Quasi-Akaike information criterion of structural equation modeling with latent variables for diffusion processes

We consider a model selection problem for structural equation modeling (SEM) with latent variables for diffusion processes. First, we define the true model of the SEM. The stochastic processes $\mathbb{X}_{1,0,t}$ and $\mathbb{X}_{2,0,t}$ are defined by the factor models as follows:

	$\displaystyle\mathbb{X}_{1,0,t}$	$\displaystyle={\bf{\Lambda}}_{x_{1},0}\xi_{0,t}+\delta_{0,t},$		(1.1)
	$\displaystyle\mathbb{X}_{2,0,t}$	$\displaystyle={\bf{\Lambda}}_{x_{2},0}\eta_{0,t}+\varepsilon_{0,t},$		(1.2)

where $\{\mathbb{X}_{1,0,t}\}_{t\geq 0}$ and $\{\mathbb{X}_{2,0,t}\}_{t\geq 0}$ are $p_{1}$ and $p_{2}$-dimensional observable vector processes, $\{\xi_{0,t}\}_{t\geq 0}$ and $\{\eta_{0,t}\}_{t\geq 0}$ are $k_{1}$ and $k_{2}$-dimensional latent common factor vector processes, $\{\delta_{0,t}\}_{t\geq 0}$ and $\{\varepsilon_{0,t}\}_{t\geq 0}$ are $p_{1}$ and $p_{2}$-dimensional latent unique factor vector processes, respectively. ${\bf{\Lambda}}_{x_{1},0}\in\mathbb{R}^{p_{1}\times k_{1}}$ and ${\bf{\Lambda}}_{x_{2},0}\in\mathbb{R}^{p_{2}\times k_{2}}$ are constant loading matrices. Both $p_{1}$ and $p_{2}$ are not zero, $p_{1}$, $p_{2}$, $k_{1}$ and $k_{2}$ are fixed, $k_{1}\leq p_{1}$ and $k_{2}\leq p_{2}$. Let $p=p_{1}+p_{2}$. Suppose that $\{\xi_{0,t}\}_{t\geq 0}$, $\{\delta_{0,t}\}_{t\geq 0}$ and $\{\varepsilon_{0,t}\}_{t\geq 0}$ satisfy the following stochastic differential equations:

	$\displaystyle\quad\mathrm{d}\xi_{0,t}$	$\displaystyle=B_{1}(\xi_{0,t})\mathrm{d}t+{\bf{S}}_{1,0}\mathrm{d}W_{1,t},\ \ \xi_{0,0}=c_{1},$		(1.3)
	$\displaystyle\quad\mathrm{d}\delta_{0,t}$	$\displaystyle=B_{2}(\delta_{0,t})\mathrm{d}t+{\bf{S}}_{2,0}\mathrm{d}W_{2,t},\ \ \delta_{0,0}=c_{2},$		(1.4)
	$\displaystyle\quad\mathrm{d}\varepsilon_{0,t}$	$\displaystyle=B_{3}(\varepsilon_{0,t})\mathrm{d}t+{\bf{S}}_{3,0}\mathrm{d}W_{3,t},\ \ \varepsilon_{0,0}=c_{3},$		(1.5)

where $B_{1}:\mathbb{R}^{k_{1}}\rightarrow\mathbb{R}^{k_{1}}$, ${\bf{S}}_{1,0}\in\mathbb{R}^{k_{1}\times r_{1}}$, $c_{1}\in\mathbb{R}^{k_{1}}$, $B_{2}:\mathbb{R}^{p_{1}}\rightarrow\mathbb{R}^{p_{1}}$, ${\bf{S}}_{2,0}\in\mathbb{R}^{p_{1}\times r_{2}}$, $c_{2}\in\mathbb{R}^{p_{1}}$, $B_{3}:\mathbb{R}^{p_{2}}\rightarrow\mathbb{R}^{p_{2}}$, ${\bf{S}}_{3,0}\in\mathbb{R}^{p_{2}\times r_{3}}$, $c_{3}\in\mathbb{R}^{p_{2}}$ and $W_{1,t}$, $W_{2,t}$ and $W_{3,t}$ are $r_{1}$, $r_{2}$ and $r_{3}$-dimensional standard Wiener processes, respectively. Moreover, we express the relationship between $\eta_{0,t}$ and $\xi_{0,t}$ as follows:

$\displaystyle\eta_{0,t}={\bf{B}}_{0}\eta_{0,t}+{\bf{\Gamma}}_{0}\xi_{0,t}+\zeta_{0,t},$

(1.6)

where ${\bf{B}}_{0}\in\mathbb{R}^{k_{2}\times k_{2}}$ is a constant loading matrix, whose diagonal elements are zero, and ${\bf{\Gamma}}_{0}\in\mathbb{R}^{k_{2}\times k_{1}}$ is a constant loading matrix. Define ${\bf{\Psi}}_{0}=\mathbb{I}_{k_{2}}-{\bf{B}}_{0}$, where $\mathbb{I}_{k_{2}}$ denotes the identity matrix of size $k_{2}$. We assume that ${\bf{\Lambda}}_{x_{1},0}$ is a full column rank matrix and ${\bf{\Psi}}_{0}$ is non-singular. $\{\zeta_{0,t}\}_{t\geq 0}$ is a $k_{2}$-dimensional latent unique factor vector process defined by the following stochastic differential equation:

$\displaystyle\quad\mathrm{d}\zeta_{0,t}=B_{4}(\zeta_{0,t})\mathrm{d}t+{\bf{S}}_{4,0}\mathrm{d}W_{4,t},\ \ \zeta_{0,0}=c_{4},$

(1.7)

where $B_{4}:\mathbb{R}^{k_{2}}\rightarrow\mathbb{R}^{k_{2}}$, ${\bf{S}}_{4,0}\in\mathbb{R}^{k_{2}\times r_{4}}$, $c_{4}\in\mathbb{R}^{k_{2}}$ and $W_{4,t}$ is an $r_{4}$-dimensional standard Wiener process. Set ${\bf{\Sigma}}_{\xi\xi,0}={\bf{S}}_{1,0}{\bf{S}}_{1,0}^{\top}$, ${\bf{\Sigma}}_{\delta\delta,0}={\bf{S}}_{2,0}{\bf{S}}_{2,0}^{\top}$, ${\bf{\Sigma}}_{\varepsilon\varepsilon,0}={\bf{S}}_{3,0}{\bf{S}}_{3,0}^{\top}$ and ${\bf{\Sigma}}_{\zeta\zeta,0}={\bf{S}}_{4,0}{\bf{S}}_{4,0}^{\top}$, where $\top$ denotes the transpose. It is supposed that ${\bf{\Sigma}}_{\delta\delta,0}$ and ${\bf{\Sigma}}_{\varepsilon\varepsilon,0}$ are positive definite matrices, and $W_{1,t}$, $W_{2,t}$, $W_{3,t}$ and $W_{4,t}$ are independent standard Wiener processes on a stochastic basis with usual conditions $(\Omega,\mathscr{F},\{\mathscr{F}_{t}\},{\bf{P}})$. Let $\mathbb{X}_{0,t}=(\mathbb{X}_{1,0,t}^{\top},\mathbb{X}_{2,0,t}^{\top})^{\top}$. Set ${\bf{\Sigma}}_{0}$ as the variance of $\mathbb{X}_{0,t}$. If there is no misunderstanding, we simply write $\mathbb{X}_{0,t}$ as $\mathbb{X}_{t}$. $\mathbb{X}_{n}=(\mathbb{X}_{t_{i}^{n}})_{0\leq i\leq n}=(\mathbb{X}_{0,t_{i}^{n}})_{0\leq i\leq n}$ are discrete observations, where $t_{i}^{n}=ih_{n}$, $h_{n}=\frac{T}{n}$, $T$ is fixed, and $p_{1}$, $p_{2}$, $k_{1}$ and $k_{2}$ are independent of $n$. We consider the situation where $h_{n}\longrightarrow 0$ as $n\longrightarrow\infty$. We cannot estimate all the elements of ${\bf{\Lambda}}_{x_{1},0}$, ${\bf{\Lambda}}_{x_{2},0}$, ${\bf{\Gamma}}_{0}$, ${\bf{\Psi}}_{0}$, ${\bf{\Sigma}}_{\xi\xi,0}$, ${\bf{\Sigma}}_{\delta\delta,0}$, ${\bf{\Sigma}}_{\varepsilon\varepsilon,0}$ and ${\bf{\Sigma}}_{\zeta\zeta,0}$. Thus, some elements may be assumed to be zero to satisfy an identifiability condition; see, e.g., Everitt [6]. Note that these constraints and the number of factors $k_{1}$ and $k_{2}$ are determined from the theoretical viewpoint of each research field.

A model selection problem among the following $M$ parametric models is considered. We define the parametric model of Model $m\in\{1,\cdots,M\}$ as follows. Set $\theta_{m}\in\Theta_{m}\subset\mathbb{R}^{q_{m}}$ as the parameter of Model $m$, where $\Theta_{m}$ is a convex compact space. It is assumed that $\Theta_{m}$ has locally Lipschitz boundary; see, e.g., Adams and Fournier [1]. The stochastic processes $\mathbb{X}^{\theta}_{1,m,t}$ and $\mathbb{X}^{\theta}_{2,m,t}$ are defined as the following factor models:

	$\displaystyle\mathbb{X}^{\theta}_{1,m,t}$	$\displaystyle={\bf{\Lambda}}^{\theta}_{x_{1},m}\xi^{\theta}_{m,t}+\delta^{\theta}_{m,t},$		(1.8)
	$\displaystyle\mathbb{X}^{\theta}_{2,m,t}$	$\displaystyle={\bf{\Lambda}}^{\theta}_{x_{2},m}\eta^{\theta}_{m,t}+\varepsilon^{\theta}_{m,t},$		(1.9)

where $\{\mathbb{X}^{\theta}_{1,m,t}\}_{t\geq 0}$ and $\{\mathbb{X}^{\theta}_{2,m,t}\}_{t\geq 0}$ are $p_{1}$ and $p_{2}$-dimensional observable vector processes, $\{\xi^{\theta}_{m,t}\}_{t\geq 0}$ and $\{\eta^{\theta}_{m,t}\}_{t\geq 0}$ are $k_{1}$ and $k_{2}$-dimensional latent common factor vector processes, $\{\delta^{\theta}_{m,t}\}_{t\geq 0}$ and $\{\varepsilon^{\theta}_{m,t}\}_{t\geq 0}$ are $p_{1}$ and $p_{2}$-dimensional latent unique factor vector processes, respectively. ${\bf{\Lambda}}^{\theta}_{x_{1},m}\in\mathbb{R}^{p_{1}\times k_{1}}$ and ${\bf{\Lambda}}^{\theta}_{x_{2},m}\in\mathbb{R}^{p_{2}\times k_{2}}$ are constant loading matrices. Assume that $\{\xi^{\theta}_{m,t}\}_{t\geq 0}$, $\{\delta^{\theta}_{m,t}\}_{t\geq 0}$ and $\{\varepsilon^{\theta}_{m,t}\}_{t\geq 0}$ satisfy the following stochastic differential equations:

	$\displaystyle\quad\mathrm{d}\xi^{\theta}_{m,t}$	$\displaystyle=B_{1}(\xi^{\theta}_{m,t})\mathrm{d}t+{\bf{S}}^{\theta}_{1,m}\mathrm{d}W_{1,t},\ \ \xi^{\theta}_{m,0}=c_{1},$		(1.10)
	$\displaystyle\quad\mathrm{d}\delta^{\theta}_{m,t}$	$\displaystyle=B_{2}(\delta^{\theta}_{m,t})\mathrm{d}t+{\bf{S}}^{\theta}_{2,m}\mathrm{d}W_{2,t},\ \ \delta^{\theta}_{m,0}=c_{2},$		(1.11)
	$\displaystyle\quad\mathrm{d}\varepsilon^{\theta}_{m,t}$	$\displaystyle=B_{3}(\varepsilon^{\theta}_{m,t})\mathrm{d}t+{\bf{S}}^{\theta}_{3,m}\mathrm{d}W_{3,t},\ \ \varepsilon^{\theta}_{m,0}=c_{3},$		(1.12)

where ${\bf{S}}^{\theta}_{1,m}\in\mathbb{R}^{k_{1}\times r_{1}}$, ${\bf{S}}^{\theta}_{2,m}\in\mathbb{R}^{p_{1}\times r_{2}}$ and ${\bf{S}}^{\theta}_{3,m}\in\mathbb{R}^{p_{2}\times r_{3}}$. Furthermore, the relationship between $\eta^{\theta}_{m,t}$ and $\xi^{\theta}_{m,t}$ is expressed as follows:

$\displaystyle\eta^{\theta}_{m,t}={\bf{B}}_{m}^{\theta}\eta^{\theta}_{m,t}+{\bf{\Gamma}}_{m}^{\theta}\xi^{\theta}_{m,t}+\zeta^{\theta}_{m,t},$

(1.13)

where ${\bf{B}}^{\theta}_{m}\in\mathbb{R}^{k_{2}\times k_{2}}$ is a constant loading matrix, whose diagonal elements are zero, and ${\bf{\Gamma}}^{\theta}_{m}\in\mathbb{R}^{k_{2}\times k_{1}}$ is a constant loading matrix. Set ${\bf{\Psi}}^{\theta}_{m}=\mathbb{I}_{k_{2}}-{\bf{B}}^{\theta}_{m}$. It is supposed that ${\bf{\Lambda}}^{\theta}_{x_{1},m}$ is a full column rank matrix and ${\bf{\Psi}}^{\theta}_{m}$ is non-singular. $\{\zeta^{\theta}_{m,t}\}_{t\geq 0}$ is a $k_{2}$-dimensional latent unique factor vector process defined by the following stochastic differential equation:

$\displaystyle\quad\mathrm{d}\zeta^{\theta}_{m,t}=B_{4}(\zeta^{\theta}_{m,t})\mathrm{d}t+{\bf{S}}^{\theta}_{4,m}\mathrm{d}W_{4,t},\ \ \zeta^{\theta}_{m,0}=c_{4},$

(1.14)

where ${\bf{S}}^{\theta}_{4,m}\in\mathbb{R}^{k_{2}\times r_{4}}$. Let ${\bf{\Sigma}}^{\theta}_{\xi\xi,m}={\bf{S}}^{\theta}_{1,m}{\bf{S}}^{\theta\top}_{1,m}$, ${\bf{\Sigma}}^{\theta}_{\delta\delta,m}={\bf{S}}^{\theta}_{2,m}{\bf{S}}^{\theta\top}_{2,m}$, ${\bf{\Sigma}}^{\theta}_{\varepsilon\varepsilon,m}={\bf{S}}^{\theta}_{3,m}{\bf{S}}^{\theta\top}_{3,m}$ and ${\bf{\Sigma}}^{\theta}_{\zeta\zeta,m}={\bf{S}}^{\theta}_{4,m}{\bf{S}}^{\theta\top}_{4,m}$. It is assumed that ${\bf{\Sigma}}^{\theta}_{\delta\delta,m}$ and ${\bf{\Sigma}}^{\theta}_{\varepsilon\varepsilon,m}$ are positive definite matrices. Define $\mathbb{X}^{\theta}_{m,t}=(\mathbb{X}_{1,m,t}^{\theta\top},\mathbb{X}_{2,m,t}^{\theta\top})^{\top}$. Set

$\displaystyle{\bf{\Sigma}}_{m}(\theta_{m})=\begin{pmatrix}{\bf{\Sigma}}_{m}^{11}(\theta_{m})&{\bf{\Sigma}}_{m}^{12}(\theta_{m})\\ {\bf{\Sigma}}_{m}^{12\top}(\theta_{m})&{\bf{\Sigma}}_{m}^{22}(\theta_{m})\end{pmatrix}$

as the variance of $\mathbb{X}^{\theta}_{m,t}$, where

	$\displaystyle\qquad\qquad{\bf{\Sigma}}^{11}_{m}(\theta_{m})$	$\displaystyle={\bf{\Lambda}}^{\theta}_{x_{1},m}{\bf{\Sigma}}^{\theta}_{\xi\xi,m}{\bf{\Lambda}}_{x_{1},m}^{\theta\top}+{\bf{\Sigma}}^{\theta}_{\delta\delta,m},$
	$\displaystyle{\bf{\Sigma}}^{12}_{m}(\theta_{m})$	$\displaystyle={\bf{\Lambda}}^{\theta}_{x_{1},m}{\bf{\Sigma}}^{\theta}_{\xi\xi,m}{\bf{\Gamma}}_{m}^{\theta\top}{\bf{\Psi}}_{m}^{\theta-1\top}{\bf{\Lambda}}_{x_{2},m}^{\theta\top},$
	$\displaystyle{\bf{\Sigma}}^{22}_{m}(\theta_{m})$	$\displaystyle={\bf{\Lambda}}^{\theta}_{x_{2},m}{\bf{\Psi}}_{m}^{\theta-1}({\bf{\Gamma}}_{m}^{\theta}{\bf{\Sigma}}^{\theta}_{\xi\xi,m}{\bf{\Gamma}}_{m}^{\theta\top}+{\bf{\Sigma}}^{\theta}_{\zeta\zeta,m}){\bf{\Psi}}_{m}^{\theta-1\top}{\bf{\Lambda}}_{x_{2},m}^{\theta\top}+{\bf{\Sigma}}^{\theta}_{\varepsilon\varepsilon,m}.$

It is supposed that there exists $\theta_{m,0}\in{\rm{Int}}\Theta_{m}$ such that ${\bf{\Sigma}}_{0}={\bf{\Sigma}}_{m}(\theta_{m,0})$, and Model $m$ satisfies an identifiability condition.

Structural equation modeling (SEM) with latent variables is a method of analyzing the relationships between latent variables that cannot be observed; see, e.g., Jöreskog [10], Everitt [6], Mueller [15] and references therein. A researcher has often some candidate models in SEM. Note that the candidate models are usually specified to express different hypotheses. The goodness-of-fit test based on the likelihood ratio is widely used for model evaluation in SEM. Akaike [4] proposed the use of the Akaike information criterion (AIC) in a factor model. Using AIC in a factor model, we can choose the optimal number of factors in terms of prediction. AIC is also widely used in SEM to choose the optimal model as well as a factor model; see e.g., Huang [9].

Thanks to the development of measuring devices, high-frequency data such as stock prices can be easily obtained these days, so that many researchers have studied parametric estimation of diffusion processes based on high-frequency data; see, e.g., Yoshida [18], Genon-Catalot and Jacod [7], Kessler [11], Uchida and Yoshida [17] and references therein. Recently, in the field of financial econometrics, the factor model based on high-frequency data has been extensively studied. Aït-Sahalia and Xiu [3] proposed a continuous-time latent factor model for a high-dimensional model using principal component analysis. Kusano and Uchida [12] suggested classical factor analysis for diffusion processes. This method enables us to analyze the relationships between low-dimensional observed variables sampled with high frequency and latent variables. For instance, based on high-frequency stock price data, we can analyze latent variables such as a world market factor and factors related to a certain industry (Figure 1). On the other hand, there have been few researchers who examine the relationships between these latent variables based on high-frequency data. Kusano and Uchida [13] proposed SEM with latent variables for diffusion processes. Using this method, one can examine the relationships between latent variables based on high-frequency data. For example, if we want to study the relationship between the world market factor and the Japanese financial factor, this method enables us to analyze the relationship (Figure 2). SEM with latent variables may be referred as the regression analysis between latent variables. While both explanatory and objective variables are observable in regression analysis, both of them are latent in SEM with latent variables. For the regression analysis and the market models based on high-frequency data, see, e.g., Aït-Sahalia et al. [2].

The model selection problem for diffusion processes based on discrete observations has been actively studied. Uchida [16] proposed the contrast-based information criterion for ergodic diffusion processes, and obtained the asymptotic result of the difference between the contrast-based information criteria. Eguchi and Masuda [5] studied the model comparison problem for semiparametric L$\acute{e}$vy driven SDE and suggested the Gaussian quasi-AIC. Since the information criterion is important in SEM as mentioned above, we propose the quasi-AIC (QAIC) of SEM with latent variables for diffusion processes and study the asymptotic properties. In this paper, we consider the non-ergodic case. For the ergodic case, see Appendix 6.3.

The paper is organized as follows. In Section 2, we introduce the notation and assumptions. In Section 3, the QAIC of SEM with latent variables for diffusion processes is considered. Moreover, the situation where the set of competing models includes some (not all) misspecified parametric models is studied. It is shown that the probability of choosing the misspecified models converges to zero. In Section 4, we give examples and simulation results. In Section 5, the results described in Section 3 are proved.

First, we prepare the following notations and definitions. For any vector $v$, $|v|=\sqrt{\mathop{\rm tr}\nolimits{vv^{\top}}}$, $v^{(i)}$ is the $i$-th element of $v$, and $\mathop{\rm Diag}\nolimits v$ is the diagonal matrix, whose $i$-th diagonal element is $v^{(i)}$. For any matrix $A$, $|A|=\sqrt{\mathop{\rm tr}\nolimits{AA^{\top}}}$, and $A_{ij}$ is the $(i,j)$-th element of $A$. For matrices $A$ and $B$ of the same size, $A[B]=\mathop{\rm tr}\nolimits(AB^{\top})$. For any matrix $A\in\mathbb{R}^{p\times p}$ and vectors $x,y\in\mathbb{R}^{p}$, we define $A[x,y]=x^{\top}Ay$. For a positive definite matrix $A$, we write $A>0$. For any symmetric matrix $A\in\mathbb{R}^{p\times p}$, $\mathop{\rm vec}\nolimits A$, $\mathop{\rm vech}\nolimits A$ and $\mathbb{D}_{p}$ are the vectorization of $A$, the half-vectorization of $A$ and the $p^{2}\times\bar{p}$ duplication matrix respectively, where $\bar{p}=p(p+1)/2$. Note that $\mathop{\rm vec}\nolimits{A}=\mathbb{D}_{p}\mathop{\rm vech}\nolimits{A}$; see, e.g., Harville [8]. For any matrix $A$, $A^{+}$ stands for the Moore-Penrose inverse of $A$. Set $\mathcal{M}_{p}^{++}$ as the sets of all $p\times p$ real-valued positive definite matrices. For any positive sequence $u_{n}$, ${\rm{R}}:[0,\infty)\times\mathbb{R}^{d}\rightarrow\mathbb{R}$ denotes the short notation for functions which satisfy $|{\rm{R}}({u_{n}},x)|\leq u_{n}C(1+|x|)^{C}$ for some $C>0$. Let $C^{k}_{\uparrow}(\mathbb{R}^{d})$ be the space of all functions $f$ satisfying the following conditions:

• (i)

  $f$ is continuously differentiable with respect to $x\in\mathbb{R}^{d}$ up to order $k$.

• (ii)

  $f$ and all its derivatives are of polynomial growth in $x\in\mathbb{R}^{d}$, i.e., $g$ is of polynomial growth in $x\in\mathbb{R}^{d}$ if $\displaystyle g(x)=R(1,x)$.

The symbols $\stackrel{{\scriptstyle p}}{{\longrightarrow}}$ and $\stackrel{{\scriptstyle d}}{{\longrightarrow}}$ denote convergence in probability and convergence in distribution, respectively. For any process $Y_{t}$, $\Delta_{i}Y=Y_{t_{i}^{n}}-Y_{t_{i-1}^{n}}$. Set

$\displaystyle\mathbb{Q}_{\mathbb{XX}}=\frac{1}{T}\sum_{i=1}^{n}(\mathbb{X}_{t_{i}^{n}}-\mathbb{X}_{t_{i-1}^{n}})(\mathbb{X}_{t_{i}^{n}}-\mathbb{X}_{t_{i-1}^{n}})^{\top}.$

${\bf{E}}$ denotes the expectation under ${\bf{P}}$. Next, we make the following assumptions.

1. [A]
   1. (a)
      1. (i)

         There exists a constant $C>0$ such that

         $\displaystyle|B_{1}(x)-B_{1}(y)|\leq C|x-y|$

         for any $x,y\in\mathbb{R}^{k_{1}}$.

      2. (ii)

         For all $\ell>0$, $\displaystyle\sup_{t}{\bf{E}}\Bigl{[}\bigl{|}\xi_{0,t}\bigr{|}^{\ell}\Bigr{]}<\infty$.

      3. (iii)

         $B_{1}\in C^{4}_{\uparrow}(\mathbb{R}^{k_{1}})$.

   2. (b)
      1. (i)

         There exists a constant $C>0$ such that

         $\displaystyle|B_{2}(x)-B_{2}(y)|\leq C|x-y|$

         for any $x,y\in\mathbb{R}^{p_{1}}$.

      2. (ii)

         For all $\ell\geq 0$, $\displaystyle\sup_{t}{\bf{E}}\Bigl{[}\bigl{|}\delta_{0,t}\bigr{|}^{\ell}\Bigr{]}<\infty$.

      3. (iii)

         $B_{2}\in C^{4}_{\uparrow}(\mathbb{R}^{p_{1}})$.

   3. (c)
      1. (i)

         There exists a constant $C>0$ such that

         $\displaystyle|B_{3}(x)-B_{3}(y)|\leq C|x-y|$

         for any $x,y\in\mathbb{R}^{p_{2}}$.

      2. (ii)

         For all $\ell\geq 0$, $\displaystyle\sup_{t}{\bf{E}}\Bigl{[}\bigl{|}\varepsilon_{0,t}\bigr{|}^{\ell}\Bigr{]}<\infty$.

      3. (iii)

         $B_{3}\in C^{4}_{\uparrow}(\mathbb{R}^{p_{2}})$.

   4. (d)
      1. (i)

         There exists a constant $C>0$ such that

         $\displaystyle|B_{4}(x)-B_{4}(y)|\leq C|x-y|$

         for any $x,y\in\mathbb{R}^{k_{2}}$.

      2. (ii)

         For all $\ell\geq 0$, $\displaystyle\sup_{t}{\bf{E}}\Bigl{[}\bigl{|}\zeta_{0,t}\bigr{|}^{\ell}\Bigr{]}<\infty$.

      3. (iii)

         $B_{4}\in C^{4}_{\uparrow}(\mathbb{R}^{k_{2}})$.

Using a locally Gaussian approximation, we obtain the following quasi-likelihood of Model $m$ from (1.8)-(1.14):

$\displaystyle\prod_{i=1}^{n}\frac{1}{(2\pi h_{n})^{\frac{p}{2}}(\mathop{\rm det}\nolimits{\bf{\Sigma}}_{m}(\theta_{m}))^{\frac{1}{2}}}\exp\Biggl{(}-\frac{1}{2h_{n}}{\bf{\Sigma}}_{m}(\theta_{m})^{-1}\Bigl{[}\bigl{(}\Delta_{i}\mathbb{X}^{\theta}_{m}\bigr{)}^{\otimes 2}\Bigr{]}\Biggr{)}.$

See Appendix 8.1 in Kusano and Uchida [14] for details of the quasi-likelihood. Define the quasi-likelihood ${\rm{L}}_{m,n}$ based on the discrete observations $\mathbb{X}_{n}$ as follows:

$\displaystyle{\rm{L}}_{m,n}\bigl{(}\mathbb{X}_{n},\theta_{m}\bigr{)}=\prod_{i=1}^{n}\frac{1}{(2\pi h_{n})^{\frac{p}{2}}(\mathop{\rm det}\nolimits{\bf{\Sigma}}_{m}(\theta_{m}))^{\frac{1}{2}}}\exp\Biggl{(}-\frac{1}{2h_{n}}{\bf{\Sigma}}_{m}(\theta_{m})^{-1}\Bigl{[}\bigl{(}\Delta_{i}\mathbb{X}\bigr{)}^{\otimes 2}\Bigr{]}\Biggr{)}.$

The quasi-maximum likelihood estimator $\hat{\theta}_{m,n}$ is defined by

$\displaystyle{\rm{L}}_{m,n}\bigl{(}\mathbb{X}_{n},\hat{\theta}_{m,n}(\mathbb{X}_{n})\bigr{)}=\sup_{\theta_{m}\in\Theta_{m}}{\rm{L}}_{m,n}\bigl{(}\mathbb{X}_{n},\theta_{m}\bigr{)}.$

Set $\mathbb{Z}_{n}$ as an i.i.d. copy of $\mathbb{X}_{n}$. Let us consider the following Kullback-Leibler divergence between the transition density $q_{n}(\mathbb{Z}_{n})$ of the true model (1.1)-(1.7) and the quasi-likelihood ${\rm{L}}_{m,n}$:

$\displaystyle\begin{split}{\rm{K_{L}}}(\mathbb{X}_{n},m)&={\bf{E}}_{\mathbb{Z}_{n}}\Biggl{[}\log\frac{q_{n}({\mathbb{Z}}_{n})}{{\rm{L}}_{m,n}\bigl{(}\mathbb{Z}_{n},\hat{\theta}_{m,n}({\mathbb{X}_{n}})\bigr{)}}\Biggr{]}\\ &={\bf{E}}_{\mathbb{Z}_{n}}\Bigl{[}\log q_{n}(\mathbb{Z}_{n})\Bigr{]}-{\bf{E}}_{\mathbb{Z}_{n}}\Bigl{[}\log{\rm{L}}_{m,n}\bigl{(}\mathbb{Z}_{n},\hat{\theta}_{m,n}({\mathbb{X}_{n}})\bigr{)}\Bigr{]},\end{split}$

where ${\bf{E}}_{\mathbb{Z}_{n}}$ is the expectation under the law of $\mathbb{Z}_{n}$. Our purpose is to know the model which minimizes ${\rm{K_{L}}}(\mathbb{X}_{n},m)$. Since ${\bf{E}}_{\mathbb{Z}_{n}}\Bigl{[}\log q_{n}(\mathbb{Z}_{n})\Bigr{]}$ does not depend on the model, it is sufficient to consider the model which maximizes

$\displaystyle{\bf{E}}_{\mathbb{Z}_{n}}\Bigl{[}\log{\rm{L}}_{m,n}\bigl{(}\mathbb{Z}_{n},\hat{\theta}_{m,n}({\mathbb{X}_{n}})\bigr{)}\Bigr{]},$

(3.1)

so that we need to estimate (3.1). Set

$\displaystyle\Delta_{m,0}=\left.\frac{\partial}{\partial\theta^{\top}}\mathop{\rm vech}\nolimits{\bf{\Sigma}}_{m}(\theta_{m})\right|_{\theta_{m}=\theta_{m,0}}$

and

$\displaystyle{\rm{Y}}_{m}(\theta_{m})=-\frac{1}{2}\Bigl{(}{\bf{\Sigma}}_{m}(\theta_{m})^{-1}-{\bf{\Sigma}}_{m}(\theta_{m,0})^{-1}\Bigr{)}\Bigl{[}{\bf{\Sigma}}_{m}(\theta_{m,0})\Bigr{]}-\frac{1}{2}\log\frac{\mathop{\rm det}\nolimits{\bf{\Sigma}}_{m}(\theta_{m})}{\mathop{\rm det}\nolimits{\bf{\Sigma}}_{m}(\theta_{m,0})}.$

Moreover, the following assumptions are made.

1. [B1]
   1. (a)

      There exists a constant $\chi>0$ such that

      $\displaystyle{\rm{Y}}_{m}(\theta_{m})\leq-\chi\bigl{|}\theta_{m}-\theta_{m,0}\bigr{|}^{2}$

      for all $\theta_{m}\in\Theta_{m}$.

   2. (b)

      $\mathop{\rm rank}\nolimits\Delta_{m,0}=q_{m}$.

By the following theorem, we obtain the asymptotically unbiased estimator of (3.1).

We define the quasi-Akaike information criterion as

$\displaystyle{\rm{QAIC}}(\mathbb{X}_{n},m)=-2\log{\rm{L}}_{m,n}\bigl{(}\mathbb{X}_{n},\hat{\theta}_{m,n}({\mathbb{X}_{n}})\bigr{)}+2q_{m}.$

(3.2)

Since it holds from Theorem 1 that ${\rm{QAIC}}(\mathbb{X}_{n},m)$ is the asymptotically unbiased estimator of

$\displaystyle-2{\bf{E}}_{\mathbb{Z}_{n}}\Bigl{[}\log{\rm{L}}_{m,n}\bigl{(}\mathbb{Z}_{n},\hat{\theta}_{m,n}({\mathbb{X}_{n}})\bigr{)}\Bigr{]},$

we select the optimal model $\hat{m}_{n}$ among competing models by

$\displaystyle{\rm{QAIC}}(\mathbb{X}_{n},\hat{m}_{n})=\min_{m\in\{1,\cdots,M\}}{\rm{QAIC}}(\mathbb{X}_{n},m).$

(3.3)

Next, we consider the situation where the set of competing models includes some (not all) misspecified parametric models; that is, there exists $m\in\{1,\cdots,M\}$ such that

$\displaystyle{\bf{\Sigma}}_{0}\neq{\bf{\Sigma}}_{m}(\theta_{m})$

for all $\theta_{m}\in\Theta_{m}$. Set

$\displaystyle\mathcal{M}=\biggl{\{}m\in\{1,\cdots,M\}\ \Big{|}\ \mbox{There exists}\ \theta_{m,0}\in\Theta_{m}\ \mbox{such that}\ {\bf{\Sigma}}_{0}={\bf{\Sigma}}_{m}(\theta_{m,0}).\biggr{\}}$

and $\mathcal{M}^{c}=\{1,\cdots,M\}\backslash\mathcal{M}$. The optimal parameter $\bar{\theta}_{m}$ is defined as

$\displaystyle{\rm{H}}_{m}(\bar{\theta}_{m})=\sup_{\theta_{m}\in\Theta_{m}}{\rm{H}}_{m}(\theta_{m}),$

where

$\displaystyle{\rm{H}}_{m}(\theta_{m})=-\frac{1}{2}\mathop{\rm tr}\nolimits\Bigl{(}{\bf{\Sigma}}_{m}(\theta_{m})^{-1}{\bf{\Sigma}}_{0}\Bigr{)}-\frac{1}{2}\log\mathop{\rm det}\nolimits{\bf{\Sigma}}_{m}(\theta_{m}).$

Note that $\bar{\theta}_{m}=\theta_{m,0}$ for $m\in\mathcal{M}$. Furthermore, we make the following assumption.

1. [B2]

   ${\rm{H}}_{m}(\theta_{m})={\rm{H}}_{m}(\bar{\theta}_{m})\Longrightarrow\theta_{m}=\bar{\theta}_{m}$.

$\bf{[B2]}$ implies that $\hat{\theta}_{m,n}\stackrel{{\scriptstyle p}}{{\longrightarrow}}\bar{\theta}_{m}$; see, Lemma 36 in Kusano and Uchida [14]. The following asymptotic result of $\hat{m}_{n}$ defined in (3.3) holds.

Theorem 2 shows that the probability of choosing the misspecified models converges to zero as $n\longrightarrow\infty$.