Test for conditional quantile change in general conditional heteroscedastic time series models

Abstract

This study aims to test for detecting a change point in the conditional quantile of general location-scale time series models. This issue is quite important in risk management because the conditional quantile is utilized to measure the value-at-risk or expected shortfall of financial assets. In this paper, we design two types of cumulative sum tests based on the conditional quantiles. Their limiting null distributions are derived under regularity conditions, together with consistency of the proposed tests under the alternative. Monte Carlo simulations demonstrate the good performance of the proposed tests in terms of both stability and power for various time series settings. A real data analysis using the daily returns of the Brent Oil futures also confirms the validity of the tests in real-world applications.

Appendix

In this section, we provide the proofs of some results in Sects. 2 and 3.

Proof of Theorem 1

It suffices to prove (9). Put

$$\begin{aligned} \varLambda _t (\vartheta ) =({{\tilde{g}}}_t (\mu )-g_t (\mu _0 ))/h_t^{1/2} (\theta _0) +\xi ({{\tilde{h}}}_t (\theta ) / h_t (\theta _0))^{1/2}-\xi _0. \end{aligned}$$

Noting that $\psi _\tau (y_t-{\hat{q}}_t)=\tau -I(\eta _t \le \xi _0 +\varLambda _t({\hat{\vartheta _n}} ) )$, we can express

$$\begin{aligned}{} & {} \frac{1}{\sqrt{n}}\max _{1\le k\le n}\Bigg |\sum _{t=1}^k |y_{t-1}|^\gamma \Big \{\psi _\tau (y_t-{\hat{q}}_t)- \psi _\tau (y_t-q_t(\vartheta _0 ))\Big \}\\{} & {} \qquad -\,\frac{k}{n}\sum _{t=1}^n |y_{t-1}|^\gamma \Big \{\psi _\tau (y_t-{\hat{q}}_t)-\psi _\tau (y_t-q_t(\vartheta _0))\Big \}\Bigg |\\{} & {} \quad \le \frac{2}{\sqrt{n}}\max _{1\le k\le n}\Bigg |\sum _{t=1}^k |y_{t-1}|^\gamma \Big \{ I(\eta _t \le \xi _0 +\varLambda _t ({\hat{\vartheta _n}}) ) -F_\eta (\xi _0 +\varLambda _t ({\hat{\vartheta _n}}) )+F_\eta (\xi _0 )-I(\eta _t \le \xi _0)\Big \}\Bigg |\\{} & {} \qquad +\,\frac{1}{\sqrt{n}}\max _{1\le k\le n}\Bigg |\sum _{t=1}^k |y_{t-1}|^\gamma \Big \{ F_\eta (\xi _0 +\varLambda _t ({\hat{\vartheta _n}}) )-F_\eta (\xi _0 )\Big \}\\{} & {} \qquad -\,\frac{k}{n}\sum _{t=1}^n |y_{t-1}|^\gamma \Big \{ F_\eta (\xi _0 +\varLambda _t ({\hat{\vartheta _n}}) )-F_\eta (\xi _0 )\Big \}\Bigg |\\{} & {} \quad :=2 A_n +B_n. \end{aligned}$$

Owing to Proposition 1, for any $\delta \in (0,1)$, there exists a (large enough) $L>0$ such that $P(\hat{\vartheta }_{n}\in \mathcal{N}_{L/\sqrt{n}})\ge 1-\delta$, where ${{{\mathcal {N}}}}_{L/\sqrt{n}}$ is a compact neighborhood of $\vartheta _{0}$ with $||\vartheta -\vartheta _{0}||\le L/\sqrt{n}$ for all $\vartheta \in {{{\mathcal {N}}}}_{L/\sqrt{n}}$. Given any fixed $\zeta >0$, we decompose ${{{\mathcal {N}}}}_{L/\sqrt{n}}$ into a finite number of subsets $\mathcal{D}_1,\ldots ,{{{\mathcal {D}}}}_N$ for some $N=N(\zeta )\ge 1$, with their diameters less than $\zeta /\sqrt{n}$. We then choose points $\vartheta _j$ from ${{{\mathcal {D}}}}_j$. Then, in case of ${\hat{\vartheta _n}} \in {{{\mathcal {D}}}}_j$, we have

$$\begin{aligned} I({\eta }_{t}\le \xi _0+ \varLambda _t (\vartheta _j )+\varLambda _{tj}^{-})\le I({\eta }_{t}\le \xi _0+\varLambda _{t} ({\hat{\vartheta _n}}))\le I({\eta }_{t}\le \xi _0+\varLambda _t (\vartheta _j)+\varLambda _{tj}^+ ) \end{aligned}$$

with $\varLambda _{tj}^{-}=\inf _{\vartheta \in {{{\mathcal {D}}}}_j} \varLambda _t (\vartheta )-\varLambda _t (\vartheta _j)$ and $\varLambda _{tj}^{+}=\sup _{\vartheta \in {{{\mathcal {D}}}}_j} \varLambda _t (\vartheta )-\varLambda _t (\vartheta _j )$. Putting

$$\begin{aligned} A_{kj}^+= & {} \frac{1}{\sqrt{n}}\sum _{t=1}^{k}|y_{t-1}|^\gamma \Big \{I({\eta }_{t}\le \xi _0+\varLambda _t (\vartheta _j)+\varLambda _{tj}^+)\\{} & {} -\,F_\eta (\xi _0+\varLambda _t (\vartheta _j)+\varLambda _{tj}^+)-F_\eta (\xi _0)+I(\eta _t\le \xi _0)\Big \},\\ A_{kj}^-= & {} \frac{1}{\sqrt{n}}\sum _{t=1}^{k}|y_{t-1}|^\gamma \Big \{I({\eta }_{t}\le \xi _0+\varLambda _t (\vartheta _j)+\varLambda _{tj}^-)\\{} & {} -\,F_\eta (\xi _0+\varLambda _t (\vartheta _j)+\varLambda _{tj}^-)-F_\eta (\xi _0)+I(\eta _t\le \xi _0)\Big \}, \end{aligned}$$

and using (A5), (A6), and the mean value theorem, we can show that

$$\begin{aligned} A_n \le \max _{1\le j\le N} \max _{1\le k\le n} |A_{kj}^+|+\max _{1\le j\le N} \max _{1\le k\le n} |A_{kj}^-|+r_n \end{aligned}$$

(12)

with

$$\begin{aligned} r_n= & {} \frac{1}{\sqrt{n}}\sum _{t=1}^{n}\max _{1\le j\le N} \sup _{\vartheta \in {{{\mathcal {D}}}}_j}|y_{t-1}|^\gamma | F_\eta (\xi _0+\varLambda _t (\vartheta _j)+\varLambda _{tj}^+)-F_\eta (\xi _0+ \varLambda _t (\vartheta ))|\nonumber \\{} & {} +\,\frac{1}{\sqrt{n}}\sum _{t=1}^{n}\max _{1\le j\le N} \sup _{\vartheta \in {{{\mathcal {D}}}}_j}|y_{t-1}|^\gamma F_\eta (\xi _0+\varLambda _t (\vartheta _j)+\varLambda _{tj}^-)-F_\eta (\xi _0+ \varLambda _t (\vartheta ))|\nonumber \\= & {} \zeta O_P (1), \end{aligned}$$

(13)

which can be made arbitrarily small by taking a small $\zeta$.

Next, putting

$$\begin{aligned} e_{tj}^+= & {} |y_{t-1}|^\gamma \Big \{I({\eta }_{t}\le \xi _0+\varLambda _t (\vartheta _j)+\varLambda _{tj}^+)\\{} & {} -\,F_\eta (\xi _0+\varLambda _t (\vartheta _j)+\varLambda _{tj}^+)-F_\eta (\xi _0)+I(\eta _t\le \xi _0)\Big \},\\ e_{tj}^-= & {} |y_{t-1}|^\gamma \Big \{I({\eta }_{t}\le \xi _0+\varLambda _t (\vartheta _j)+\varLambda _{tj}^-)\\{} & {} -\,F_\eta (\xi _0+\varLambda _t (\vartheta _j)+\varLambda _{tj}^-)-F_\eta (\xi _0)+I(\eta _t\le \xi _0)\Big \}, \end{aligned}$$

we set $S_{kj}^+=\sum _{t=1}^{k}e_{tj}^+$ and $S_{kj}^-=\sum _{t=1}^{k}e_{tj}^-$, Moreover, for $B>0$, we set

$$\begin{aligned} {{\tilde{S}}}_{kj}^+=\sum _{t=1}^{k}{{\tilde{e}}}_{tj}^+, \ \ \tilde{S}_{kj}^-=\sum _{t=1}^{k}{{\tilde{e}}}_{tj}^- \end{aligned}$$

with

$$\begin{aligned} {{\tilde{e}}}_{tj}^+= & {} e_{tj}^+ I \Bigg (\sum _{s=1}^{t} |y_{s}|^{2\gamma } \sup _{\vartheta \in \varTheta }\Big |\Big | \frac{\partial \varLambda _s (\vartheta )}{\partial \vartheta }\Big |\Big |\le B n\Bigg ),\\ {{\tilde{e}}}_{tj}^-= & {} e_{tj}^- I \Bigg (\sum _{s=1}^{t} |y_{s}|^{2\gamma } \sup _{\vartheta \in \varTheta }\Big |\Big | \frac{\partial \varLambda _s (\vartheta )}{\partial \vartheta }\Big |\Big |\le B n\Bigg ). \end{aligned}$$

Then, we get martingale arrays $\{({{\tilde{S}}}_{kj}^+, {{{\mathcal {F}}}}_t); k=1,\ldots ,n\}$ and $\{({{\tilde{S}}}_{kj}^-, {{{\mathcal {F}}}}_t); k=1,\ldots , n\}$, where ${{{\mathcal {F}}}}_t=\sigma (\eta _t, \eta _{t-1},\ldots )$.

Note that $P ({{\tilde{e}}}_{tj}^+ \ne e_{tj}^+ \ \ \mathrm{for\ some}\ t=1,\ldots , n)$ can be made arbitrarily small by taking a large B, which can be seen using the mean value theorem, Hölder’s inequality, (A5) and (A6). Then, for any $\zeta ^{'}, \lambda >0$, $0<\delta <1$ and $p>1$, using the sub-martingale inequality and taking a sufficiently large $B:= B(\zeta )$, we can get

$$\begin{aligned}{} & {} \limsup _{n\rightarrow \infty }P(A_n^+\ge \lambda )\nonumber \\{} & {} \quad \le \limsup _{n\rightarrow \infty }P\Big (\max _{1\le j\le N}\max _{1\le k\le n} |A_{kj}^+|\ge \lambda \Big )\le \limsup _{n\rightarrow \infty } \sum _{j=1}^N P\Big (\max _{n^\delta \le k\le n}|S_{kj}^+|\ge \sqrt{n}\lambda \Big )\nonumber \\{} & {} \quad \le \limsup _{n\rightarrow \infty }\sum _{j=1}^NP\Big (\max _{1 \le k\le n}|{{\tilde{S}}}_{kj}^+|\ge \sqrt{n}\lambda \Big )+\zeta ^{'} \le \limsup _{n\rightarrow \infty } \sum _{j=1}^N E \tilde{S}_{nj}^{+2p}/n^{p}\lambda ^{2p}+\zeta ^{'}. \end{aligned}$$

(14)

Moreover, using Rosenthal’s inequality (Hall and Heyde 1980), we get

$$\begin{aligned} E{{\tilde{S}}}_{nj}^{+2p}\le & {} C\Big (E\Big [\sum _{t=1}^{n}E({{\tilde{e}}}_{tj}^{+2}|\mathcal{F}_{t-1})\Big ]^{p}+\sum _{t=1}^{n}E({{\tilde{e}}}_{tn}^{+2p})\Big ), \ C>0, \end{aligned}$$

and further,

$$\begin{aligned} E\Bigg [\sum _{t=1}^{n}E({{\tilde{e}}}_{tj}^{+2}|\mathcal{F}_{t-1})\Bigg ]^{p}=O(n^{p/2})\quad \textrm{and}\quad \sum _{t=1}^{n}E(\tilde{e}_{tj}^{+2p})= O(n), \end{aligned}$$

which leads to $\sum _{j=1}^N E{{\tilde{S}}}_{nj}^{+2p}=O(n^{p/2}+n )$. Then, in view of (14), we obtain

$$\begin{aligned} \max _{1\le j\le N}\max _{1\le k\le n} |A_{kj}^+|=o_P (1). \end{aligned}$$

Analogously, we have $\max _{1\le j\le N}\max _{1\le k\le n} |A_{kj}^-|=o_P (1).$ Hence, from (12) and (13), we have $A_n= o_P (1)$.

Next, we handle $B_n$. Set

$$\begin{aligned} B_{nk}= & {} \Bigg |\sum _{t=1}^k |y_{t-1}|^\gamma \Big \{ F_\eta (\xi _0 +\varLambda _t ({\hat{\vartheta _n}}) )-F_\eta (\xi _0 )\Big \}\\{} & {} -\,\frac{k}{n}\sum _{t=1}^n |y_{t-1}|^\gamma \Big \{ F_\eta (\xi _0 +\varLambda _t ({\hat{\vartheta _n}}) )-F_\eta (\xi _0 )\Big \}\Bigg |. \end{aligned}$$

Note that for $k_n = n^{1/3}$, $\frac{1}{\sqrt{n}}\max _{1\le k\le k_n} B_{nk}=o_P (1).$ Thus it suffices to show that

$$\begin{aligned} B_n^{'}:=\frac{1}{\sqrt{n}}\max _{k_n\le k\le n}B_{nk}=o_P (1). \end{aligned}$$

To this end, we use Taylor’s theorem to get

$$\begin{aligned} F_\eta (\xi _0 +\varLambda _t ({\hat{\vartheta _n}}) )-F_\eta (\xi _0 )=\varLambda _t ({\hat{\vartheta _n}}) f_\eta (\xi _0) +\frac{1}{2} \varLambda _t^2 ({\hat{\vartheta _n}}) f^{'}_\eta (\xi _t^*) \end{aligned}$$

for some number $\xi _t^*$ lying between $\xi _0$ and $\xi _0 +\varLambda _t ({\hat{\vartheta _n}})$. Then, using this, (A2), (A5), (A6), and our assumption on $f_\eta ^{'}$, we can show that for some $K>0$,

$$\begin{aligned} B_n^{'}\le & {} K \sqrt{n}||{\hat{\mu _n}} -\mu _0||\max _{k_n\le k\le n} \Bigg |\Bigg | \frac{1}{k}\sum _{t=1}^k |y_{t-1}|^\gamma \frac{\partial g_t (\mu _0 )}{\partial \mu }\frac{1}{h^{1/2}_t (\theta _0 )}\\{} & {} -\,\frac{1}{n}\sum _{t=1}^n |y_{t-1}|^\gamma \frac{\partial g_t (\mu _0 )}{\partial \mu }\frac{1}{h^{1/2}_t (\theta _0 )}\Bigg |\Bigg |\\{} & {} +\, K\sqrt{n}||{\hat{\theta _n}} -\theta _0|| \max _{k_n\le k\le n}\Bigg |\Bigg | \frac{1}{k}\sum _{t=1}^k |y_{t-1}|^\gamma \frac{\partial h_t (\theta _0 )}{\partial \theta }\frac{1}{\sqrt{2}h_t (\theta _0 )}\\{} & {} -\,\frac{1}{n}\sum _{t=1}^n |y_{t-1}|^\gamma \frac{\partial h_t (\theta _0 )}{\partial \theta }\frac{1}{\sqrt{2}h_t (\theta _0 )}\Bigg |\Bigg |\\{} & {} +\, K\sqrt{n}|{\hat{\xi _n}} -\xi _0| \max _{k_n\le k\le n}\Big |\frac{1}{k}\sum _{t=1}^k |y_{t-1}|^\gamma -\frac{1}{n}\sum _{t=1}^n |y_{t-1}|^\gamma \Big |+o_P (1)\\= & {} o_P (1), \end{aligned}$$

due to the ergodicity of $\{y_t\}$, which entails $B_n=o_P (1)$. As we can check ${\hat{\tau }}_{1}^2=\tau _1^2+o_P (1)$, we finally get ${\hat{T}}_{n}- T_{n}=o_P (1)$, which in turn implicates the weak convergence of ${\hat{T}}_{n}$ to $\sup _{0\le s\le 1} | B^\circ (s)|$. $\square$

Proof of Theorem 4

For $i=1, 2$, we set

$$\begin{aligned} \varLambda _{ti}^* (\vartheta )= & {} \frac{{{\tilde{g}}}_{ti} (\mu )-g_{ti} (\mu _0^* )}{h_{ti}^{1/2} (\theta _{i})} +\frac{ \xi \tilde{h}_{ti}^{1/2}(\theta ) -\xi _{0}^* h_{ti}^{1/2}(\theta _{0}^*)}{ h_{ti}^{1/2} (\theta _{i})}, \\ \xi _{ti}= & {} \frac{ g_{ti}(\mu _0^* )-g_{ti}(\mu _{i})+\xi _0^* h_{ti}^{1/2}(\theta _0^*)}{h_{ti}^{1/2}(\theta _{i})}. \end{aligned}$$

Noting that $\psi _\tau (y_{t1}-{\hat{q}}_{t1})=\tau -I(\eta _t \le \varLambda ^*_t ({\hat{\vartheta _n}} )+\xi _{t1} )$, we can express for $k\le k_0$,

$$\begin{aligned} \frac{1}{n}\sum _{t=1}^k |y_{t-1,1}|^\gamma \psi _\tau (y_{t1}-{\hat{q}}_{t1})=\frac{1}{n}\sum _{t=1}^k |y_{t-1,1}|^\gamma \psi _\tau (y_{t1}- q_{t1}(\vartheta _0^*))+I_{nk} +II_{nk}, \end{aligned}$$

where

$$\begin{aligned} I_{nk}= & {} -\frac{1}{n}\sum _{t=1}^k |y_{t-1,1}|^\gamma \Big \{ I(\eta _t \le \varLambda _{t1}^*({\hat{\vartheta _n}})+\xi _{t1} ) -F_\eta (\varLambda _{t1}^*({\hat{\vartheta _n}})\\{} & {} +\,\xi _{t1} )+F_\eta (\xi _{t1} )-I(\eta _t \le \xi _{t1})\Big \},\\ II_{nk}= & {} -\frac{1}{n}\sum _{t=1}^k |y_{t-1,1}|^\gamma \Big \{ F_\eta ( \varLambda _{t1}^*({\hat{\vartheta _n}})+\xi _{t1} )-F_\eta (\xi _{t1} )\Big \}. \end{aligned}$$

Then, following the lines similar to those used for verifying $A_n=o_P (1)$ in the proof of Theorem 1, with $\mathcal{N}_{L/\sqrt{n}}$, $\varLambda _t (\vartheta )$, and $\xi _0$, therein, replaced by ${{{\mathcal {N}}}}_L^*=\{\vartheta : ||\vartheta -\vartheta _0^*||\le L\}$, $\varLambda _{t1}^{*}(\vartheta )$, and $\xi _{t1}$, respectively, where L can be taken to be arbitrarily small, ${{{\mathcal {N}}}}_L^*$ and L play the role of the partition ${{{\mathcal {D}}}}_1$ and $\zeta$ (namely, ${{{\mathcal {N}}}}_L^*=\mathcal{D}_1$, $L=\zeta$, and $N(\zeta )=1$), and both $A_{k1}^+$ and $A_{k1}^-$ are accordingly reformulated with the denominator $\sqrt{n}$ replaced by n, we can similarly verify that $\max _{1\le k\le k_0} | I_{nk}|= o_P (1)$. Also, using (A2), (A5), (A6) (modified with $g_{ti}, h_{ti}, \tilde{g}_{ti}, {{\tilde{h}}}_{ti}$), (B1), the ergodicity, and the mean value theorem, we can readily check that $\max _{1\le k\le k_0} | II_{nk}|= o_P (1)$. Subsequently, we get

$$\begin{aligned} \frac{1}{n}\sum _{t=1}^k |y_{t-1,1}|^\gamma \psi _\tau (y_{t1}-{\hat{q}}_{t1})= & {} \frac{1}{n}\sum _{t=1}^k |y_{t-1,1}|^\gamma \psi _\tau (y_{t1}- q_{t1}(\vartheta _0^*))\nonumber \\{} & {} +\,o_P (1), \end{aligned}$$

(15)

uniformly in $k\le k_0$. Similarly, additionally harnessing (11), we can verify

$$\begin{aligned} \frac{1}{n}\sum _{t=k_0+1}^k |y_{t-1,2}|^\gamma \psi _\tau (y_{t2}-{\hat{q}}_{t2})= & {} \frac{1}{n}\sum _{t=k_0+1}^n |y_{t-1,2}|^\gamma \psi _\tau (y_{t2}- q_{t2}(\vartheta _0^*))\nonumber \\{} & {} +\,o_P (1), \end{aligned}$$

(16)

uniformly in $k>k_0$. Therefore, we can have

$$\begin{aligned} \frac{|{\hat{\tau _n}} {\hat{T}}_n |}{\sqrt{n}} &\ge \frac{1}{n}\Big |\sum _{t=1}^{k_0} |y_{t-1,1}|^\gamma \psi _\tau (y_{t1}-{\hat{q}}_{t1})\nonumber \\{} & {} \qquad - \frac{k_0}{n} \Big \{ \sum _{t=1}^{k_0} |y_{t-1,1}|^\gamma \psi _\tau (y_{t1}-{\hat{q}}_{t1})+\sum _{t=k_0 +1}^n |y_{t-1,2}|^\gamma \psi _\tau (y_{t2}-{\hat{q}}_{t2})\Big \}\Big |\nonumber \\{} & =\frac{1}{n}\Big |\sum _{t=1}^{k_0} |y_{t-1,1}|^\gamma \psi _\tau (y_{t1}- q_{t1}(\vartheta _0^*))\nonumber \\{} & {} \qquad -\frac{k_0}{n} \Big \{ \sum _{t=1}^{k_0} |y_{t-1,1}|^\gamma \psi _\tau (y_{ti}- q_{t1}(\vartheta _0^*))\nonumber \\{} & {} \qquad +\sum _{t=k_0 +1}^n |y_{t-1,2}|^\gamma \psi _\tau (y_{t2}- q_{t2}(\vartheta _0^*))\Big \}\Big |+o_P (1)\nonumber \\{} & {} \quad {\mathop {\longrightarrow }\limits ^{P}} \kappa _0(1-\kappa _0)|d_1-d_2| > 0, \end{aligned}$$

due to the ergodicity and (B2), so that ${\hat{T}}_n \rightarrow \infty$ in probability as ${\hat{\tau _n}}=\tau _0+ o_P(1)$ for some $\tau _0>0$.

Moreover, putting

$$\begin{aligned} \hat{{{\mathcal {L}}}}_n (k)= & {} \frac{1}{n}\Big |\sum _{t=1}^{k} |y_{t-1,1}|^\gamma \psi _\tau (y_{t1}-{\hat{q}}_{t1})\\{} & {} -\, \frac{k}{n} \Big \{ \sum _{t=1}^{k_0} |y_{t-1,1}|^\gamma \psi _\tau (y_{t1}-{\hat{q}}_{t1})+\sum _{t=k_0 +1}^n |y_{t-1,2}|^\gamma \psi _\tau (y_{t2}-{\hat{q}}_{t2})\Big \}\Big |, \end{aligned}$$

we can see that $\max _{0\le s\le 1}|\hat{{{\mathcal {L}}}}_n ([ns])- \mathcal{L}_n ([ns])|= o_P (1)$ by virtue of (15) and (16), where

$$\begin{aligned} {{{\mathcal {L}}}}_n (k)= & {} \frac{1}{n}\Big |\sum _{t=1}^{k} |y_{t-1,1}|^\gamma \psi _\tau (y_{t1}- q_{t1}(\vartheta _0^*))I(k\le k_0) \\{} & {} +\, \Big \{\sum _{t=1}^{k_0} |y_{t-1,1}|^\gamma \psi _\tau (y_{t1}- q_{t1}(\vartheta _0^*))+ \sum _{t=k_0+1}^{k} |y_{t-1,2}|^\gamma \psi _\tau (y_{t2}- q_{t2}(\vartheta _0^*))\Big \}I(k>k_0)\\{} & {} -\, \frac{k}{n} \Big \{ \sum _{t=1}^{k_0} |y_{t-1,1}|^\gamma \psi _\tau (y_{ti}- q_{t1}(\vartheta _0^*))+\sum _{t=k_0 +1}^n |y_{t-1,2}|^\gamma \psi _\tau (y_{ti}- q_{t2}(\vartheta _0^*))\Big \}\Big |. \end{aligned}$$

Furthermore, using the ergodicity, we can easily check that $\sup _{0\le s\le 1}|{{{\mathcal {L}}}}_n ([ns])-{{{\mathcal {L}}}}(s)|=o_P (1),$ where

$$\begin{aligned} {{{\mathcal {L}}}}(s)= s (1-\kappa _0)|d_1-d_2|I (s\le \kappa _0)+\kappa _0 (1-s) |d_1-d_2|I (\kappa _0 < s\le 1), \end{aligned}$$

which has a maximum value $\kappa _0(1-\kappa _0)|d_1-d_2|$ uniquely at $s=\kappa _0$. This with the argmax theorem yields ${\hat{k}}_n /n \rightarrow \kappa _0$ in probability, which validates the theorem. $\square$

On the issue regarding (B1) in Remark 4. For $t\le k_0$, put

$$\begin{aligned} \varLambda _t^{\dagger } (\vartheta )= & {} ({{\tilde{g}}}_{t1} (\mu )-g_{t1} (\mu _1 ))/h_{t1}^{1/2} (\theta _1) +\xi ({{\tilde{h}}}_{t1} (\theta ) / h_{t1} (\theta _1))^{1/2},\\ \varLambda _t^\circ (\vartheta )= & {} ( g_{t1} (\mu )-g_{t1} (\mu _1 ))/h_{t1}^{1/2} (\theta _1) +\xi ( h_{t1} (\theta ) / h_{t1} (\theta _1))^{1/2}. \end{aligned}$$

Then, we can express

$$\begin{aligned} \rho _\tau (y_t - {{\tilde{q}}}_{t}(\vartheta ))= & {} (y_{t1} - \tilde{q}_{t1}(\vartheta ))\psi _\tau (y_{t1}-{{\tilde{q}}}_{t1}(\vartheta )),\\ \rho _\tau (y_{t} - q_{t}(\vartheta ))= & {} (y_{t1} - q_{t1}(\vartheta ))\psi _\tau (y_{t1}- q_{t1}(\vartheta )) \end{aligned}$$

with

$$\begin{aligned} \psi _\tau (y_t-{{\tilde{q}}}_{t}(\vartheta ))=\tau -I(\eta _t \le \varLambda _t^\dagger (\vartheta ) ),\quad \psi _\tau (y_t- q_{t1}(\vartheta ))=\tau -I(\eta _t \le \varLambda _t^\circ (\vartheta ) ). \end{aligned}$$

We first verify

$$\begin{aligned} \sup _{\vartheta \in \varTheta ^*}\Big |\frac{1}{n}\sum _{t=1}^{k_0} \rho _\tau (y_{t} - {{\tilde{q}}}_{t}(\vartheta ))- \frac{1}{n}\sum _{t=1}^{k_0} \rho _\tau (y_{t1} - q_{t1}(\vartheta ))\Big |=o_P (1). \end{aligned}$$

(17)

Due to (A5) and (A6), it suffices to show that

$$\begin{aligned} \sup _{\vartheta \in \varTheta ^*}\Big |\frac{1}{n}\sum _{t=1}^{k_0} (y_t - {{\tilde{q}}}_{t}(\vartheta ))I(\eta _t \le \varLambda _t^\dagger (\vartheta ) )- \frac{1}{n}\sum _{t=1}^{k_0} (y_{t1} - q_{t1}(\vartheta ))I(\eta _t \le \varLambda _t^\circ (\vartheta ) )\Big |=o_P (1), \end{aligned}$$

or equivalently, for $\delta \in (0,1)$,

$$\begin{aligned} \sup _{\vartheta \in \varTheta ^*}\Big |\frac{1}{n}\sum _{t=n^\delta }^{k_0} (y_t - q_{t}(\vartheta ))I(\eta _t \le \varLambda _t^\dagger (\vartheta ) )- \frac{1}{n}\sum _{t=n^\delta }^{k_0} (y_{t1} - q_{t1}(\vartheta ))I(\eta _t \le \varLambda _t^\circ (\vartheta ) )\Big |=o_P (1), \end{aligned}$$

which we can check to hold via using the Cauchy–Schwarz inequality, provided

$$\begin{aligned} \sup _{\vartheta \in \varTheta ^*}\frac{1}{n}\sum _{t=n^\delta }^{k_0} (I(\eta _t \le \varLambda _t^\dagger (\vartheta ) )-I(\eta _t \le \varLambda _t^\circ (\vartheta ) )^2=o_P (1). \end{aligned}$$

(18)

To verify (18), we partition $\varTheta ^*$ into ${{{\mathcal {D}}}}_j$ with diameter less than $\zeta$, $j=1,\ldots , N=N(\zeta )$, and choose points $\vartheta _j$ from ${{{\mathcal {D}}}}_j$. Then, in case of $\vartheta \in {{{\mathcal {D}}}}_j$, we have

$$\begin{aligned} I({\eta }_{t}\le \varLambda _t^\circ (\vartheta _j )+\varLambda _{tj}^{\dagger (1)})\le I({\eta }_{t}\le \varLambda _t^{\dagger } (\vartheta ))\le I({\eta }_{t}\le \varLambda _t^\circ (\vartheta _j)+\varLambda _{tj}^{\dagger (2)} ) \end{aligned}$$

with $\varLambda _{tj}^{\dagger (1)}=\inf _{\vartheta \in {{{\mathcal {D}}}}_j} \varLambda _t^\dagger (\vartheta )-\varLambda _t^\circ (\vartheta _j)$ and $\varLambda _{tj}^{\dagger (2)}=\sup _{\vartheta \in {{{\mathcal {D}}}}_j} \varLambda _t^\dagger (\vartheta )-\varLambda _t^\circ (\vartheta _j )$. Also,

$$\begin{aligned} I({\eta }_{t}\le \varLambda _t^\circ (\vartheta _j )+\varLambda _{tj}^{\circ (1)})\le I({\eta }_{t}\le \varLambda _t^{\circ } (\vartheta ))\le I({\eta }_{t}\le \varLambda _t^\circ (\vartheta _j)+\varLambda _{tj}^{\circ (2)} ) \end{aligned}$$

with $\varLambda _{tj}^{\circ (1)}=\inf _{\vartheta \in {{{\mathcal {D}}}}_j} \varLambda _t^\circ (\vartheta )-\varLambda _t^\circ (\vartheta _j)$ and $\varLambda _{tj}^{\circ (2)}=\sup _{\vartheta \in {{{\mathcal {D}}}}_j} \varLambda _t^\circ (\vartheta )-\varLambda _t^\circ (\vartheta _j )$.

By simple algebras using (A5) and (A6), it can be seen that

$$\begin{aligned} C_n:= & {} \max _{1\le j\le N}\max _{1\le k\ne l\le 2 }\frac{1}{n}\sum _{t=n^\delta }^{k_0} \Big [I(\eta _t \le \varLambda _t^\circ (\vartheta _j)+\varLambda _{tj}^{\dagger (k)} )-I(\eta _t \le \varLambda _t^\circ (\vartheta _j)+\varLambda _{tj}^{\circ (l)}) \Big ]^2\\= & {} \zeta O_P (1)+o_P (1). \end{aligned}$$

As $\zeta$ can be taken to be arbitrarily small, $C_n$ becomes $o_P(1)$, which ensures (18) and thereby (17) as the quantity in (18) is no more than $C_n$. Moreover, following the lines similar to the above, we can also check

$$\begin{aligned} \sup _{\vartheta \in \varTheta ^*}\Big |\frac{1}{n}\sum _{t=1}^{k_0} \rho _\tau (y_{t1} - q_{t1}(\vartheta ))- \kappa _0 E\rho _\tau (y_{t1} - q_{t1}(\vartheta ))\Big |=o_P (1), \end{aligned}$$

and thus, due to (17),

$$\begin{aligned} \sup _{\vartheta \in \varTheta ^*}\Big |\frac{1}{n}\sum _{t=1}^{k_0} \rho _\tau (y_{t} - {{\tilde{q}}}_{t}(\vartheta ))- \kappa _0 E\rho _\tau (y_{t1} - q_{t1}(\vartheta ))\Big |=o_P (1). \end{aligned}$$

(19)

Similarly, it can be shown that

$$\begin{aligned}{} & {} \sup _{\vartheta \in \varTheta ^*}\Big |\frac{1}{n}\sum _{t=k_0+1}^{n} \rho _\tau (y_{t} - {{\tilde{q}}}_{t}(\vartheta ))- (1-\kappa _0)\rho _\tau (y_{t2} - q_{t2}(\vartheta ))\Big |\nonumber \\{} & {} \quad \le \sup _{\vartheta \in \varTheta ^*}\Big |\frac{1}{n}\sum _{t=k_0+1}^{n} \rho _\tau (y_{t} - {{\tilde{q}}}_{t}(\vartheta ))- \frac{1}{n}\sum _{t=k_0+1}^{n} \rho _\tau (y_{t2} - q_{t2}(\vartheta ))\Big |\nonumber \\{} & {} \qquad + \sup _{\vartheta \in \varTheta ^*}\Big |\frac{1}{n}\sum _{t=k_0+1}^{n} \rho _\tau (y_{t2} - q_{t2}(\vartheta ))- (1-\kappa _0) E \rho _\tau (y_{t2} - q_{t2}(\vartheta ))\Big |\nonumber \\{} & {} \quad =o_P (1), \end{aligned}$$

(20)

then combining (19) and (20), we have

$$\begin{aligned}{} & {} \sup _{\vartheta \in \varTheta ^*}\Big |\frac{1}{n}\sum _{t=1}^{n} \rho _\tau (y_t - {{\tilde{q}}}_{t}(\vartheta ))- \big \{\kappa _0 E\rho _\tau (y_{t1}\\{} & {} \quad -\, q_{t1}(\vartheta ))+ (1-\kappa _0)E \rho _\tau (y_{t2} - q_{t2}(\vartheta ))\big \}\Big |= o_P (1), \end{aligned}$$

which validates the conclusion in Remark 4. Namely, (B1) holds true if $\kappa _0 E\rho _\tau (y_{t1} - q_{t1}(\vartheta ))+ (1-\kappa _0)E \rho _\tau (y_{t2} - q_{t2}(\vartheta ))$ has its maximum value uniquely at an interior point $\vartheta _0^*$ in $\varTheta ^*$. $\square$