Flexible bivariate Poisson integer-valued GARCH model

Abstract

Integer-valued time series models have been widely used, especially integer-valued autoregressive models and integer-valued generalized autoregressive conditional heteroscedastic (INGARCH) models. Recently, there has been a growing interest in multivariate count time series. However, existing models restrict the dependence structures imposed by the way they constructed. In this paper, we consider a class of flexible bivariate Poisson INGARCH(1,1) model whose dependence is established by a special multiplicative factor. Stationarity and ergodicity of the process are discussed. The maximization by parts algorithm and its modified version together with the alternative method by using R package Template Model Builder are employed to estimate the parameters of interest. The consistency and asymptotic normality for estimates are obtained, and the finite sample performance of estimators is given via simulations. A real data example is also provided to illustrate the model.

Appendices

Appendix A

Let X and Y be two independent random variables from Poisson distribution with cdf $F_1$ and $F_2$. Then, the joint cdf of newly proposed BP distribution $C(x,y)=\mathbb {P}(X\le x,Y\le y)$ should satisfy the definition of bivariate probability distribution and the following properties:

$$\begin{aligned} (a)&~\text {Right continuity}:~C(x,y+0)=C(x,y);~C(x+0,y)=C(x,y);\\ (b)&~\text {Boundedness}:~\lim _{x\rightarrow 0^{-}}C(x,y)=\lim _{y\rightarrow 0^{-}}C(x,y)=0,\\&{}~~~~~\qquad ~~~~~~~~~\lim _{x,y\rightarrow +\infty }C(x,y)=1;\\ (c)&~\text {Monotonicity}:~C(x,y)~\text {is monotonically non-decreased for}~x~\text {and}~y,\\&{}~~~~~~\qquad ~~~~~~~~~~\text {respectively};\\ (d)&~\text {Nonnegativity}:~\text {for any}~x_1\le x_2,~y_1\le y_2~\text {such that}\\&~~~\qquad ~~~~C(x_2,y_2)-C(x_1,y_2)-C(x_2,y_1)+C(x_1,y_1)\ge 0. \end{aligned}$$

It is worth mentioning that from a different prospective, our model (6) can be viewed as a bivariate Poisson distribution constructed by linking function, copula. By imposing a positive multiplicative factor $Z^{-1}$, the properties (a) and (b) can be easily verified by relation of events and nature of probability. Now, we turn to consider properties (c) and (d). By the form of equation (6) and the definition of Eqs. (3)–(5), we find that the multiplicative factors $c(\cdot ,\cdot )$ are all nonnegative bounded functions on $[0,1]^2$ for suitably chosen range of values of the dependency parameters. Hence, when $x_2\ge x_1$, we have

$$\begin{aligned} C(x_2,y)-C(x_1,y)=Z^{-1}\sum _{k=x_1+1}^{x_2}\sum _{s=0}^{y}\frac{\lambda _1^k\lambda _2^s}{k!s!} \exp \{ -(\lambda _1 + \lambda _2)\}c(F_1(k),F_2(s))\ge 0. \end{aligned}$$

The case for $y_2\ge y_1$ is similar, so (c) is verified. As for nonnegativity, we have the following result by definition of the cdf, for any $x_1\le x_2$, $y_1\le y_2$,

$$\begin{aligned}&C(x_2,y_2)-C(x_1,y_2)-C(x_2,y_1)+C(x_1,y_1)\\&\quad =\mathbb {P}(x_1\le X\le x_2,y_1\le Y\le y_2)\ge 0. \end{aligned}$$

Hence, property (d) is also verified.$\square$

In fact, $Z(\lambda _1,\lambda _2,\theta )$ is a bounded factor because the series converges by its definition. It also can be viewed as a weight, which makes the sum of our model’s pdf equals to one. One can find similar methods to define some distribution families, for example, Efron (1986) considered the double exponential families by adding a nonlinear constant $c(\mu ,\theta ,n)$; Shmueli et al. (2005) tackled the important task of characterizing the Conway–Maxwell–Poisson distribution by dividing a series as weight. In our bivariate case, it is difficult to give the explicit approximation to $Z^{-1}(\lambda _1,\lambda _2,\theta )$ due to the copula structure. Instead, we could use the truncated double summation at some $k_1, k_2$ to compute $Z^{-1}$, which will give the reasonable approximation with a small error. More specifically, we write $Z(\lambda _1,\lambda _2,\theta )=\widehat{Z}_{k_1,k_2}+R_{k_1,k_2}$, where

$$\begin{aligned} \widehat{Z}_{k_1,k_2}=&\sum _{y_1=0}^{k_1}\sum _{y_2=0}^{k_2}\frac{\lambda _1^{y_1}\lambda _2^{y_2}}{y_1! y_2!} \exp \{ -(\lambda _1 + \lambda _2)\}c(F_1(y_1),F_2(y_2)),\\ R_{k_1,k_2}=&\left( \sum _{y_1=k_1+1}^\infty \sum _{y_2=0}^{k_2}+ \sum _{y_1=0}^{k_1}\sum _{y_2=k_2+1}^\infty +\sum _{y_1=k_1+1}^\infty \sum _{y_2=k_2+1}^ \infty \right) \frac{\lambda _1^{y_1}\lambda _2^{y_2}}{y_1! y_2!} \\&{}~~~\times \exp \{ -(\lambda _1 + \lambda _2)\}c(F_1(y_1),F_2(y_2)). \end{aligned}$$

Similar to Shmueli et al. (2005), we define the relative truncation error (TE) as

$$\begin{aligned} \mathrm{TE}=\frac{\widehat{Z}_{k_1,k_2}^{-1}-Z^{-1}}{\widehat{Z}_{k_1,k_2}^{-1}} =\frac{R_{k_1,k_2}}{Z}. \end{aligned}$$

(14)

Here, we show some numerical simulations of different combinations with our proposed model. We choose the Poisson intensity pairs as $(\lambda _1,\lambda _2)=(1,1),~(1,2)$ and (3,2). For sake of illustration, we let $k=k_1=k_2$ be the same truncated number, which will specified in different cases.

$$\begin{aligned}&(a)~\mathrm{BPG}: \rho =-0.5,-0.2,0.2,0.5;\\&(b)~\mathrm{BPF}: \gamma =-1,-0.5,0.5,1;\\&(c)~\mathrm{BGFGM}: \sigma =-0.5,-0.2,0.2,0.5. \end{aligned}$$

Table 11 TE for the above three cases

Based on the fact that the kurtosis of univariate Poisson distribution is $\lambda ^{-1}$ ($\lambda$ as the Poisson intensity), leading to the flatter curve plot of pmf as $\lambda$ increases. Therefore, we know that the truncated number k needs to be increased to approximate Z as $\lambda _1, \lambda _2$ increase. In the above three scenarios, we choose the largest value of k to be 6 as $\lambda _1$ and $\lambda _2$ are relative small. If we increase $\lambda _1$ or $\lambda _2$, then k may be 10 or much larger to approximate the infinite sum. From Table 11, we conclude that the truncated sum can be an approximate of Z in practice.

Furthermore, following by the discussion of Appendix B in Shmueli et al. (2005) and elementary analysis, we know that Z converges and both Z and $Z^{-1}$ are bounded by some positive constant. Therefore, going back to our model (6), we can view $Z^{-1}$ as a regularity constant weight. The rest terms by removing $Z^{-1}$ at the right-hand side of (6) will dominate the pmf and can be convenient to use, thus we only consider the rest dominated terms for estimation and inference parts. As mentioned by Efron (1986), he suggested to leave out the highly nonlinear multiplicative factor when estimating the parameters.

Appendix B

Proof of Theorem 1

When considering BPFGM$(\lambda _{t,1},\lambda _{t,2},\sigma )$ model, this theorem can be proved using arguments similar to Cui and Zhu (2018, Theorem 1). So we only consider BPG$(\lambda _{t,1},\lambda _{t,2},\rho )$ and BPF$(\lambda _{t,1},\lambda _{t,2},\gamma )$ models. The proof employs the theory of Markov chain again, but a little difference from those in Cui and Zhu (2018) is that we introduce a uniform constant bound for the corresponding multiplicative function. First note that $\{\varvec{\lambda }_t\}$ has at least one stationary distribution, refer to Liu (2012) for more details. From (8), it is easy to see that $(\varvec{I} -\varvec{A})^{-1}\varvec{\omega }$ is a reachable state if $\varvec{Y}_{t-1} = \varvec{Y}_{t-2} =\cdots = \varvec{0}$ for some $t \in \mathbb {N}$ large enough. Then we only have to show that $\{\varvec{\lambda }_t\}$ is an e-chain, which can naturally guarantee the existence of a unique invariant probability measure by the main virtue of Meyn and Tweedie (2009, Theorem 18.8.4). To see this, we recall the definition of e-chain, i.e., for any continuous function f with compact support defined on $[0,\infty ) \times [0,\infty )$ and $\varepsilon > 0$, there exists an $\eta > 0$ such that $|\mathbb {P}_{\varvec{x}_1}^kf - \mathbb {P}_{\varvec{z}_1}^kf| < \varepsilon$, for $\left||\varvec{x}_1 - \varvec{z}_1\right||_p < \eta$ and all $k \ge 1$, where $\varvec{x}_1 = (x_{1,1},x_{1,2})^{\top }, \varvec{z}_1 = (z_{1,1},z_{1,2})^{\top },~\mathbb {P}_{\varvec{x}_1}^kf= \mathbb {E}\{f(\varvec{\lambda }_k)|\varvec{\lambda }_0 = \varvec{x}\}$. Without loss of generality, assume $|f| \le 1$, M is a finite constant and take $\varepsilon '$ and $\eta$ sufficiently small such that $\varepsilon ' + 8M\eta / (1 - \left||\varvec{A} \right||_p) < \varepsilon$ and $|f(\varvec{x}_1) - f(\varvec{z}_1)| < \varepsilon '$ whenever $\left||\varvec{x}_1 - \varvec{z}_1\right||_p < \eta$, for some $p \in [1,\infty ]$. When we choose Gaussian factor, denote $c_1 =Z^{-1}(x_{1,1},x_{1,2},\rho )c_\rho (F_1(x_{1,1}),F_2(x_{1,2}))$, $c_2 =Z^{-1}(z_{1,1},z_{1,2},\rho )c_\rho (F_1(z_{1,1}),F_2(z_{1,2}))$. When it comes to Frank factor, they turn to be $c_1 =Z^{-1}(x_{1,1},x_{1,2},\gamma )c_\gamma (F_1(x_{1,1}),F_2(x_{1,2})),~c_2 =Z^{-1}(z_{1,1},z_{1,2},\gamma ) c_\gamma (F_1(z_{1,1}),F_2(z_{1,2}))$. Hence, according to (3), (4) and the fact $Z^{-1}$ is bounded, one can easily see that there exists a finite constant M, such that $|c_1|, |c_2| \le M$.

For the case $k = 1$,

$$\begin{aligned}&~\quad |\mathbb {P}_{\varvec{x}_1}f - \mathbb {P}_{\varvec{z}_1}f|\\&= \left| \sum _{m=0}^\infty \sum _{n=0}^\infty [f(\varvec{\omega } + \varvec{A x}_1 + \varvec{B}(m,n)^{\top }) p(m,n|\varvec{x}_1)\right. \\&\quad - \left. |f(\varvec{\omega } + \varvec{A z}_1+ \varvec{B}(m,n)^{\top }) p(m,n|\varvec{z}_1)]\right| \\&\le \sum _{m=0}^\infty \sum _{n=0}^\infty p(m,n|\varvec{x}_1)|f(\varvec{\omega } + \varvec{A x}_1 + \varvec{B}(m,n)^{\top }) - f(\varvec{\omega } + \varvec{A z}_1+ \varvec{B}(m,n)^{\top })|\\&\quad + \sum _{m=0}^\infty \sum _{n=0}^\infty |p(m,n|\varvec{x}_1) - p(m,n|\varvec{z}_1)||f(\varvec{\omega } + \varvec{A z}_1 + \varvec{B}(m,n)^{\top })|\\&= I_1 + I_2, \end{aligned}$$

where $p(m,n|\varvec{x}_1)$ and $p(m,n|\varvec{z}_1)$ are the pmfs of BPG$(x_{1,1},x_{1,2},\rho )$ (or BPF$(x_{1,1},x_{1,2},\gamma )$) and BPG $(z_{1,1},z_{1,2},\rho )$ (or BPF$(z_{1,1},z_{1,2},\gamma )$) given by (6). We start to formulate the main part of $I_2$, first suppose $c_1\le c_2$,

$$\begin{aligned}&\sum _{m=0}^\infty \sum _{n=0}^\infty |p(m,n|\varvec{x}_1) - p(m,n|\varvec{z}_1)|\\ \le&\sum _{m=0}^\infty \sum _{n=0}^\infty \left| \frac{x_{1,1}^m x_{1,2}^n}{m!n!}{\mathrm{e}}^{-(x_{1,1}+x_{1,2})}c_1 - \frac{z_{1,1}^m z_{1,2}^n}{m!n!} {\mathrm{e}}^{-(z_{1,1}+z_{1,2})}c_2\right| \\ \le&\sum _{m=0}^\infty \sum _{n=0}^\infty M\left| \frac{x_{1,1}^m}{m!} {\mathrm{e}}^{-x_{1,1}} - \frac{z_{1,1}^m}{m!} {\mathrm{e}}^{-z_{1,1}}\right| \frac{x_{1,2}^n}{n!} {\mathrm{e}}^{-x_{1,2}} \\&{}+ \sum _{m=0}^\infty \sum _{n=0}^\infty M\left| \frac{x_{1,2}^n}{n!} {\mathrm{e}}^{-x_{1,2}} - \frac{z_{1,2}^n}{n!} {\mathrm{e}}^{-z_{1,2}}\right| \frac{z_{1,1}^m}{m!} {\mathrm{e}}^{-z_{1,1}} \\ \le&M\sum _{i=0}^\infty |p(i|x_{1,1})-p(i|z_{1,1})| + M\sum _{i=0}^\infty |p(i|x_{1,2})-p(i|z_{1,2})|. \end{aligned}$$

By the proof of Wang et al. (2014, Lemma 6.4), we know that $\sum _{i=0}^\infty |p(i|x_1) - p(i|z_1)| \le 2(1 - {\mathrm{e}}^{-|x_1-z_1|})$, where p(i|x) is the pmf of a univariate Poisson distribution with intensity x evaluated at i. And since $|x_{1,i} - z_{1,i}| \le \left||\varvec{x}_1-\varvec{z}_1\right||_1 \le c_p\left||\varvec{x}_1 - \varvec{z}_1\right||_p$, for $i=1,2$ and any $1\le p \le \infty$, where $c_p = 2^{1-1/p} \le 2$, so for any $\varvec{x}_1, \varvec{z}_1$ and $p \in [1,\infty ]$, we have

$$\begin{aligned} \sum _{m=0}^\infty \sum _{n=0}^\infty |p(m,n|\varvec{x}_1) - p(m,n|\varvec{z}_1)| \le 4M(1 - {\mathrm{e}}^{-2\left||\varvec{x}_1 - \varvec{z}_1\right||_p}). \end{aligned}$$

(A.1)

When $c_1>c_2$, we can obtain the same results. So it follows from $|f| \le 1$ that $I_2 \le 4M(1 - {\mathrm{e}}^{-2\left||\varvec{x}_1 - \varvec{z}_1\right||_p})$. As for $I_1$, since

$$\begin{aligned}&\left||\varvec{\omega } + \varvec{A x}_1 + \varvec{B}(m , n)^{\top } -(\varvec{\omega } + \varvec{A z}_1 + \varvec{B}(m , n)^{\top })\right||_p \\ =&\left||\varvec{A}(\varvec{x}_1 - \varvec{z}_1)\right||_p \le \left||\varvec{A}\right||_p\left||\varvec{x}_1- \varvec{z}_1\right||_p \le \eta , \end{aligned}$$

so $I_1 \le \varepsilon '$. Therefore, we have

$$\begin{aligned} |\mathbb {P}_{\varvec{x}_1}f - \mathbb {P}_{\varvec{z}_1}f| \le \varepsilon '+ 4M(1 - {\mathrm{e}}^{-2\left||\varvec{x}_1 - \varvec{z}_1\right||_p}). \end{aligned}$$

(A.2)

For the case that $k = 2$, it follows from

$$\begin{aligned} \mathbb {E}\{f(\varvec{\lambda }_2)|\varvec{\lambda }_0= \varvec{x}\} = \mathbb {E}\{\mathbb {E}[f(\varvec{\lambda }_2)|\varvec{\lambda }_1]|\varvec{\lambda }_0= \varvec{x}\}, \end{aligned}$$

then

$$\begin{aligned}&|\mathbb {P}_{\varvec{x}_1}^2f - \mathbb {P}_{\varvec{z}_1}^2f|= \left| \sum _{m=0}^\infty \sum _{n=0}^\infty [p(m,n|\varvec{x}_1)\mathbb {P}_{\varvec{x}_2}f - p(m,n|\varvec{z}_1)\mathbb {P}_{\varvec{z}_2}f]\right| \\&\quad \le \sum _{m=0}^\infty \sum _{n=0}^\infty p(m,n|\varvec{x}_1)|\mathbb {P}_{\varvec{x}_2}f - \mathbb {P}_{\varvec{z}_2}f| + \sum _{m=0}^\infty \sum _{n=0}^\infty |p(m,n|\varvec{x}_1) - p(m,n|\varvec{z}_1)||\mathbb {P}_{\varvec{z}_2}f|, \end{aligned}$$

where $\varvec{x}_2 = \varvec{\omega } + \varvec{Ax}_1+ \varvec{B}(m , n)^{\top }$ and $\varvec{z}_2 = \varvec{\omega } + \varvec{Az}_1 + \varvec{B}(m , n)^{\top }$. Since $\left||\varvec{x}_2 - \varvec{z}_2\right||_p= \left||\varvec{A}(\varvec{x}_1 - \varvec{z}_1)\right||_p \le \left||\varvec{A}\right||_p\left||\varvec{x}_1 - \varvec{z}_1\right||_p \le \eta$, so it follows from (A.1) and (A.2) that

$$\begin{aligned} |\mathbb {P}_{\varvec{x}_1}^2f - \mathbb {P}_{\varvec{z}_1}^2f|\le & {} \varepsilon ' + 4M(1 - {\mathrm{e}}^{-2\left||\varvec{x}_2 - \varvec{z}_2\right||_p}) + 4M(1 - {\mathrm{e}}^{-2\left||\varvec{x}_1 - \varvec{z}_1\right||_p})\\\le & {} \varepsilon '+ 4M(1 - {\mathrm{e}}^{{-2\left||\varvec{A}\right||_p}{\left||\varvec{x}_1 - \varvec{z}_1\right||_p}}) + 4M(1 - {\mathrm{e}}^{-2\left||\varvec{x}_1 - \varvec{z}_1\right||_p}). \end{aligned}$$

Hence by induction, we have for any $k \ge 1$ that

$$\begin{aligned} |\mathbb {P}_{\varvec{x}_1}^kf - \mathbb {P}_{\varvec{z}_1}^kf|\le & {} \varepsilon ' + 4M\sum _{s=0}^{k-1}(1 - {\mathrm{e}}^{{-2\left||\varvec{A}\right||_p^s}{\left||\varvec{x}_1 - \varvec{z}_1\right||_p}})\\\le & {} \varepsilon ' + 8M\sum _{s=0}^\infty \left||\varvec{A}\right||_p^s\left||\varvec{x}_1 - \varvec{z}_1\right||_p \le \varepsilon ' + \frac{8M\eta }{1 - \left||\varvec{A}\right||_p} \le \varepsilon , \end{aligned}$$

which proves that $\{\varvec{\lambda }_t\}$ is an e-chain. Therefore there exists a unique stationary distribution to $\{\varvec{\lambda }_t\}$.

As for (b), it holds similar arguments to the proof of in Liu (2012, Proposition 4.2.1). $\square$

Proof of Theorem 2

The proof follows the technique from Song et al. (2005, Theorem 1) and we just rewrite them in component form. Note that we only prove the consistency of $\hat{\varvec{\theta }}_n^2$, because the estimators $\hat{\varvec{\theta }}_n^k$ for $k>2$ can be derived from it in the same manner. Suppose that $\hat{\varvec{\theta }}_n^1 \xrightarrow {\mathbb {P}} \varvec{\theta }^0$, then $\hat{\varvec{\theta }}_n^1=\varvec{\theta }^0+o_{\mathbb {P}}(1)$. Because $\hat{\varvec{\theta }}_n^2$ satisfies equations

$$\begin{aligned} \dot{l}_{m(1)}(\hat{\varvec{\theta }}_{1,n}^2)+\dot{l}_{c(1)} (\hat{\varvec{\theta }}_{1,n}^1,\hat{\theta }_{2,n}^1)=\varvec{0}, \text { and } \dot{l}_{c(2)}(\hat{\varvec{\theta }}_{1,n}^1,\hat{\theta }_{2,n}^2)=0. \end{aligned}$$

By Taylor’s expansion, we have

$$\begin{aligned} \varvec{0}&=\dot{l}_{m(1)}(\hat{\varvec{\theta }}_{1,n}^2)+\dot{l}_{c(1)} (\hat{\varvec{\theta }}_{1,n}^1,\hat{\theta }_{2,n}^1)\\&=\dot{l}_{m(1)}(\varvec{\theta }_1^0)+\ddot{l}_{m(11)}(\varvec{\theta }_1^0) (\hat{\varvec{\theta }}_{1,n}^2-\varvec{\theta }_1^0){+} o(||\hat{\varvec{\theta }}_{1,n}^2-\varvec{\theta }_1^0||^2) +\dot{l}_{c(1)}(\hat{\varvec{\theta }}_{1,n}^1,\hat{\theta }_{2,n}^1),\\ 0&=\dot{l}_{c(2)}(\hat{\varvec{\theta }}_{1,n}^1,\hat{\theta }_{2,n}^2)= \dot{l}_{c(2)}(\hat{\varvec{\theta }}_{1,n}^1,\theta _2^0)+ \ddot{l}_{c(22)}(\hat{\varvec{\theta }}_{1,n}^1,\theta _2^0)(\hat{\theta }_{2,n}^2-\theta _2^0) +o(||\hat{\theta }_{2,n}^2-\theta _2^0||^2). \end{aligned}$$

Then we can obtain

$$\begin{aligned} \hat{\varvec{\theta }}_{1,n}^2-\varvec{\theta }_1^0= & {} [-n^{-1}\ddot{l}_{m(11)} (\varvec{\theta }_{1,n}^*)]^{-1}n^{-1}\nonumber \\&\times [\dot{l}_{m(1)}(\varvec{\theta }_1^0)+\dot{l}_{c(1)} (\hat{\varvec{\theta }}_{1,n}^1,\hat{\theta }_{2,n}^1)+ o(||\hat{\varvec{\theta }}_{1,n}^2-\varvec{\theta }_1^0||^2)], \end{aligned}$$

(A.3)

$$\begin{aligned} \hat{\theta }_{2,n}^2-\theta _2^0= & {} [-n^{-1}\ddot{l}_{c(22)} (\hat{\varvec{\theta }}_{1,n}^1,\theta _{2,n}^*)]^{-1} n^{-1}[\dot{l}_{c(2)}(\hat{\varvec{\theta }}_{1,n}^1,\theta _2^0)+ o(||\hat{\theta }_{2,n}^2-\theta _2^0||^2)].\nonumber \\ \end{aligned}$$

(A.4)

Under the regularity conditions, $[-n^{-1}\ddot{l}_{m(11)}(\varvec{\theta }_{1,n}^*)]$ and $[-n^{-1}\ddot{l}_{c(22)}(\hat{\varvec{\theta }}_{1,n}^1,\theta _{2,n}^*)]$ are bounded and due to the consistency of $\hat{\varvec{\theta }}_n^1$, it follows that

$$\begin{aligned}&\lim _{n\rightarrow \infty }[n^{-1}\dot{l}_{m(1)}(\varvec{\theta }_1^0)+n^{-1}\dot{l}_{c(1)} (\hat{\varvec{\theta }}_{1,n}^1,\hat{\theta }_{2,n}^1)+ n^{-1}o(||\hat{\varvec{\theta }}_{1,n}^2-\varvec{\theta }_1^0||^2)]\\ =&\lim _{n\rightarrow \infty }n^{-1}\dot{l}(\varvec{\theta }^0)=\varvec{0},\\&\lim _{n\rightarrow \infty }[n^{-1}\dot{l}_{c(2)}(\hat{\varvec{\theta }}_{1,n}^1,\theta _2^0)+ n^{-1}o(||\hat{\theta }_{2,n}^2-\theta _2^0||^2)]\\ =&\lim _{n\rightarrow \infty }n^{-1} \dot{l}_{c(2)}(\varvec{\theta }_1^0,\theta _2^0)=0. \end{aligned}$$

It is easy to find that $\hat{\varvec{\theta }}_n^2 \xrightarrow {\mathbb {P}} \varvec{\theta }^0$ according to (A.3) and (A.4). $\square$

Proof of Theorem 3

We employ the similar arguments of Song et al. (2005, Theorem 3). First, it is important to mention that under the regularity conditions, the following result is obvious:

$$\begin{aligned} n^{-1/2}\left( \begin{array}{c} \dot{l}_m(\varvec{\theta }^0) \\ \dot{l}_c(\varvec{\theta }^0)\\ \end{array} \right) \rightarrow N(0,\varvec{\Omega }), \end{aligned}$$

where

$$\begin{aligned} \varvec{\Omega }=\lim _{n\rightarrow \infty }n^{-1}\left( \begin{array}{cc} \mathbb {E}\dot{l}_m(\varvec{\theta }^0)\dot{l}_m^\top (\varvec{\theta }^0) &{} \mathbb {E}\dot{l}_m(\varvec{\theta }^0)\dot{l}_c^\top (\varvec{\theta }^0) \\ \mathbb {E}\dot{l}_c(\varvec{\theta }^0)\dot{l}_m^\top (\varvec{\theta }^0) &{} \mathbb {E}\dot{l}_c(\varvec{\theta }^0)\dot{l}_c^\top (\varvec{\theta }^0) \\ \end{array} \right) . \end{aligned}$$

Then according to Theorem 2, the consistency of $\varvec{\theta }_n^k$ holds and satisfies equations

$$\begin{aligned} \dot{l}_{m(1)}(\hat{\varvec{\theta }}_{1,n}^k)+\dot{l}_{c(1)} (\hat{\varvec{\theta }}_{1,n}^{k-1},\hat{\theta }_{2,n}^{k-1})=\varvec{0}, \text { and } \dot{l}_{c(2)}(\hat{\varvec{\theta }}_{1,n}^{k-1},\hat{\theta }_{2,n}^k)=0. \end{aligned}$$

By Taylor’s expansion without the remainder terms, we have at Step k

$$\begin{aligned}&\dot{l}_{m(1)}(\varvec{\theta }_{1}^0)+\ddot{l}_{m(11)}(\varvec{\theta }_{1}^0) (\hat{\varvec{\theta }}_{1,n}^k-\varvec{\theta }_{1}^0)\\&{}+\dot{l}_{c(1)}(\varvec{\theta }^0)+\ddot{l}_{c(11)}(\varvec{\theta }^0) (\hat{\varvec{\theta }}_{1,n}^{k-1}-\varvec{\theta }_{1}^0) +\ddot{l}_{c(12)}(\varvec{\theta }^0)(\hat{\theta }_{2,n}^{k-1}-\varvec{\theta }_{2}^0)=\varvec{0},\\&\dot{l}_{c(2)}(\varvec{\theta }^0)+\ddot{l}_{c(21)}(\varvec{\theta }^0) (\hat{\varvec{\theta }}_{1,n}^{k-1}-\varvec{\theta }_{1}^0) +\ddot{l}_{c(22)}(\varvec{\theta }^0)(\hat{\theta }_{2,n}^k-\varvec{\theta }_{2}^0)=0. \end{aligned}$$

Rewriting these in a matrix form, we can obtain

$$\begin{aligned} \sqrt{n}(\hat{\varvec{\theta }}_{n}^k-\varvec{\theta }^0)= \varvec{D}_n^{-1}\varvec{T}_n\sqrt{n}(\hat{\varvec{\theta }}_{n}^{k-1}-\varvec{\theta }^0) +\varvec{D}_n^{-1}[n^{-1/2}\dot{l}(\varvec{\theta }^0)]. \end{aligned}$$

(A.5)

Hence by recursion, (A.5) turns to be

$$\begin{aligned} \sqrt{n}(\hat{\varvec{\theta }}_{n}^k-\varvec{\theta }^0)=(\varvec{D}_n^{-1}\varvec{T}_n)^{k-1}\sqrt{n} (\hat{\varvec{\theta }}_{n}^{1}-\varvec{\theta }^0)+\sum _{s=0}^{k-2}(\varvec{D}_n^{-1}\varvec{T}_n)^s\varvec{D}_n^{-1} [n^{-1/2}\dot{l}(\varvec{\theta }^0)]. \end{aligned}$$

Because $\hat{\varvec{\theta }}_{1,n}^{1}$ is used to define $\hat{\theta }_{2,n}^{1}$, a Taylor’s expansion at Step 1 leads to

$$\begin{aligned} \sqrt{n}(\hat{\varvec{\theta }}_{n}^{1}-\varvec{\theta }^0)&=\left( \begin{array}{cc} -n^{-1}\ddot{l}_{m(11)} &{} 0 \\ -n^{-1}\ddot{l}_{c(21)} &{} -n^{-1}\ddot{l}_{c(22)} \\ \end{array} \right) ^{-1}\left( \begin{array}{c} -n^{-1/2}\dot{l}_{m(1)}\\ -n^{-1/2}\dot{l}_{c(2)} \\ \end{array} \right) \\&=\varvec{D}_n^{-1}\left( \begin{array}{c} -n^{-1/2}\dot{l}_{m(1)}\\ -n^{-1/2}\dot{l}_{c(2)} \\ \end{array} \right) +\varvec{V}_n\left( \begin{array}{c} -n^{-1/2}\dot{l}_{m(1)}\\ -n^{-1/2}\dot{l}_{c(2)} \\ \end{array} \right) . \end{aligned}$$

Thus,

$$\begin{aligned} \sqrt{n}(\hat{\varvec{\theta }}_{n}^{k}-\varvec{\theta }^0)&=\sum _{s=0}^{k-1} (\varvec{D}_n^{-1}\varvec{T}_n)^s\varvec{D}_n^{-1} \left( \begin{array}{c} -n^{-1/2}\dot{l}_{m(1)} \\ -n^{-1/2}\dot{l}_{c(2)} \\ \end{array} \right) \\&~~+\sum _{s=0}^{k-2}(\varvec{D}_n^{-1}\varvec{T}_n)^s\varvec{D}_n^{-1} \left( \begin{array}{c} -n^{-1/2}\dot{l}_{c(1)} \\ 0 \\ \end{array} \right) \\&~~+(\varvec{D}_n^{-1}\varvec{T}_n)^{k-1}\varvec{V}_n\left( \begin{array}{c} -n^{-1/2}\dot{l}_{m(1)} \\ -n^{-1/2}\dot{l}_{c(2)} \\ \end{array} \right) \\&=[\varvec{I}-(\varvec{D}_n^{-1}\varvec{T}_n)^k][\varvec{I}-\varvec{D}_n^{-1}\varvec{T}_n]^{-1} \varvec{D}_n^{-1}\left( \begin{array}{c} -n^{-1/2}\dot{l}_{m(1)} \\ -n^{-1/2}\dot{l}_{c(2)} \\ \end{array} \right) \\&~~+[\varvec{I}-(\varvec{D}_n^{-1}\varvec{T}_n)^{k-1}][\varvec{I}-\varvec{D}_n^{-1}\varvec{T}_n]^{-1} \varvec{D}_n^{-1}\left( \begin{array}{c} -n^{-1/2}\dot{l}_{c(1)} \\ 0 \\ \end{array} \right) \\&~~+(\varvec{D}_n^{-1}\varvec{T}_n)^{k-1}\varvec{V}_n\left( \begin{array}{c} -n^{-1/2}\dot{l}_{m(1)} \\ -n^{-1/2}\dot{l}_{c(2)} \\ \end{array} \right) . \end{aligned}$$

Note that $[\varvec{I}-\varvec{D}_n^{-1}\varvec{T}_n]^{-1}\varvec{D}_n^{-1}=(\varvec{D}_n^{-1}-\varvec{T}_n)^{-1}=[-n^{-1}\ddot{l} (\varvec{\theta }^0)]^{-1}$, then it follows that

where $\varvec{\Gamma }_k$ is defined in the statement of Theorem 3.

Furthermore, when Assumption 2 holds, i.e., the marginal function $l_m$ satisfies the information dominance condition. Then we have $\varvec{\Gamma }^k \rightarrow 0$ as $k \rightarrow \infty$, thus the asymptotic variance–covariance matrix becomes

$$\begin{aligned} \varvec{\Sigma }_{\infty }&=\varvec{\mathcal {I}}^{-1}\left( \begin{array}{cc} n^{-1}\mathbb {E}(\dot{l}_{m(1)}+\dot{l}_{c(1)})(\dot{l}_{m(1)}+\dot{l}_{c(1)})^\top &{} n^{-1}\mathbb {E}\dot{l}_{c(2)}(\dot{l}_{m(1)}+\dot{l}_{c(1)})^\top \\ n^{-1}\mathbb {E}(\dot{l}_{m(1)}+\dot{l}_{c(1)})\dot{l}_{c(2)}^\top &{} n^{-1}\dot{l}_{c(2)}\dot{l}_{c(2)}^\top \\ \end{array} \right) \varvec{\mathcal {I}}^{-1}\\&=\varvec{\mathcal {I}}^{-1}\{\lim _{n\rightarrow \infty }n^{-1}\mathbb {E}\dot{l}(\varvec{\theta }^0)\dot{l}(\varvec{\theta }^0)^\top \} \varvec{\mathcal {I}}^{-1}=\varvec{\mathcal {I}}^{-1}. \end{aligned}$$

$\square$