))+Op(1n).

Similarly, we can show that

β~2(λ)β2=δi=1nxi2[Ψi(γλ¯)Ψi(max{γ0+ui,γλ¯})]i=1nxi2Ψi(γλ¯)+Op(1n).

### 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:

|i=1nxi2[Ψi(γ0+ui)Ψi(max{γ0+ui,γ^})]i=1nxi2(1Ψi(γ^))|>|i=1nxi2[Ψi(γ0+ui)Ψi(max{γ0+ui,γλ_})]i=1nxi2(1Ψi(γλ_))|.

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

(19)i=1nxi2[Ψi(γ0+ui)Ψi(max{γ0+ui,γ^})]i=1nxi2(1Ψi(γ^))>i=1nxi2[Ψi(γ0+ui)Ψi(max{γ0+ui,γλ_})]i=1nxi2(1Ψi(γλ_)).

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

i=1nxi2[Ψi(γ0+ui)Ψi(max{γ0+ui,γ^})]i=1nxi2(1Ψi(γ^))=i=1nxi2I(γ0+ui<ziγ^)i=1nxi2I(ziγ^)=i=1nxi2I(γ0+ui<ziγ^,γ0+uiγλ_)+i=1nxi2I(γ0+ui<ziγ^,γ0+ui>γλ_)i=1nxi2I(ziγλ_)+i=1nxi2I(γλ_<ziγ^).

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^).

Thus,

(20)i=1nxi2[Ψi(γ0+ui)Ψi(max{γ0+ui,γ^})]i=1nxi2(1Ψi(γ^))i=1nxi2I(γ0+ui<ziγλ_)+i=1nxi2I(γλ_<ziγ^)i=1nxi2I(ziγλ_)+i=1nxi2I(γλ_<ziγ^)>i=1n[xi2I(γ0+ui<ziγλ_]i=1nxi2I(ziγλ_).

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

(21)i=1nxi2[Ψi(γ0+ui)Ψi(max{γ0+ui,γλ_})]i=1nxi2(1Ψi(γλ_))=i=1n[xi2I(γ0+ui<ziγλ_]i=1nxi2I(ziγλ_).

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

i=1nxi2[Ψi(γ0+ui)Ψi(max{γ0+ui,γ^})]i=1nxi2(1Ψi(γ^))>i=1nxi2[Ψi(γ0+ui)Ψi(max{γ0+ui,γλ_})]i=1nxi2(1Ψi(γλ_))

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

|i=1nxi2[Ψi(γ^)Ψi(max{γ0+ui,γ^})]i=1nxi2Ψi(γ^)|>|i=1nxi2[Ψi(γλ¯)Ψi(max{γ0+ui,γλ¯})]i=1nxi2Ψi(γλ¯)|.

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

i=1nxi2[Ψi(γ^)Ψi(max{γ0+ui,γ^})]i=1nxi2Ψi(γ^)>i=1nxi2[Ψi(γλ¯)Ψi(max{γ0+ui,γλ¯})]i=1nxi2Ψi(γλ¯).

Given γ^<γλ¯, we have

i=1nxi2[Ψi(γ^)Ψi(max{γ0+ui,γ^})]i=1nxi2Ψi(γ^)=i=1nxi2I(γ^<zi<γ0+ui)i=1nxi2I(γ^<zi)i=1nxi2I(γλ¯<zi<γ0+ui)+i=1nxi2I(γ^<ziγλ¯)i=1nxi2I(γλ¯<zi)+i=1nxi2I(γ^<ziγλ¯)>i=1nxi2I(γλ¯<zi<γ0+ui)i=1nxi2I(γλ¯<zi)

and

i=1nxi2[Ψi(γλ¯)Ψi(max{γ0+ui,γλ¯})]i=1nxi2Ψi(γλ¯)=i=1nxi2I(γλ¯<zi<γ0+ui)i=1nxi2I(γλ¯<zi).

Thus, we have

i=1nxi2[Ψi(γ^)Ψi(max{γ0+ui,γ^})]i=1nxi2Ψi(γ^)>i=1nxi2[Ψi(γλ¯)Ψi(max{γ0+ui,γλ¯})]i=1nxi2Ψi(γλ¯)

which completes the proof.

### Proof of Theorem 2

Consider a general threshold regression with multiple regressors

yi=β1xi+(β2β1)xiI(zi0>γ0)+εi,

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:

Y=[II0(γ0)]Xβ1+I0(γ0)Xβ2+ε,

where

I0(γ0)=diag{Ψ10(γ0),Ψ20(γ0),,Ψn0(γ0)}.

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

zi=zi0+ui.

Let

Ψi(γ)=I(zi>γ),

and

I(γ)=diag{Ψ1(γ),Ψ2(γ),,Ψn(γ)}.

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

(22)β^1(γ)=[X(II(γ))X]1X[II(γ)]Y=[X(II(γ))X]1X(II(γ))[Xβ1+I(γ0+u)Xδ+ε]=β1+φ1+op(1),

and

(23)β^2(γ)=(XI(γ)X)1XI(γ)Y=(XI(γ)X)1XI(γ)[Xβ2I(γ0+u)Xδ+ε]=β2φ2+op(1)

where

φ1=[X(II(γ))X]1X(II(γ))I(γ0+u)Xδ

and

φ2=(XI(γ)X)1XI(γ)I(γ0+u)Xδ.

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

(24)β~1(λ)=[X(II(γλ_))X]1X[II(γλ_)]Y,

and

(25)β~2(λ)=[XI(γλ¯)X]1XI(γλ¯)Y.

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

β~1(λ)=[X(II(γλ_))X]1X[II(γλ_)]Y=β1+[X(II(γλ_))X]1X[II(γλ_)]I(γ0)Xβ2+[X(II(γλ_))X]1X[II(γλ_)]ε=β1+[X(II(γλ_))X]1X[II(γλ_)]ε.

The equation (22) can be written as

β^1(γ0)=β1+[X(II(γ0))X]1X(II(γ0))I(γ0)Xδ+(X(II(γ0))X)1X(II(γ0))ε=β1+[X(II(γ0))X]1X[II(γ0)]ε.

Thus,

n(β~1(λ)β^1(γ0))=n[(X(II(γλ_))X)1X(II(γλ_))(X(II(γ0))X)1X(II(γ0))]ε.

Similarly, we have

n(β~2(λ)β^2(γ0))=n[(X(I(γλ¯))X)1X(I(γλ¯))(X(I(γ0))X)1X(I(γ0))]ε.

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

M1(γ)=E(xixiI(ziγ)),M2(γ)=E(xixiI(zi>γ)).

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

Ω11(γ1,γ2)=E(xiI(ziγ1)εiεiI(ziγ2)xi),Ω12(γ1,γ2)=E(xiI(ziγ1)εiεiI(zi>γ2)xi),Ω22(γ1,γ2)=E(xiI(zi>γ1)εiεiI(zi>γ2)xi).

The corresponding sample moment estimators are defined as

M^1(γ)=X(II(γ))Xn,M^2(γ)=XI(γ)Xn,

and

Ω^11(γ1,γ2)=X(II(γ1))ε^ε^(II(γ2))XnΩ^12(γ1,γ2)=X(II(γ1))ε^ε^(I(γ2))XnΩ^22(γ1,γ2)=X(I(γ1))ε^ε^(I(γ2))Xn

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

Var^[n(β~1(λ)β^1(γ0))]=[(X(II(γλ_))Xn)1X(II(γλ_))ε^n(X(II(γ0))Xn)1X(II(γ0))ε^n][(X(II(γλ_))Xn)1X(II(γλ_))ε^n(X(II(γ0))Xn)1X(II(γ0))ε^n]=(X(II(γλ_))Xn)1X(II(γλ_))ε^ε^(II(γλ_))Xn(X(II(γλ_))Xn)1+(X(II(γ0))Xn)1X(II(γ0))ε^ε^(II(γ0))Xn(X(II(γ0))Xn)1(X(II(γ0))Xn)1X(II(γ0))ε^ε^(II(γλ_))Xn(X(II(γλ_))Xn)1(X(II(γλ_))Xn)1X(II(γλ_))ε^ε^(II(γ0))Xn(X(II(γ0))Xn)1=M^1(γλ_))1Ω^11(γλ_,γλ_)M^1(γλ_))1+M^1(γ0))1Ω^11(γ0,γ0)M^1(γ0))1M^1(γλ_))1Ω^11(γλ_,γ0)M^1(γ0))1M^1(γ0))1Ω^11(γ0,γλ_)M^1(γλ_))1Π^11(γλ_,γ0),say.

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

Π^11(γλ_,γ0)pM1(γλ_)1Ω11(γλ_)M1(γλ_)1+M1(γ0)1Ω11(γ0)M1(γ0)1M1(γλ_)1Ω11(γλ_,γ0)M1(γ0)1M1(γ0)1Ω11(γ0,γλ_)M1(γλ_)1Π11(γλ_,γ0),say.

Similarly, we have

Var^(n(β~2(λ)β^2(γ0)))=[(X(I(γλ¯))Xn)1X(I(γλ¯))ε^n(X(I(γ0))Xn)1X(I(γ0))ε^n][(X(I(γλ¯))Xn)1X(I(γλ¯))ε^n(X(I(γ0))Xn)1X(I(γ0))ε^n]=M^2(γλ¯)1Ω^22(γλ¯,γλ¯)M^2(γλ¯)1+M^2(γ0)1Ω^22(γ0,γ0)M^2(γ0)1M^2(γλ¯)1Ω^22(γλ¯,γ0)M^2(γ0)1M^2(γ0)1Ω^22(γ0,γλ¯)M^2(γλ¯)1Π^22(γλ¯,γ0)pΠ22(γλ¯,γ0).

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

Cov^(n(β~1(λ)β^1(γ0)),n(β^2(γ0)β~2(λ)))=[(X(II(γλ_))Xn)1X(II(γλ_))ε^n(X(II(γ0))Xn)1X(II(γ0))ε^n][(X(I(γλ¯))Xn)1X(I(γλ¯))ε^n(X(I(γ0))Xn)1X(I(γ0))ε^n]=M^1(γλ_)1Ω^12(γλ_,γλ¯)M^2(γλ¯)1+M^1(γ0)1Ω^12(γ0,γ0)M^2(γ0)1M^1(γλ_)1Ω^12(γλ_,γ0)M^2(γ0)1M^1(γ0))1Ω^12(γ0,γλ¯)M^2(γλ¯)1Π^12(γλ_,γλ¯,γ0)pΠ12(γλ_,γλ¯,γ0).

Let

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

Π^(γλ_,γλ¯,γ0)pΠ(γλ_,γλ¯,γ0).

Applying the central limiting theorem for martingale processes, we have

n(β^1(γ0)β~1(λ)β^2(γ0)β~2(λ))N(0,Π(γλ_,γλ¯,γ0)).

Therefore

T0(λ)=n(β^1(γ0)β~1(λ)β^2(γ0)β~2(λ))Π^(γλ_,γλ¯,γ0)1(β^1(γ0)β~1(λ)β^2(γ0)β~2(λ))dχ2(2k).

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

T(λ)=n(β^1(γ^)β~1(λ)β^2(γ^)β~2(λ))Π^(γλ_,γλ¯,γ^)1(β^1(γ^)β~1(λ)β^2(γ^)β~2(λ))=T0(λ)+op(1)χ2(2k).

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