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**Class: **ssm

Forward recursion of state-space models

`X = filter(Mdl,Y)`

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

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

returns filtered states (`X`

= filter(`Mdl`

,`Y`

)`X`

)
from performing forward recursion of the fully specified state-space model `Mdl`

.
That is, `filter`

applies the standard Kalman filter using `Mdl`

and
the observed responses `Y`

.

uses
additional options specified by one or more `X`

= filter(`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 square-root
filter.

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`

]
= filter(___)`logL`

) and an output structure array (`Output`

)
using any of the input arguments in the previous syntaxes. `Output`

contains:

Filtered and forecasted states

Estimated covariance matrices of the filtered and forecasted states

Loglikelihood value

Forecasted observations and its estimated covariance matrix

Adjusted Kalman gain

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,

`filter`

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.