where $ x_{t} $ is an unknown $ n_{x}\times1 $ state vector, $ v_{t} $ is $ n_{v}\times1 $ system noise vector with a density function $ q(\mathbf{v}|\cdot)^{17} $ , $ \epsilon_{t} $ is $ n_{\epsilon}\times1 $ observation noise vector with a density function $ r(\mathbf{\epsilon}|\cdot) $ . The function $ f_{t}:R^{n_{x}}\timesR^{n_{v}}\toR^{n_{x}} $ is a possibly nonlinear time-varying function and the function $ h_{t}:R^{n_{x}}\timesR^{n_{\epsilon}}\toR^{n_{y}} $ is a possibly nonlinear time-varying function. The first equation of (28) is called a system equation and the second equation of (28) is called an observation equation. We would like to emphasize the functions, $ f_{t} $ and $ h_{t} $ , are possibly time dependent. A system equation depends on a possibly unknown $ n_{s}\times1 $ parameter vector, $ \xi_{s} $ , and an observation equation depends on a possibly unknown $ n_{o}\times1 $ parameter vector, $ \xi_{o} $ . This NNSS specifies the two following conditional density functions.

$$ \begin{aligned}&p(\boldsymbol{x}_{t}|\boldsymbol{x}_{t-1},\boldsymbol{\xi}_{s}),\\&p(\boldsymbol{Y}_{t}|\boldsymbol{x}_{t},\boldsymbol{\xi}_{o}).\\ \end{aligned} $$

We define a parameter vector $ \theta $ as follows.

$$ \boldsymbol{\theta}=\left[\xi_{s}\atop\xi_{o}\right]. $$

We denote that $ \theta_{j} $ is the jth element of $ \theta $ and $ J(=n_{s}+n_{o}) $ is the number of elements of $ \theta $ . This type of state space model (28) contains a broad class of linear, nonlinear, Gaussian, or non-Gaussian time series models. In state space modeling, estimating the state space vector $ x_{t} $ is the most important problem. For the linear Gaussian state space model, the Kalman filter, which is proposed by Kalman (1960), is the most popular algorithm to estimate the state vector $ x_{t} $ . For nonlinear or non-Gaussian state space models, there are many algorithms. For example, the extended Kalman filter (Jazwinski (1970)) is the most popular algorithm; other examples are the Gaussian-sum filter (Alspach and Sorenson (1972)), the dynamic generalized model (West et al. (1985)), and the non-Gaussian filter and smoother (Kitagawa (1987)). In recent years, MCPF for NNNSS has been a popular algorithm because it is easily applicable to various time series models $ ^{18} $ .

In econometric analysis, generally, we don’t know the parameter vector $ \theta $ . In our framework, the unknown parameter vectors are $ \xi_{o} $ and $ \xi_{s} $ $ ^{19} $ . In traditional parameter estimation, maximizing the log-likelihood function of $ \theta $ is often used. The log-likelihood of $ \theta $ in MCPF is proposed by Kitagawa (1996). However, MCPF is problematic to estimate the parameter vector $ \theta $ because the likelihood of the filter contains errors from the Monte Carlo method. Thus, you cannot use nonlinear optimizing algorithm like Newton’s method $ ^{20} $ . To solve the problem, Kitagawa (1998) proposes a self-organizing state space model. In Kitagawa (1998), an augmented state vector is defined as follows.

$$ \boldsymbol{z}_{t}=\left[\begin{aligned}\boldsymbol{x}_{t}\\ \boldsymbol{\theta}\end{aligned}\right]. $$

An augmented system equation and an augmented measurement equation are defined as

$$ \boldsymbol{z}_{t}=F_{t}(\boldsymbol{z}_{t-1},\boldsymbol{v}_{t},\boldsymbol{\xi}_{s}), $$

$$ \mathbf{Y}_{t}=H_{t}(\mathbf{z}_{t},\boldsymbol{\epsilon}_{t},\boldsymbol{\xi}_{o}), $$