Goodness-of-fit testing of a count time series’ marginal distribution

Abstract

Popular goodness-of-fit tests like the famous Pearson test compare the estimated probability mass function with the corresponding hypothetical one. If the resulting divergence value is too large, then the null hypothesis is rejected. If applied to i. i. d. data, the required critical values can be computed according to well-known asymptotic approximations, e. g., according to an appropriate $\chi ^2$-distribution in case of the Pearson statistic. In this article, an approach is presented of how to derive an asymptotic approximation if being concerned with time series of autocorrelated counts. Solutions are presented for the case of a fully specified null model as well as for the case where parameters have to be estimated. The proposed approaches are exemplified for (among others) different types of CLAR(1) models, INAR(p) models, discrete ARMA models and Hidden-Markov models.

Appendices

Specific models for count processes

The subsequent models are used in the main part of this article to illustrate the derivation of the asymptotic distributions of the considered goodness-of-fit tests.

1.1 CLAR(1) model

Many important models for count Markov chains belong to the class of conditional linear autoregressive models of order one (CLAR(1) models) as discussed by Grunwald et al. (2000). The homogeneous count Markov chain $(X_t)_{\mathbb {Z}}$ is said to have CLAR(1) structure if the conditional mean of $X_t$ is linear in $X_{t-1}$, i. e., if

$$\begin{aligned} E[X_t\ |\ X_{t-1}] =\ \alpha \cdot X_{t-1} + \beta \qquad \text {for some } \alpha \in \mathbb {R}\text { and } \beta >0. \end{aligned}$$

(A.1)

The condition $|\alpha |<1$ guarantees a finite stationary mean given by $\mu :=E[X_t]=\beta /(1-\alpha )$. If also the stationary variance of $X_t$ is finite, i. e., $\sigma ^2:=V[X_t]<\infty$, then the autocorrelation function (ACF) is of AR(1)-type, i. e., it altogether holds that

$$\begin{aligned} \rho (h):=Corr[X_t,X_{t-h}] =\ \alpha ^h,\qquad E[X_t\ |\ X_{t-h}]\ =\ \alpha ^h\cdot X_{t-h} + (1-\alpha ^h)\,\mu . \end{aligned}$$

(A.2)

1.2 INAR models

If X is a discrete random variable with range $\mathbb {N}_0$ and if $\alpha \in (0;1)$, then the random variable $\alpha \circ X := \sum _{i=1}^{X} Z_i$ is said to arise from X by binomial thinning (Steutel and Harn 1979). Here, the $Z_i$ are i. i. d. binary random variables with $P(Z_i=1)=\alpha$, which are also independent of X. Hence, $\alpha \circ X$ has a conditional binomial distribution given the value of X, i. e., $\alpha \circ X|X\ \sim {\text{ Bin }}(X,\alpha )$. The boundary values $\alpha =0$ and $\alpha =1$ might be included into this definition by setting $0\circ X := 0$ and $1\circ X := X$.

Using the random operator “$\circ$”, McKenzie (1985) defined the INAR(1) model in the following way.

Definition A.2.1

(INAR(1) Model) Let the innovations$(\epsilon _t)_{\mathbb {Z}}$ be an i. i. d. process with range $\mathbb {N}_0$, denote $E[\epsilon _t]=\mu _{\epsilon }$, $V[\epsilon _t]=\sigma _{\epsilon }^2$. Let $\alpha \in (0;1)$. A process $(X_t)_{\mathbb {Z}}$ of observations, which follows the recursion

$$\begin{aligned} X_t =\ \alpha \circ X_{t-1}\ +\ \epsilon _t, \end{aligned}$$

is said to be an INAR(1) process if all thinning operations are performed independently of each other and of $(\epsilon _t)_{\mathbb {Z}}$, and if the thinning operations at each time t as well as $\epsilon _t$ are independent of $(X_s)_{s<t}$.

The most popular instance of the INAR(1) family is the Poisson INAR(1) model (McKenzie 1985), which assumes the innovations $(\epsilon _t)_{\mathbb {Z}}$ to be i. i. d. according to the Poisson distribution ${\text{ Poi }}(\lambda )$. A Poisson INAR(1) process is an irreducible and aperiodic Markov chain with a unique stationary marginal distribution for $(X_t)_{\mathbb {Z}}$, the Poisson distribution ${\text{ Poi }}(\mu )$ with $\mu =\frac{\lambda }{1-\alpha }$. It is also $\alpha$-mixing with geometrically decreasing weights (Schweer and Weiß 2014). Furthermore, the Poisson INAR(1) model constitutes the only instance within the INAR(1) family, which is time reversible (McKenzie 1985; Schweer 2015).

The (Poisson) INAR(1) model belongs to the class of CLAR(1) models, so it satisfies (A.2). The h-step-ahead transition probabilities are given by (Freeland and McCabe 2004)

$$\begin{aligned} p_{i|j}^{(h)} =\ \sum \limits _{m=0}^{\min {\{i,j\}}}\ \left( {\begin{array}{c}j\\ m\end{array}}\right) \alpha ^{h\,m}\,(1-\alpha ^h)^{j-m}\cdot e^{-\mu \,(1-\alpha ^h)}\,\frac{\big (\mu \,(1-\alpha ^h)\big )^{i-m}}{(i-m)!}, \end{aligned}$$

(A.3)

and $(X_t,X_{t-h})$ are bivariately Poisson distributed (Johnson et al. 1997) according to ${\text{ BPoi }}\big (\alpha ^h\,\mu ;\ (1-\alpha ^h)\,\mu , (1-\alpha ^h)\,\mu \big )$ (Alzaid and Al-Osh 1988). Here, ${\text{ BPoi }}(\lambda _0;\ \lambda _1, \lambda _2)$ refers to the joint distribution of $(Y_0+Y_1, Y_0+Y_2)^{\top }$ with independent Poisson variates $Y_i\sim {\text{ Poi }}(\lambda _i)$ for $i=0,1,2$.

A simple example of an INAR(1) process not being time reversible is the geometric INAR(1) process (McKenzie 1985, 1986), which has marginal distribution ${\text{ Geom }}(\pi )$, i. e., $P(X_t=x)=\pi \,(1-\pi )^x$. Here, the innovations stem from a zero-inflated geometric distribution,

$$\begin{aligned} \epsilon _t :=\ B_t\,G_t \qquad \text {with independent } B_t\sim {\text{ Bin }}(1,1-\alpha ),\ G_t\sim {\text{ Geom }}(\pi ), \end{aligned}$$

(A.4)

with pmf $P(\epsilon =x) = \delta _{x,0}\,\alpha + (1-\alpha )\,\pi \,(1-\pi )^x$. Hence, the 1-step-ahead transition probabilities are computed as

$$\begin{aligned} p_{i|j} =\ \sum \limits _{m=0}^{\min {\{i,j\}}}\ \left( {\begin{array}{c}j\\ m\end{array}}\right) \alpha ^{m}\,(1-\alpha )^{j-m}\cdot P(\epsilon =i-m). \end{aligned}$$

(A.5)

It is also possible to obtain any other member of the family of negative binomial distributions as a marginal distribution by choosing an appropritae innovations’ distribution, see McKenzie (1986) for details.

It is also possible to extend the INAR(1) recursion in Definition A.2.1 to a pth-order autoregression of the form

$$\begin{aligned} X_t =\ \alpha _1\circ X_{t-1}\ +\cdots +\ \alpha _p\circ X_{t-p}\ +\ \epsilon _t \qquad \text {with } \alpha _{\bullet }:=\sum \limits _{j=1}^p \alpha _j\ <1. \end{aligned}$$

(A.6)

Due to the stochastic nature of the thinnings involved in (A.6), however, additional assumptions concerning the thinnings $(\alpha _1\circ X_{t},\ldots , \alpha _p\circ X_{t})$ are required. While the INAR(p) model by Du and Li (1991) assumes the conditional independence of $(\alpha _1\circ X_{t},\ldots , \alpha _p\circ X_{t})$ given $X_t$, the one by Alzaid and Al-Osh (1990) supposes a conditional multinomial distribution. As shown by Schweer (2015), only the latter model continues the INAR(1)’s property that we have time reversibility exactly in the case of Poisson innovations (then also the observations are Poisson-distributed), while the INAR(p) model by Du and Li (1991) is neither time reversible nor does it have Poisson marginals. For this reason, we shall focus here on the time reversible Poisson INAR(p) model according to Alzaid and Al-Osh (1990), where the innovations are i. i. d. ${\text{ Poi }}(\lambda )$ and, hence, the observations have the stationary marginal distribution ${\text{ Poi }}(\mu )$ with $\mu =\lambda /(1-\alpha _{\bullet })$.

Example A.2.2

(Poisson INAR(2) Model) Solving the Yule–Walker-type equations (3.6) and (3.8) in Alzaid and Al-Osh (1990), the ACF of the Poisson INAR(2) model becomes

$$\begin{aligned} \rho (1)\,=\,\alpha _1,\qquad \rho (h) =\ \alpha _1\,\rho (h-1)\,+\,\alpha _2\,\rho (h-2)\quad \text {for } h\ge 2. \end{aligned}$$

As shown in Appendix B, the lagged observations $X_t$ and $X_{t-h}$ with $h\in \mathbb {N}$ are bivariately Poisson distributed,

$$\begin{aligned} (X_t,X_{t-h})\ \sim \ {\text{ BPoi }}\Big (\rho (h)\,\mu ;\ \big (1-\rho (h)\big )\,\mu ,\ \big (1-\rho (h)\big )\,\mu \Big ), \end{aligned}$$

with conditional mean $E[X_t\ |\ X_{t-h}]\, =\, \rho (h)\,X_{t-h}+\big (1-\rho (h)\big )\,\mu$.

1.3 Binomial AR(1) model

In many applications, it is known that the observed count data cannot become arbitrarily large, but their range has a natural upper bound $n\in \mathbb {N}$ that can never be exceeded. For the case of such time series of counts supported on $\{0,\ldots ,n\}$, McKenzie (1985) proposed the binomial AR(1) model.

Definition A.3.1

(Binomial AR(1) Model) Let $\rho \in \big (\max {\{-\frac{\pi }{1-\pi }, -\frac{1-\pi }{\pi }\}}\ ;\ 1\big )$ and $\pi \in (0;1)$. Define $\beta :=\pi \cdot (1-\rho )$ and $\alpha :=\beta +\rho$. Fix $n\in \mathbb {N}$. The process $(X_t)_{\mathbb {Z}}$, defined by the recursion

$$\begin{aligned} X_t =\ \alpha \circ X_{t-1}\ +\ \beta \circ (n-X_{t-1}), \end{aligned}$$

where all thinnings are performed independently of each other, and where the thinnings at time t are independent of $(X_s)_{s<t}$, is referred to as a binomial AR(1) process.

The condition on $\rho$ guarantees that the derived parameters $\alpha ,\beta$ satisfy $\alpha ,\beta \in (0;1)$, i. e., these parameters can indeed serve as thinning probabilities.

It is known that $(X_t)_{\mathbb {Z}}$ is a stationary, ergodic and $\phi$-mixing finite Markov chain (again with geometrically decreasing weights), the marginal distribution of which is ${\text{ Bin }}(n,\pi )$ (McKenzie 1985; Kim and Weiß 2015). The binomial AR(1) model belongs to the class of CLAR(1) models, so it satisfies (A.2), and it is time reversible (McKenzie 1985). The h-step-ahead transition probabilities are given by (Weiß and Pollett 2012)

$$\begin{aligned} p_{i|j}^{(h)} =\! \sum \limits _{m=\max {\{0,i+j-n\}}}^{\min {\{i,j\}}}\ \left( {\begin{array}{c}j\\ m\end{array}}\right) \ \left( {\begin{array}{c}n-j\\ i-m\end{array}}\right) \ \alpha _h^m (1-\alpha _h)^{j-m}\ \beta _h^{i-m} (1-\beta _h)^{n-j+m-i}, \end{aligned}$$

(A.7)

where $\beta _h:=\pi \cdot (1-\rho ^h)$ and $\alpha _h:=\beta _h +\rho ^h$.

1.4 NDARMA model for counts

The “new” discrete ARMA (NDARMA) models have been proposed by Jacobs and Lewis (1983). They generate an ARMA-like dependence structure through some kind of random mixture.

Definition A.4.1

(NDARMA Model for Counts) Let the observations $(X_t)_{\mathbb {Z}}$ and the innovations $(\epsilon _t)_{\mathbb {Z}}$ be count processes, where $(\epsilon _t)_{\mathbb {Z}}$ is i. i. d. with $P(\epsilon _t=i)=p_i$, and where $\epsilon _t$ is independent of $(X_s)_{s<t}$. The random mixture is obtained through the i. i. d. multinomial random vectors

$$\begin{aligned} (\alpha _{t,1},\ldots ,\alpha _{t,{\text{ p }}},\beta _{t,0},\ldots ,\beta _{t,{\text{ q }}}) \quad \sim \ {\text{ MULT }}(1;\ \phi _1, \ldots ,\phi _{{\text{ p }}},\varphi _0,\ldots ,\varphi _{{\text{ q }}}), \end{aligned}$$

which are independent of $(\epsilon _t)_{\mathbb {Z}}$ and of $(X_s)_{s<t}$. Then $(X_t)_{\mathbb {Z}}$ is said to be an NDARMA(p, q) process if it follows the recursion

$$\begin{aligned} X_t =\ \alpha _{t,1}\cdot X_{t-1}+\cdots +\alpha _{t,{\text{ p }}}\cdot X_{t-{\text{ p }}}\ +\ \beta _{t,0}\cdot \epsilon _{t}+\cdots +\beta _{t,{\text{ q }}}\cdot \epsilon _{t-{\text{ q }}}. \end{aligned}$$

The stationary marginal distribution of $X_t$ is identical to that of $\epsilon _{t}$, i. e., $P(X_t=i)=p_i=P(\epsilon _t=i)$, and we always have

$$\begin{aligned} p_{i|j}^{(h)} =\ p_i\cdot \big (1-\rho (h)\big )\ +\ \delta _{i,j}\cdot \rho (h). \end{aligned}$$

(A.8)

The autocorrelations are non-negative and can be determined from the Yule–Walker equations (Jacobs and Lewis 1983)

$$\begin{aligned} \rho (h) =\ \sum \limits _{j=1}^{{\text{ p }}}\ \phi _j\cdot \rho (|h-j|)\ +\ \sum \limits _{i=0}^{{\text{ q }}-h}\ \varphi _{i+h}\cdot r(i)\qquad \text {for } h\ge 1, \end{aligned}$$

where the r(i) satisfy

$$\begin{aligned} r(i) =\ \sum \limits _{j=\max {\{0,i-{\text{ p }}\}}}^{i-1}\ \phi _{i-j}\cdot r(j)\ +\ \varphi _i\,\mathbb {1}(0\le i\le {\text{ q }}), \end{aligned}$$

which implies $r(i)=0$ for $i<0$, and $r(0)=\varphi _0$. Mixing properties have been established by Weiß (2013).

Bivariate distributions of INAR(2) model

We pick up the derivations of Alzaid and Al-Osh (1990) and extend them to obtain the bivariate distribution of $X_t$ and $X_{t-h}$ for $h\in \mathbb {N}$. Let $(X_t)_{\mathbb {Z}}$ follow an INAR(2) model, where we first do not further specify the innovations’ distribution. Define the sequence of weights $(w_j)_{j\ge -1}$ by

$$\begin{aligned} w_{-1}=0,\quad w_0=1,\quad w_j =\ \alpha _1\,w_{j-1}+\alpha _2\,w_{j-2}\quad \text {for } j=1,2,\ldots , \end{aligned}$$

(B.1)

which satisfy $\sum _{j=0}^{\infty } w_j\,=\,1/(1-\alpha _{\bullet })$ (Alzaid and Al-Osh 1990 [p. 317]). Let us introduce a further sequence of coefficients $({\varvec{a}}_j)_{j\ge -1}$ with ${\varvec{a}}_j=(a_{j,1},a_{j,2})^{\top }$:

$$\begin{aligned} {\varvec{a}}_{-1}=(0,1)^{\top },\quad {\varvec{a}}_{0}=(1,0)^{\top },\quad {\varvec{a}}_j =\ \alpha _1\,{\varvec{a}}_{j-1}+\alpha _2\,{\varvec{a}}_{j-2}\quad \text {for } j=1,2,\ldots \end{aligned}$$

(B.2)

Obviously, the first components are identical to the weights (B.1), $a_{j,1}=w_j$ for all $j\ge -1$, while the second components satisfy $a_{j,2}=\alpha _2\,a_{j-1,1}=\alpha _2\,w_{j-1}$ for $j\ge 0$.

Following Alzaid and Al-Osh (1990), p. 320, we define the bivariate process $({\varvec{X}}_t)_{\mathbb {Z}}$ by ${\varvec{X}}_t=(X_t, \alpha _2\circ X_{t-1})^{\top }$, which is a Markov chain satisfying

$$\begin{aligned} { E\big [z_1^{X_{t,1}}\,z_2^{X_{t,2}}\ \big |\ {\varvec{X}}_{t-1},\ldots \big ] =\ {\text{ pgf }}_{\epsilon }(z_1)\,\big (1+\alpha _1\,(z_1-1)+\alpha _2\,(z_2-1)\big )^{X_ {t-1,1}}\,z_1^{X_{t-1,2}}. } \end{aligned}$$

(B.3)

In a first step, we extend this result to arbitrary time lags $h\in \mathbb {N}$. Using the coefficients (B.2), we rewrite (B.3) as

$$\begin{aligned}&E\Big [\big (1+a_{0,1}\,(z_1-1)+a_{0,2}\,(z_2-1)\big )^{X_{t,1}}\,\nonumber \\&\quad \big (1+a_{-1,1}\,(z_1-1)+a_{-1,2}\,(z_2-1)\big )^{X_{t,2}}\ \Big |\ {\varvec{X}}_{t-1},\ldots \Big ]\nonumber \\&\quad =\ {\text{ pgf }}_{\epsilon }\big (1+a_{0,1}\,(z_1-1)+a_{0,2}\,(z_2-1)\big )\nonumber \\&\qquad \cdot \big (1+a_{1,1}\,(z_1-1)+a_{1,2}\,(z_2-1)\big )^{X_{t-1,1}}\,\nonumber \\&\qquad \big (1+a_{0,1}\,(z_1-1)+a_{0,2}\,(z_2-1)\big )^{X_{t-1,2}}. \end{aligned}$$

(B.4)

Using the law of total expectation, we apply (B.4) and obtain

$$\begin{aligned}&E\big [z_1^{X_{t,1}}\,z_2^{X_{t,2}}\ \big |\ {\varvec{X}}_{t-h}\big ] =\ {\text{ pgf }}_{\epsilon } \quad \big (1+a_{0,1}\,(z_1-1)+a_{0,2}\,(z_2-1)\big )\nonumber \\&\quad \cdots E\Big [\big (1+a_{1,1}\,(z_1-1)+a_{1,2}\,(z_2-1)\big )^{X_{t-1,1}}\,\nonumber \\&\qquad \big (1+a_{0,1}\,(z_1-1)+a_{0,2}\,(z_2-1)\big )^{X_{t-1,2}}\ \Big |\ {\varvec{X}}_{t-h}\Big ]\nonumber \\&=\cdots =\ \prod _{j=0}^{h-1}\ {\text{ pgf }}_{\epsilon }\big (1+a_{j,1}\,(z_1-1)+a_{j,2}\,(z_2-1)\big )\nonumber \\&\qquad \cdot \big (1+a_{h,1}\,(z_1-1)+a_{h,2}\,(z_2-1)\big )^{X_{t-h,1}}\,\nonumber \\&\qquad \big (1+a_{h-1,1}\,(z_1-1)+a_{h-1,2}\,(z_2-1)\big )^{X_{t-h,2}}. \end{aligned}$$

(B.5)

On the one hand, this implies the marginal pgf as

$$\begin{aligned} \begin{array}{l} {\text{ pgf }}_{{\varvec{X}}}({\varvec{z}}) =\ \prod _{j=0}^{\infty }\ {\text{ pgf }}_{\epsilon }\big (1+w_j\,(z_1-1)+\alpha _2\,w_{j-1}\,(z_2-1)\big ),\\ {\text{ pgf }}_{X}(z) =\ {\text{ pgf }}_{{\varvec{X}}}(z,1) =\ \prod _{j=0}^{\infty }\ {\text{ pgf }}_{\epsilon }\big (1+w_j\,(z-1)\big ), \end{array} \end{aligned}$$

(B.6)

also see (4.4) and Theorem 2.1 in Alzaid and Al-Osh (1990). On the other hand, we compute from (B.5) the lagged bivariate pgf as

$$\begin{aligned}&{\text{ pgf }}_{{\varvec{X}}_t,{\varvec{X}}_{t-h}}({\varvec{z}},{\varvec{y}}) =\\&\prod _{j=0}^{h-1}\ {\text{ pgf }}_{\epsilon }\big (1+w_j\,(z_1-1)+\alpha _2\,w_{j-1}\,(z_2-1)\big )\cdot E\Big [y_1^{X_{t-h,1}}\,y_2^{X_{t-h,2}}\\&\qquad \cdot \big (1+w_{h}\,(z_1-1)+\alpha _2\,w_{h-1}\,(z_2-1)\big )^{X_{t-h,1}}\,\\&\qquad \big (1+w_{h-1}\,(z_1-1)+\alpha _2\,w_{h-2}\,(z_2-1)\big )^{X_{t-h,2}}\Big ]\\&=\ \prod _{j=0}^{h-1}\ {\text{ pgf }}_{\epsilon }\big (1+w_j\,(z_1-1)+\alpha _2\,w_{j-1}\,(z_2-1)\big )\\&\qquad \cdot {\text{ pgf }}_{{\varvec{X}}}\Big (y_1\,\big (1+w_{h}\,(z_1-1)+\alpha _2\,w_{h-1}\,(z_2-1)\big )\ ,\ \\&\qquad y_2\,\big (1+w_{h-1}\,(z_1-1)+\alpha _2\,w_{h-2}\,(z_2-1)\big )\Big ). \end{aligned}$$

This implies that

$$\begin{aligned} {\text{ pgf }}_{X_t,X_{t-h}}(z,y)= & {} {\text{ pgf }}_{{\varvec{X}}}\Big (y\,\big (1+w_{h}\,(z-1)\big )\ ,\ 1+w_{h-1}\,(z-1)\Big ) \, \nonumber \\&\quad \prod _{j=0}^{h-1} {\text{ pgf }}_{\epsilon }\big (1+w_j\,(z-1)\big ). \end{aligned}$$

(B.7)

Now, we turn to the special case of the Poisson INAR(2) model (Example A.2.2), i. e., where the innovations $(\epsilon _t)_{\mathbb {Z}}$ satisfy ${\text{ pgf }}_{\epsilon }(z)\,=\,\exp {\big (\lambda \,(z-1)\big )}$. Using that $\sum _{j=0}^{\infty } w_j\,=\,1/(1-\alpha _{\bullet })$ and that $\mu =\lambda /(1-\alpha _{\bullet })$, (B.6) simplifies to

$$\begin{aligned} {\text{ pgf }}_{{\varvec{X}}}({\varvec{z}}) =\ \exp {\Big (\mu \,\big ((z_1-1)+\alpha _2\,(z_2-1)\big )\Big )},\quad {\text{ pgf }}_{X}(z) =\ \exp {\big (\mu \,(z-1)\big )}, \end{aligned}$$

(B.8)

also see (5.1) in Alzaid and Al-Osh (1990). In particular, the stationary marginal distribution is ${\text{ Poi }}(\mu )$. The bivariate pgf (B.7) becomes

$$\begin{aligned}&{\text{ pgf }}_{X_t,X_{t-h}}(z,y) =\ \exp {\Big (\mu \,\big (y-1+w_{h}\,y\,(z-1)+\alpha _2\,w_{h-1}\,(z-1)\big )\Big )} \, \\&\quad \exp {\Big (\lambda \,(z-1)\,\mathop {\sum }\limits _{j=0}^{h-1} w_j\Big )}. \end{aligned}$$

This expression can be further simplified by considering (B.1). It follows that

$$\begin{aligned} (1-\alpha _{\bullet })\,\mathop {\sum }\limits _{j=0}^{h-1} w_j= & {} \mathop {\sum }\limits _{j=0}^{h-1} w_j\ -\ \alpha _2\,w_{h-1} - \alpha _1\,w_0 \\&- \mathop {\sum }\limits _{j=2}^h\, (\alpha _1\,w_{j-1}+\alpha _2\,w_{j-2})\\= & {} 1+\alpha _1+\mathop {\sum }\limits _{j=2}^{h-1} w_j\ -\ \alpha _2\,w_{h-1} - \alpha _1 - \mathop {\sum }\limits _{j=2}^h w_j\ \\= & {} 1 - \alpha _2\,w_{h-1} - w_h, \end{aligned}$$

so we continue

$$\begin{aligned} { \begin{array}{rl} {\text{ pgf }}_{X_t,X_{t-h}}(z,y) =&{} \exp {\Big (\mu \,\big (y-1\ +\ z-1\ +\ w_{h}\,(y-1)(z-1)\big )\Big )}\\ =&{} \exp {\Big (\mu \,\big ((1-w_h)\,(y-1)\ +\ (1-w_h)\,(z-1)\ +\ w_{h}\,(yz-1)\big )\Big )}. \end{array} } \end{aligned}$$

(B.9)

Obviously, this pgf is symmetric in z and y, confirming the time reversibility. For $h=1$, it simplifies to (5.3) in Alzaid and Al-Osh (1990). In particular, (B.9) shows that $(X_t,X_{t-h})$ are bivariately Poisson distributed according to ${\text{ BPoi }}\big (w_h\,\mu ;\ (1-w_h)\,\mu , (1-w_h)\,\mu \big )$, see Johnson et al. (1997). This implies the conditional mean

$$\begin{aligned} E[X_t\ |\ X_{t-h}]\ =\ w_h\,X_{t-h}+(1-w_h)\,\mu . \end{aligned}$$

The proof of Example A.2.2 is completed by noting that the weights in (B.1) follow the same recursion as the ACF of the Poisson INAR(2) model, so $w_h=\rho (h)$ for $h\ge 0$ in this special case.