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**Superclasses: **

Create diffuse state-space model

`dssm`

creates a linear diffuse state-space model with
independent Gaussian state disturbances and observation innovations.
A diffuse state-space model contains diffuse states, and variances
of the initial distributions of diffuse states are `Inf`

.
All diffuse states are independent of each other and all other states.
The software implements the diffuse Kalman filter for filtering, smoothing,
and parameter estimation.

You can:

Specify a time-invariant or time-varying model.

Specify whether states are stationary, static, or nonstationary.

Specify the state-transition, state-disturbance-loading, measurement-sensitivity, or observation-innovation matrices:

Explicitly by providing the matrices

Implicitly by providing a function that maps the parameters to the matrices, that is, a parameter-to-matrix mapping function

After creating a diffuse state-space model containing unknown
parameters, you can estimate its parameters by passing the created `dssm`

model
object and data to `estimate`

.
The `estimate`

function builds the likelihood function
using the diffuse Kalman filter.

Use a completely specified model (that is, all parameter values of the model are known) to:

creates
a diffuse
state-space model (`Mdl`

= dssm(`A`

,`B`

,`C`

)`Mdl`

) using state-transition
matrix `A`

, state-disturbance-loading matrix `B`

,
and measurement-sensitivity matrix `C`

.

creates
a diffuse state-space model using state-transition matrix `Mdl`

= dssm(`A`

,`B`

,`C`

,`D`

)`A`

,
state-disturbance-loading matrix `B`

, measurement-sensitivity
matrix `C`

, and observation-innovation matrix `D`

.

uses
any of the input arguments in the previous syntaxes and additional
options that you specify by one or more `Mdl`

= dssm(___,`Name,Value`

)`Name,Value`

pair
arguments.

creates
a diffuse state-space model using a parameter-to-matrix mapping function
(`Mdl`

= dssm(`ParamMap`

)`ParamMap`

) that you write. The function maps a
vector of parameters to the matrices `A`

, `B`

,
and `C`

. Optionally, `ParamMap`

can
map parameters to `D`

, `Mean0`

, `Cov0`

.
To specify the types of states, the function can return `StateType`

.
To accommodate a regression component in the observation equation, `ParamMap`

can
also return deflated observation data.

converts
a state-space model object (`Mdl`

= dssm(`SSMMdl`

)`SSMMdl`

) to a diffuse
state-space model object (`Mdl`

). `dssm`

sets
all initial variances of diffuse states in `SSMMdl.Cov0`

to `Inf`

.

disp | Display summary information for diffuse state-space model |

estimate | Maximum likelihood parameter estimation of diffuse state-space models |

filter | Forward recursion of diffuse state-space models |

forecast | Forecast states and observations of diffuse state-space models |

refine | Refine initial parameters to aid diffuse state-space model estimation |

smooth | Backward recursion of diffuse state-space models |

Value. To learn how value classes affect copy operations, see Copying Objects (MATLAB).

Specify

`ParamMap`

in a more general or complex setting, where, for example:The initial state values are parameters.

In time-varying models, you want to use the same parameters for more than one period.

You want to impose parameter constraints.

You can create a

`dssm`

model object that does not contain any diffuse states. However, subsequent computations, for example, filtering and parameter estimation, can be inefficient. If all states have stationary distributions or are the constant 1, then create an`ssm`

model object instead.

Default values for

`Mean0`

and`Cov0`

:If you explicitly specify the state-space model (that is, you provide the coefficient matrices

`A`

,`B`

,`C`

, and optionally`D`

), then:For stationary states, the software generates the initial value using the stationary distribution. If you provide all values in the coefficient matrices (that is, your model has no unknown parameters), then

`dssm`

generates the initial values. Otherwise, the software generates the initial values during estimation.For states that are always the constant 1,

`dssm`

sets`Mean0`

to 1 and`Cov0`

to`0`

.For diffuse states, the software sets

`Mean0`

to 0 and`Cov0`

to`Inf`

by default.

If you implicitly specify the state-space model (that is, you provide the parameter vector to the coefficient-matrices-mapping function

`ParamMap`

), then the software generates the initial values during estimation.

For static states that do not equal 1 throughout the sample, the software cannot assign a value to the degenerate, initial state distribution. Therefore, set static states to

`2`

using the name-value pair argument`StateType`

. Subsequently, the software treats static states as nonstationary and assigns the static state a diffuse initial distribution.It is best practice to set

`StateType`

for each state. By default, the software generates`StateType`

, but this behavior might not be accurate. For example, the software cannot distinguish between a constant 1 state and a static state.The software cannot infer

`StateType`

from data because the data theoretically comes from the observation equation. The realizations of the state equation are unobservable.`dssm`

models do not store observed responses or predictor data. Supply the data wherever necessary using the appropriate input or name-value pair arguments.Suppose that you want to create a diffuse state-space model using a parameter-to-matrix mapping function with this signature:

and you specify the model using an anonymous function[A,B,C,D,Mean0,Cov0,StateType,DeflateY] = paramMap(params,Y,Z)

The observed responsesMdl = dssm(@(params)paramMap(params,Y,Z))

`Y`

and predictor data`Z`

are not input arguments in the anonymous function. If`Y`

and`Z`

exist in the MATLAB Workspace before you create`Mdl`

, then the software establishes a link to them. Otherwise, if you pass`Mdl`

to`estimate`

, the software throws an error.The link to the data established by the anonymous function overrides all other corresponding input argument values of

`estimate`

. This distinction is important particularly when conducting a rolling window analysis. For details, see Rolling-Window Analysis of Time-Series Models.

Create an `ssm`

model
object instead of a `dssm`

model object when:

The model does not contain any diffuse states.

The diffuse states are correlated with each other or to other states.

You want to implement the standard Kalman filter.

[1] Durbin J., and S. J. Koopman. *Time Series
Analysis by State Space Methods*. 2nd ed. Oxford: Oxford
University Press, 2012.

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