On bivariate threshold Poisson integer-valued autoregressive processes

Abstract

To capture the bivariate count time series showing piecewise phenomena, we introduce a first-order bivariate threshold Poisson integer-valued autoregressive process. Basic probabilistic and statistical properties of the model are discussed. Conditional least squares and conditional maximum likelihood estimators, as well as their asymptotic properties, are obtained for both the cases that the threshold parameter is known or not. A new algorithm to estimate the threshold parameter of the model is also provided. Moreover, the nonlinearity test and forecasting problems are also addressed. Finally, some numerical results of the estimates and a real data example are presented.

Appendices

Appendix A: Proofs

We begin with some lemmas.

Lemma A1

Let ${\varvec{X}}$ be a bivariate non-negative integer-valued random vector. Let “${\varvec{A}}\circ$" and “${\varvec{B}}\circ$" be the 2$\times$2-matricial thinning operation defined in Definition 2.1. Then

• (i) ${\varvec{I}}\circ {\varvec{X}}={\varvec{X}}$ where ${\varvec{I}}$ denotes the unit matrix.

• (ii) ${\varvec{A}}\circ ({\varvec{B}}\circ {\varvec{X}})\overset{d}{=}({\varvec{A}}{\varvec{B}})\circ {\varvec{X}}$, where “$\overset{d}{=}$" stands for equal in distribution.

• (iii) $E({\varvec{A}}\circ {\varvec{X}})={\varvec{A}}E({\varvec{X}})$.

Proof

The above statements are straightforward to verify. See, for example, Franke and Rao (1993) and Latour (1997). $\square$

Lemma A2

Let ${\varvec{\varepsilon }}= (\varepsilon _1,\varepsilon _2)^{\textsf {T}} \in {\mathbb {N}}_0^2$ be a bivariate random vector, ${\varvec{A}}=diag(\alpha _1,\alpha _2)$ with $\alpha _i \in (0,1)$, i=1,2. Denote $h(\alpha _1,\alpha _2):=P({\varvec{A}}\circ {\varvec{\varepsilon }}={\varvec{0}})$ with “${\varvec{A}}\circ$" stands for the matricial thinning operation defined in Definition 2.1. Then $h(\alpha _1,\alpha _2)$ is a monotonic decreasing function with respect to $\alpha _i$ for a fixed $\alpha _j$, $i,j \in \{1,2\}$ and $i\ne j$.

Proof

Calculate to see that

$$\begin{aligned} h(\alpha _1,\alpha _2)&=P({\varvec{A}}\circ {\varvec{\varepsilon }}={\varvec{0}})\\&=\sum _{k=0}^\infty \sum _{s=0}^\infty P({\varvec{\varepsilon }}=(k,s)^{\textsf {T}}) P(\alpha _1\circ \varepsilon _1=0|{\varvec{\varepsilon }}=(k,s)^{\textsf {T}}) P(\alpha _2\circ \varepsilon _2=0|{\varvec{\varepsilon }}=(k,s)^{\textsf {T}})\\&=\sum _{k=0}^\infty \sum _{s=0}^\infty P({\varvec{\varepsilon }}=(k,s)^{\textsf {T}})(1-\alpha _1)^k(1-\alpha _2)^s. \end{aligned}$$

Note that $P({\varvec{\varepsilon }}=(k,s)^{\textsf {T}})$ is always nonnegative. This implies the conclusion is correct. $\square$

The proofs of Proposition 2.1 (i) Note that $\{{\varvec{X}}_t\}_{t \in {\mathbb {Z}}}$ is a Markov chain on ${\mathbb {N}}_0^2$. The irreducible and aperiodic properties of $\{{\varvec{X}}_t\}_{t \in {\mathbb {Z}}}$ are obviously since the transition probabilities given in (2.4) are positive for all the states.

Now we are going to show that $\{{\varvec{X}}_t\}_{t \in {\mathbb {Z}}}$ is positive recurrent. To show this, it is sufficient to prove that $\sum _{t=1}^\infty P^t({\varvec{0}}|{\varvec{0}})=+\infty$ with $P^t({\varvec{y}}|{\varvec{x}}):=P({\varvec{X}}_t={\varvec{y}}|{\varvec{X}}_{0}={\varvec{x}})$ denotes the t-steps transition probabilities of $\{{\varvec{X}}_t\}_{t \in {\mathbb {Z}}}$. Let $s_t:=I\{Z_t > r\}$, i.e., $s_t$ takes value 1 when $Z_t > r$ and 0 otherwise. Then, (2.3) can be written as

$$\begin{aligned} {\varvec{X}}_t = {\varvec{A}}_{s_t+1} \circ {\varvec{X}}_{t-1} + {\varvec{\varepsilon }}_t,~t \in {\mathbb {Z}}. \end{aligned}$$

(A.1)

By iterating (A.1) $t-1$ times, we have

$$\begin{aligned} {\varvec{X}}_t&={\varvec{A}}_{s_t+1} \circ {\varvec{A}}_{s_{t-1}+1} \circ \cdots \circ {\varvec{A}}_{s_1+1}\circ {\varvec{X}}_0+ \sum _{i=1}^{t-1}{\varvec{A}}_{s_t+1} \circ {\varvec{A}}_{s_{t-1}+1} \circ \cdots \circ {\varvec{A}}_{s_{i+1}+1}\circ {\varvec{\varepsilon }}_{i}+{\varvec{\varepsilon }}_{t}, \end{aligned}$$

implying

$$\begin{aligned} P^t({\varvec{0}}|{\varvec{0}})&=P\left( \sum _{i=1}^{t-1}{\varvec{A}}_{s_t+1} \circ {\varvec{A}}_{s_{t-1}+1} \circ \cdots \circ {\varvec{A}}_{s_{i+1}+1} \circ {\varvec{\varepsilon }}_{i}+{\varvec{\varepsilon }}_{t}={\varvec{0}}|{\varvec{X}}_0={\varvec{0}}\right) \\&=P({\varvec{\varepsilon }}_{t}={\varvec{0}},{\varvec{A}}_{s_t+1}\circ {\varvec{\varepsilon }}_{t-1}={\varvec{0}},\cdots , {\varvec{A}}_{s_t+1}\circ \cdots \circ {\varvec{A}}_{s_2+1}\circ {\varvec{\varepsilon }}_{1}={\varvec{0}}|{\varvec{X}}_0={\varvec{0}})\\&=\sum _{i_2=1}^2\sum _{i_3=1}^2\cdots \sum _{i_t=1}^2 P(s_2+1=i_2,s_3+1=i_3,\cdots ,s_t+1=i_t|{\varvec{X}}_0={\varvec{0}})\\&~~~~\times P({\varvec{\varepsilon }}_{t}={\varvec{0}},{\varvec{A}}_{i_t}\circ {\varvec{\varepsilon }}_{t-1}={\varvec{0}},\cdots , {\varvec{A}}_{i_t}\circ \cdots \circ {\varvec{A}}_{i_2}\circ {\varvec{\varepsilon }}_{1}={\varvec{0}}|{\varvec{X}}_0={\varvec{0}})\\&=\sum _{i_2=1}^2\sum _{i_3=1}^2\cdots \sum _{i_t=1}^2 P(s_2+1=i_2,s_3+1=i_3,\cdots ,s_t+1=i_t|{\varvec{X}}_0={\varvec{0}})\\&~~~~\times P({\varvec{\varepsilon }}_{t}={\varvec{0}})P({\varvec{A}}_{i_t}\circ {\varvec{\varepsilon }}_{t-1}={\varvec{0}})\cdots P(({\varvec{A}}_{i_t} \cdots {\varvec{A}}_{i_2})\circ {\varvec{\varepsilon }}_{1}={\varvec{0}}). \end{aligned}$$

Denote ${\varvec{A}}_{\max }=diag(\max (\alpha _{1,1},\alpha _{2,1}),\max (\alpha _{1,2},\alpha _{2,2}))$. It follows by Lemmas A1 and A2 that

$$\begin{aligned} P^t({\varvec{0}}|{\varvec{0}})\ge \prod _{j=0}^{t-1}P({\varvec{A}}_{\max }^j\circ {\varvec{\varepsilon }}_{t-j}={\varvec{0}}). \end{aligned}$$

(A.2)

By the proof of Theorem 1 in Franke and Rao (1993), we known that the right side of (A.2) is bounded from below by a positive number as $t \rightarrow \infty$, which implies that $\lim _{t \rightarrow \infty } P^t({\varvec{0}}|{\varvec{0}})\ne 0$. Therefore, we conclude that $\sum _{t=1}^{+\infty }P^t({\varvec{0}}|{\varvec{0}})=+\infty$. Thus, it follows by Proposition 4.2.3 and Corollary 4.2.4 in Ross (1996) that $\{{\varvec{X}}_t\}_{t \in {\mathbb {Z}}}$ is a recurrent Markov chain. We go on to show that state ${\varvec{0}}$ is positive recurrent. Note that ${\varvec{0}}$ is currently a recurrent state and $\lim _{t \rightarrow \infty } P^t({\varvec{0}}|{\varvec{0}})\ne 0$, it follows by Theorem 4.3.3 in Ross (1996) that state ${\varvec{0}}$ is positive recurrent. Since $\{{\varvec{X}}_t\}_{t \in {\mathbb {Z}}}$ is irreducible, it follows that all states of $\{{\varvec{X}}_t\}$ are positive recurrent. This proves that $\{{\varvec{X}}_t\}_{t \in {\mathbb {Z}}}$ is a positive recurrent Markov chain and hence ergodic.

(ii) The existence of a strictly stationary solution of (2.3) is followed by (i) immediately, since it is a classical conclusion. For completeness and reader’s convenience, we refer to Theorem 1.2.1 in Rosenblatt (1971) for a proof.                                                                                                             $\Box$

The proofs of Proposition 2.2 The proofs of this Proposition are simple (for (i)–(iii)) or tedious calculations (for (iv)), so we omit the details here. However, the full version proof is available upon request. $\square$

The proof of Theorem 3.1 For sake of simplicity, we only prove (3.8) holds for $\hat{{\varvec{\theta }}}_{1,CLS}$. Denote $S_{i,0}=0$, $S_{i,n}=-\frac{1}{2}\frac{\partial Q({\varvec{\theta }}_1,{\varvec{\theta }}_2)}{\partial \alpha _{i,1}}=\sum _{t=1}^n(X_{1,t}-\alpha _{1,1}X_{1,t-1}I_{1,t}-\alpha _{2,1}X_{1,t-1}I_{2,t}-\lambda _1)X_{1,t-1}I_{i,t}$, $i=1,2$. Then we have

$$\begin{aligned} E(S_{i,n}|{\mathcal {F}}_{n-1})&=E(S_{i,n-1}+(X_{1,n}-\alpha _{1,1}X_{1,n-1}I_{1,n}-\alpha _{2,1}X_{1,n-1}I_{2,n}-\lambda _1)X_{1,n-1}I_{i,n}|{\mathcal {F}}_{n-1})\nonumber \\&=S_{i,n-1}+E((X_{1,n}-\alpha _{1,1}X_{1,n-1}I_{1,n}-\alpha _{2,1}X_{1,n-1}I_{2,n}-\lambda _1)X_{1,n-1}I_{i,n}|{\mathcal {F}}_{n-1})\nonumber \\&=S_{i,n-1},~i=1,2. \end{aligned}$$

(A.6)

Thus, $\{S_{i,n},{\mathcal {F}}_{n},n\ge 0\}$ ($i=1,2$) is a martingale. Note that $E\Vert {\varvec{\varepsilon }}_t\Vert ^4<\infty$ implies $E\Vert {\varvec{X}}_t\Vert ^4$. Thus, the sequence $\{(X_{1,n}-\alpha _{1,1}X_{1,n-1}I_{1,n}-\alpha _{2,1}X_{1,n-1}I_{2,n}-\lambda _1)^2X_{1,n-1}^2I_{i,n},n\ge 1\}$ is uniformly integrable. By the Theorem 1.1 of Billingsley (1961), as $n\rightarrow \infty$,

$$\begin{aligned} \frac{1}{n}\sum _{t=1}^n(X_{1,t}-\alpha _{1,1}X_{1,t-1}I_{1,t}-\alpha _{2,1}X_{1,t-1}I_{2,t}-\lambda _1)^2X_{1,t-1}^2I_{i,t}\overset{a.s}{\longrightarrow } \sigma _{ii},~i=1,2, \end{aligned}$$

where $\sigma _{ii}:=E[(X_{1,1}-\alpha _{1,1}X_{1,0}I_{1,1}-\alpha _{2,1}X_{1,0}I_{2,1}-\lambda _1)^2X_{1,0}^2I_{i,1}]$. Thus, by the Corollary 3.2 in Hall and Heyde (1980), and the martingale central limit theorem, we have, as $n\rightarrow \infty ,$

$$\begin{aligned} \frac{1}{\sqrt{n}}S_{i,n}\overset{L}{\longrightarrow }N(0,\sigma _{ii})~i=1,2. \end{aligned}$$

Let $S_{3,n}=-\frac{1}{2}\frac{\partial Q({\varvec{\theta }}_1,{\varvec{\theta }}_2)}{\partial \lambda _1} =\sum _{t=1}^n(X_{1,t}-\alpha _{1,1}X_{1,t-1}I_{1,t}-\alpha _{2,1}X_{1,t-1}I_{2,t}-\lambda _1)$. Using similar arguments as the proof of (A.6), we can prove that $\{S_{3,n},{\mathcal {F}}_{n},n\ge 0\}$ is a martingale. Then, as $n\rightarrow \infty ,$ we have $\frac{1}{n}\sum _{t=1}^n(X_{1,t}-\alpha _{1,1}X_{1,t-1}I_{1,t}-\alpha _{2,1}X_{1,t-1}I_{2,t}-\lambda _1)^2\overset{a.s}{\longrightarrow } \sigma _{33}$ and $\frac{1}{\sqrt{n}}S_{3,n}\overset{L}{\longrightarrow }N(0,\sigma _{33})$, where $\sigma _{33}:=E[(X_{1,1}-\alpha _{1,1}X_{1,0}I_{1,1}-\alpha _{2,1}X_{1,0}I_{2,1}-\lambda _1)^2]$. In the same way, for any ${\varvec{c}}=(c_1,c_2,c_3)^{\textsf {T}}\in {\mathbb {R}}^3\backslash (0,0,0)^{\textsf {T}}$, we have that $\{{\varvec{c}}^{\textsf {T}}{\varvec{S}}_n$, ${\mathcal {F}}_{n},n\ge 1\}$ is a martingale, where ${\varvec{S}}_n=(S_{1,n},S_{2,n},S_{3,n})^{\textsf {T}}$. By the ergodic and stationary properties of $\{{\varvec{X}}_t\}$, we have that as $n\rightarrow \infty$,

$$\begin{aligned}{} & {} \frac{1}{n}\sum _{t=1}^n (X_{1,t}-\alpha _{1,1}X_{1,t-1}I_{1,t}-\alpha _{2,1}X_{1,t-1}I_{2,t}-\lambda _1)^2 (c_1X_{1,t-1}I_{1,t}\\ {}{} & {} \qquad \quad +c_2X_{1,t-1}I_{2,t}+c_3)^2 \overset{a.s}{\longrightarrow }\sigma ^2, \end{aligned}$$

where $\sigma ^2:=E[(X_{1,1}-\alpha _{1,1}X_{1,0}I_{1,1}-\alpha _{2,1}X_{1,0}I_{2,1}-\lambda _1)^2 (c_1X_{1,0}I_{1,1}+c_2X_{1,0}I_{2,1}+c_3)^2]$. Then, we have $\frac{1}{\sqrt{n}}{\varvec{c}}^{\textsf {T}}{\varvec{S}}_n \overset{L}{\longrightarrow }N(0,\sigma ^2).$ Thus, by the $\mathrm{Cram\acute{e}r}$-$\textrm{Wold}$ device (see Chapter 2.3 in van der Vaart (1998)), we obtain

$$\begin{aligned} \frac{1}{\sqrt{n}} {\varvec{S}}_n \overset{L}{\longrightarrow }N({\varvec{0}},{\varvec{W}}_1). \end{aligned}$$

Denote ${\varvec{U}}_{t}=(X_{1,t-1}I_{1,t},X_{1,t-1}I_{2,t},1)^{\textsf {T}}$. Thus, we have ${\varvec{M}}_1=\sum _{t=1}^n {\varvec{U}}_{t}{\varvec{U}}_{t}^{\textsf {T}}$ and the CLS-estimators $\hat{{\varvec{\theta }}}_{1,CLS}=(\sum _{t=1}^n {\varvec{U}}_{t}{\varvec{U}}_{t}^{\textsf {T}})^{-1}(\sum _{t=1}^n X_{1,t}{\varvec{U}}_{t})$. Since $\{{\varvec{X_{t}}}\}$ is a stationary and ergodic process, by the ergodic theorem, we have

$$\begin{aligned} \frac{1}{n} {\varvec{M}}_1=\frac{1}{n} \sum _{t=1}^n{\varvec{U}}_{t}{\varvec{U}}_{t}^{\textsf {T}}\overset{P}{\longrightarrow } E\left( \frac{\partial g_1({\varvec{\theta }}_{1,0},X_{1,0})}{\partial {\varvec{\theta }}_1} \frac{\partial g_1({\varvec{\theta }}_{1,0},X_{1,0})}{\partial {\varvec{\theta }}_1^{\textsf {T}}}\right) ={\varvec{V}}_1. \end{aligned}$$

After some algebra, we have $\hat{{\varvec{\theta }}}_{1,CLS}-{\varvec{\theta }}_{1,0}={\varvec{M}}_1^{-1}{\varvec{S}}_n.$ Therefore,

$$\begin{aligned} \sqrt{n}(\hat{{\varvec{\theta }}}_{1,CLS}-{\varvec{\theta }}_{1,0})=\left( \frac{1}{n} {\varvec{M}}_1\right) ^{-1}\frac{1}{\sqrt{n}}{\varvec{S}}_n\overset{L}{\longrightarrow }N({\varvec{0}},{\varvec{V}}_1^{-1}{\varvec{W}}_1{\varvec{V}}_1^{-1}). \end{aligned}$$

This ends the proof. $\square$

The proof of Theorem 3.2 The proof of Theorem 3.2 is similar to Theorem 3.1, so omitted.$\square$

The proof of Theorem 3.3 To prove Theorem 3.3, we want to apply the results of Theorems 2.1 and 2.2 in Billingsley (1961) on estimates for the parameters of Markov processes. For this purpose, we need to check that conditions (C1)–(C6) in Sect. 3.2 hold, which implies the regularity conditions of Theorems 2.1 and 2.2 in Billingsley (1961) hold.

Conditions (C1)–(C3) hold by the properties of $BP(\lambda _1,\lambda _2,\phi )$ distribution. We go on to prove (C4) holds. Note that for any ${\varvec{\vartheta }}' \in {\mathcal {B}}$ and the neighborhood U of ${\varvec{\vartheta }}'$, $\sum _{{\varvec{k}}\ge {\varvec{0}}} \sup _{{\varvec{\vartheta }} \in U} f({\varvec{k}},{\varvec{\vartheta }})<\infty$ holds trivially. Furthermore, we can calculate to see that

$$\begin{aligned} \left( -1 +\frac{k_j}{\lambda _j-\phi }\right) f({\varvec{k}},{\varvec{\vartheta }}) \le \frac{\partial f({\varvec{k}},{\varvec{\vartheta }})}{\partial \lambda _j} \le \left( 1 +\frac{2k_j}{\lambda _j-\phi }\right) f({\varvec{k}},{\varvec{\vartheta }}),~j=1,2,\nonumber \\ \end{aligned}$$

(A.7)

and

$$\begin{aligned} \left( 1-\sum _{j=1}^2\frac{k_j}{\lambda _j-\phi }\right) f({\varvec{k}},{\varvec{\vartheta }}) \le \frac{\partial f({\varvec{k}},{\varvec{\vartheta }})}{\partial \phi }&\le \left( 1+\sum _{j=1}^2\frac{2k_j}{\lambda _j-\phi }+\frac{k_j}{\phi }\right) f({\varvec{k}},{\varvec{\vartheta }}), \end{aligned}$$

(A.8)

where $k_j$ is the jth component of ${\varvec{k}}$. Thus, we have $\sum _{{\varvec{k}}\ge {\varvec{0}}} \sup _{{\varvec{\vartheta }} \in U}|f_u({\varvec{k}},{\varvec{\vartheta }})|\le C_1\cdot E\Vert {\varvec{\varepsilon }}_1\Vert <\infty$, for some suitable constant $C_1$, $u=1,2,3$. Thus, by using similar arguments, we can see that $\sum _{{\varvec{k}}\ge {\varvec{0}}} \sup _{{\varvec{\vartheta }} \in U}|f_{uv}({\varvec{k}},{\varvec{\vartheta }})|\le C_2\cdot E\Vert {\varvec{\varepsilon }}_1\Vert ^2<\infty$, for some suitable constant $C_2$, $u,v=1,2,3$. Therefore, condition (C4) holds true.

To prove (C5), we denote ${\varvec{1}}=(1,1)^{\textsf {T}}$, $\delta =\min (\phi ,\lambda _1-\phi ,\lambda _2-\phi )/3$. Thus, we have

$$\begin{aligned} 1+\sum _{j=1}^2\frac{2k_j}{\lambda _j-\phi }+\frac{k_j}{\phi } \le 1+ \frac{{\varvec{1}}^{\textsf {T}}{\varvec{k}}}{\delta }. \end{aligned}$$

Therefore, let

$$\begin{aligned} {\psi }_{u}({\varvec{n}})=1+ \frac{{\varvec{1}}^{\textsf {T}}{\varvec{n}}}{\delta },~ {\psi }_{uv}({\varvec{n}})=\left( 1+ \frac{{\varvec{1}}^{\textsf {T}}{\varvec{n}}}{\delta }\right) ^2,~ {\psi }_{uvw}({\varvec{n}})=\left( 1+ \frac{{\varvec{1}}^{\textsf {T}}{\varvec{n}}}{\delta }\right) ^3,~u,v,w=1,2,3, \end{aligned}$$

then we have that $\left| f_u({\varvec{k}},{\varvec{\vartheta }})\right| \le {\psi }_{u}({\varvec{n}})f({\varvec{k}},{\varvec{\vartheta }})$, $\left| f_{uv}({\varvec{k}},{\varvec{\vartheta }})\right| \le {\psi }_{uv}({\varvec{n}})f({\varvec{k}},{\varvec{\vartheta }})$, and $\left| f_{uvw}({\varvec{k}},{\varvec{\vartheta }})\right| \le {\psi }_{uvw}({\varvec{n}})f({\varvec{k}},{\varvec{\vartheta }})$. Furthermore, note that $E\Vert {\varvec{\varepsilon }}_t\Vert ^3<\infty$ implies $E\Vert {\varvec{X}}_t\Vert ^3$. We have that (C5) holds.

Now, let us verify the last condition also holds. To this end, we need to check the following statements are all true.

• (S1) $E\left| \frac{\partial }{\partial \alpha _{i,j}}\log p({\varvec{X}}_2|{\varvec{X}}_1,{\varvec{\theta }})\frac{\partial }{\partial \alpha _{u,v}}\log p({\varvec{X}}_2|{\varvec{X}}_1,{\varvec{\theta }})\right| < \infty ,~i,j,u,v=1,2$;

• (S2) $E\left| \frac{\partial }{\partial \phi } \log p({\varvec{X}}_2|{\varvec{X}}_1,{\varvec{\theta }})\right| ^2 < \infty$, $E\left| \frac{\partial }{\partial \lambda _{i}} \log p({\varvec{X}}_2|{\varvec{X}}_1,{\varvec{\theta }})\right| ^2 < \infty ,~i=1,2$;

• (S3) $E\left| \frac{\partial }{\partial \alpha _{i,j}}\log p({\varvec{X}}_2|{\varvec{X}}_1,{\varvec{\theta }})\frac{\partial }{\partial \lambda _{u}}\log p({\varvec{X}}_2|{\varvec{X}}_1,{\varvec{\theta }})\right| < \infty ,~i,j,u=1,2$;

• (S4) $E\left| \frac{\partial }{\partial \alpha _{i,j}}\log p({\varvec{X}}_2|{\varvec{X}}_1,{\varvec{\theta }})\frac{\partial }{\partial \phi }\log p({\varvec{X}}_2|{\varvec{X}}_1,{\varvec{\theta }})\right| < \infty ,~i,j=1,2$;

• (S5) $E\left| \frac{\partial }{\partial \lambda _j}\log p({\varvec{X}}_2|{\varvec{X}}_1,{\varvec{\theta }})\frac{\partial }{\partial \phi }\log p({\varvec{X}}_2|{\varvec{X}}_1,{\varvec{\theta }})\right| < \infty ,~j=1,2$.

We shall first prove statement (S1). Recall that $p({\varvec{X}}_t|{\varvec{X}}_{t-1},{\varvec{\theta }})=P({\varvec{X}}_t={\varvec{x}}_t|{\varvec{X}}_{t-1}={\varvec{x}}_{t-1})$ is the transition probability, i.e.,

$$\begin{aligned} p({\varvec{X}}_t|{\varvec{X}}_{t-1},{\varvec{\theta }})=\sum _{k_1=0}^{x_{1,t} \wedge x_{1,t-1}}\sum _{k_2=0}^{x_{2,t} \wedge x_{2,t-1}}p_1(k_1)p_2(k_2)f({\varvec{x}}_t-{\varvec{k}},{\varvec{\vartheta }}), \end{aligned}$$

where $p_j(x) =\sum _{i=1}^2 I_{i,t} \left( {\begin{array}{c}x_{j,t-1}\\ x\end{array}}\right) \alpha _{i,j}^{x}(1-\alpha _{i,j})^{x_{j,t-1}-x}$, $j=1,2$. Denote $\alpha _{\max }=\max (\alpha _{1,1},\alpha _{1,2},\alpha _{2,1},\alpha _{2,2})$ and $\alpha _{\min }=\min (\alpha _{1,1},\alpha _{1,2},\alpha _{2,1},\alpha _{2,2})$. Thus, a direct calculation and properly scaling gives

$$\begin{aligned} -p_j(x)\frac{x_{j,t-1}}{1-\alpha _{\max }} \le \frac{\partial p_j(x)}{\partial \alpha _{i,j}} \le p_j(x) \frac{x_{j,t-1}}{\alpha _{\min }},~i,j=1,2, \end{aligned}$$

(A.9)

which implies $E\left| \frac{\partial }{\partial \alpha _{i,j}}\log P({\varvec{X}}_1,{\varvec{X}}_2)\right| ^2< C_3\cdot E\Vert {\varvec{X}}_1\Vert ^2 < \infty ,$ for some suitable constant $C_3$.

Next we will prove statement (S2). Recall that by (C4), we have $\left| f_u({\varvec{k}},{\varvec{\vartheta }})\right| \le {\psi }_{u}({\varvec{n}})f({\varvec{k}},{\varvec{\vartheta }})$ ($u=1,2,3$), which implies $E\left| \frac{\partial }{\partial \lambda _{i}} \log P({\varvec{X}}_1,{\varvec{X}}_2)\right| ^2 \le E \psi _i^2({\varvec{X}}_1) < \infty ,~i=1,2,$ and

$$\begin{aligned} E\left| \frac{\partial }{\partial \phi } \log P({\varvec{X}}_1,{\varvec{X}}_2)\right| ^2 \le E \psi _3^2({\varvec{X}}_1) < \infty . \end{aligned}$$

Therefore, (S2) holds.

Lastly, by (A.9) and (C4) we can verify that statements (S3)–(S5) hold. Therefore, the Fisher information matrix ${\varvec{I}}({\varvec{\theta }})$ is well-defined. Finally, some elementary but tedious calculation shows that (C6) is satisfied, too.

The proof is complete. $\square$

Appendix B: Figures

The time series plots of Poisson BINAR(1) model (Pedeli and Karlis 2011) with following parameter settings are given in Fig. 9.

• Scenario D. $(\alpha _{1},\alpha _{2},\lambda _1,\lambda _2,\phi )$ $=(0.5,0.25,4,2,1)$.

• Scenario E. $(\alpha _{1},\alpha _{2},\lambda _1,\lambda _2,\phi )$ $=(0.2,0.4,2,3,1)$.

• Scenario F. $(\alpha _{1},\alpha _{2},\lambda _1,\lambda _2,\phi )$ $=(0.75,0.4,2,3,1)$.

Fig. 10

Time series plots of Poisson BINAR(1) model and Poisson BTINAR(1) model

As a comparison, the time series plots of Scenarios A–C are also drawn in Fig. 9. We can see from Fig. 9 that there is no piecewise characteristics in the time series of Scenarios D, E, and F, while there are obvious piecewise characteristics for Scenarios A, B, and C. Therefore, it is clearly to see that the BTINAR(1) models can capture the time series with piecewise component.