**Class: **ssm

Backward recursion of state-space models

`X = smooth(Mdl,Y)`

`X = smooth(Mdl,Y,Name,Value)`

```
[X,logL,Output]
= smooth(___)
```

returns smoothed states (`X`

= smooth(`Mdl`

,`Y`

)`X`

)
by performing backward recursion of the fully-specified state-space model `Mdl`

.
That is, `smooth`

applies the standard 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.

If `Mdl`

is not fully specified, then you must
set the unknown parameters to 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`

)
containing:

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.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,

`smooth`

applies all predictors to each response series. However, each response series has its own set of regression coefficients.

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