Residual-based tests for cointegration with three-regime TAR adjustment

Abstract

This paper proposes residual-based tests for cointegration with three-regime threshold autoregressive (TAR) adjustment. We propose Wald-type and $t$-type tests that have the null hypothesis of linear no cointegration and the alternative of cointegration with three-regime TAR adjustment and also derive the asymptotic distributions. Monte Carlo simulations show that the proposed tests perform better than the Engle–Granger cointegration test and the cointegration test in a two-regime TAR model introduced by Enders and Siklos (J Bus Econ Stat 19:166–176, 2001), under cointegration with three-regime TAR adjustment, particularly when the threshold and sample size increase. When we apply these tests to the money demand of the U.S., the proposed tests reject the null of no cointegration whereas other tests do not.

Appendices

Appendix 1

Proof of Theorem 1

From (21), we obtain that $T^{-1/2}\ell _{11}^{-1}\hat{u}_t \Rightarrow W^{*}(r)$. We have

$$\begin{aligned} \begin{aligned} \mathbf {1} \{ \hat{u}_{t-1} \le \lambda _1 \}&=\mathbf {1} \{ T^{-1/2}\hat{\eta }^{\prime } z_{t-1} \le T^{-1/2} \lambda _1 \} \\&\Rightarrow \mathbf {1} \{ \eta ^{\prime }B(r) \le \bar{\lambda }_{1} \} = \mathbf {1} \{ W^{*}(r) \le \tilde{\lambda }_1 \}, \end{aligned} \end{aligned}$$

(38)

where $\bar{\lambda }_1=\lim _{T \rightarrow \infty } T^{-1/2}\lambda _1$ and $\tilde{\lambda }_1=\ell _{11}^{-1} \bar{\lambda }_{1}$. From $\epsilon _t=\sum _{j=0}^{\infty }D_j\xi _{t-j}^{\prime }\eta =D(L)\xi _t^{\prime }\eta$ and Lemma 2.1 of Phillips and Ouliaris (1990), we have

$$\begin{aligned} T^{-1/2}\sum _{t=1}^{[Tr]} \epsilon _t \Rightarrow D(1)\eta ^{\prime }B(r) , \end{aligned}$$

(39)

where $D(1)=\sum _{j=0}^{\infty }D_j$. (38), (39), and the continuous mapping theorem (CMT) and Theorem 2.2 of Kurtz and Protter (1991) yield

$$\begin{aligned} \begin{aligned}&T^{-1} \sum _{t=1}^T \hat{u}_{t-1} \mathbf {1} \{ \hat{u}_{t-1}\le \lambda _1 \} \epsilon _t\\&\quad =T^{-1/2}\hat{\eta }^{\prime }\sum _{t=1}^T T^{-1/2} z_{t-1} \mathbf {1} \{ T^{-1/2}\hat{\eta }^{\prime } z_{t-1} \le T^{-1/2} \lambda _1 \}D(L)\xi _t^{\prime }\hat{\eta } \\&\quad \Rightarrow D(1) \eta ^{\prime } \int \limits _0^1 B \mathrm{d}B^{\prime }\eta \mathbf {1} \{\eta ^{\prime } B \le \bar{\lambda }_{1} \} \\&\quad = D(1) \ell _{11}^2 \int \limits _0^1 \mathbf {1} \{W^{*} \le \tilde{\lambda }_1 \} W^{*}\mathrm{d}W^{*} \end{aligned} \end{aligned}$$

(40)

and

$$\begin{aligned} \begin{aligned}&T^{-2} \sum _{t=1}^T \hat{u}_{t-1}^2 \mathbf {1} \{ \hat{u}_{t-1} \le \lambda _1 \}\\&\quad = T^{-1} \sum _{t=1}^{T} (T^{-1/2}z_{t-1}^{\prime }\eta )^{\prime }(T^{-1/2}z_{t-1}^{\prime }\eta ) \mathbf {1} \{ T^{-1/2}\hat{\eta }^{\prime } z_{t-1} \le T^{-1/2} \lambda _1 \} \\&\quad \Rightarrow \eta ^{\prime } \int \limits _0^1 B B^{\prime }\eta \mathbf {1} \{\eta ^{\prime } B \le \bar{\lambda }_{1} \} \\&\quad = \ell _{11}^2 \int \limits _0^1 \mathbf {1} \{W^{*} \le \tilde{\lambda }_1 \} W^{*2}. \end{aligned} \end{aligned}$$

(41)

A similar type of analysis can be applied to the term for $\mathbf {1} \{ \hat{u}_{t-1} > \lambda _2 \}$. Therefore, it can be also shown that

$$\begin{aligned} T^{-1} \sum _{t=1}^T \hat{u}_{t-1}\mathbf {1} \{ \hat{u}_{t-1} > \lambda _2 \} \epsilon _t \Rightarrow D(1)\ell _{11}^2 \int \limits _0^1 \mathbf {1} \{W^{*} > \tilde{\lambda }_2 \} W^{*}\mathrm{d}W^{*} \end{aligned}$$

(42)

and

$$\begin{aligned} T^{-2} \sum _{t=1}^T \hat{u}_{t-1}^2 \mathbf {1} \{ \hat{u}_{t-1} > \lambda _2 \} \Rightarrow \ell _{11}^2 \int \limits _0^1 \mathbf {1} \{W^{*} > \tilde{\lambda }_2 \} W^{*2}. \end{aligned}$$

(43)

Proof of Theorem 2

Part of (2a) The statistic $W_T(\lambda )$ can be written as

$$\begin{aligned} W_T(\lambda )=\frac{1}{\hat{\sigma }^2}\hat{\rho }^{\prime }(\mathbf {U}^{\prime } \mathbf {Q}_p \mathbf {U})\hat{\rho }, \end{aligned}$$

(44)

where

$$\begin{aligned} \mathbf {U}= \begin{pmatrix} \hat{u}_0 \mathbf {1}\{\hat{u}_0 \le \lambda _1 \} &{}\quad \hat{u}_0 \mathbf {1}\{\hat{u}_0 > \lambda _2 \} \\ \hat{u}_1 \mathbf {1}\{\hat{u}_1 \le \lambda _1 \} &{}\quad \hat{u}_1 \mathbf {1}\{\hat{u}_1 > \lambda _2 \} \\ \vdots &{} \vdots \\ \hat{u}_{T-1} \mathbf {1}\{\hat{u}_{T-1} \le \lambda _1 \} &{} \hat{u}_{T-1} \mathbf {1}\{\hat{u}_{T-1} > \lambda _2 \} \end{pmatrix} \end{aligned}$$

and $\mathbf {Q}_p=\mathbf {I}-\mathbf {M}_p(\mathbf {M}_p^{\prime }\mathbf {M}_p)^{-1} \mathbf {M}_p^{\prime }$ is a $T \times T$ idempotent matrix. $\mathbf {M}_p$ is the matrix of observations on $\Delta \hat{u}_{t-p}^p$. Since under $H_0$ with $\rho _1=\rho _2=0$, $\hat{\rho }$ is given by $\hat{\rho }=(\mathbf {U}^{\prime } \mathbf {Q}_p \mathbf {U})^{-1} \mathbf {U}^{\prime } \mathbf {Q}_p \varvec{\epsilon }$, where $\varvec{\epsilon }=(\epsilon _1,\ldots , \epsilon _T)^{\prime }$, (44) is expressed by

$$\begin{aligned} W_T(\lambda ) =\frac{1}{\hat{\sigma }^2} \varvec{\epsilon }^{\prime }\mathbf {Q}_p \mathbf {U} (\mathbf {U}^{\prime } \mathbf {Q}_p \mathbf {U})^{-1} \mathbf {U}^{\prime }\mathbf {Q}_p \varvec{\epsilon }. \end{aligned}$$

(45)

Let $\mathbf {U}_1=(\hat{u}_0 \mathbf {1}\{\hat{u}_0 \le \lambda _1 \}, \ldots , \hat{u}_{T-1} \mathbf {1}\{\hat{u}_{T-1} \le \lambda _1 \} )^{\prime }$ and $\mathbf {U}_2=(\hat{u}_0 \mathbf {1}\{\hat{u}_0 > \lambda _2 \}, \ldots , \hat{u}_{T-1} \mathbf {1}\{\hat{u}_{T-1} > \lambda _2 \} )^{\prime }$. From the orthogonality between $\mathbf {1} \{ \hat{u}_{t-1} \le \lambda _1 \}$ and $\mathbf {1} \{ \hat{u}_{t-1}>\lambda _2 \}$, we can decompose (45) as

$$\begin{aligned} W_T(\lambda )&= \frac{1}{\hat{\sigma }^2} (\varvec{\epsilon }^{\prime } \mathbf {Q}_p \mathbf {U}_1, \varvec{\epsilon }^{\prime } \mathbf {Q}_p \mathbf {U}_2) \begin{pmatrix} \mathbf {U}_1^{\prime } \mathbf {Q}_p \mathbf {U}_1 &{} 0 \\ 0 &{} \mathbf {U}_2^{\prime } \mathbf {Q}_p \mathbf {U}_2 \end{pmatrix}^{-1} \begin{pmatrix} \mathbf {U}_1^{\prime } \mathbf {Q}_p \varvec{\epsilon } \\ \mathbf {U}_2^{\prime } \mathbf {Q}_p \varvec{\epsilon } \\ \end{pmatrix}\nonumber \\&= \frac{1}{\hat{\sigma }^2} \{ \varvec{\epsilon }^{\prime } \mathbf {Q}_p \mathbf {U}_1(\mathbf {U}_1^{\prime } \mathbf {Q}_p \mathbf {U}_1)^{-1} \mathbf {U}_1^{\prime } \mathbf {Q}_p \varvec{\epsilon } + \varvec{\epsilon }^{\prime } \mathbf {Q}_p \mathbf {U}_2(\mathbf {U}_2^{\prime } \mathbf {Q}_p \mathbf {U}_2)^{-1} \mathbf {U}_2^{\prime } \mathbf {Q}_p \varvec{\epsilon } \}.\qquad \quad \end{aligned}$$

(46)

Consider $\varvec{\epsilon }^{\prime } \mathbf {Q}_p \mathbf {U}_1(\mathbf {U}_1^{\prime } \mathbf {Q}_p \mathbf {U}_1)^{-1} \mathbf {U}_1^{\prime } \mathbf {Q}_p \varvec{\epsilon }$. It follows from CMT that $T^{-1}\mathbf {U}_1^{\prime }\mathbf {M}_p=O_p(1)$, $T^{-1}\mathbf {M}_p^{\prime }\mathbf {M}_p=O_p(1)$, and $T^{-1/2}\mathbf {M}_p^{\prime }\varvec{\epsilon }=O_p(1)$. Combining these results and Theorem 1, we have

$$\begin{aligned} T^{-1} \mathbf {U}_1^{\prime } \mathbf {Q}_p \varvec{\epsilon }&= T^{-1} \mathbf {U}_1^{\prime }\varvec{\epsilon }-T^{-1/2}\cdot T^{-1} \mathbf {U}_1^{\prime }\mathbf {M}_p(T^{-1}\mathbf {M}_p^{\prime }\mathbf {M}_p)^{-1}T^{-1/2}\mathbf {M}_p^{\prime }\varvec{\epsilon } \nonumber \\&= T^{-1} \mathbf {U}_1^{\prime }\varvec{\epsilon }+o_p(1) \Rightarrow D(1) \ell _{11}^2 \int \limits _0^1 \mathbf {1}\{ W^{*} \le \tilde{\lambda }_1 \} W^{*}\mathrm{d}W^{*} \end{aligned}$$

(47)

and

$$\begin{aligned} T^{-2} \mathbf {U}_1^{\prime } \mathbf {Q}_p \mathbf {U}_1&= T^{-2} \mathbf {U}_1^{\prime }\mathbf {U}_1 -T^{-1}\cdot T^{-1}\mathbf {U}_1^{\prime }\mathbf {M}_p(T^{-1}\mathbf {M}_p^{\prime }\mathbf {M}_p)^{-1} T^{-1}\mathbf {M}_p^{\prime }\mathbf {U}_1 \nonumber \\&= T^{-2} \mathbf {U}_1^{\prime }\mathbf {U}_1+o_p(1) \Rightarrow \ell _{11}^2 \int \limits _0^1 \mathbf {1}\{ W^{*} \le \tilde{\lambda }_1 \} W^{*2}. \end{aligned}$$

(48)

Similarly, it can be shown that

$$\begin{aligned} T^{-1} \mathbf {U}_2^{\prime } \mathbf {Q}_p \varvec{\epsilon } = T^{-1} \mathbf {U}_2^{\prime }\varvec{\epsilon }+o_p(1) \Rightarrow D(1) \ell _{11}^2 \int \limits _0^1 \mathbf {1}\{ W^{*} > \tilde{\lambda }_2 \} W^{*}\mathrm{d}W^{*} \end{aligned}$$

(49)

and

$$\begin{aligned} T^{-2} \mathbf {U}_2^{\prime } \mathbf {Q}_p \mathbf {U}_2 = T^{-2} \mathbf {U}_2^{\prime }\mathbf {U}_2+o_p(1) \Rightarrow \ell _{11}^2 \int \limits _0^1 \mathbf {1}\{ W^{*} > \tilde{\lambda }_2 \} W^{*2}. \end{aligned}$$

(50)

Next, we consider the variance $\hat{\sigma }^2$. Note that $\hat{\rho }_1=O_p(T^{-1}),\, \hat{\rho }_2=O_p(T^{-1})$, and $(\hat{\alpha }_j-\alpha _j)=O_p(T^{-1/2})$. Then, using Lemma 2.2 of Phillips and Ouliaris (1990), we obtain

$$\begin{aligned} \hat{\sigma }^2&= T^{-1} \sum _{t=1}^T \left( \Delta \hat{u}_t-\hat{\rho }_1 \hat{u}_{t-1}\mathbf {1}\{\hat{u}_{t-1} \le \lambda _1 \} \right. \nonumber \\&\qquad \left. -\hat{\rho }_2 \hat{u}_{t-1}\mathbf {1}\{\hat{u}_{t-1} > \lambda _2 \} -\sum \limits _{j=1}^p (\hat{\alpha }_j-\alpha _j) \Delta \hat{u}_{t-j} \right) ^2 \nonumber \\&= T^{-1} \sum _{t=1}^T \epsilon _t^2+o_p(1) = T^{-1} \sum _{t=1}^T D(L)^2 \hat{\eta }^{\prime } \xi _t^{\prime } \xi _t \hat{\eta }^{\prime } \nonumber \\&\Rightarrow D(1)^2 \eta ^{\prime }\varOmega \eta =D(1)^2 \ell _{11}^2 k^{\prime }k. \end{aligned}$$

(51)

Therefore, by (47)–(51),

$$\begin{aligned} W_T(\lambda )&= \frac{1}{\hat{\sigma }^2} \{ T^{-1}\varvec{\epsilon }^{\prime } \mathbf {U}_1(T^{-2} \mathbf {U}_1^{\prime } \mathbf {U}_1)^{-1} T^{-1}\mathbf {U}_1^{\prime } \varvec{\epsilon } \nonumber \\&+ T^{-1}\varvec{\epsilon }^{\prime } \mathbf {U}_2(T^{-2} \mathbf {U}_2^{\prime } \mathbf {U}_2)^{-1} T^{-1}\mathbf {U}_2^{\prime } \varvec{\epsilon } \}+o_p(1)\nonumber \\&\Rightarrow \frac{1}{D(1)^2 \ell _{11}^2k^{\prime }k} \Bigg [ \frac{\Big ( D(1) \ell _{11}^2 \int _0^1 \mathbf {1}\{W^{*} \le \tilde{\lambda }_1 \}W^{*}\mathrm{d}W^{*} \Big )^2}{ \ell _{11}^2 \int _0^1 \mathbf {1}\{W^{*} \le \tilde{\lambda }_1 \} W^{*2} }\nonumber \\&+\frac{\Big ( D(1) \ell _{11}^2 \int _0^1 \mathbf {1}\{W^{*} > \tilde{\lambda }_2 \}W^{*}\mathrm{d}W^{*} \Big )^2}{ \ell _{11}^2 \int _0^1 \mathbf {1}\{W^{*} > \tilde{\lambda }_2 \} W^{*2} } \Bigg ] \nonumber \\&= \frac{\Big ( \int _0^1 \mathbf {1}\{W^{*} \le \tilde{\lambda }_1 \}W^{*}\mathrm{d}W^{*} \Big )^2}{ (k^{\prime }k) \int _0^1 \mathbf {1}\{W^{*} \le \tilde{\lambda }_1 \} W^{*2} } +\frac{\Big ( \int _0^1 \mathbf {1}\{W^{*} > \tilde{\lambda }_2 \}W^{*}\mathrm{d}W^{*} \Big )^2}{ (k^{\prime }k) \int _0^1 \mathbf {1}\{W^{*} > \tilde{\lambda }_2 \} W^{*2} }. \end{aligned}$$

(52)

We can deduce the required results from the asymptotic distribution of $W_T(\lambda )$ and Assumption 2.

Under the alternative hypothesis, $W_T(\lambda )$ can be written as

$$\begin{aligned} W_T(\lambda )=\frac{1}{\hat{\sigma }^2} \rho ^{\prime }(\mathbf {U}^{\prime } \mathbf {Q}_p \mathbf {U})\rho +\frac{1}{\hat{\sigma }^2} \varvec{\epsilon }^{\prime }\mathbf {Q}_p \mathbf {U} (\mathbf {U}^{\prime } \mathbf {Q}_p \mathbf {U})^{-1} \mathbf {U}^{\prime }\mathbf {Q}_p \varvec{\epsilon }. \end{aligned}$$

(53)

Note that under the alternative hypothesis, $T^{-1}\mathbf {U}^{\prime }\mathbf {Q}_p \mathbf {U}=O_p(1)$, $T^{-1/2}\mathbf {U}^{\prime } \mathbf {Q}_p \varvec{\epsilon }=O_p(1)$, and $\hat{\sigma }^2=O_p(1)$. We can see that

$$\begin{aligned} W_T(\lambda )&= T \frac{1}{\hat{\sigma }^2} \rho ^{\prime }(T^{-1}\mathbf {U}^{\prime } \mathbf {Q}_p \mathbf {U})\rho +\frac{1}{\hat{\sigma }^2} T^{-1/2} \varvec{\epsilon }^{\prime }\mathbf {Q}_p \mathbf {U} (T^{-1}\mathbf {U}^{\prime } \mathbf {Q}_p \mathbf {U})^{-1} T^{-1/2} \mathbf {U}^{\prime }\mathbf {Q}_p \varvec{\epsilon } \nonumber \\&= T \frac{1}{\hat{\sigma }^2} \rho ^{\prime }(T^{-1}\mathbf {U}^{\prime } \mathbf {Q}_p \mathbf {U})\rho +O_p(1) \nonumber \\&= O_p(T). \end{aligned}$$

(54)

Therefore, $W_T(\lambda )$ diverges to infinity as $T \rightarrow \infty$. This also implies that the test statistic diverges to infinity as $T \rightarrow \infty$.

Part of (2b) From the proof of part (2a), the statistic $t_T(\lambda )_{\max }$ can be written as

$$\begin{aligned} t_T(\lambda )_{\max }=\max \Bigg [\frac{\hat{\rho }_1}{\{ \hat{\sigma }^2 (\mathbf {U}_1^{\prime } \mathbf {Q}_p \mathbf {U}_1)^{-1} \}^{1/2}}, \frac{\hat{\rho }_2}{\{ \hat{\sigma }^2 (\mathbf {U}_2^{\prime } \mathbf {Q}_p \mathbf {U}_2)^{-1} \}^{1/2}} \Bigg ]. \end{aligned}$$

(55)

Since $\hat{\rho }_i\,(i=1,2)$ is given by $\hat{\rho }_i=(\mathbf {U}_i^{\prime } \mathbf {Q}_p \mathbf {U}_i)^{-1} \mathbf {U}_i^{\prime } \mathbf {Q}_p \varvec{\epsilon }$ under $H_0$ with $\rho _1=\rho _2=0,\, t_T(\lambda )_{\max }$ is expressed as

$$\begin{aligned} t_T(\lambda )_{\max }=\max \Bigg [\frac{ \mathbf {U}_1^{\prime } \mathbf {Q}_p \varvec{\epsilon } }{\{ \hat{\sigma }^2 (\mathbf {U}_1^{\prime } \mathbf {Q}_p \mathbf {U}_1) \}^{1/2}}, \frac{ \mathbf {U}_2^{\prime } \mathbf {Q}_p \varvec{\epsilon } }{\{ \hat{\sigma }^2 (\mathbf {U}_2^{\prime } \mathbf {Q}_p \mathbf {U}_2) \}^{1/2}} \Bigg ]. \end{aligned}$$

(56)

Using from (46) to (50), we have

$$\begin{aligned} t (\lambda )_{\max }&\Rightarrow \max \Bigg [ \frac{ D(1)\ell _{11}^2 \int _0^1 \mathbf {1} \{W^{*} \le \tilde{\lambda }_1 \}W^{*}\mathrm{d}W^{*} }{ \big (D(1)^2 \ell _{11}^2 k^{\prime }k \ell _{11}^2 \int _0^1 \mathbf {1} \{W^{*} \le \tilde{\lambda }_1 \} W^{*2} \big )^{1/2} },\nonumber \\&\qquad \qquad \frac{ D(1)\ell _{11}^2 \int _0^1 \mathbf {1} \{W^{*} > \tilde{\lambda }_2 \}W^{*}\mathrm{d}W^{*} }{ \big (D(1)^2 \ell _{11}^2 k^{\prime }k \ell _{11}^2 \int _0^1 \mathbf {1} \{W^{*} > \tilde{\lambda }_2 \} W^{*2} \big )^{1/2} } \Bigg ]\nonumber \\&= \max \Bigg [ \frac{\int _0^1 \mathbf {1} \{W^{*} \le \tilde{\lambda }_1 \}W^{*}\mathrm{d}W^{*} }{ \big ( k^{\prime }k \int _0^1 \mathbf {1} \{W^{*} \le \tilde{\lambda }_1 \} W^{*2} \big )^{1/2} }, \frac{ \int _0^1 \mathbf {1} \{W^{*} > \tilde{\lambda }_2 \}W^{*}\mathrm{d}W^{*} }{ \big ( k^{\prime }k \int _0^1 \mathbf {1} \{W^{*} > \tilde{\lambda }_2 \} W^{*2} \big )^{1/2} } \Bigg ].\nonumber \\ \end{aligned}$$

(57)

We can deduce the required results from the asymptotic distribution of $t_T(\lambda )_{\max }$ and Assumption 2. Next, we consider the properties of the test statistic under the alternative hypothesis. Under the alternative hypothesis, $t_T(\lambda )_{\max }$ can be written as

$$\begin{aligned} t_T(\lambda )_{\max }&= \max \Bigg [\frac{\rho _1}{\{ \hat{\sigma }^2 (\mathbf {U}_1^{\prime } \mathbf {Q}_p \mathbf {U}_1)^{-1} \}^{1/2}} + \frac{1}{\hat{\sigma }} \mathbf {U}_1^{\prime }\mathbf {Q}_p \varvec{\epsilon } (\mathbf {U}_1^{\prime } \mathbf {Q}_p \mathbf {U}_1)^{-1/2},\nonumber \\&\qquad \quad \frac{\rho _2}{\{ \hat{\sigma }^2 (\mathbf {U}_2^{\prime } \mathbf {Q}_p \mathbf {U}_2)^{-1} \}^{1/2}} + \frac{1}{\hat{\sigma }} \mathbf {U}_2^{\prime }\mathbf {Q}_p \varvec{\epsilon } (\mathbf {U}_2^{\prime } \mathbf {Q}_p \mathbf {U}_2)^{-1/2} \Bigg ]. \end{aligned}$$

(58)

Then, we have

$$\begin{aligned} t_T(\lambda )_{\max }&= \max \Bigg [ \frac{T^{1/2} \rho _1}{\{ \hat{\sigma }^2 (T^{-1} \mathbf {U}_1^{\prime } \mathbf {Q}_p \mathbf {U}_1)^{-1} \}^{1/2}} + \frac{1}{\hat{\sigma }}T^{-1/2} \mathbf {U}_1^{\prime }\mathbf {Q}_p \varvec{\epsilon } (T^{-1}\mathbf {U}_1^{\prime } \mathbf {Q}_p \mathbf {U}_1)^{-1/2}, \nonumber \\&\frac{T^{1/2} \rho _2}{\{ \hat{\sigma }^2 (T^{-1} \mathbf {U}_2^{\prime } \mathbf {Q}_p \mathbf {U}_2)^{-1} \}^{1/2}} + \frac{1}{\hat{\sigma }}T^{-1/2} \mathbf {U}_2^{\prime }\mathbf {Q}_p \varvec{\epsilon } (T^{-1}\mathbf {U}_2^{\prime } \mathbf {Q}_p \mathbf {U}_2)^{-1/2} \Bigg ]\nonumber \\&= \max [ O_p(T^{1/2}), O_p(T^{1/2}) ]. \end{aligned}$$

(59)

from $T^{-1} \mathbf {U}_1^{\prime } \mathbf {Q}_p \mathbf {U}_1=O_p(1),\, T^{-1/2} \mathbf {U}_1^{\prime }\mathbf {Q}_p \varvec{\epsilon }=O_p(1),\, T^{-1} \mathbf {U}_2^{\prime } \mathbf {Q}_p \mathbf {U}_2=O_p(1), T^{-1/2} \mathbf {U}_2^{\prime }\mathbf {Q}_p \varvec{\epsilon }=O_p(1)$, and $\hat{\sigma }^2=O_p(1)$. Therefore, $t_T(\lambda )_{\max }$ diverges to minus infinity as $T \rightarrow \infty$. This also implies that the test statistic diverges to minus infinity under the alternative hypothesis as $T \rightarrow \infty$.

Proof of Theorem 3

Part of (3a). Under $H_0,\, \hat{\rho }$ is given by $\hat{\rho }=(\mathbf {U}^{\prime } \mathbf {S}_p \mathbf {U})^{-1} \mathbf {U}^{\prime } \mathbf {S}_p \varvec{\epsilon }$, where $\mathbf {S}_p=\mathbf {I}-\mathbf {N}_p(\mathbf {N}_p^{\prime }\mathbf {N}_p)^{-1}\mathbf {N}_p^{\prime }$ is a $T \times T$ idempotent matrix and

$$\begin{aligned} \mathbf {N}_p= \begin{pmatrix} \mathbf {1}\{\hat{u}_0 \le \lambda _1 \} &{}\quad \mathbf {1}\{\hat{u}_0 > \lambda _2 \} &{}\quad \Delta \hat{u}_{1-p}^p \\ \mathbf {1}\{\hat{u}_1 \le \lambda _1 \} &{}\quad \mathbf {1}\{\hat{u}_1 > \lambda _2 \} &{}\quad \Delta \hat{u}_{2-p}^p \\ \vdots &{}\quad \vdots &{}\quad \vdots \\ \mathbf {1}\{\hat{u}_{T-1} \le \lambda _1 \} &{}\quad \mathbf {1}\{\hat{u}_{T-1} > \lambda _2 \}&{}\quad \Delta \hat{u}_{T-p}^p \end{pmatrix}. \end{aligned}$$

Similar to (46), we can rewrite $W_T^B(\lambda )$ as

$$\begin{aligned} W_T^B(\lambda ) = \frac{1}{\hat{\sigma }^2} \{ \varvec{\epsilon }^{\prime } \mathbf {S}_p \mathbf {U}_1(\mathbf {U}_1^{\prime } \mathbf {S}_p \mathbf {U}_1)^{-1} \mathbf {U}_1^{\prime } \mathbf {S}_p \varvec{\epsilon } + \varvec{\epsilon }^{\prime } \mathbf {S}_p \mathbf {U}_2(\mathbf {U}_2^{\prime } S_p \mathbf {U}_2)^{-1} \mathbf {U}_2^{\prime } \mathbf {S}_p \varvec{\epsilon } \}. \end{aligned}$$

(60)

Note that

$$\begin{aligned} T^{-1} \mathbf {U}_1^{\prime } \mathbf {S}_p \varvec{\epsilon } = T^{-1} \mathbf {U}_1^{\prime }\varvec{\epsilon } -T^{-3/2}\mathbf {U}_1^{\prime }\mathbf {N}_p(T^{-1}\mathbf {N}_p^{\prime }\mathbf {N}_p)^{-1} T^{-1/2}\mathbf {N}_p^{\prime }\varvec{\epsilon }. \end{aligned}$$

(61)

Consider $(T^{-1} N_p N_p)^{-1}$. Since we obtain $T^{-1}\sum _{t=1}^T \mathbf {1}\{\hat{u}_{t-1} \le \lambda _1 \} \Rightarrow \int _0^1 \mathbf {1} \{ W^*\le \tilde{\lambda }_1 \} \mathrm{d}r$, it can be shown that

$$\begin{aligned} T^{-1}\mathbf {N}_p^{\prime }\mathbf {N}_p \Rightarrow \begin{pmatrix} \int _0^1 I_1 \mathrm{d}r &{}\quad 0 &{}\quad \mathbf {0} \\ 0 &{}\quad \int _0^1 I_2 \mathrm{d}r &{}\quad \mathbf {0} \\ \mathbf {0} &{}\quad \mathbf {0} &{}\quad \varGamma \end{pmatrix}, \end{aligned}$$

where $I_1=\mathbf {1} \{ W^*\le \tilde{\lambda }_1 \}$, $I_2=\mathbf {1} \{ W^*> \tilde{\lambda }_2 \}$, and $\varGamma =\lim _{T \rightarrow \infty } T^{-1}\sum E(\Delta \hat{u}_{t-p}^{p\prime } \Delta \hat{u}_{t-p}^p)$. Therefore, we have

$$\begin{aligned} (T^{-1}\mathbf {N}_p^{\prime }\mathbf {N}_p )^{-1} \Rightarrow \begin{pmatrix} \left( \int _0^1 I_1 \mathrm{d}r\right) ^{-1} &{}\quad 0 &{}\quad \mathbf {0} \\ 0 &{}\quad \left( \int _0^1 I_2 \mathrm{d}r\right) ^{-1} &{}\quad \mathbf {0} \\ \mathbf {0} &{} \quad \mathbf {0} &{}\quad \varGamma ^{-1}\\ \end{pmatrix}. \end{aligned}$$

(62)

We also have

$$\begin{aligned} \begin{aligned} T^{-3/2}\mathbf {U}_1^{\prime }\mathbf {N}_p&= T^{-3/2} \left( \sum \hat{u}_{t-1}\mathbf {1} \{ \hat{u}_{t-1} \le \lambda _1 \}, 0, \sum \hat{u}_{t-1}\mathbf {1} \{ \hat{u}_{t-1} \le \lambda _1 \}\Delta \hat{u}_{t-p}^p \right) \\&\Rightarrow \left( \ell _{11} \int \limits _0^1 W^{*}I_1, 0, 0\right) , \end{aligned} \end{aligned}$$

(63)

and

$$\begin{aligned} T^{-1/2}\mathbf {N}_p^{\prime }\varvec{\epsilon }&= T^{-1/2} \left( \sum \mathbf {1} \{ \hat{u}_{t-1} \le \lambda _1 \}\epsilon _t, \sum \mathbf {1} \{ \hat{u}_{t-1} > \lambda _2 \}\epsilon _t, \sum \Delta \hat{u}_{t-p}^p \epsilon _t \right) ^{\prime } \nonumber \\&\Rightarrow (D(1) \ell _{11}\int \limits _0^1 I_1 \mathrm{d}W^{*}, D(1) \ell _{11}\int \limits _0^1 I_2 \mathrm{d}W^{*}, O_p(1))^{\prime }. \end{aligned}$$

(64)

It follows from Theorem 1, (62), (63), and (64) that

$$\begin{aligned} T^{-1}\mathbf {U}_1^{\prime } \mathbf {S}_p \varvec{\epsilon }&\Rightarrow D(1)\ell _{11}^2 \int \limits _0^1 I_1 W^{*} \mathrm{d}W^{*} - D(1)\ell _{11}^2 \int \limits _0^1 W^{*}I_1 \left( \int \limits _0^1 I_1\right) ^{-1} \int \limits _0^1 I_1 \mathrm{d}W^{*} \nonumber \\&= D(1)\ell _{11}^2 \left\{ \int \limits _0^1 I_1 \int _0^1 I_1 W^{*} \mathrm{d}W^{*} - \int \limits _0^1 W^{*}I_1 \int \limits _0^1 I_1 \mathrm{d}W^{*} \right\} . \end{aligned}$$

(65)

Next, consider $\mathbf {U}_1^{\prime } \mathbf {S}_p \mathbf {U}_1$. It is easily seen that

$$\begin{aligned} T^{-2} \mathbf {U}_1^{\prime } \mathbf {S}_p \mathbf {U}_1 = T^{-2} \mathbf {U}_1^{\prime }\mathbf {U}_1 -T^{-3/2}\mathbf {U}_1^{\prime }\mathbf {N}_p(T^{-1}\mathbf {N}_p^{\prime }\mathbf {N}_p)^{-1} T^{-3/2}\mathbf {N}_p^{\prime }\mathbf {U}_1. \end{aligned}$$

(66)

Using (62) and (63), we obtain

$$\begin{aligned} -T^{-3/2}\mathbf {U}_1^{\prime }\mathbf {N}_p(T^{-1}\mathbf {N}_p^{\prime }\mathbf {N}_p)^{-1}T^{-3/2}\mathbf {N}_p^{\prime }\mathbf {U}_1 \Rightarrow \ell _{11}^2 \left( \int \limits _0^1 W^{*} I_1\right) ^2 \left( \int \limits _0^1 I_1\right) ^{-1}.\qquad \quad \end{aligned}$$

(67)

Therefore, we have

$$\begin{aligned} T^{-2}\mathbf {U}_1^{\prime }\mathbf {S}_p\mathbf {U}_1&\Rightarrow \ell _{11}^2 \int \limits _0^1 W^{*2} I_1-\ell _{11}^2 \left( \int \limits _0^1 W^{*}I_1\right) ^2\left( \int \limits _0^1 I_1\right) ^{-1} \nonumber \\&= \ell _{11}^2 \left\{ \int \limits _0^1 I_1 \int \limits _0^1 W^{*2}I_1-\left( \int \limits _0^1 W^{*}I_1\right) ^2 \right\} . \end{aligned}$$

(68)

Since similar analysis can be applied to $\varvec{\epsilon }^{\prime } \mathbf {S}_p \mathbf {U}_2(\mathbf {U}_2^{\prime } S_p \mathbf {U}_2)^{-1} \mathbf {U}_2^{\prime } \mathbf {S}_p \varvec{\epsilon }$, it can also be shown that

$$\begin{aligned} T^{-1}\mathbf {U}_2^{\prime } \mathbf {S}_p \varvec{\epsilon } \Rightarrow D(1)\ell _{11}^2 \left\{ \int \limits _0^1 I_2 \int \limits _0^1 I_2 W^{*} \mathrm{d}W^{*} - \int \limits _0^1 W^{*}I_2 \int \limits _0^1 I_2 \mathrm{d}W^{*} \right\} \end{aligned}$$

(69)

and

$$\begin{aligned} T^{-2}\mathbf {U}_2^{\prime }\mathbf {S}_p\mathbf {U}_2 \Rightarrow \ell _{11}^2 \left\{ \int \limits _0^1 I_2 \int \limits _0^1 W^{*2}I_2-\left( \int \limits _0^1 W^{*}I_2\right) ^2 \right\} . \end{aligned}$$

(70)

Noting that $(\hat{\mu }_1-\mu _1)=O_p(T^{-1/2})$ and $(\hat{\mu }_2-\mu _2)=O_p(T^{-1/2})$, similar to (51), we obtain

$$\begin{aligned} \hat{\sigma }^2&= T^{-1} \sum _{t=1}^T \Big \{ \Delta \hat{u}_t-(\hat{\mu }_1-\mu _1+\hat{\rho }_1 )\hat{u}_{t-1} \mathbf {1}\{\hat{u}_{t-1} \le \lambda _1 \} \nonumber \\&-(\hat{\mu }_2-\mu _2+\hat{\rho }_2 )\hat{u}_{t-1} \mathbf {1}\{\hat{u}_{t-1} > \lambda _2 \}-\sum \limits _{j=1}^p (\hat{\alpha }_j-\alpha _j) \Delta \hat{u}_{t-j} \Big \}^2\nonumber \\&= T^{-1} \sum _{t=1}^T \epsilon _t^2+o_p(1) \Rightarrow D(1)^2 \ell _{11}^2 k^{\prime }k. \end{aligned}$$

(71)

Therefore, by (65)–(71),

$$\begin{aligned} W_T^B(\lambda )&\Rightarrow \frac{1}{D(1)^2 \ell _{11}^2 k^{\prime }k} \left[ \frac{\left\{ D(1) \ell _{11}^2 \left( \int _0^1 I_1 \int _0^1 I_1 W^{*} \mathrm{d}W^{*} -\int _0^1 W^{*}I_1 \int _0^1 I_1 \mathrm{d}W^{*}\right) \right\} ^2}{ \ell _{11}^2 \left\{ \int _0^1 I_1 \int _0^1 W^{*2}I_1-\left( \int _0^1 W^{*}I_1\right) ^2 \right\} }\right. \nonumber \\&\left. +\frac{\left\{ D(1) \ell _{11}^2 \left( \int _0^1 I_2 \int _0^1 I_2 W^{*} \mathrm{d}W^{*} -\int _0^1 W^{*}I_2 \int _0^1 I_2 \mathrm{d}W^{*}\right) \right\} ^2}{ \ell _{11}^2 \left\{ \int _0^1 I_2 \int _0^1 I_2 W^{*2} \mathrm{d}W^{*}-\left( \int _0^1 W^{*}I_2\right) ^2 \right\} } \right] \nonumber \\&= \frac{1}{k^{\prime }k} \frac{\left( \int _0^1 I_1 \int _0^1 W^{*}I_1-\int _0^1 W^{*}I_1 \int _0^1 I_1 \mathrm{d}W^{*} \right) ^2}{\int _0^1 I_1 \int _0^1 W^{*2}I_1-\left( \int _0^1 W^{*}I_1\right) ^2 } \nonumber \\&+ \frac{1}{k^{\prime }k} \frac{\left( \int _0^1 I_2 \int _0^1 W^{*}I_2-\int _0^1 W^{*}I_2 \int _0^1 I_2 \mathrm{d}W^{*} \right) ^2}{ \int _0^1 I_2 \int _0^1 W^{*2}I_2-\left( \int _0^1 W^{*}I_2\right) ^2 }. \end{aligned}$$

(72)

We can deduce the required results from the asymptotic distribution of $W_B^T(\lambda )$ and Assumption 2.

Under the alternative hypothesis, we can write $W_T^B(\lambda )$ as

$$\begin{aligned} W_T^B(\lambda )=\frac{1}{\hat{\sigma }^2} \rho ^{\prime }(\mathbf {U}^{\prime } \mathbf {S}_p \mathbf {U})\rho +\frac{1}{\hat{\sigma }^2} \varvec{\epsilon }^{\prime }\mathbf {S}_p \mathbf {U} (\mathbf {U}^{\prime } \mathbf {S}_p \mathbf {U})^{-1} \mathbf {U}^{\prime }\mathbf {S}_p \varvec{\epsilon }. \end{aligned}$$

(73)

Since under the alternative hypothesis, we have $T^{-1}\mathbf {U}^{\prime }\mathbf {S}_p \mathbf {U}=O_p(1)$, $T^{-1/2}\mathbf {U}^{\prime } \mathbf {S}_p \varvec{\epsilon }=O_p(1)$, and $\hat{\sigma }^2=O_p(1)$, it can be shown that

$$\begin{aligned} W_T^B(\lambda )&= T \frac{1}{\hat{\sigma }^2} \rho ^{\prime }(T^{-1}\mathbf {U}^{\prime } \mathbf {S}_p \mathbf {U})\rho +\frac{1}{\hat{\sigma }^2} T^{-1/2} \varvec{\epsilon }^{\prime }\mathbf {S}_p \mathbf {U} (T^{-1}\mathbf {U}^{\prime } \mathbf {S}_p \mathbf {U})^{-1} T^{-1/2} \mathbf {U}^{\prime }\mathbf {S}_p \varvec{\epsilon } \nonumber \\&= T \frac{1}{\hat{\sigma }^2} \rho ^{\prime }(T^{-1}\mathbf {U}^{\prime } \mathbf {S}_p \mathbf {U})\rho +O_p(1) \nonumber \\&= O_p(T). \end{aligned}$$

(74)

Therefore, $W_T^B(\lambda )$ diverges to infinity as $T \rightarrow \infty$. This also implies that the test statistic diverges to infinity as $T \rightarrow \infty$.

Part of (3b) From the proof of part (2b) and (3a), under $H_0$, the statistic $t_T^B(\lambda )_{\max }$ can be written as

$$\begin{aligned} t_T^B(\lambda )_{\max }=\max \Bigg [\frac{ \mathbf {U}_1^{\prime } \mathbf {S}_p \varvec{\epsilon } }{\{ \hat{\sigma }^2 (\mathbf {U}_1^{\prime } \mathbf {S}_p \mathbf {U}_1) \}^{1/2}}, \frac{ \mathbf {U}_2^{\prime } \mathbf {S}_p \varvec{\epsilon } }{\{ \hat{\sigma }^2 (\mathbf {U}_2^{\prime } \mathbf {S}_p \mathbf {U}_2) \}^{1/2}} \Bigg ]. \end{aligned}$$

(75)

Using from (65) to (71), we have

$$\begin{aligned} t^B(\lambda )_{\max }&= \max \Bigg [ \frac{\int _0^1 I_1 \int _0^1 I_1 W^{*} \mathrm{d}W^{*} - \int _0^1 W^{*}I_1 \int _0^1 I_1 \mathrm{d}W^{*} }{ \big ( k^{\prime }k \big \{ \int _0^1 I_1 \int _0^1 W^{*2} I_1 - \big (\int _0^1 W^{*2} I_1\big )^2 \big \} \big )^{1/2} }, \nonumber \\&\frac{\int _0^1 I_2 \int _0^1 I_2 W^{*} \mathrm{d}W^{*} - \int _0^1 W^{*}I_2 \int _0^1 I_2 \mathrm{d}W^{*}}{ \big ( k^{\prime }k \big \{ \int _0^1 I_2 \int _0^1 W^{*2} I_2 - (\int _0^1 W^{*2} I_2)^2 \big \} \big )^{1/2}} \Bigg ] \end{aligned}$$

(76)

We can deduce the required results from the asymptotic distribution of $t_T^B(\lambda )_{\max }$ and Assumption 2. Next, we consider the properties of the test statistic under the alternative hypothesis. Under the alternative hypothesis, $t_T^B(\lambda )_{\max }$ can be expressed as

$$\begin{aligned} t_T^B(\lambda )_{\max }&= \max \Bigg [\frac{\rho _1}{\{ \hat{\sigma }^2 (\mathbf {U}_1^{\prime } \mathbf {S}_p \mathbf {U}_1)^{-1} \}^{1/2}} + \frac{1}{\hat{\sigma }} \mathbf {U}_1^{\prime }\mathbf {S}_p \varvec{\epsilon } (\mathbf {U}_1^{\prime } \mathbf {S}_p \mathbf {U}_1)^{-1/2}, \nonumber \\&\frac{\rho _2}{\{ \hat{\sigma }^2 (\mathbf {U}_2^{\prime } \mathbf {S}_p \mathbf {U}_2)^{-1} \}^{1/2}} + \frac{1}{\hat{\sigma }} \mathbf {U}_2^{\prime }\mathbf {S}_p \varvec{\epsilon } (\mathbf {U}_2^{\prime } \mathbf {S}_p \mathbf {U}_2)^{-1/2} \Bigg ]. \end{aligned}$$

(77)

Using $T^{-1} \mathbf {U}_1^{\prime } \mathbf {S}_p \mathbf {U}_1=O_p(1),\, T^{-1/2} \mathbf {U}_1^{\prime }\mathbf {S}_p \varvec{\epsilon }=O_p(1),\, T^{-1} \mathbf {U}_2^{\prime } \mathbf {S}_p \mathbf {U}_2=O_p(1), T^{-1/2} \mathbf {U}_2^{\prime }\mathbf {S}_p \varvec{\epsilon }=O_p{1}$, and $\hat{\sigma }^2=O_p(1)$, we obtain

$$\begin{aligned} t_T^B(\lambda )_{\max }&= \max \Bigg [ \frac{T^{1/2} \rho _1}{\{ \hat{\sigma }^2 (T^{-1} \mathbf {U}_1^{\prime } \mathbf {S}_p \mathbf {U}_1)^{-1} \}^{1/2}} + \frac{1}{\hat{\sigma }}T^{-1/2} \mathbf {U}_1^{\prime }\mathbf {S}_p \varvec{\epsilon } (T^{-1}\mathbf {U}_1^{\prime } \mathbf {S}_p \mathbf {U}_1)^{-1/2}, \nonumber \\&\frac{T^{1/2} \rho _2}{\{ \hat{\sigma }^2 (T^{-1} \mathbf {U}_2^{\prime } \mathbf {S}_p \mathbf {U}_2)^{-1} \}^{1/2}} + \frac{1}{\hat{\sigma }}T^{-1/2} \mathbf {U}_2^{\prime }\mathbf {S}_p \varvec{\epsilon } (T^{-1}\mathbf {U}_2^{\prime } \mathbf {S}_p \mathbf {U}_2)^{-1/2} \Bigg ]\nonumber \\&= \max [ O_p(T^{1/2}), O_p(T^{1/2}) ], \end{aligned}$$

(78)

which shows that $t_T^B(\lambda )_{\max }$ diverges to minus infinity as $T \rightarrow \infty$. This also implies that the test statistic diverges to minus infinity under the alternative hypothesis as $T \rightarrow \infty$.

Appendix 2: Heteroskedasticity-robust test statistics

For (7), the heteroskedasticity-robust test statistic is given by

$$\begin{aligned} \displaystyle \sup _{\lambda \in \varLambda _T} W^{*}_T (\lambda ) \equiv \sup _{\lambda \in [\lambda _\mathrm{min}, \lambda _\mathrm{max}]}W^{*}_T(\lambda ), \end{aligned}$$

(79)

where $W^{*}_T$ is the test statistics (6) corrected by the White estimator. $W^{*}_T$ is given by

$$\begin{aligned} W^{*}_T(\lambda )=\hat{\rho }^{\prime } \bigg [ R\bigg (\sum _{t=1}^T h_t h_t^{\prime }\bigg )^{-1} \bigg (\sum _{t=1}^T \hat{\epsilon }^2_t h_t h_t^{\prime } \bigg )\bigg (\sum _{t=1}^T h_t h_t^{\prime }\bigg )^{-1}R^{\prime } \bigg ]^{-1}\hat{\rho }, \end{aligned}$$

(80)

where $\hat{\epsilon }$ is the residuals in regression (3). For the heteroskedasticity-robust test statistic of (10), we use White correction to calculate the standard error. The heteroskedasticity-robust versions for (15), (16), EG-$t$, and ES-$\phi$ are obtained similar to those above. See also Bec et al. (2010) for heteroskedasticity-robust unit root tests in smooth transition autoregressive (STAR) models.