Estimation and inference of threshold regression models with measurement errors

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 Ψ_i(⋅), the left side of the inequality (19) can be written as

Given γ^>γλ_, we have ∑i=1nxi2I(γ0+ui<zi≤γ^,γ0+ui≤γλ_)≥∑i=1nxi2I(γ0+ui<zi≤γλ_) and ∑i=1nxi2I(γ0+ui<zi≤γ^,γ0+ui>γλ_)≥∑i=1nxi2I(γλ_<zi≤γ^).

Thus,

Using the definition of the indicator function Ψ_i(⋅), 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

and

Thus, we have

which completes the proof.

Proof of Theorem 2

Consider a general threshold regression with multiple regressors

where x_i 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:

where

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

Let

and

Note that Ψi(γ)=I(zi>γ)=I(zi0>γ−ui)=Ψi0(γ−ui) 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

and

where

and

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

and

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

Thus,

Similarly, we have

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

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

The corresponding sample moment estimators are defined as

and

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