Introduction

Let s_t be the vector of state (predetermined) variables in a DSGE model (we implicitly assume that the set of state variables is finite) and y_t be the vector of observed variables. This last vector may contain predetermined and non predetermined variables. We assume that the reduced form DSGE model can be cast into the following state space model:

\begin{aligned}
  & s_t = f( s_{t-1}, \varepsilon_t; \bm\theta ) \\
  & y_t = g( s_t; \bm\theta) + e_t \\
\end{aligned}

with f(.) the set of state equations, g(.) the set of measurement equations, \bm\theta\in\Theta\subseteq \mathbb R^m a vector of m parameters, which are assumed to be known, \varepsilon_t and e_t respectively the set of structural shocks and additive measurement errors. These innovations are assumed to be Gaussian.

⚠ Measurement errors are mandatory You must have as many measurement errors as observed variables to estimate a model with a nonlinear filter, otherwise you will obtain a singularity error (for the covariance matrix of the expectation errors).

This reduced form can be obtained using local or global approximation methods. Functions f(.) and g(.) need not to be explicitly defined, the only requirement is to have an algorithm that updates the state variables and determines the observed variables from the state variables. In Dynare, a k-order perturbation approach is used to build these functions. Given the aforementioned state-space model, it is obvious that the state variables, s_t, are driven by a first order Markov process:

  p\left(s_t | s_{0:t-1} \right) = p\left(s_t | s_{t-1} \right)

meaning that all the information about s_t is embodied in s_{t-1}, and that the observations are conditionally independent:

  p\left( y_t | y_{1:t-1}, s_{0:t} \right) = p\left(y_t | s_t \right)

These two properties imply many simplifications that considerably alleviate the derivation of the nonlinear filters.

The likelihood of the model is the density of the sample y_{1:T} = \left\{ y_t \right\}_{t=1}^{T} conditional on the parameters \bm\theta, which, in principle, can be written as a product of conditional densities of y_t|y_{1:t-1}. The evaluation of these densities requires the knowledge of the state variables, s_t, but in general not all of them are observed. We need to track (infer) the unobserved (latent) state variables, and this is where the nonlinear filters come in. A nonlinear filter is a recursive Bayesian algorithm that allows us to track the dynamic of the distribution of the latent variables. We describe the different filtering algorithms assuming that the parameters \bm\theta are known, keeping the issues related to estimation for latter. For the sake of simplicity, we do not express the distributions in function of \bm\theta. This vector will appear explicitly when inference about the parameters is discussed.

Using all the available information, the model and the sample, we need to infer the distribution of the latent variables s_t. More formally, we need to build the density of s_t conditional on the sample up to time t, y_{1:t}. This can be done recursively using the Bayes theorem. We have:

  p\left( s_t| y_{1:t} \right) =
  \frac{ p\left( y_t | s_t \right) p\left( s_t | y_{1:t-1}
  \right)}{p\left(y_t | y_{1:t-1} \right)}

with

  p\left( y_t | y_{1:t-1}\right) = \int p\left( y_t  | s_t
  \right)p\left( s_t | y_{1:t-1} \right)\mathrm d s_t

where the density of y_t conditional on s_t, p(y_t|s_t), is interpreted as the likelihood of s_t, while the density of s_t conditional on the sample up to time t-1, p(s_t|y_{1:t-1}), can be interpreted as a prior belief about the state variables at time t given the information available at time t-1.

These equations are not analytically tractable, except for continuous state variables in linear/Gaussian model or for discrete state variables in linear/conditionally Gaussian model. In these two cases, the equations allow to derive respectively the Kalman filter and the Hamilton filter. In all other cases, approximations are required, like sequential importance sampling.

Numerical approximations and sequential importance sampling

Assume that the continuous distribution of s_t conditional on y_{1:t} can be approximated by a set of particles \left\{s_t^i\right\}_{i=1:N} and associated weights \left\{w_t^i\right\}_{i=1:N} summing-up to one. Any moments of this conditional distribution can be approximated by a weighted average as follows:

  \begin{aligned}
    \mathbb E_{p(s_t|y_{1:t})} \left[ h(s_t) \right]
    &= \int h(s_t)p(s_t | y_{1:t} ) \mathrm d s_t\\
    &\approx \sum_{i=1}^N w_t^i h\left(s_t^i\right)
  \end{aligned}

If we were able to sample the state variables directly from the distribution of s_t conditional on y_{1:t}, we would then approximate the moment with uniform weights: w_t^i=\frac{1}{N} for all i. Since the density of s_t conditional on the sample up to time t is unknown, we cannot do that. An importance sampling algorithm can be used to recover this information and build the non uniform weights. It consists in choosing an easy-to-sample proposal distribution q\left(s_t| y_{1:t} \right), ideally not too different from p\left(s_t| y_{1:t} \right), and correct the weights for the difference between the targeted and proposal conditional distributions.

Choose an easy-to-sample proposal distribution q(s_t| y_{1:t} ):

  \begin{aligned}
    \mathbb E_{p(s_{t}| y_{1:t})}\left[ h(s_t) \right]
    & =\int{h({{s}_{t}})\frac{p({{s}_{t}}\left| {{y}_{1:t}}
    \right)}{q({{s}_{t}}\left| {{y}_{1:t}} \right)}{q({{s}_{t}}\left|  
{{y}_{1:t}} \right)}\mathrm d{{s}_{t}}}\\
    &=\mathbb E_{q({{s}_{t}}\left| {{y}_{1:t}} \right)}\left[
    {\tilde{w}_{t}}({{s}_{t}})h({{s}_{t}}) \right]
   \end{aligned}

with normalized weights defined as:

\tilde{w}_t(s_t) \equiv \frac{p(s_t | y_{1:t})}{q(s_t| y_{1:t} )}.

These new weights can be viewed as importance ratios, namely the correction to implement due to the sampling from the proposal distribution. These weights can be computed recursively if the proposal distribution satisfies the following condition:

q\left(s_t | y_{1:t} \right) = q(s_t|s_{t-1},y_t)q\left(s_{t-1}|
y_{1:t-1} \right)

In this case, s_t is simply drawn in q(s_t|s_{t-1},y_t) and the (unormalized) weights can be recursively computed as:

\hat{w}_t(s_t) \propto \tilde{w}_{t-1}(s_{t-1})\frac{p\left(y_t | s_t
\right)p\left(s_t | s_{t-1} \right)}{q\left(s_t| s_{t-1},y_t \right)}

However, in this case, a degeneracy problem occurs. As t increases, all-but-one particles have negligible weights (essentially in large samples). That is the reason why systematic resampling was initially proposed in the literature. It consists in randomly drawing with replacement particles in their empirical distribution \left\{\tilde{s}_t^i,\tilde{w}_t^i\right\}_{i=1:N}. It amounts to discard particles with low weights and replicate particles with high weights to focus on interesting areas of the distribution using a constant number of particles. Doucet proposed measures indicating when resampling is necessary. Particles are resampled when a degeneracy measure:

N_{eff,t} = \frac{1}{\sum_{i=1}^{N}{{{(\tilde{w}_t^i)}^2}}}

is below a particular threshold (generally \frac{N}{2}).

We can then obtain the recursive iterations for a general particle filter with systematic resampling: \forall t=1,\ldots,T and \forall i=1,\ldots,N, knowing \left\{s_{t-1}^i,w_{t-1}^i\right\}_{i=1:N} that approximates p(s_{t-1}|y_{1:t-1}).

• Draw \left\{\tilde{s}_t^i\right\}_{i=1:N} from q(s_t|s_{t-1}^i,y_t).

• Evaluate the weights: \hat{w}_t^i \propto w_{t-1}^i\frac{p(y_t | \tilde{s}_t^i)p(\tilde{s}_t^i | s_{t-1}^i )}{q(\tilde{s}_t^i | s_{t-1}^i,y_t)}.

• In case of systematic resampling or if N_{eff,t} is lesser than the chosen threshold, resample particles and replace \left\{\tilde{s}_t^i,\tilde{w}_t^i=\frac{{\hat{w}_t^i}}{\sum_{i=1}^{N}{{\hat{w}_t^i}}}\right\} with \left\{ s_t^i,w_t^i=\frac{1}{N}\; \right\} that approximates p(s_t|y_{1:t}).

The choice for the proposal explains the diversity of filters implemented in the literature.

Estimation

The likelihood expression

From now on, the parameters set \bm\theta reappears explicitly in the distribution expressions. In the general framework, we can derive the sample likelihood expression:

p\left(y_{1:T} |\bm \theta \right.)  = p\left(y_1 | s_0;\bm\theta
\right.)p\left(s_0 |\bm\theta \right.) \prod_{t=2}^{T}{p\left(y_t |  
y_{1:t-1};\bm\theta \right.)}

with the evidence (or unconditional likelihood) p\left(y_t | y_{1:t-1} ; \bm\theta \right.) that can be approximated for all the filters discussed below with:

\begin{aligned}
p\left(y_t | y_{1:t-1};\bm\theta \right)
& \approx \sum_{i=1}^{N}{\hat{w}_t^i} \\
& = \sum_{i=1}^{N} {w_{t-1}^i \frac{ p(y_t |\tilde{s}_t^i ; \bm\theta
) p(\tilde{s}_t^i | s_{t-1}^i ; \bm\theta  
)}{q(\tilde{s}_t^i|s_{t-1}^i,y_t ; \bm\theta)}}
\end{aligned}

In the case of the standard proposal where q(s_t|s_{t-1},y_t;\bm\theta)=p(s_t | s_{t-1};\bm\theta):

p(y_t | y_{1:t-1};\bm\theta )\approx \sum_{i=1}^{N}{ w_{t-1}^i} p(y_t
| \tilde{s}_t^i ; \bm\theta)

In case of systematic resampling, as w_{t-1}^i = 1/N, we simply get:

p(y_t | y_{1:t-1} ; \bm\theta ) \approx \frac{1}{N} \sum_{i=1}^{N}{
p(y_t | \tilde{s}_t^i ; \bm\theta ) }

While necessary to avoid degeneracy, an important issue with resampling is that it renders the maximization of the likelihood or posterior kernel quite difficult (Pitt (2002), Kantas et al. (2015)). Even when the seed for random draws is fixed across the simulations, the traditional likelihood estimator depends on both resampled particles and the unknown parameters. A small change in the parameters value will cause a small change in the importance weights that will potentially generate a different set of resampled particles. This produces a discontinuity in the likelihood criterion and explains why applied approaches depart from the usual likelihood-based approach. A first alternative consists in resampling but maximizing the estimation criterion with no gradient-based methods (such as downhill simplex, S.A., CMAES, ...). A second possibility consists in using a MCMC approach to build the posterior distribution of parameters, using the unbiased likelihood estimator provided by a particle filter. The generic denomination of these methods is P-MCMC (Andrieu et al. (2010)). A third alternative consists in considering the structural parameters as extra state variables. Parameters are then estimated with a specific particle filter at each date, which allows to treat issues such as structural breaks or change in behavior (see Yano(2010), Yano et al (2010) or Chen et al (2010)). Besides, it is relatively faster since it requires only one pass over the sample. This is the online approach proposed by Liu et al (2001).

Particle MCMC

The idea is to build the posterior distribution of parameters \bm\theta. From the Bayes rule, we know that the posterior density is proportional to the product of the sample likelihood p\left(y_{1:T} | \bm\theta \right) and the prior density over the parameters p(\bm\theta):

p\left(\bm\theta  | y_{1:T} \right) \propto p\left(y_{1:T} | \bm\theta
\right) p(\bm\theta)

In the case of a linear model, the sample likelihood is evaluated with a Kalman filter. In the case of a nonlinear model, an unbiased estimator of p\left(y_{1:T} | \bm\theta \right) is provided by particle filtering ({Delmoral2004}). Except for this difference, we can follow the same approach. In a MCMC framework, like the random walk Particle Marginal Metropolis-Hastings (PMMH) algorithm, a candidate is drawn from a proposal distribution:

\bm\theta^\star_j= \bm\theta_{j-1} + \epsilon_j

with \epsilon_j \sim \mathcal{N}(0,\gamma_{RW} V(\Theta_0)), where the scale parameter \gamma_{RW} is set in order to obtain an acceptance ratio around 25%. The posterior distribution can then be approximated using the following acceptance rule:

\bm\theta_j =
\begin{cases}
\bm\theta^\star_j & \text{if $U_{[0,1]} \leq \min
\left\{1,\frac{p\left(\bm\theta^\star_j | y_{1:T}  
\right)}{p\left(\bm\theta_{j-1} | y_{1:T} \right)}\right\} $} \\
\bm\theta_{j-1} & \text{otherwise}
\end{cases}

For further developments, see Andrieu et al. (2010) (with discussions).

The online approach

This approach has been proposed by Liu et al (2001) as an alternative to estimate the parameters of nonlinear/non gaussian models. Instead of maximizing a likelihood function or posterior kernel, they consider the unknown parameters as extra state variables and reveal these parameters as any state variable using a nonlinear filter. For that purpose, they combine the auxiliary particle filter introduced by Pitt et al (1999) with an assumed transition distribution for the parameters that serves as extra state equations and importance density:

p\left(\bm\theta_t | \bm\theta_{t-1} \right) = \mathcal{N}(m_{t-1},b^2V_{t-1})

This equation produces time-varying parameters (note however that this is not internalized by the agents in the DSGE model who consider that the parameters are fixed when solving for the expectations in each period). The authors adopt a kernel shrinkage technique based on a parameter a to produce slowly time-varying parameters and also to control the variability. Suppose we have a particle swarm \left\{s_{t-1}^i,\bm\theta_{t-1}^i, w_{t-1}^i\right\}_{i=1:N} from the preceeding period. This particle swwarm is updated using:

\begin{aligned}
& \bar{\bm\theta}_{t-1} = \sum_{i=1}^{N}{ w_{t-1}^i \bm\theta_{t-1}^i }\\
& m_{t-1}^i = a\bm\theta_{t-1}^i + (1-a)\bar{\bm\theta}_{t-1}  \\
& V_{t-1} = \sum_{i=1}^{N}{ w_{t-1}^i
(\bm\theta_{t-1}^i-\bar{\bm\theta}_{t-1} )
(\bm\theta_{t-1}^i-\bar{\bm\theta}_{t-1} )' }\\
& \bar{s}_t^i = f(s_{t-1}^i,0;m_{t-1}^i )
\end{aligned}

As in the auxiliary particle filter, we build a resampling index k^l. The proposal for parameters \left\{\tilde{\bm\theta}_t^l\right\}_{l=1:N} are drawn from \mathcal{N}(m_{t-1}^{k^l},b^2V_{t-1}). The proposal for state variables \left\{\tilde{s}_t^l\right\}_{l=1:N} are drawn from p(s_t | s_{t-1}^{k^l},\tilde{\bm\theta}_t^l). Finally, the weights are updated as: \hat{w}_t^l \propto p(y_t | \tilde{s}_t^l,\tilde{\bm\theta}_t^l)\frac{{w}_{t-1}^{k^l}}{\tilde{\tau}_{t-1}^{k^l}} and normalized. The new particles swarm is then \left\{\tilde{s}_t^{l},\tilde{\bm\theta}_t^{l},\tilde{w}_t^l\right\}_{l=1:N}. An extra resampling step can be added.

Parameter \delta is key to this approach, since it controls the shrinkage and the smoothness parameters a and b:

\begin{aligned}
& b^2 = 1-\left( \frac{3\delta-1}{2\delta} \right)^2\\
& a = \sqrt{1-b^2}
\end{aligned}

In the literature, \delta is generally chosen in the range [0.9;0.99]. Two outputs can be exploited in this framework: the evolution of parameters along the sample and the distribution of parameters at the last observation (incorporating full sample information), that can be summarized by the usual statistics as the mean, the median and some empirical quantiles.

The estimation command and its options

Use for instance the following instruction:

estimation(datafile=extreme,order=2,mode_compute=0,particle_filter_options = ('pruning', true, ...),...);

For order=1, Dynare estimates the linearized model using the Kalman filter. When order>1, Dynare switches automatically to the nonlinear estimation routines. Here, as mode_compute=0, P-MCMC will build the posterior distribution using 20,000 draws (by default) starting from the initial conditions, the likelihood being calculated with the nonlinear filter by default, namely the Bootstrap particle filter with systematic resampling using standard Kitagawa's approach and 5,000 particles and using pruning for the particle filter-related simulations. The particle_filter_options = (NAME, VALUE, ...) syntax allows setting some fine-grained options.

Contrarily to linear estimation, it is not possible to calculate accurately the posterior mode in the presence of resampling because it induces discontinuities in the likelihood function. However, a non gradient-based method (such as the Nelder and Mead Downhill Simplex) can be used (mode_compute=7 or 8 or 9 for instance). For any other choice, a warning message appears and asks the user to confirm his/her choice.

The following table summarizes the options included in estimation.

Option names	Values ([default])
filter_algorithm	[sis], apf, nlkf, gf, gmf, cpf
proposal_approximation	[cubature], unscented, montecarlo
distribution_approximation	[cubature], unscented, montecarlo
number_of_particles	[5000]
resampling	[systematic], none, generic
resampling_method	[kitagawa], residual
mode_compute	7, 8, 9
mh_replic	[20000], 0
online_particle_filter
nonlinear_filter_initialization	[1],2,3

First of all, the choice of the filter is operated with the keyword filter_algorithm. The sequential importance sampling (sis) is the filter by default but one can also choose the auxiliary particle filter (apf), the nonlinear Kalman filter (nlkf), the Gaussian filter (gf), the Gaussian-mixture filter (gmf), and the conditional particle filter (cpf).

Keyword online_particle_filter triggers the online estimation of the model, using the method developed by Liu and West. It works for order=1 as well as order>1. particle_filter_options = ('liu_west_delta', VALUE, ...) controls the value of the \delta parameter (set equal to 0.99 by default).

Some dependencies among other keywords should be clarified. They are summarized by the following table.

Keyword	Options	[sis]	apf	nlkf	gf	gmf	cpf	online
number_of_particles	[5000]	x	x	x	x	x	x	x
proposal_approximation	[cubature], unscented, montecarlo			x
distribution_approximation	[cubature], unscented, montecarlo
resampling	[systematic], none, generic	x	x		x	x	x	x
resampling_method	[kitagawa], residual	x	x		x	x	x	x
mode_compute	7, 8, 9	x	x		if resampling	if resampling	x	0

Some filters require Gaussian approximations, that can be done with whether sparse grids methods (cubature or unscented transform) or MC methods. One should notice that unscented transform is controled with three parameters: \alpha and \kappa that determine the spread of the sigma-points and \beta that characterizes the (non-gaussian) distribution. By default, we set \alpha = \kappa=1 and \beta=2. They can be modified by using the particle_filter_options = ('unscented_alpha', VALUE, 'unscented_beta', VALUE,'unscented_kappa', VALUE, ...) syntax.

• The number of particles can be chosen with the keyword number_of_particles.

• The keyword proposal_approximation allows to choose the Gaussian approximation for the proposal distribution. One can choose sparse grids approximations (cubature by default, or unscented) or an approximation using particles (by choosing montecarlo). In this former case, the number of particles should be specified. It is only compatible with filter_algorithm=nlkf, gf, gmf, or cpf that use the posterior state distributions from a nonlinear Kalman filter as proposal.

• distribution_approximation allows to choose the Gaussian approximation for the state variables. It is only effective with the marginal filters (gf or gmf). One can choose sparse grids approximations (cubature by default, or unscented) or an approximation using particles (by choosing montecarlo). For other filters, state distributions are approximated with particles by default.

• The resampling keyword controls the resampling step of the current particles. It is set by default to systematic which is highly recommended in the literature for filter_algorithm=sis,cpf. For methods based on a pre-selection step (like the auxiliary particle filter and the online filter), the literature advises to choose resampling=none. In these cases, it won't affect the pre-selection step that is compulsory but only skip the second one that is generally considered as optional. At last, there is no clear consensus concerning marginal filters (Gaussian and Gaussian-mixture filters). If resampling=none, the current means and variances of the states that will feed the next time iteration are calculated using the normalized weights of particles drawn in the NLKF posterior distribution. With the default, they will be calculated as the empirical moments of resampled particles. If resampling=generic, the option Neff_threshold can be defined as a fraction of the sample (generally between 0 and 0.5).

• Two resampling methods are available for the moment. They can be chosen with the keyword resampling_method when resampling is used.

• nonlinear_filter_initialization controls the initial states distribution of the filter. Three possibilities are offered to the user. If nonlinear_filter_initialization=1 (the default), the initial state vector covariance is the ergodic variance associated to the first order Taylor-approximation of the model. If it equals to 2, the initial state vector covariance is a monte-carlo based estimate of the ergodic variance (consistent with a k-order Taylor-approximation of the model). At last, if it equals to 3, the covariance is a diagonal matrix, with diagonal values that can be set with particle_filter_options = ('initial_state_prior_std', double, ...).

• particle_filter_options = ('pruning', true, ...) allows to enable pruning for particle-filter related simulations.

• particle_filter_options = ('mixture_state_variables', Integer, 'mixture_structural_shocks', Integer,'mixture_measurement_shocks', Integer,... ) allows to set the number of mixture components for the states (default: 5), structural shocks (default: 1), and measurement errors (default: 1), respectively for the Gaussian-mixture filter (gmf).

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