2.15.3. Nonlinear filtering

When a DSGE model is solved with a higher-order perturbation (or otherwise has a nonlinear state-space representation), the regime-switching Kalman/Hamilton filter of Filtering and decompositions no longer applies. RISE then uses a sigma-point filter for regime-switching nonlinear state-space models, following [Binning and Maih, 2015].

When the nonlinear filter is used

filter (and, through it, estimate, forecast with conditioning, the historical decomposition, …) picks the algorithm automatically from the solution:

solve_order == 1 – the linear regime-switching Kalman/Hamilton filter (a fast constant-parameter variant when there is a single regime, no occasionally-binding constraint, and no real-time information);
solve_order >= 2 – a nonlinear sigma-point filter (the default is the switching divided-difference filter).

Nonlinear filtering with real-time information is not supported (RISE raises an error).

Choosing the filter

Three sigma-point filters are available; the divided-difference filter is the default, the other two are selected by passing them as the filtering algorithm via kf_user_algo (they all share the standard filter signature, so they can be plugged in directly):

% default: switching divided-difference filter
[fkst, LogLik] = filter(m);

% switching unscented Kalman filter
[fkst, LogLik] = filter(m, 'kf_user_algo', 'switching_unscented_kalman_filter');

% switching cubature Kalman filter
[fkst, LogLik] = filter(m, 'kf_user_algo', 'switching_cubature_kalman_filter');

The same option can be passed to estimate (it changes which filter computes the likelihood that is maximised).

Options

The filtering options apply equally to the linear and nonlinear cases (see Filtering and decompositions and help dsge/filter); the ones most relevant here:

kf_filtering_level [0|1|2|{3}] – 0 likelihood only; 1 + the filtered (one-step-ahead) series; 2 + the updated series; 3 + the smoothed series.
kf_ergodic [{true}|false], kf_init_variance, kf_user_init – how the state and regime probabilities are initialised.
kf_presample [{0}|integer] – observations to discard before evaluating the likelihood.
kf_householder_chol [{false}|true] – return the Cholesky factor when taking the Householder transformation; primarily used by the divided-difference filter.

User-defined nonlinear filter

A filter of your own can be supplied through kf_user_algo – a function (or function name, or cell {name, extra-args...}) with signature [LogLik, Incr, retcode, Filters] = myFilter(syst, y, U, z, options). If it needs information beyond what RISE passes, prefix the name with * so RISE first calls it as myFilter(modelObject, struct) to let it fetch what it needs. See Extending the DSGE functionalities.

The available sigma-point filters

Switching divided-difference filter

switching_divided_difference_filter - filter for nonlinear regime-swiching models
[loglik,Incr,retcode,Filters]=switching_divided_difference_filter(...
 syst,y,U,z,options)
Args:

syst [struct]: structure containing:

PAI00 [vector]: initial probability distributions of regimes

a [cell]: initial conditions in each regime

Qfunc [function handle]: transition matrix generator

ff [function handle]: ft=ff(rt,xt,et), where rt is the regime, xt is the vector of state variables and et the vector of shocks

P [cell]: initial covariance matrix of the states in each regime

H [cell]: Measurement error covariance matrices in each regime

SIGeta [cell]: Covariance matrix of structural shocks.

y [matrix]: ny x T matrix of data

U [[]|matrix]: ndx x T matrix of exogenous data

z [function handle|logical|vector]: linear connection of the observables to the state.

include_in_likelihood [logical]: selector of increments to include in the likelihood calculation

options [struct]: structure with various options

Returns:

loglik [scalar]: log likelihood

Incr [vector]: increments of elements going into the likelihood

retcode [{0}|integer]: flag for problems.

Filters [struct]: Filtered, updated and smoothed variables

Note:

Example:

See also

switching_cubature_kalman_filter, switching_unscented_kalman_filter

Switching unscented Kalman filter

switching_unscented_kalman_filter - filter for nonlinear regime-swiching models
[loglik,Incr,retcode,Filters]=switching_unscented_kalman_filter(...
 syst,data_y,U,z,options)
Args:

syst [struct]: structure containing:

PAI00 [vector]: initial probability distributions of regimes

a [cell]: initial conditions in each regime

Qfunc [function handle]: transition matrix generator

ff [function handle]: ft=ff(rt,xt,et), where rt is the regime, xt is the vector of state variables and et the vector of shocks

P [cell]: initial covariance matrix of the states in each regime

H [cell]: Measurement error covariance matrices in each regime

SIGeta [cell]: Covariance matrix of structural shocks.

data_y [matrix]: ny x T matrix of data

U [[]|matrix]: ndx x T matrix of exogenous data

z [function handle|logical|vector]: linear connection of the observables to the state.

include_in_likelihood [logical]: selector of increments to include in the likelihood calculation (temporarily disabled)

options [struct]: structure with various options

Returns:

loglik [scalar]: log likelihood

Incr [vector]: increments of elements going into the likelihood

retcode [{0}|integer]: flag for problems.

Filters [struct]: Filtered, updated and smoothed variables

Note:

Example:

See also

switching_cubature_kalman_filter, switching_divided_difference_filter

Switching cubature Kalman filter

switching_unscented_kalman_filter - filter for nonlinear regime-swiching models
[loglik,Incr,retcode,Filters]=switching_unscented_kalman_filter(...
 syst,data_y,U,z,options)
Args:

syst [struct]: structure containing:

PAI00 [vector]: initial probability distributions of regimes

a [cell]: initial conditions in each regime

Qfunc [function handle]: transition matrix generator

ff [function handle]: ft=ff(rt,xt,et), where rt is the regime, xt is the vector of state variables and et the vector of shocks

P [cell]: initial covariance matrix of the states in each regime

H [cell]: Measurement error covariance matrices in each regime

SIGeta [cell]: Covariance matrix of structural shocks.

data_y [matrix]: ny x T matrix of data

U [[]|matrix]: ndx x T matrix of exogenous data

z [function handle|logical|vector]: linear connection of the observables to the state.

include_in_likelihood [logical]: selector of increments to include in the likelihood calculation (temporarily disabled)

options [struct]: structure with various options

Returns:

loglik [scalar]: log likelihood

Incr [vector]: increments of elements going into the likelihood

retcode [{0}|integer]: flag for problems.

Filters [struct]: Filtered, updated and smoothed variables

Note:

Example:

See also

switching_divided_difference_filter, switching_unscented_kalman_filter