A bivariate integer-valued bilinear autoregressive model with random coefficients

Abstract

This paper introduces a new bivariate autoregressive model with random coefficients for the time series of counts. It is composed of two components, the survival and the innovation component. The dependence between two series that comprise the bivariate model stems from both of these components. The introduced model is achieved by defining a bilinear model and the existence of a unique strict stationarity of it is proved. The method of moments is examined for parameters estimation. The practical aspect of the model is discussed by using a real-life data example.

Appendix

Proof of Theorem 1

Proof

Let us define a random sequence $\big \{\varvec{Z}_t^{(n)}=(X_{1,t}^{(n)}, X_{2,t}^{(n)})'\big \}$ such that

$$\begin{aligned} \varvec{Z}_t^{(n)}=\left\{ \begin{array}{ll}\varvec{0}, &{} \quad n<0,\\ \varvec{e}_t, &{} \quad n=0,\\ \varvec{A}_{(t)}\circ (\varvec{B}_{t-1}\varvec{Z}_{t-1}^{(n-1)})+\varvec{e}_t, &{} \quad n>0,\end{array}\right. \end{aligned}$$

where n is the index of the sequence element and $\varvec{A}_{(t)}$ means that the counting series which defines the binomial thinning operator is fixed at time t, i.e. at time t, when n tends to infinity, the counting series remains unchanged. Also, similarly as in Popović et al. (2016), define the Hilbert space $L^2(\Omega , \mathcal {F}, P)=\{\varvec{Z}:E(\varvec{Z}\varvec{Z}')<\infty \}$, where the measure between two random variables $\varvec{Y}$ and $\varvec{W}$ is defined as $E(\varvec{Y}\varvec{W}')$. The idea is to prove that the sequence $\{\varvec{Z}_t^{(n)}\}$ is strictly stationary and that it converges in probability to the solution of Eq. (3).

The first step is to show that the sequence $\{\varvec{Z}_t^{(n)}\}$ is non-decreasing. From the definition of the sequence, we have

$$\begin{aligned} \varvec{Z}_{t}^{(0)}=\varvec{e}_t\le \varvec{A}_{(t)}\circ (\varvec{B}_{t-1}\varvec{Z}_{t-1}^{(0)})+\varvec{e}_t=\varvec{Z}_t^{(1)}. \end{aligned}$$

Following the induction hypotheses, we have that $\varvec{Z}_{t-1}^{(k-1)}\le \varvec{Z}_{t-1}^{(k)}$, which implies that

$$\begin{aligned} \varvec{A}_{(t)}\circ (\varvec{B}_{t-1}\varvec{Z}_{t-1}^{(k-1)})+\varvec{e}_t\le \varvec{A}_{(t)}\circ (\varvec{B}_{t-1}\varvec{Z}_{t-1}^{(k)})+\varvec{e}_t. \end{aligned}$$

Thus, we have $\varvec{Z}_{t}^{(k)}\le \varvec{Z}_{t}^{(k+1)}$. By induction, we can conclude that the sequence $\{\varvec{Z}_t^{(n)}\}$ is non-decreasing.

The proof of the strict stationarity of the sequence $\{\varvec{Z}_t^{(n)}\}$ is similar to the one in Doukhan et al. (2006) so we will omit it here.

Now, let us focus on the convergence of the sequence $\{\varvec{Z}_t^{(n)}\}$ in the defined Hilbert space. Having in mind strict stationarity, we have that $\varvec{Z}_t^{(n)}-\varvec{Z}_{t}^{(n-1)}{\mathop {=}\limits ^{d}}\varvec{A}_{(t)}\circ (\varvec{B}_{t-1}(\varvec{Z}_{t-1}^{(n-1)}-\varvec{Z}_{t-1}^{(n-2)}))$. Taking the expectation of both sides, we obtain

$$\begin{aligned}&E(\varvec{Z}_t^{(n)}-\varvec{Z}_{t}^{(n-1)})=\varvec{A}\varvec{\Lambda }E(\varvec{Z}_{t-1}^{(n-1)}-\varvec{Z}_{t-1}^{(n-2)}) =\ldots \\&=\varvec{A}^{n-1}\varvec{\Lambda }^{n-1}E(\varvec{A}_{(t-n+1)}\circ (\varvec{B}_{t-n}\varvec{e}_{t-n}))= \varvec{A}^{n}\varvec{\Lambda }^{n-1}E\left[ \begin{array}{c}\varepsilon _{1,t-n}^2\\ \varepsilon _{2,t-n}^2\end{array}\right] \\&= \left[ \begin{array}{cc}(\alpha _1p_1)^n &{} 0\\ 0 &{} (\alpha _2p_2)^n\end{array}\right] \cdot \left[ \begin{array}{cc}\lambda _1^{n-1} &{} 0\\ 0 &{} \lambda _2^{n-1}\end{array}\right] \cdot E\left[ \begin{array}{c}\varepsilon _{1,t-n}^2\\ \varepsilon _{2,t-n}^2\end{array}\right] . \end{aligned}$$

The right side converges to zero matrix as n tends to infinity if and only if $\alpha _jp_j\lambda _j<1$, $j=1,2$. Note that if we define a random array $\varvec{z}_n=E(\varvec{Z}_t^{(n)}-\varvec{Z}_{t}^{(n-1)})$, then we have

$$\begin{aligned} \varvec{z}_n=\varvec{A}\varvec{\Lambda }\varvec{z}_{n-1}. \end{aligned}$$

Thus, when $\alpha _jp_j\lambda _j<1$, $j=1,2$, the series $\sum _{n=1}^{\infty }\varvec{z}_n$ is convergent.

Since the sequence $\{\varvec{Z}_t^{(n)}\}$ is non-negative and non-decreasing, we should prove that the limit of such sequence is almost surely finite. By using a similar approach as in Doukhan et al. (2006), we define sets $B_{\infty }=\{\omega : \varvec{Z}_t(\omega )=\infty \}$ and $B_{n}=\{\omega : \varvec{Z}_t^{(n)}(\omega )-\varvec{Z}_t^{(n-1)}(\omega )>\varvec{0}\}$. Then, $B_{\infty }=\{\omega : \varvec{Z}_t(\omega )=\infty \}=\underset{n\rightarrow \infty }{\limsup }B_n$. Also, we have that

$$\begin{aligned} E(\varvec{Z}_t^{(n)}-\varvec{Z}_t^{(n-1)})\ge \sum _{i,j=1}^\infty P(\omega : \varvec{Z}_t^{(n)}(\omega )-\varvec{Z}_t^{(n-1)}(\omega )=(i,j)')=P(B_n). \end{aligned}$$

Having in mind previous results, we can conclude that the series $\sum _{n=1}^{\infty }P(B_n)$ is convergent when $\alpha _jp_j\lambda _j<1$, $j=1,2$. A direct implication of the Borel–Cantelli lemma is that $\underset{n\rightarrow \infty }{\limsup }B_n=0$, i.e. $P(B_{\infty })=0$. Since the sequence $\{\varvec{Z}_t^{(n)}\}$ is strictly stationary, then its almost sure limit $\varvec{Z}_t$ is strictly stationary as well.

The uniqueness of the solution of Eq. (3) can be proved by using the same technique as in Doukhan et al. (2006). $\square$