Estimation and Inference for the Threshold Model with Hybrid Stochastic Local Unit Root Regressors

Abstract

In this paper, we study the estimation and inference of the threshold model with hybrid local stochastic unit root regressors. Our main contribution is to propose an estimator that generalizes the threshold model with various forms of nonstationary regressors and to obtain its limiting distribution theory. In particular, our proposed model generalizes the threshold model with unit root, local-to-unity, and stochastic unit root regressors. We provide the estimation strategy for the least squares estimator and derive the asymptotic results for the proposed estimator. Depending on the diminishing rate of the threshold effect, we find that the limiting distribution of the threshold estimator takes different forms. Monte Carlo simulations are used to assess our proposed estimator’s finite sample performance, which is found to perform well.

1. Introduction

Widely used to capture nonlinearities in economic relationships, the parametric threshold model has received much attention over the past two decades. Many empirical studies use threshold models in time series applications and examples include the pricing asymmetry of oil prices and the nonlinear effect of public debt-to-GDP ratio on the per capita GDP growth of a given country over time.

The statistical inference of the threshold regression model is well established with stationary regressors. Under the fixed threshold effect assumption, Chan (1993) develops a threshold estimator which converges to a function of a compound Poisson process. Hansen (2000) establishes the limiting results under the diminishing threshold effect assumption and shows that the limiting distribution of the threshold parameter estimator is in the form of two independent Brownian motions. Extending Hansen (2000), Caner and Hansen (2004) further allow for the endogenous slope regressors by using a two-step least squares estimation method. In the spirit of Heckman (1979), Kourtellos et al. (2016) propose using a parametric control function method to allow for the endogenous threshold variable under the joint normality assumption. Yu et al. (2021) correct and expand Kourtellos et al. (2016)’s method, assuming the endogeneity is known with a finite functional form. Kourtellos et al. (2021) further relax Yu et al. (2021)’s assumption on the known endogenous form and use a nonparametric approach to control for the unknown form of endogeneity. Chen et al. (2012) consider a threshold regression model with two separate exogenous threshold variables, while Chen et al. (2021) further allow two thresholds to be endogenous. Yang et al. (2021) extend Hansen (2000) by allowing for a time-varying threshold. Miao et al. (2020) study a panel threshold model with latent group structures. Seo and Linton (2007) explore a linear index threshold model that allows multiple stationary threshold variables to be a linear index of regressors, and Yu and Fan (2020) provide the distributional theory of a least squares estimator for that model. In light of Seo and Shin (2016), Chen and Stengos (2021) employ a generalized method of moment approach to allow for the endogeneity in the linear index threshold model.

As for the threshold regression model with nonstationary regressors, surprisingly, the asymptotic properties have not been fully developed. Among the few, Caner and Hansen (2001) study a threshold autoregressive model with unit root regressors, and they derive the limiting distribution of Wald tests for a threshold, under the null of a linear model to be nonstandard and different from the stationary case. Gonzalo and Pitarakis (2012) extend the nonlinearity testing to the case of nearly unit root regressors, while Chen (2014) develops the estimation and inference of a threshold model with integrated regressors. One of the findings is that the model may be identified only weakly if the diminishing rate of the threshold effect is fast enough.

Nevertheless, the studies mentioned above restrict the regressors to either unit root or nearly unit root, though many series, particularly from empirical macroeconomics and finance, display various forms of dependence in different periods. Recent empirical work also sheds light on the advantages of linking multiple regimes of stationary and non-stationary behaviors and transition mechanisms between them (e.g., Phillips et al. 2011; Phillips and Yu 2011). Hence, a statistical inference of a threshold regression model that accommodates various forms of nonstationary regressors is needed.

Our paper aims to fill this gap in the literature. Our main contribution in this paper is to propose an estimator that generalizes the threshold model with various forms of nonstationary regressors and to obtain its limiting distribution theory. Following Lieberman and Phillips (2020), we propose a threshold regression model with hybrid local stochastic unit root (HLSUR) regressors. The proposed model generalizes the threshold model well, with unit root, local-to-unity, and stochastic unit root regressors. We provide a least squares estimation strategy and derive the limiting results of the threshold estimator. We also demonstrate the good finite sample performance of our proposed estimator via Monte Carlo simulations.

The remainder of the paper is organized as follows. In Section 2, we introduce the threshold model with HLSUR regressors and provide an estimation strategy, while in Section 3, we derive the limiting results. Section 4 provides a heuristic example to discuss the limiting results. In Section 5, we show the inference for the threshold effect. Section 6 reports the Monte Carlo results for the proposed estimators. Finally, Section 7 concludes the paper. Technical proofs are relegated to Appendix A.

To proceed, we adopt the following notation. The indicator function is denoted as $I(·)$. We use $\parallel ·\parallel$ to denote the Euclidean norm. $\left[np\right]$ denotes the integer part of $np$. $\stackrel{p}{\to }$, $\stackrel{d}{\to }$, ⟹, and ∼ denote convergence in probability, convergence in distribution, weak convergence, and equivalent to, respectively. ⋀ and ⋁ denote the minimum and maximum operators. Following Caner and Hansen (2001), we make $W(s,\mu )$ a two-parameter Brownian motion on $(s,\mu )\in {[0,1]}^2$.

2. The Model and Estimators

Consider the following threshold model,

$$\begin{array}{c}\hfill y_t={\beta }^T{x}_t+{\delta }_n^T{x}_{t}I(q_t\le {\gamma }_0)+{\mu }_t,\end{array}$$

(1)

where $x_t$ is a $d_x\times 1$ vector of hybrid stochastic unit root processes, for all $j=1,\dots ,d_x$, with the form

$$\begin{array}{c}\hfill x_{j,t}={\beta }_{nt}{x}_{j,t-1}+v_{j,t},\\ \hfill {\beta }_{nt}=e^{\frac{c_j}{n}+\frac{{\alpha }_j^T{\epsilon }_{jt}}{\sqrt{n}}},\end{array}$$

(2)

where $c_j$ is a localizing constant, ${\epsilon }_{jt}$ is a $d_{{\epsilon }_j}\times 1$ vector, and $x_{j1}=v_{j1}$. Note that the specification of ${\beta }_{nt}$ allows for different types of dependence. For example, if all ${\alpha }_j$ and $c_j$ are zero, we will have the standard unit root case.

Let ${\epsilon }_t={[{\epsilon }_{1t}^T,\dots ,{\epsilon }_{d_{x}t}^T]}^T$ and $v_t={[v_{1t},\dots ,v_{d_{x}t}]}^T$. We assume the vector ${\eta }_t={({\mu }_t,v_t^T,{\epsilon }_t^T)}^T$ to be a strictly stationary martingale difference sequence, and the partial sums satisfy the following invariance principle

$$\begin{array}{c}\hfill \frac{1}{\sqrt{n}}\sum _{t=1}^{\left[np\right]}{\eta }_t⟹\mathbf{B}\left(p\right)={\left[B_{\mu }\left(p\right),{\mathbf{B}}_v^T\left(p\right),{\mathbf{B}}_{\epsilon }^T\left(p\right)\right]}^T,\end{array}$$

(3)

where ${\mathbf{B}}_v\left(p\right)={[B_{v_1}\left(p\right),\dots ,B_{v_{d_x}}\left(p\right)]}^T$, ${\mathbf{B}}_{\epsilon }\left(p\right)={[B_{{\epsilon }_1}^T\left(p\right),\dots ,B_{{\epsilon }_{d_x}}^T\left(p\right)]}^T$, and $\mathbf{B}\left(p\right)$ is a vector of Brownian motions with a long run positive definite covariance matrix

$$\begin{array}{c}\hfill \mathsf{\Sigma }=\left[\begin{array}{ccc}{\sigma }_{\mu }^2& 0& 0\\ 0& {\mathsf{\Sigma }}_v& 0\\ 0& 0& {\mathsf{\Sigma }}_{\epsilon }\end{array}\right],\end{array}$$

(4)

and ${\sigma }_{\mu }^2>0$, ${\mathsf{\Sigma }}_v>0$, and ${\mathsf{\Sigma }}_{\epsilon }>0$.

We can rewrite model (1) in a stacked form

$$\begin{array}{c}\hfill \mathbf{y}=X\left(\gamma \right)\theta +\mu ,\end{array}$$

(5)

where $\theta ={\left[{\beta }^T,{\delta }_n^T\right]}^T$, $\mathbf{y}$ stacks up $y_t$, $\mu$ stacks up ${\mu }_t$, $X\left(\gamma \right)$ stacks up $X_t\left(\gamma \right)$, and $X_t\left(\gamma \right)={\left[x_t^T,x_t^{T}I(q_t\le \gamma )\right]}^T$.

Hence, given a fixed $\gamma \in [\underline{\gamma },\overline{\gamma }]$, model (5) is linear in $\theta$. We obtain

$$\begin{array}{c}\hfill \widehat{\theta }\left(\gamma \right)={\left[X{\left(\gamma \right)}^{T}X\left(\gamma \right)\right]}^{-1}X{\left(\gamma \right)}^T\mathbf{y}.\end{array}$$

(6)

The least squares estimator of ${\gamma }_0$ is

$$\begin{array}{c}\hfill \widehat{\gamma }=arg\underset{\gamma \in [\underline{\gamma },\overline{\gamma }]}{min}SSR\left(\gamma \right),\end{array}$$

(7)

where $SSR\left(\gamma \right)={\sum }_{t=1}^n{\left(y_t-\widehat{\theta }{\left(\gamma \right)}^T{X}_t\left(\gamma \right)\right)}^2$.

Due to the non-smooth nature of the objective function, following Hansen (2000), we employ a grid-search method to estimate the model empirically.

3. Asymptotic Properties for the Estimators

We make the following assumptions to support model (1).

Assumption 1.

$\left\{q_t\right\}$ is a strictly stationary mixing sequence with the mixing coefficient of size $-r/(r-2)$ for some $r>2$.

Assumption 2.

Let ${\mathcal{F}}_{n,t}$ be the smallest sigma-field generated by $\{q_{s+1},{\eta }_s^T:1\le s<t\le n\}$. ${\left\{({\eta }_t,{\mathcal{F}}_{n,t})\right\}}_{t=1}^n$ is a strictly martingale difference sequence (MDS) with positive definite covariance $E\left({\eta }_t{\eta }_t^T\right|{\mathcal{F}}_{n,t})$, whose partial sums satisfy the invariance principle, as in (3).

Assumption 3.

The threshold variable $q_t$ has a continuous distribution $F(·)$. $f(·)$ is the corresponding density function with $0<f\left(\gamma \right)<\overline{f}<\infty$ for all $\gamma \in [\underline{\gamma },\overline{\gamma }]$.

Define

$$\begin{array}{c}\hfill G_{{\alpha }_j,c_j}\left(s\right)=e^{s{c}_j+{\alpha }_j^T{B}_{{\epsilon }_j}\left(s\right)}\left({\int }_0^s{e}^{-p{c}_j-{\alpha }_j^T{B}_{{\epsilon }_j}\left(p\right)}d{B}_{v_j}\left(p\right)\right),\end{array}$$

(8)

and

$$\begin{array}{c}\hfill G_{\alpha ,c}\left(s\right)={\left[G_{{\alpha }_1,c_1}\left(s\right),\dots ,G_{{\alpha }_{d_x},c_{d_x}}\left(s\right)\right]}^T.\end{array}$$

(9)

Assumption 4.

${\int }_0^1{G}_{\alpha ,c}\left(s\right)G_{\alpha ,c}^T\left(s\right)ds$ is positive definite.

Assumption 1 ensures the threshold variable $q_t$ is strictly stationary. Assumption 2 states that $q_t$ is contemporaneously exogenous in model (1) and ensures the multivariate invariance principle for the partial sum of a martingale difference array. Assumption 3 is common in the threshold regression literature. This assumption assumes the threshold variable is continuous with positive density everywhere, ensuring they are dense near the true threshold level. Assumption 4 is a regular full rank condition.

Theorem 1.

Under Assumptions 1–4, with ${\delta }_n=n^{-1/2-\rho }{\delta }_0$ and $\rho \in (-1/2,1/2)$, we have $\widehat{\gamma }-{\gamma }_0=o_p\left(1\right)$.

Theorem 2.

Under Assumptions 1–4, with ${\delta }_n=n^{-1/2-\rho }{\delta }_0$, the following limiting results hold:

(i). 

if $\rho \in (0,\frac{1}{2})$,

$$\begin{array}{c}\hfill n^{1-2\rho }\left(\widehat{\gamma }-{\gamma }_0\right)\stackrel{d}{\to }\lambda T,\end{array}$$

(10)

where $\lambda =\frac{{\sigma }_{\mu }^2}{f\left({\gamma }_0\right){\delta }_0^T{\int }_0^1{G}_{\alpha ,c}\left(s\right)G_{\alpha ,c}^T\left(s\right)ds{\delta }_0}$, and $T=\underset{r\in (-\infty ,+\infty )}{argmax}T\left(r\right)$ with $T\left(r\right)$ denotes a two-sided Brownian motion on the real line

$$\begin{array}{c}\hfill T\left(r\right)=\left\{\begin{array}{cc}{W}_1\left(r\right)-\frac{1}{2}r,\hfill & \mathrm{if}r>0,\hfill \\ 0,\hfill & \mathrm{if}r=0,\hfill \\ W_2(-r)-\frac{1}{2}\left|r\right|,\hfill & \mathrm{if}r<0,\hfill \end{array}\right.\end{array}$$

(11)

and $W_1\left(r\right)$ and $W_2$ (r) are two independent standard Brownian motion processes defined on $[0,\infty )$;

(ii). 

if $\rho =0$,

$$\begin{array}{c}\hfill n\left(\widehat{\gamma }-{\gamma }_0\right)\stackrel{d}{\to }arg\underset{v}{min}D\left(v\right),\end{array}$$

(12)

where

$$\begin{array}{c}\hfill D\left(v\right)=\left\{\begin{array}{cc}{\delta }_0^T\sum _{t=1}^{N_1\left(v\right)}\left(x_t{x}_t^T{\delta }_0-2x_t{\mu }_t\right),\hfill & \mathrm{if}v>0,\hfill \\ {\delta }_0^T\sum _{t=1}^{N_2\left(\right|v\left|\right)}\left(x_t{x}_t^T{\delta }_0+2x_t{\mu }_t\right),\hfill & \mathrm{if}v\le 0,\hfill \end{array}\right.\end{array}$$

and $N_1\left(v\right)$ and $N_2\left(v\right)$ are two independent Poisson process with intensity $f\left({\gamma }_0\right)$.

Theorems 1 and 2 show the threshold estimator’s consistency and asymptotic distribution, respectively. We make the following remarks for Theorem 2.

Remark 1.

The convergence rate of $\widehat{\gamma }$ depends only on the diminishing rate of the threshold effect, ρ. It does not relate to both the localizing constant, c, and the coefficient of the stochastic term, α.

Remark 2.

If the diminishing rate of the threshold effect is fast enough $\left(\rho \in (0,1/2)\right)$, the asymptotic distribution of $\widehat{\gamma }$ is in the form of the argmin of the drifted two-sided Brownian motion and is essentially in the same form of Hansen (2000) and Chen (2014), with a different scaling factor, λ. Intuitively, this is because we have infinite threshold variables in the local neighborhood of ${\gamma }_0$ asymptotically. Thus, we can apply the functional central limit theorem to derive the limiting results. We notice Chen (2014) also allows a “slowly” diminishing rate ($\rho \in (-1/2,0)$). We exclude that case because, as we show in Lemma A4 of the appendix, the uniform law of large numbers and the functional central limit theorem can be applied if and only if $\frac{a_n}{n}$ goes to zero, where $a_n$ is the convergence rate of the threshold estimator. This condition is well documented in the change point model. For example, Kejriwal and Perron (2008) assume the jump size is diminishing at a faster rate with the nonstationary regressors than with the stationary regressors.

Remark 3.

If $\rho =0$, only a finite number of $q_t$ are involved asymptotically in the local neighborhood of the true threshold level. Under this case, that $P\left(N\right(v)⟶\infty )=0$ for any finite v holds since the threshold variable is strictly stationarity. Therefore, we can extend Chan (1993)’s distributional theory to the nonstationary regressors. As a result, the limiting distribution of the threshold estimator is given as the argmin of the stochastic process $D\left(v\right)$. A similar argument can be found in the comment of proposition 2 of Bai (1997). Note that, if $x_t$ is also strictly stationary, $D\left(v\right)$ reduces to a compound Poisson process, as shown in Theorem 2 of Chan (1993). 1

Remark 4.

If $\rho <0$, the threshold estimator converges very fast, such that we have zero information around the local neighborhood of the true threshold level asymptotically. As a result, we cannot obtain a limiting distribution.

We consider some extensions to Theorem 2. The following provides the limiting results of the threshold estimator for model (1) with some special HLSUR regressors.

Corollary 1.

Under Assumptions 1–4, and $c_j=0$ for $j=1,\dots ,d_x$, with $\rho \in (0,\frac{1}{2})$, we have $n^{1-2\rho }\left(\widehat{\gamma }-{\gamma }_0\right)\stackrel{d}{\to }{\lambda }^{\prime }T$, where ${\lambda }^{\prime }=\frac{{\sigma }_{\mu }^2}{f\left({\gamma }_0\right){\delta }_0^T{\int }_0^1{G}_{\alpha }\left(s\right)G_{\alpha }^T\left(s\right)ds{\delta }_0}$, $G_{\alpha }\left(s\right)={\left[G_{{\alpha }_1}\left(s\right),\dots ,G_{{\alpha }_{d_x}}\left(s\right)\right]}^T$, and $G_{{\alpha }_j}\left(s\right)=e^{{\alpha }_j^T{B}_{{\epsilon }_j}\left(s\right)}\left({\int }_0^s{e}^{-{\alpha }_j^T{B}_{{\epsilon }_j}\left(p\right)}d{B}_{v_j}\left(p\right)\right)$.

Corollary 2.

Under Assumptions 1–4, and ${\alpha }_j=0$ for $j=1,\dots ,d_x$, with $\rho \in (0,\frac{1}{2})$, we have $n^{1-2\rho }\left(\widehat{\gamma }-{\gamma }_0\right)\stackrel{d}{\to }{\lambda }^{″}T$, where ${\lambda }^{″}=\frac{{\sigma }_{\mu }^2}{f\left({\gamma }_0\right){\delta }_0^T{\int }_0^1{G}_c\left(s\right)G_c^T\left(s\right)ds{\delta }_0}$, $G_c\left(s\right)={\left[G_{c_1}\left(s\right),\dots ,G_{c_{d_x}}\left(s\right)\right]}^T$, and $G_{c_j}\left(s\right)=e^{s{c}_j}\left({\int }_0^s{e}^{-p{c}_j}d{B}_{v_j}\left(p\right)\right)$.

Corollaries 1 and 2 show the limiting results of the threshold estimator in the case of a stochastic unit root and local-to-unity regressors, respectively. If we assume both $c_j=0$ and ${\alpha }_j=0$ for all $j=1,\dots ,d_x$, the regressors are unit root processes. The limiting distribution further reduces to Theorem 3.1 of Chen (2014).

4. Heuristic Arguments and an Illustrative Example

In this section, we use a simple setup of model (1) to provide more intuitions on the limiting result in Theorem 2. Suppose $x_t$ and $\alpha$ are of dimension one, ${\delta }_0=1$, and $\beta =0$. In addition, we further assume ${\sigma }_{\epsilon }^2={\sigma }_{\mu }^2={\sigma }_v^2=1$. Hence, model (1) becomes $y_t=n^{-1/2-\rho }{x}_{t}I(q_t\le {\gamma }_0)+{\mu }_t$, where $E\left(G_{\alpha ,c}\left(s\right)\right)=0$ and $Var\left(G_{{\alpha }_c,c}\left(s\right)\right)=\frac{e^{2(c+{\alpha }^2)s}-1}{2(c+{\alpha }^2)}$.

Assuming the knowledge of the consistency and the convergence rate, we examine the asymptotic behavior of $SSR({\gamma }_0+\frac{v}{a_n})-SSR\left({\gamma }_0\right)$, where $a_n=n^{1-2\rho }$. For any $v\in (0,\infty )$, we have,

$$\begin{array}{cc}\hfill & SSR({\gamma }_0+\frac{v}{a_n})-SS{R}_n\left({\gamma }_0\right)=n^{-1-2\rho }\sum _{t=1}^n{x}_t^{2}I\left(\frac{v}{a_n}\right)-2n^{-1/2-\rho }\sum _{t=1}^n\left[x_{t}I\left(\frac{v}{a_n}\right){\mu }_t\right]\hfill \\ \hfill & =S_{n1}\left(v\right)-2S_{n2}\left(v\right),\hfill \end{array}$$

where $I\left(\frac{v}{a_n}\right)=I({\gamma }_0<q_t\le {\gamma }_0+\frac{v}{a_n})$.

Note that $S_{n1}\left(v\right)=\frac{1}{n}{\sum }_{t=1}^n{\left(\frac{x_t}{\sqrt{n}}\right)}^2{a}_{n}I\left(\frac{v}{a_n}\right)$. Applying Lemma 1 of Hansen (2000), we have

$$\begin{array}{c}\hfill E{\left|\frac{a_n}{n}\sum _{t=1}^n\left[I\left(\frac{v}{a_n}\right)-E\left(I\left(\frac{v}{a_n}\right)\right)\right]\right|}^2=\frac{a_n^2}{n}E{\left|\frac{1}{\sqrt{n}}\sum _{t=1}^n\left[I\left(\frac{v}{a_n}\right)-E\left(I\left(\frac{v}{a_n}\right)\right)\right]\right|}^2\le \frac{a_n^2}{n}C\frac{v}{a_n}⟶0,\end{array}$$

where C is a constant. Thus, for $\rho \in (0,\frac{1}{2})$, we can verify, for any finite positive v, $\frac{a_n}{n}{\sum }_{t=1}^{n}I\left(\frac{v}{a_n}\right)\stackrel{p}{\to }f\left({\gamma }_0\right)v$. Note that $\frac{1}{n}{\sum }_{t=1}^n{\left(\frac{x_t}{\sqrt{n}}\right)}^2⟹\int G_{\alpha ,c}{\left(s\right)}^{2}ds$. Therefore, we have $S_{n1}\left(v\right)⟹vf\left({\gamma }_0\right){\int }_0^1{G}_{\alpha ,c}{\left(s\right)}^{2}ds$.

For $\rho =0$, $E{\left|\frac{a_n}{n}{\sum }_{t=1}^n\left(I\left(\frac{v}{a_n}\right)-E\left(I\left(\frac{v}{a_n}\right)\right)\right)\right|}^2=O\left(1\right)$. This implies $\frac{a_n}{n}{\sum }_{t=1}^{n}I\left(\frac{v}{a_n}\right)\stackrel{p}{\nrightarrow }f\left({\gamma }_0\right)v$, which is well explored in the case of the fixed threshold effect in a threshold regression model with stationary regressors. Following Chan (1993), we can show ${\sum }_{t=1}^{n}I\left(\frac{v}{a_n}\right)\sim Binomial\left(n,p\left(v\right)\right)⟹N\left(v\right)$,2 where $p\left(v\right)=f\left({\gamma }_0\right)\frac{v}{a_n}$ and $N\left(v\right)$ is a Poisson distribution with intensity $f\left({\gamma }_0\right)$ due to the fact that $np\left(v\right)=f\left({\gamma }_0\right)<\infty$. As a result, $S_{n1}\left(v\right)⟹{\sum }_{t=1}^{N\left(v\right)}{x}_t^2$.

Next, we can rewrite $S_{n2}\left(v\right)$ as $S_{n2}\left(v\right)=\frac{1}{\sqrt{n}}{\sum }_{t=1}^n\frac{x_t}{\sqrt{n}}\sqrt{a_n}I\left(\frac{v}{a_n}\right){\mu }_t$. For $\rho \in (0,\frac{1}{2})$, we have

$$\begin{array}{c}\hfill E{\left|\frac{a_n}{n}\sum _{t=1}^n\left[{\mu }_t^{2}I\left(\frac{v}{a_n}\right)-E\left({\mu }_t^{2}I\left(\frac{v}{a_n}\right)\right)\right]\right|}^2=\frac{a_n^2}{n}E{\left|\frac{1}{\sqrt{n}}\sum _{t=1}^n\left[{\mu }_t^{2}I\left(\frac{v}{a_n}\right)-E\left({\mu }_t^{2}I\left(\frac{v}{a_n}\right)\right)\right]\right|}^2⟶0,\end{array}$$

Thus, we can show $\frac{1}{n}{\sum }_{t=1}^n{a}_{n}I\left(\frac{v}{a_n}\right){\mu }_t^2\stackrel{p}{⟶}vf\left({\gamma }_0\right)$ uniformly for all v. Then, we can apply the functional central limit theorem on $\frac{1}{\sqrt{n}}{\sum }_{t=1}^n\sqrt{a_n}I\left(\frac{v}{a_n}\right){\mu }_t$. As $\frac{x_{t=\left[ns\right]}}{\sqrt{n}}⟹G_{\alpha ,c}\left(s\right)$, we have $S_{n2}\left(v\right)⟹\int G_{\alpha ,c}\left(s\right)dW(s,f\left({\gamma }_0\right)v)$.

By contrast, for $\rho =0$,

$$\begin{array}{c}\hfill E{\left|\frac{a_n}{n}\sum _{t=1}^n\left[{\mu }_t^{2}I\left(\frac{v}{a_n}\right)-E\left({\mu }_t^{2}I\left(\frac{v}{a_n}\right)\right)\right]\right|}^2=O\left(1\right).\end{array}$$

Therefore, $\frac{1}{n}{\sum }_{t=1}^n{a}_{n}I\left(\frac{v}{a_n}\right){\mu }_t^2\stackrel{p}{\nrightarrow }vf\left({\gamma }_0\right)$. Following Chan (1993), we have $S_{n2}\left(v\right)⟹{\sum }_{t=1}^{N\left(v\right)}{x}_t{\mu }_t$.

5. Test for the Linearity

The threshold effect of model (1) disappears if ${\delta }_0=0$. Thus, for a given $\gamma \in [\underline{\gamma },\overline{\gamma }]$, we can construct the Wald statistic to test the null hypothesis of ${\delta }_0=0$ as

$$\begin{array}{c}\hfill W_n\left(\gamma \right)={\widehat{\delta }}^T\left[X_{\gamma }^T\left(I_n-P_x\right)X_{\gamma }\right]\widehat{\delta }/{\widehat{\sigma }}_{\mu }^2.\end{array}$$

(13)

Following Hansen (1996), we define $SupW=\underset{\gamma \in [\underline{\gamma },\overline{\gamma }]}{sup}{W}_n\left(\gamma \right)$.

Theorem 3.

Under Assumptions 1–4 and ${\delta }_0\ne 0$, we have $W_n\left(\gamma \right)=O_p\left(n^{1/2-\rho }\right)$.

Corollary 3.

Under the null and Assumptions 1–4, we have

$$\begin{array}{cc}\hfill & SupW⟹\underset{\gamma \in [\underline{\gamma },\overline{\gamma }]}{sup}({\left\{{\int }_0^1{G}_{\alpha ,c}\left(s\right)dW(s,F\left(\gamma \right))-F\left(\gamma \right){\int }_0^1{G}_{\alpha ,c}\left(s\right)dW\left(s\right)\right\}}^T\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & \times {\left\{\left[F\left(\gamma \right)-F{\left(\gamma \right)}^2\right]{\int }_0^1{G}_{\alpha ,c}\left(s\right)G_{\alpha ,c}{\left(s\right)}^{T}ds\right\}}^{-1}\hfill \\ \hfill & \times \left\{{\int }_0^1{G}_{\alpha ,c}\left(s\right)dW(s,F\left(\gamma \right))-F\left(\gamma \right){\int }_0^1{G}_{\alpha ,c}\left(s\right)dW\left(s\right)\right\})/{\sigma }_{\mu }^2.\hfill \end{array}$$

(15)

Theorem 3 shows $SupW⟹\infty$ under alternative assumptions. Under the null hypothesis, the limiting random variable of $SupW$ depends on some unknown parameters. Note that if $c=0$ and $\alpha =0$, $W_n\left(\gamma \right)$ is in the same form as Chen (2014). If $c<0$ and $\alpha =0$, Corrollary 3 reduces to Proposition 1 of Gonzalo and Pitarakis (2012).

6. Monte Carlo Studies

To evaluate the finite sample performance of our proposed estimator, we provide Monte Carlo simulation results in this section. We generate our data from following a simple structure:

$$\begin{array}{cc}\hfill & y_t=0.2x_t+2n^{-1/2-\rho }{x}_{t}I(q_t\le 0.2)+{\mu }_t,\hfill \\ \hfill & x_t={\beta }_{nt}{x}_{t-1}+v_t,\hfill \\ \hfill & {\beta }_{nt}=e^{\frac{c}{n}+\frac{\alpha {\epsilon }_t}{\sqrt{n}}},\hfill \end{array}$$

(16)

where $q_t,{\mu }_t,v_t$, and ${\epsilon }_t$ are independently normally distributed with mean zero and variance one.

We vary $c\in (-0.5,0,0.5)$ and $\alpha \in (0,0.07,0.2)$. For each data-generation process, we replicate 2000 times. We consider sample sizes of $n=300$, 500, and 1000. Table 1, Table 2 and Table 3 report the simulation results with $\rho =0$, $1/4$, and $1/3$, respectively.

Table 1. Simulation results with $\rho =0$.

Panel A: $\alpha =0$
	$c=-0.5$			$c=0$			$c=0.5$
n	Bias	MSE	Std	Bias	MSE	Std	Bias	MSE	Std
300	−0.0367	0.0635	0.2494	−0.0270	0.0595	0.2425	−0.0204	0.0473	0.2166
500	−0.0179	0.0448	0.2109	−0.0175	0.0353	0.1870	−0.0119	0.0299	0.1727
1000	−0.0099	0.0170	0.1300	−0.0029	0.0129	0.1136	−0.0021	0.0095	0.0977
Panel B: $\alpha =0.07$
	$c=-0.5$			$c=0$			$c=0.5$
n	Bias	MSE	Std	Bias	MSE	Std	Bias	MSE	Std
300	−0.0228	0.0532	0.2297	−0.0197	0.0493	0.2212	−0.0243	0.0399	0.1982
500	−0.0163	0.0292	0.1700	−0.0093	0.0235	0.1531	−0.0105	0.0180	0.1338
1000	−0.0059	0.0101	0.1004	−0.0053	0.0089	0.0942	−0.0037	0.0058	0.0758
Panel C: $\alpha =0.2$
	$c=-0.5$			$c=0$			$c=0.5$
n	Bias	MSE	Std	Bias	MSE	Std	Bias	MSE	Std
300	−0.0141	0.0339	0.1835	−0.0077	0.0250	0.1580	−0.0167	0.0234	0.1519
500	−0.0119	0.0174	0.1316	−0.0094	0.0114	0.1062	−0.0047	0.0077	0.0879
1000	−0.0018	0.0039	0.0624	−0.0012	0.0023	0.0476	−0.0022	0.0017	0.0417

This table reports the simulation results with $\rho =0$. The first column shows the sample size. The second to the fourth columns report the results with c = −0.5. The fifth to the seventh columns report the results with c = 0. The last three columns report the results with c = 0.5. Panel A shows the results with α = 0. Panel B reports the results with α = 0.07, and Panel C reports the results with α = 0.2.

Table 2. Simulation results with $\rho =1/4$.

Panel A: $\alpha =0$
	$c=-0.5$			$c=0$			$c=0.5$
n	Bias	MSE	Std	Bias	MSE	Std	Bias	MSE	Std
300	−0.1519	0.2711	0.4981	−0.1304	0.2649	0.4980	−0.1448	0.2501	0.4788
500	−0.1380	0.2550	0.4859	−0.1201	0.2344	0.4691	−0.1109	0.2160	0.4514
1000	−0.1347	0.2494	0.4810	−0.1077	0.2241	0.4611	−0.1148	0.1951	0.4267
Panel B: $\alpha =0.07$
	$c=-0.5$			$c=0$			$c=0.5$
n	Bias	MSE	Std	Bias	MSE	Std	Bias	MSE	Std
300	−0.1450	0.2502	0.4788	−0.1350	0.2311	0.4615	−0.1001	0.2066	0.4435
500	−0.1195	0.2264	0.4607	−0.1106	0.2149	0.4504	−0.1076	0.1894	0.4218
1000	−0.1100	0.1947	0.4274	−0.0640	0.1518	0.3845	−0.0675	0.1391	0.3669
Panel C: $\alpha =0.2$
	$c=-0.5$			$c=0$			$c=0.5$
n	Bias	MSE	Std	Bias	MSE	Std	Bias	MSE	Std
300	−0.1095	0.2001	0.4338	−0.0841	0.1686	0.4021	−0.0723	0.1372	0.3633
500	−0.0644	0.1389	0.3671	−0.0619	0.1255	0.3489	−0.0452	0.0902	0.2970
1000	−0.0422	0.0779	0.2759	−0.0211	0.0526	0.2285	−0.0203	0.0433	0.2070

This table reports the simulation results with $\rho =1/4$. The first column shows the sample size. The second to the fourth columns report the results with c = −0.5. The fifth to the seventh columns report the results with c = 0. The last three columns report the results with c = 0.5. Panel A shows results with α = 0. Panel B reports results with α = 0.07, and Panel C reports results with α = 0.2.

Table 3. Simulation results with $\rho =1/3$.

Panel A: $\alpha =0$
	$c=-0.5$			$c=0$			$c=0.5$
n	Bias	MSE	Std	Bias	MSE	Std	Bias	MSE	Std
300	−0.1903	0.3193	0.5322	−0.1859	0.3105	0.5254	−0.1629	0.2868	0.5102
500	−0.2009	0.3091	0.5185	−0.1798	0.3019	0.5194	−0.1731	0.2886	0.5087
1000	−0.1615	0.2859	0.5099	−0.1524	0.2853	0.5121	−0.1553	0.2712	0.4972
Panel B: $\alpha =0.07$
	$c=-0.5$			$c=0$			$c=0.5$
n	Bias	MSE	Std	Bias	MSE	Std	Bias	MSE	Std
300	−0.1549	0.2930	0.5188	−0.1527	0.2804	0.5071	−0.1541	0.2586	0.4848
500	−0.1483	0.2831	0.5111	−0.1551	0.2639	0.4899	−0.1233	0.2385	0.4727
1000	−0.1470	0.2549	0.4831	−0.1201	0.2324	0.4670	−0.1126	0.2162	0.4513
Panel C: $\alpha =0.2$
	$c=-0.5$			$c=0$			$c=0.5$
n	Bias	MSE	Std	Bias	MSE	Std	Bias	MSE	Std
300	−0.1494	0.2624	0.4901	−0.1175	0.2288	0.4638	−0.1168	0.2038	0.4361
500	−0.1326	0.2229	0.4533	−0.0834	0.1782	0.4139	−0.0742	0.1512	0.3818
1000	−0.0786	0.1449	0.3725	−0.0633	0.1132	0.3305	−0.0468	0.0783	0.2759

This table reports the simulation results with $\rho =1/3$. The first column shows the sample size. The second to the fourth columns report the results with c = −0.5. The fifth to the seventh columns report the results with c = 0. The last three columns report the results with c = 0.5. Panel A shows results with α = 0. Panel B reports results with α = 0.07, and Panel C reports results with α = 0.2.

Overall, we see that the performance of the threshold estimator improves as the sample size increase for all cases. In addition, the finite sample performance slightly changes over various $\alpha$ and c. We observe more considerable bias and MSE as the threshold effect diminishes faster. These observations are consistent with our limiting results, suggesting that the convergence rate of the threshold estimator only relates to the diminishing rate of the threshold effect, $\rho$.

7. Concluding Remarks

The estimation and statistical theories of the threshold model with stationary or integrated regressors are well explored (e.g., Hansen 2000; Caner and Hansen 2001; Chen 2014). Yet, in empirical macroeconomics and finance, many explanatory variables follow either near unit root or stochastic unit root processes (e.g., Phillips et al. 2011; Phillips and Yu 2011; Lieberman and Phillips 2017, 2020). This paper proposes estimating a threshold regression model with HLSUR regressors that has both local-to-unity and stochastic unit root components. The estimation of the threshold parameter is based on a concentrated least squares method. We develop estimations and inferences for the threshold estimator under the diminishing threshold effect assumption, including a sup-Wald statistic to test for the existence of the threshold effect. Finally, we assess the performance of the proposed estimator using Monte Carlo simulations.

There is a wide range of directions open for future work. First, the current paper assumes the threshold variable is strictly stationary. It may be interesting to extend it to be nonstationary. Second, our model relies on exogenous regressors and the threshold variable. Future works can relax this assumption by applying a control function approach. Moreover, a more general model with nonstationary regressors and multiple threshold variables may be interesting. These extensions are left for future studies.

Appendix B.1. Proof of Theorem 1

Define the projection matrix $P\left(\gamma \right)=X_{\gamma }^{*}$$(X_{\gamma }^{*}$${}^T$$X_{\gamma }^{*})$${}^{-1}$$X_{\gamma }^{*}$${}^T$, where $X_{\gamma }^{*}=\left[X_{\gamma },X-X_{\gamma }\right]$.

By simple calculation, we have

$$\begin{array}{cc}\hfill & SSR\left(\gamma \right)={\mathbf{y}}^T\left(I_n-P\left(\gamma \right)\right)\mathbf{y}\hfill \\ \hfill & ={\delta }_n^T{X}_{{\gamma }_0}^T\left(I_n-P\left(\gamma \right)\right)X_{{\gamma }_0}{\delta }_n+2{\delta }_n^T{X}_{{\gamma }_0}^T\left(I_n-P\left(\gamma \right)\right)\mu +{\mu }^T\left(I_n-P\left(\gamma \right)\right)\mu .\hfill \end{array}$$

(A13)

Hence, the centered process is as follows:

$$\begin{array}{cc}\hfill & n^{2\rho -1}\left(SSR\left(\gamma \right)-SSR\left({\gamma }_0\right)\right)=n^{-2}{\delta }_0^T{X}_{{\gamma }_0}^T\left(I_n-P\left(\gamma \right)\right)X_{{\gamma }_0}{\delta }_0+n^{\rho -3/2}2{\delta }_0^T{X}_{{\gamma }_0}^T\left(I_n-P\left(\gamma \right)\right)\mu \hfill \\ \hfill & +n^{2\rho -1}\left[{\mu }^T\left(I_n-P\left(\gamma \right)\right)\mu -{\mu }^T\left(I_n-P\left({\gamma }_0\right)\right)\mu \right]=A_{n1}+A_{n2}+A_{n3}.\hfill \end{array}$$

For all $\gamma \in ({\gamma }_0,\overline{\gamma }]$, closely following the same line of argument in the proof of Lemma A.5 of Chen (2014), and by applying Lemma A2, we can show that

$$\begin{array}{cc}\hfill & \left(1\right).A_{n1}⟹\left(F\left({\gamma }_0\right)-F\left({\gamma }_0\right)F^{-1}\left(\gamma \right)F\left({\gamma }_0\right)\right){\delta }_0^T{\int }_0^1{G}_{\alpha ,c}\left(s\right)G_{\alpha ,c}^T\left(s\right)ds,\hfill \\ \hfill & \left(2\right).A_{n2}=O_p\left(n^{\rho -1/2}\right),\hfill \\ \hfill & \left(3\right).A_{n3}=O_p\left(n^{2\rho -1}\right),\hfill \end{array}$$

where we use the fact that $I(q_t\le {\gamma }_0)I(q_t\le \gamma )=I(q_t\le {\gamma }_0)$.

Therefore, we have

$$\begin{array}{c}\hfill n^{2\rho -1}\left(SSR\left(\gamma \right)-SSR\left({\gamma }_0\right)\right)⟹\left(F\left({\gamma }_0\right)-F\left({\gamma }_0\right)F^{-1}\left(\gamma \right)F\left({\gamma }_0\right)\right){\delta }_0^T{\int }_0^1{G}_{\alpha ,c}\left(s\right)G_{\alpha ,c}^T\left(s\right)ds.\end{array}$$

Note that, for all $\gamma \in ({\gamma }_0,\overline{\gamma }]$, we have $F\left(\gamma \right)>F\left({\gamma }_0\right)$. Thus, we have $n^{2\rho -1}{S}_{n1}\left(SSR\left(\gamma \right)-SSR\left({\gamma }_0\right)\right)>0$ uniformly for all $\gamma \in ({\gamma }_0,\overline{\gamma }]$. Similarly, we can show that $n^{2\rho -1}{S}_{n1}\left(SSR\left(\gamma \right)-SSR\left({\gamma }_0\right)\right)<0$ if $\gamma \in [\underline{\gamma },{\gamma }_0)$, which completes the proof of this Theorem.

Appendix B.2. Proof of Theorem 2

First, closely following the proof of Lemma A.6 of Chen (2014), we can show $a_n\left(\widehat{\gamma }-{\gamma }_0\right)=\underset{v\in (-\infty ,+\infty )}{argmin}{Q}_n\left(v\right)=O_p\left(1\right)$, where $Q_n\left(v\right)=SSR({\gamma }_0+\frac{v}{a_n})-SSR\left({\gamma }_0\right)$.

Next, note that

$$\begin{array}{c}\hfill \mathbf{y}-X\left(\gamma \right)\widehat{\theta }=\mu +X\left(\beta -\widehat{\beta }\right)+X_{{\gamma }_0}({\delta }_n-{\widehat{\delta }}_n)-\left(X_{\gamma }-X_{{\gamma }_0}\right){\widehat{\delta }}_n.\end{array}$$

Therefore, we can show

$$\begin{array}{cc}\hfill & SSR\left(\gamma \right)-SSR\left({\gamma }_0\right)\hfill \\ \hfill & ={\widehat{\delta }}_n^T{\left(X_{\gamma }-X_{{\gamma }_0}\right)}^T\left(X_{\gamma }-X_{{\gamma }_0}\right){\widehat{\delta }}_n-2{\left[\mu +X\left(\beta -\widehat{\beta }\right)+X_{{\gamma }_0}\left({\delta }_n-{\widehat{\delta }}_n\right)\right]}^T\left[\left(X_{\gamma }-X_{{\gamma }_0}\right){\widehat{\delta }}_n\right]\hfill \\ \hfill & ={\delta }_n^T\sum _{t=1}^n{x}_t{x}_t^T\left|I(q_t\le \gamma )-I(q_t\le {\gamma }_0)\right|{\delta }_n\hfill \\ \hfill & -2{\widehat{\delta }}_n^T\sum _{t=1}^n{\mu }_t\left(x_{t,\gamma }-x_{t,{\gamma }_0}\right)\hfill \\ \hfill & -2{\widehat{\delta }}_n^T\sum _{t=1}^n\left[{\left(\beta -\widehat{\beta }\right)}^T{x}_t+{\left({\delta }_n-{\widehat{\delta }}_n\right)}^T{x}_{t,{\gamma }_0}\right]\left(x_{t,\gamma }-x_{t,{\gamma }_0}\right)\hfill \\ \hfill & +{\left({\widehat{\delta }}_n+{\delta }_n\right)}^T\sum _{t=1}^n{x}_t{x}_t^T\left|I(q_t\le \gamma )-I(q_t\le {\gamma }_0)\right|\left({\widehat{\delta }}_n-{\delta }_n\right)\hfill \\ \hfill & =S_{n1}\left(\gamma \right)-2S_{n2}\left(\gamma \right)-2S_{n3}\left(\gamma \right)+S_{n4}\left(\gamma \right).\hfill \end{array}$$

Now, we consider the limiting behavior of $S_{nj}\left(\gamma \right)$, for $j=1,2,3,4$ with $\rho \in (0,\frac{1}{2})$. For any $v\in [\underline{v},\overline{v}]$, a finite interval, by Lemmas A2–A4 and the Taylor expansion, we have

$$\begin{array}{cc}\hfill & S_{n1}({\gamma }_0+\frac{v}{a_n})=\hfill \\ \hfill & n^{1-2\rho }{\delta }_0^T\left|F({\gamma }_0+\frac{v}{a_n})-F\left({\gamma }_0\right)\right|{\int }_0^1{G}_{\alpha ,c}\left(s\right)G_{\alpha ,c}^T\left(s\right)ds{\delta }_0+o_p\left(1\right)\stackrel{p}{\to }\left|v\right|f_{{\gamma }_0}{\delta }_0^T{\int }_0^1{G}_{\alpha ,c}\left(s\right)G_{\alpha ,c}^T\left(s\right)ds{\delta }_0.\hfill \end{array}$$

For $S_{n2}$ and for $v>0$, by Lemmas A2–A3, we have

$$\begin{array}{cc}\hfill & S_{n2}({\gamma }_0+\frac{v}{a_n})\hfill \\ \hfill & ={\delta }_n^T\sum _{t=1}^n{\mu }_t{x}_t\left[I(q_t\le {\gamma }_0+\frac{v}{a_n})-I(q_t\le {\gamma }_0)\right]+{\left({\widehat{\delta }}_n-{\delta }_n\right)}^T\sum _{t=1}^n{\mu }_t{x}_t\left[I(q_t\le {\gamma }_0+\frac{v}{a_n})-I(q_t\le {\gamma }_0)\right]\hfill \\ \hfill & ⟹n^{1/2-\rho }{\delta }_0^T{\sigma }_{\mu }{\int }_0^1{G}_{\alpha ,c}\left(s\right)d\left\{W\left(s,F({\gamma }_0+\frac{v}{a_n})\right)-W\left(s,F\left({\gamma }_0\right)\right)\right\}\left[1+O_p\left(n^{-1/2}\right)\right].\hfill \end{array}$$

Let $\mathbf{H}\left(\eta \right)=\int G_{\alpha ,c}\left(s\right)dW(s,\eta )$. Note that $E\left[\mathbf{H}\left(\eta \right)\right]=0$, $Var\left(\mathbf{H}\left(\eta \right)\right)=\eta {\int }_0^1{G}_{\alpha ,c}\left(s\right)G_{\alpha ,c}^T\left(s\right)ds$, and $E\left(\mathbf{H}\left({\eta }_1\right)\mathbf{H}\left({\eta }_2\right)\right)=\left({\eta }_1\bigwedge {\eta }_2\right){\int }_0^1{G}_{\alpha ,c}\left(s\right)G_{\alpha ,c}^T\left(s\right)ds$. Thus, $ϵ\left(v\right)=\sqrt{a_n}{\delta }_0^T{\sigma }_{\mu }\left[\mathbf{H}({\gamma }_0+\frac{v}{a_n})-\mathbf{H}\left({\gamma }_0\right)\right]$ is a Brownian motion with

$$\begin{array}{c}\hfill Var\left(ϵ\left(v\right)\right)={\sigma }_{\mu }^{2}f\left({\gamma }_0\right)v{\delta }_0^T{\int }_0^1{G}_{\alpha ,c}\left(s\right)G_{\alpha ,c}^T\left(s\right)ds{\delta }_0+o_p\left(1\right),\end{array}$$

where we use the first-order Taylor expansion of $F({\gamma }_0+\frac{v}{a_n})$ around ${\gamma }_0$.

Next, we can show that, for all $v\in [\underline{v},\overline{v}]$, $S_{n3}\left(v\right)=o_p\left(S_{n1}\left(v\right)\right)$ and $S_{n4}\left(v\right)=o_p\left(S_{n1}\left(v\right)\right)$. Thus, combining the above results together, we have

$$\begin{array}{cc}\hfill & SSR({\gamma }_0+\frac{v}{a_n})-SSR\left({\gamma }_0\right)\hfill \\ \hfill & ⟹\left\{\begin{array}{cc}vf\left({\gamma }_0\right){\delta }_0^T{\int }_0^1{G}_{\alpha ,c}{G}_{\alpha ,c}^T\left(s\right)ds{\delta }_0-2{\sigma }_{\mu }\sqrt{f\left({\gamma }_0\right)v{\delta }_0^T{\int }_0^1{G}_{\alpha ,c}\left(s\right)G_{\alpha ,c}^T\left(s\right)ds{\delta }_0}{W}_1\left(v\right),\hfill & \mathrm{if}v>0\hfill \\ 0,\hfill & \mathrm{if}v=0\hfill \\ \left|v\right|f\left({\gamma }_0\right){\delta }_0^T{\int }_0^1{G}_{\alpha ,c}{G}_{\alpha ,c}^T\left(s\right)ds{\delta }_0-2{\sigma }_{\mu }\sqrt{f\left({\gamma }_0\right)\left|v\right|{\delta }_0^T{\int }_0^1{G}_{\alpha ,c}\left(s\right)G_{\alpha ,c}^T\left(s\right)ds{\delta }_0}{W}_2(-v),\hfill & \mathrm{if}v<0,\hfill \end{array}\right.\hfill \end{array}$$

where $W_1\left(v\right)$ and $W_2(-v)$ are two independent standard Brownian motion process defined on $[0.\infty )$.

Following the proof of Theorem 1 of Hansen (2000) and Lemma A.7 in Chen (2014), by making the change of variables, we complete the proof of (i).

Now, we consider the limiting behavior with $\rho =0$. Note that, for any positive finite v, ${\sum }_{t=1}^{n}I\left(\frac{v}{a_n}\right)\sim Binomial(n,p\left(v\right))$, where $p\left(v\right)\approx f\left({\gamma }_0\right)\frac{v}{n}$ and $np\left(v\right)$ is finite. Hence, ${\sum }_{t=1}^{n}I\left(\frac{v}{a_n}\right)⟹$$N_1\left(v\right)$ where $N\left(v\right)$ is a Poisson process with intensity $f\left({\gamma }_0\right)$. Similarly, for any negative finite v, we have ${\sum }_{t=1}^{n}I(-\frac{v}{a_n})⟹$$N_2\left(\right|v\left|\right)$, which completes the proof of (ii).

Appendix B.3. Proof of Theorem 3

Note that, given $\gamma \in [\underline{\gamma },\overline{\gamma }]$, $\widehat{\delta }={\left[X_{\gamma }^T\left(I_n-P_x\right)X_{\gamma }\right]}^{-1}{X}_{\gamma }^T\left(I_n-P_x\right)\mathbf{y}$. Hence, we have

$$\begin{array}{cc}\hfill & W_n\left(\gamma \right)={\widehat{\delta }}^T\left[X_{\gamma }^T\left(I_n-P_x\right)X_{\gamma }\right]\widehat{\delta }{\widehat{\sigma }}_{\mu }^{-2}\hfill \\ \hfill & ={\mathbf{y}}^T\left(I_n-P_x\right)X_{\gamma }{\left[X_{\gamma }^T\left(I_n-P_x\right)X_{\gamma }\right]}^{-1}{X}_{\gamma }^T\left(I_n-P_x\right)\mathbf{y},\hfill \end{array}$$

where, by Lemma A2, we can show

$$\begin{array}{cc}\hfill & \left(i\right).n^{-2}{X}_{\gamma }^T\left(I_n-P_x\right)⟹\left[F\left(\gamma \right)-F{\left(\gamma \right)}^2\right]{\int }_0^1{G}_{\alpha ,c}\left(s\right)G_{\alpha ,c}{\left(s\right)}^{T}ds,\hfill \\ \hfill & \left(ii\right).n^{-1}{X}_{\gamma }^T(I_n-P_x)\mathbf{y}=n^{-1}\left[X_{\gamma }^{\mu }-X_{\gamma }^T{P}_x\mu +X_{\gamma }^T{X}_{{\gamma }_0}{\delta }_n-X_{\gamma }^T{P}_x{X}_{{\gamma }_0}{\delta }_n\right]\hfill \\ \hfill & ⟹{\int }_0^1{G}_{\alpha ,c}\left(s\right)dW\left(s,F\left(\gamma \right)\right)-F\left(\gamma \right){\int }_0^1{G}_{\alpha ,c}\left(s\right)dW\left(s\right)+n^{1-\rho }F(\gamma \bigwedge {\gamma }_0){\int }_0^1{G}_{\alpha ,c}\left(s\right)G_{\alpha ,c}{\left(s\right)}^{T}ds{\delta }_0\hfill \\ \hfill & -n^{1-\rho }F\left(\gamma \right)F\left({\gamma }_0\right){\int }_0^1{G}_{\alpha ,c}\left(s\right)G_{\alpha ,c}{\left(s\right)}^{T}ds{\delta }_0.\hfill \end{array}$$

Thus, $W_n\left(\gamma \right)=O_p\left(n^{1-\rho }\right)$. Under the null of ${\delta }_0=0$, we can show

$$\begin{array}{cc}\hfill & W_n\left(\gamma \right)⟹{\left\{{\int }_0^1{G}_{\alpha ,c}\left(s\right)dW(s,F\left(\gamma \right))-F\left(\gamma \right){\int }_0^1{G}_{\alpha ,c}\left(s\right)dW\left(s\right)\right\}}^T\hfill \\ \hfill & \times {\left\{\left[F\left(\gamma \right)-F{\left(\gamma \right)}^2\right]{\int }_0^1{G}_{\alpha ,c}\left(s\right)G_{\alpha ,c}{\left(s\right)}^{T}ds\right\}}^{-1}\hfill \\ \hfill & \times \left\{{\int }_0^1{G}_{\alpha ,c}\left(s\right)dW(s,F\left(\gamma \right))-F\left(\gamma \right){\int }_0^1{G}_{\alpha ,c}\left(s\right)dW\left(s\right)\right\},\hfill \end{array}$$

which completes the proof of this Theorem.