First-order random coefficients integer-valued threshold autoregressive processes

Abstract

In this paper, we introduce a first-order random coefficient integer-valued threshold autoregressive process, which is based on binomial thinning. Basic probabilistic and statistical properties of this model are discussed. Conditional least squares and conditional maximum likelihood estimators are derived for both the cases that the threshold variable is known or not. The asymptotic properties of the estimators are established. Moreover, forecasting problem is addressed. Finally, some numerical results of the estimates and a real data example are presented.

Appendix

Proof of Lemma 2.1

By the definition of the thinning operator “$\circ$” defined in (1.2), we have

$$\begin{aligned} P(\phi _t\circ X=0)&=\sum _{m=0}^\infty P(X=m)P(\phi _t \circ X=0|X=m)\\&=\sum _{m=0}^\infty P(X=m)\int f(\phi _t)P(\phi _t\circ X=0|\phi _t,X=m)d\phi _t\\&=\sum _{m=0}^\infty P(X=m)\frac{\varGamma (k)}{\varGamma (k-q)}\frac{\varGamma (k-q+m)}{\varGamma (k+m)}\\&=\sum _{m=0}^\infty P(X=m)g(m,q), \end{aligned}$$

where $f(\cdot )$ denotes the density function of $\phi _t$, $\varGamma (\cdot )$ is the Gamma function, and

$$\begin{aligned} g(m,q)=\frac{(k-q)(k-q+1)\cdots (k-q+m-2)(k-q+m-1)}{k(k+1)\cdots (k+m-2)(k+m-1)}. \end{aligned}$$

It is easy to verify that (i) $0<g(m,q)<1$ and, (ii) for fixed m, g(m, q) monotonously decreases with respect to q, which implies that $P(\phi _t\circ X=0)$ is a monotonic decreasing function with respect to q when $0<q<k$. This completes the proof. $\square$

Proof of Proposition 2.1

It is easy to see that $\{X_t\}_{t \in \mathbb {Z}}$ is a Markov chain with state space $\mathbb {N}_0$ and transition probabilities:

$$\begin{aligned}&P(X_t=j|X_{t-1}=i)\nonumber \\&\quad =P((\phi _{1,t}\circ X_{t-1}+Z_{t})I_{1,t}+(\phi _{2,t}\circ X_{t-1}+Z_{t})I_{2,t}=j|X_{t-1}=i)\nonumber \\&\quad =p(i,j,q_{1},\lambda )I_{1,t}+p(i,j,q_{2},\lambda )I_{2,t}\nonumber \\&\quad =p(i,j,q_{1}I_{1,t}+q_{2}I_{2,t},\lambda ), \end{aligned}$$

(7.1)

where

$$\begin{aligned} \ p(i,j,q_{l},\lambda )= & {} \sum _{m=0}^{\min (i,j)} \left( \begin{array}{cc} i\\ m \end{array}\right) e^{-\lambda }\frac{\lambda ^{j-m}}{(j-m)!} \frac{\varGamma (k)\varGamma (q_{l}+m)\varGamma (k+i-q_{l}-m)}{\varGamma (q_{l})\varGamma (k-q_{l})\varGamma (k+i)}>0,\nonumber \\&\quad ~l=1,2. \end{aligned}$$

(7.2)

From the expression above, it follows that the chain is irreducible and aperiodic. Furthermore, to show that $\{X_t\}_{t \in \mathbb {Z}}$ is positive recurrent it is sufficient to prove that $\sum _{t=1}^{\infty }P^t(0,0)=+\infty$ (since $\{X_t\}_{t \in \mathbb {Z}}$ is irreducible) with $P^t(x,y):=P(X_t=y|X_0=x)$. For convenience, we denote

$$\begin{aligned} S_t={\left\{ \begin{array}{ll} 0, &{}\quad \text {if}\ X_{t-1} {\le } r,\\ 1, &{}\quad \text {if} \ X_{t-1} {>} r. \end{array}\right. } \end{aligned}$$

Then (2.1) can be rewritten as

$$\begin{aligned} X_{t}=\phi _{S_{t}+1,t}\circ X_{t-1}+Z_{t}. \end{aligned}$$

(7.3)

By iterating (7.3) $t-1$ times, we have

$$\begin{aligned} X_t= & {} \phi _{S_t+1,t}\circ \phi _{S_{t-1}+1,t-1} \circ \cdots \circ \phi _{S_1+1,1}\circ X_0\\&+\left( \sum _{i=1}^{t-1}\phi _{S_t+1,t} \circ \phi _{S_{t-1}+1,t-1} \circ \cdots \circ \phi _{S_{i+1}+1,i+1}\circ Z_{i}\right) +Z_{t}. \end{aligned}$$

This allows us to write

$$\begin{aligned} P^t(0,0)&=P\left( \sum _{i=1}^{t-1}\phi _{S_t+1,t}\circ \phi _{S_{t-1}+1,t-1}\circ \cdots \circ \phi _{S_{i+1}+1,i+1}\circ Z_{i}+Z_{t}=0|X_0=0\right) \\&=P\left( Z_{t}=0,\phi _{S_t+1,t}\circ Z_{t-1}=0,\ldots , \phi _{S_t+1,t}\circ \cdots \circ \phi _{S_2+1,2}\circ Z_{1}=0|X_0=0\right) \\&=\sum _{i_2=1}^2\sum _{i_3=1}^2\cdots \sum _{i_t=1}^2 P\left( S_2+1=i_2,S_3+1=i_3,\ldots ,S_t+1=i_t|X_0=0\right) \nonumber \\&\quad \times \, P\left( Z_{t}=0,\phi _{i_t,t}\circ Z_{t-1}=0,\ldots , \phi _{i_t,t}\circ \cdots \circ \phi _{i_2,2}\circ Z_{1}=0|X_0=0\right) \\&=\sum _{i_2=1}^2\sum _{i_3=1}^2\cdots \sum _{i_t=1}^2 P(S_2+1=i_2,S_3+1=i_3,\ldots ,S_t+1=i_t|X_0=0)\\&\quad \times \, P(Z_{t}=0)\cdot P(\phi _{i_t,t}\circ Z_{t-1}=0)\cdots P(\phi _{i_t,t}\circ \phi _{i_{t-1},t-1}\circ \cdots \circ \phi _{i_2,2}\circ Z_{1}=0). \end{aligned}$$

Denote $q_{\max }=\max \{q_1,q_2\}$, let $\phi _{L,t}\sim Beta(q_{\max },k-q_{\max })$. By Lemma 2.1 and the properties of binomial distribution, we have

$$\begin{aligned} P^t(0,0)&\ge \sum _{i_2=1}^2\sum _{i_3=1}^2\cdots \sum _{i_t=1}^2 P\left( S_2+1=i_2,S_3+1=i_3,\ldots ,S_t+1=i_t|X_0=0\right) \\&~~~~\times P(Z_{t}=0)\cdot P(\phi _{L,t}\circ Z_{t-1}=0)\cdot P\left( \phi _{L,t}\circ \phi _{i_{t-1},t-1}\circ Z_{t-2}=0\right) \\&~~~~\cdots P\left( \phi _{L,t}\circ \phi _{i_t-1,t-1}\circ \cdots \circ \phi _{i_2,2}\circ Z_{1}=0\right) \\&\ge \sum _{i_2=1}^2\sum _{i_3=1}^2\cdots \sum _{i_t=1}^2 P\left( S_2+1=i_2,S_3+1=i_3,\ldots ,S_t+1=i_t|X_0=0\right) \\&~~~~\times P(Z_{t}=0)\cdot P(\phi _{L,t}\circ Z_{t-1}=0)\cdot P(\phi _{L,t}\circ \phi _{L,t-1}\circ Z_{t-2}=0)\\&~~~~\cdots P\left( \phi _{L,t}\circ \phi _{L,t-1}\circ \cdots \circ \phi _{i_2,2}\circ Z_{1}=0\right) \\&\ge \sum _{i_2=1}^2\sum _{i_3=1}^2\cdots \sum _{i_t=1}^2 P(S_2+1=i_2,S_3+1=i_3,\ldots ,S_t+1=i_t|X_0=0)\\&~~~~\times P(Z_{t}=0)\cdot P(\phi _{L,t}\circ Z_{t-1}=0)\cdot P(\phi _{L,t}\circ \phi _{L,t-1}\circ Z_{t-2}=0)\\&~~~~\cdots P(\phi _{L,t}\circ \phi _{L,t-1}\circ \cdots \circ \phi _{L,2}\circ Z_{1}=0)\\&=P(Z_{t}=0)\cdot P(\phi _{L,t}\circ Z_{t-1}=0)\cdot \cdots P(\phi _{L,t}\circ \phi _{L,t-1}\circ \cdots \circ \phi _{L,2}\circ Z_{1}=0)\\&=L_t. \end{aligned}$$

By the proof of proposition 2.2 in Zheng et al. (2007), we know that $\lim _{t\rightarrow \infty }L_t\ne 0$, which implies that $\lim _{t\rightarrow \infty }P^t(0,0)\ne 0$. Therefore, we conclude that $\sum _{t=1}^{\infty }P^t(0,0)=+\infty .$ This proves that $\{X_t\}$ is a positive recurrent Markov chain (and hence ergodic) which ensures the existence of a strictly stationary distribution of (2.1). $\square$

Proof of Proposition 2.2

Compute to see that, under the stationary distribution,

$$\begin{aligned} E(X_{t})&=E[(\phi _{1,t}\circ X_{t-1})I_{1,t}]+E[(\phi _{2,t}\circ X_{t-1})I_{2,t}]+\lambda \nonumber \\&=E[E((\phi _{1,t}\circ X_{t-1})I_{1,t}|X_{t-1})]+E[E((\phi _{2,t}\circ X_{t-1})I_{2,t}|X_{t-1})]+\lambda \nonumber \\&=\sum _{i=1}^2E\left( \int f(\phi _{i,t})E((\phi _{i,t}\circ X_{t-1})I_{i,t}|X_{t-1},\phi _{i,t})d\phi _{i,t}\right) \nonumber \\&\le \mu _{\max }E(X_{t-1})+\lambda \nonumber \\&\le \cdots \nonumber \\&\le (\mu _{\max })^{t}E(X_{0})+\lambda \sum _{i=0}^{t-1}(\mu _{\max })^i\le \infty . \end{aligned}$$

(7.4)

Similarly, we have

$$\begin{aligned} E(X_t^2)&\le \ (1+\lambda )\left( \sum _{k=0}^{t-1}(\mu _{\max })^{t-k}(\mu _{\max }^{(2)})^{k}\right) E(X_{0})+(\mu _{\max }^{(2)})^{t}E(X_{0}^{2})\nonumber \\&\quad +(\lambda ^{2}+\lambda )\sum _{j=0}^{t-1}(\mu _{\max }^{(2)})^{j} +\lambda (1+2\lambda )\sum _{j=1}^{t}\left( \sum _{k=0}^{j-1}(\mu _{\max })^{j-k}(\mu _{\max }^{(2)})^k\right) <\infty . \end{aligned}$$

(7.5)

Some similar but tedious calculations show that $E(X_t^3) \le \infty$ and $E(X_t^4) \le \infty$. Combining (7.4) and (7.5), one can see that $E(X_t^k) < \infty$ for $k = 1, 2, 3,4$. $\square$

Proof of Proposition 2.3

The results (i) to (iii) are straightforward to verify. We give the proof of (iv) and (v) only. (iv) The variance of $X_t$ is given by

$$\begin{aligned} \mathrm{Var}(X_{t})&=\sum _{i=1}^2\mathrm{Var}[I_{i,t}(\phi _{i,t}\circ X_{t-1})]+ 2\mathrm{Cov}(I_{1,t}(\phi _{1,t}\circ X_{t-1}),I_{2,t}(\phi _{2,t}\circ X_{t-1}))+\lambda . \end{aligned}$$

(7.6)

A direct calculation shows

$$\begin{aligned}&\mathrm{Var}[I_{1,t}(\phi _{1,t}\circ X_{t-1})]\nonumber \\&\quad =\mathrm{Var}\left[ E(I_{1,t}(\phi _{1,t}\circ X_{t-1}))|X_{t-1}]\right) + E\left[ \mathrm{Var}(I_{1,t}(\phi _{1,t}\circ X_{t-1}))|X_{t-1}]\right) \nonumber \\&\quad =\mathrm{Var}[I_{1,t}\int f(\phi _{1,t})\phi _{1,t}X_{t-1}d\phi _{1,t}]+E\{E[I_{1,t}\phi _{1,t}(1-\phi _{1,t})X_{t-1}]\nonumber \\&\quad \quad +\,\mathrm{Var}[I_{1,t}\phi _{1,t}X_{t-1}]\}\nonumber \\&\quad =\mathrm{Var}[I_{1,t}X_{t-1}E(\phi _{1,t})]+E\{I_{1,t}X_{t-1}E(\phi _{1,t}(1-\phi _{1,t}))+I_{1,t}X_{t-1}^2\mathrm Var(\phi _{1,t})\}\nonumber \\&\quad =\mathrm{Var}[I_{1,t}X_{t-1}\phi _{1}]+(\phi _1-\sigma _{\phi _1}^2-\phi _1^2)E(I_{1,t}X_{t-1})+\sigma _{\phi _1}^2E(I_{1,t}X_{t-1}^2)\nonumber \\&\quad =\phi _1^2 [p_1(\sigma _1^2+u_1^2)-p_1^2u_1^2]+p_1u_1(\phi _1-\sigma _{\phi _1}^2-\phi _1^2)+p_1\sigma _{\phi _1}^2(\sigma _1^2+u_1^2). \end{aligned}$$

(7.7)

Similarly, we have

$$\begin{aligned} \mathrm{Var}[I_{2,t}(\phi _{2,t}\circ X_{t-1})]&=\phi _2^2 [p_2(\sigma _2^2+u_2^2)-p_2^2u_2^2]+p_2u_2(\phi _2-\sigma _{\phi _2}^2-\phi _2^2)\nonumber \\&\quad +p_2\sigma _{\phi _2}^2(\sigma _2^2+u_2^2). \end{aligned}$$

(7.8)

and

$$\begin{aligned} 2\mathrm{Cov}(I_{1,t}(\phi _{1,t}\circ X_{t-1}),I_{2,t}(\phi _{2,t}\circ X_{t-1}))&=-2\prod _{j=1}^{2}E(I_{j,t}(\phi _{j,t}\circ X_{t-1}))\nonumber \\&=-2p_1p_2(\phi _1\phi _2 u_1 u_2). \end{aligned}$$

(7.9)

Then, (iv) follows by replacing (7.7), (7.8) and (7.9) in (7.6) and some algebra.

(v) By the law of total covariance, we have

$$\begin{aligned} \mathrm{Cov}(X_{t},X_{t-h})&=\mathrm{Cov}(E(X_t|X_{t-1},\cdots ),E(X_{t-h}|X_{t-1},\cdots ))+0\\&=\sum _{i=1}^2\phi _{i}[E(X_{t-1}I_{i,t}\cdot X_{t-h})-E(X_{t-1}I_{i,t})\cdot E(X_{t-h})]\\&=\sum _{i=1}^2\phi _{i}\{E[E(X_{t-1}I_{i,t}\cdot X_{t-h}|I_{i,t})]-E[E(X_{t-1}I_{i,t}|I_{i,t})]\cdot E(X_{t-h}))]\}\\&=(\phi _{1}p_1+\phi _{2}p_2)\cdot \mathrm{Cov}(X_{t-1},X_{t-h})\\&=\cdots \\&=(\phi _{1}p_1+\phi _{2}p_2)^h\cdot \mathrm{Var}(X_{t-h}), \end{aligned}$$

Thus, the autocorrelation function is $\rho (k)=\mathrm{Corr}(X_{t},X_{t-h})=(\phi _{1}p_1+\phi _{2}p_2)^h.$$\square$

Proof of Theorem 3.1

This theorem follows from Theorem 3.1 in Klimko and Nelson (1978). $\square$

Proof of Theorems 3.2 and 3.3

Theorems 3.2 and 3.3 are special cases of Theorems 2.1 and 2.2 in Billingsley (1961). As discussed in Franke and Seligmann (1993), we only have to check that the conditions (C1)–(C6) hold, which implies the regularity conditions of Theorems 2.1 and 2.2 in Billingsley (1961) hold.

(C1) The set $\{k:P(Z_t=m)=f(m,\lambda )=\frac{\lambda ^m}{m!}e^{-\lambda }>0\}$ does not depend on $\lambda$;

(C2)$E[Z_t^3]=\lambda ^3+3\lambda ^2+\lambda <\infty$;

(C3)$P(Z_t=m)$ is three times continuously differentiable with respect to $\lambda$;

(C4) For any $\lambda '\in B$, where B is an open subset of $\mathbb {R}$, there exists a neighborhood U of $\lambda '$ such that:

1. $\sum _{k=0}^\infty \sup _{\lambda \in U}f(k,\lambda )<\infty$,

2. $\sum _{k=0}^\infty \sup _{\lambda \in U}|\frac{\partial f(k,\lambda )}{\partial \lambda }|<\infty$,

3. $\sum _{k=0}^\infty \sup _{\lambda \in U}|\frac{\partial ^2 f(k,\lambda )}{\partial \lambda ^2}|<\infty$;

(C5) For any $\lambda '\in B$ there exists a neighborhood U of $\lambda '$ and the sequences $\psi _1(n)=const1 \cdot n$, $\psi _{11}(n)=const2 \cdot n^2$, and $\psi _{111}(n)=const3 \cdot n^3$, with suitable constants const1, const2, const3 and $n\ge 0$ such that $\forall \lambda \in U$ and $\forall m\le n$, with $P(Z_t)$ nonvanishing $f(m,\lambda )$,

$$\begin{aligned}&\left| \frac{\partial f(m,\lambda )}{\partial \lambda }\right| \le \psi _{1}(n)f(m,\lambda ),~ \left| \frac{\partial ^2 f(m,\lambda )}{\partial \lambda ^2}\right| \le \psi _{11}(n)f(m,\lambda ),~\\&\left| \frac{\partial ^3 f(m,\lambda )}{\partial \lambda ^3}\right| \le \psi _{111}(n)f(m,\lambda ), \end{aligned}$$

and with respect to the stationary distribution of the process $\{X_t\}$,

$$\begin{aligned}\begin{array}{l@{\quad }l} E[\psi _1^3(X_1)]<\infty , &{} E[X_1\psi _{11}(X_2)]<\infty , \\ E[\psi _1(X_1)\psi _{11}(X_2)]<\infty , &{} E[\psi _{111}(X_1)]<\infty . \end{array} \end{aligned}$$

(C6) Let $\varvec{I}(\varvec{\theta })=(\sigma _{ij})_{3\times 3}$ denote the Fisher information matrix, i.e.,

$$\begin{aligned} \sigma _{ii}&=E\left( \frac{\partial }{\partial q_i}\log P(X_1,X_2)\right) ^2,~i=1,2;\\ \sigma _{33}&=E\left( \frac{\partial }{\partial \lambda }\log P(X_1,X_2)\right) ^2;\\ \sigma _{i3}&=\sigma _{3i}=E\left( \frac{\partial }{\partial q_i}\log P(X_1,X_2)\frac{\partial }{\partial \lambda }\log P(X_1,X_2)\right) ,~i=1,2;\\ \sigma _{12}&=\sigma _{21}=0. \end{aligned}$$

$\varvec{I}(\varvec{\theta })$ is nonsingular, where $P(X_1,X_2)$ denotes the transition probability given in (7.1).

Franke and Seligmann (1993) proved, also can be seen in Monteiro et al. (2012), that the above conditions (C1)–(C4) are all hold. (C5) follows by proposition 2.2 and the properties of Poisson distribution. Therefore, for the RCTINAR(1) model it is only necessary to verify the last condition (analogous to condition (C6) in Franke and Seligmann 1993) also holds. To this end, we need to check the following statements are all true.

• (S1) $E\left| \frac{\partial }{\partial q_i}\log P(X_1,X_2)\right| ^2 < \infty ,~i=1,2$;

• (S2) $E\left| \frac{\partial }{\partial \lambda }\log P(X_1,X_2)\right| ^2 < \infty$;

• (S3) $E\left| \frac{\partial }{\partial q_i}\log P(X_1,X_2)\frac{\partial }{\partial \lambda }\log P(X_1,X_2)\right| < \infty ,~i=1,2$.

We shall first prove Statement (S1). Recall that for $i=1,2$,

$$\begin{aligned}&p(x_{t-1},x_t,q_{i},\lambda )\nonumber \\&\quad =\sum _{m=0}^{\min (x_t,x_{t-1})} \left( \begin{array}{cc} x_{t-1}\\ m \end{array}\right) e^{-\lambda }\frac{\lambda ^{x_t-m}}{(x_t-m)!} \frac{\varGamma (k)\varGamma (q_{i}+m)\varGamma (k+x_{t-1}-q_i-m)}{\varGamma (q_i)\varGamma (k-q_i)\varGamma (k+x_{t-1})}\nonumber \\&\quad =\sum _{m=0}^{\min (x_t,x_{t-1})} \left( \begin{array}{cc} x_{t-1}\\ m \end{array}\right) e^{-\lambda }\frac{\lambda ^{x_t-m}}{(x_t-m)!}h(q_i,m,x_{t-1}), \end{aligned}$$

(7.10)

where $h(q_i,0,0)=1$ and

$$\begin{aligned}&h(q_i,m,x_{t-1})\\&\quad ={\left\{ \begin{array}{ll} {\prod _{l=0}^{x_{t-1}-1}(k-q_i+l)}/{\prod _{w=0}^{x_{t-1}-1}(k+w)},~m=0,~x_{t-1}\ge 1,\\ {\prod _{j=0}^{m-1}(q_i+j)\prod _{l=0}^{x_{t-1}-m-1}(k-q_i+l)}/{\prod _{w=0}^{x_{t-1}-1}(k+w)},~x_{t-1}>m\ge 1. \end{array}\right. } \end{aligned}$$

With the convention $\sum _{j=0}^{-1}=0$, we conclude that for $i=1,2,$

$$\begin{aligned} \frac{\partial h(q_i,m,x_{t-1})}{\partial q_i}=\left( \sum _{j=0}^{m-1}\frac{1}{q_i+j} -\sum _{j=0}^{x_{t-1}-m-1}\frac{1}{k-q_i+j} \right) h(q_i,m,x_{t-1}), \end{aligned}$$

(7.11)

yielding

$$\begin{aligned} -\frac{x_{t-1}}{k-q_i}h(q_i,m,x_{t-1}) \le \frac{\partial h(q_i,m,x_{t-1})}{\partial q_i} \le \frac{x_{t-1}}{q_i}h(q_i,m,x_{t-1}). \end{aligned}$$

(7.12)

By (7.10), (7.11) and (7.12), an immediate consequence is that

$$\begin{aligned} -\frac{x_{t-1}}{k-\max (q_1,q_2)}\le \sum _{i=1}^2\frac{\partial \log p(x_{t-1},x_t,q_i,\lambda )}{\partial q_i}I_{i,t} \le \frac{x_{t-1}}{\min (q_1,q_2)}, \end{aligned}$$

(7.13)

which implies

$$\begin{aligned} E\left| \frac{\partial }{\partial q_i}\log P(X_1,X_2)\right| ^2< C\cdot EX_1^2 < \infty ,~i=1,2,~(\text {by~(C5)}) \end{aligned}$$

for some suitable constant C.

Next we will prove Statement (S2). Through direct calculation, we obtain

$$\begin{aligned} \left| \frac{\partial \log p(x_{t-1},x_t,q_i,\lambda )}{\partial \lambda }\right|&=\frac{1}{p(x_{t-1},x_t,q_i,\lambda )}\sum _{m=0}^{\min (x_t,x_{t-1})}\left| \frac{\partial f(m,\lambda )}{\partial \lambda }\right| h(q_i,m,x_{t-1})\nonumber \\&\le \psi _1(x_{t}),~(\text {by}~(C5)) \end{aligned}$$

(7.14)

and therefore,

$$\begin{aligned} E\left| \frac{\partial }{\partial \lambda }\log P(X_1,X_2)\right| ^2< E \psi _1^2(X_1) < \infty ,~(\text {by~(C5)}). \end{aligned}$$

Lastly, by (7.13), (7.14) and (C5) we can conclude that Statement (S3) holds. Therefore, the Fisher information matrix $\varvec{I}(\varvec{\theta })$ is well defined. Finally, some elementary but tedious calculations show that (C6) is satisfied, too. $\square$

Proof of Theorems 4.1

See the proof of Theorem 2 in Freeland and McCabe (2004). $\square$