Class: dssm
Backward recursion of diffuse state-space models
returns smoothed states (X
= smooth(Mdl
,Y
)X
)
by performing backward recursion of the fully-specified diffuse state-space
model Mdl
. That is, smooth
applies
the diffuse Kalman filter using Mdl
and
the observed responses Y
.
uses
additional options specified by one or more X
= smooth(Mdl
,Y
,Name,Value
)Name,Value
pair
arguments. For example, specify the regression coefficients and predictor
data to deflate the observations, or specify to use the univariate
treatment of a multivariate model.
If Mdl
is not fully specified, then you must
specify the unknown parameters as known scalars using the '
Params
'
Name,Value
pair
argument.
[
uses any of the input arguments
in the previous syntaxes to additionally return the loglikelihood
value (X
,logL
,Output
]
= smooth(___)logL
) and an output structure array (Output
)
using any of the input arguments in the previous syntaxes. The fields
of Output
include:
Smoothed states and their estimated covariance matrix
Smoothed state disturbances and their estimated covariance matrix
Smoothed observation innovations and their estimated covariance matrix
The loglikelihood value
The adjusted Kalman gain
And a vector indicating which data the software used to filter
Mdl
does not store the response
data, predictor data, and the regression coefficients. Supply the
data wherever necessary using the appropriate input or name-value
pair arguments.
It is a best practice to allow smooth
to
determine the value of SwitchTime
. However, in
rare cases, you might experience numerical issues during estimation,
filtering, or smoothing diffuse state-space models. For such cases,
try experimenting with various SwitchTime
specifications,
or consider a different model structure (e.g., simplify or reverify
the model). For example, convert the diffuse state-space model to
a standard state-space model using ssm
.
To accelerate estimation for low-dimensional, time-invariant
models, set 'Univariate',true
. Using this specification,
the software sequentially updates rather then updating all at once
during the filtering process.
The Kalman filter accommodates missing data by not updating filtered state estimates corresponding to missing observations. In other words, suppose there is a missing observation at period t. Then, the state forecast for period t based on the previous t – 1 observations and filtered state for period t are equivalent.
For explicitly defined state-space models, filter
applies
all predictors to each response series. However, each response series
has its own set of regression coefficients.
The diffuse Kalman filter requires presample data. If missing observations begin the time series, then the diffuse Kalman filter must gather enough nonmissing observations to initialize the diffuse states.
For diffuse state-space models, filter
usually
switches from the diffuse Kalman filter to the standard Kalman filter
when the number of cumulative observations and the number of diffuse
states are equal. However, if a diffuse state-space model has identifiability
issues (e.g., the model is too complex to fit to the data), then filter
might
require more observations to initialize the diffuse states. In extreme
cases, filter
requires the entire sample.
[1] Durbin J., and S. J. Koopman. Time Series Analysis by State Space Methods. 2nd ed. Oxford: Oxford University Press, 2012.