## What Is the Kalman Filter?

### Standard Kalman Filter

In the state-space model framework, the Kalman filter estimates the values of a latent, linear, stochastic, dynamic process based on possibly mismeasured observations. Given distribution assumptions on the uncertainty, the Kalman filter also estimates model parameters via maximum likelihood.

Starting with initial values for states
(*x*_{0|0}), the initial state
variance-covariance matrix (*P*_{0|0}), and
initial values for all unknown parameters
(*θ*_{0}), the simple Kalman filter:

Estimates, for

*t*= 1,...,*T*:The 1-step-ahead vector of state forecasts vector for period

*t*($${\widehat{x}}_{t|t-1}$$) and its variance-covariance matrix ($${P}_{t|t-1}$$)The 1-step-ahead vector of observation forecasts for period

*t*($${\widehat{y}}_{t|t-1}$$) and its estimated variance-covariance matrix ($${V}_{t|t-1}$$)The filtered states for period

*t*($${\widehat{x}}_{t|t}$$) and its estimated variance-covariance matrix ($${P}_{t|t}$$)

Feeds the forecasted and filtered estimates into the data likelihood function

$$\mathrm{ln}p({y}_{T},\mathrm{...},{y}_{1})={\displaystyle \sum _{t=1}^{T}\mathrm{ln}\varphi ({y}_{t};{\widehat{y}}_{t|t-1},{V}_{t|t-1})},$$

where $$\varphi ({y}_{t};{\widehat{y}}_{t|t-1},{V}_{t|t-1})$$ is the multivariate normal probability density function with mean $${\widehat{y}}_{t|t-1}$$ and variance $${V}_{t|t-1}$$.

Feeds this procedure into an optimizer to maximize the likelihood with respect to the model parameters.

### State Forecasts

*s*-step-ahead, state forecasts are estimates of the states at
period *t* using all information (for example, the observed
responses) up to period *t* – *s*.

The *m _{t}*-by-1 vector of 1-step-ahead,
state forecasts at period

*t*is $${x}_{t|t-1}=E\left({x}_{t}|{y}_{t-1},\mathrm{...},{y}_{1}\right)$$. The estimated vector of state forecasts is

$${\widehat{x}}_{t|t-1}={A}_{t}{\widehat{x}}_{t-1|t-1},$$

where $${\widehat{x}}_{t-1|t-1}$$ is the *m*_{t –
1}-by-1 filtered state vector at
period *t* – 1.

At period *t*, the 1-step-ahead, state forecasts have the
variance-covariance matrix

$${P}_{t|t-1}={A}_{t}{P}_{t-1|t-1}{A}_{t}^{\prime}+{B}_{t}{B}_{t}^{\prime},$$

where$${P}_{t-1|t-1}$$ is the estimated variance-covariance matrix of the filtered states
at period *t* – 1, given all information up to period
*t* – 1.

The corresponding 1-step-ahead forecasted observation is $${\widehat{y}}_{t|t-1}={C}_{t}{\widehat{x}}_{t|t-1},$$, and its variance-covariance matrix is $${V}_{t|t-1}=Var\left({y}_{t}|{y}_{t-1},\mathrm{...},{y}_{1}\right)={C}_{t}{P}_{t|t-1}{C}_{t}^{\prime}+{D}_{t}{D}_{t}^{\prime}.$$

In general, the *s*-step-ahead, forecasted state vector is $${x}_{t|t-s}=E\left({x}_{t}|{y}_{t-s},\mathrm{...},{y}_{1}\right)$$. The *s*-step-ahead, vector of state forecasts is

$${\widehat{x}}_{t+s|t}=\left({\displaystyle \prod _{j=t+1}^{t+s}{A}_{j}}\right){x}_{t|t}$$

and the *s*-step-ahead, forecasted observation
vector is

$${\widehat{y}}_{t+s|t}={C}_{t+s}{\widehat{x}}_{t+s|t}.$$

### Filtered States

State forecasts at
period *t*, updated using all information (for example the observed
responses) up to period *t*.

The *m _{t}*-by-1 vector of filtered states at
period

*t*is $${x}_{t|t}=E\left({x}_{t}|{y}_{t},\mathrm{...},{y}_{1}\right)$$. The estimated vector of filtered states is

$${\widehat{x}}_{t|t}={\widehat{x}}_{t|t-1}+{K}_{t}{\widehat{\epsilon}}_{t},$$

where:

$${\widehat{x}}_{t|t-1}$$ is the vector of state forecasts at period

*t*using the observed responses from periods 1 through*t*– 1.*K*is the_{t}*m*-by-_{t}*h*raw Kalman gain matrix for period_{t}*t*.$${\widehat{\epsilon}}_{t}={y}_{t}-{C}_{t}{\widehat{x}}_{t|t-1}$$ is the

*h*-by-1 vector of estimated observation innovations at period_{t}*t*.

In other words, the filtered states at period *t* are the
forecasted states at period *t* plus an adjustment based on the
trustworthiness of the observation. Trustworthy observations have very little
corresponding observation innovation variance (for example, the maximum eigenvalue
of *D _{t}D_{t}′* is
relatively small). Consequently, for a given estimated observation innovation, the
term $${K}_{t}{\widehat{\epsilon}}_{t}$$ has a higher impact on the values of the filtered states than
untrustworthy observations.

At period *t*, the filtered states have variance-covariance
matrix

$${P}_{t|t}={P}_{t|t-1}-{K}_{t}{C}_{t}{P}_{t|t-1},$$

where $${P}_{t|t-1}$$ is the estimated variance-covariance matrix of the state forecasts
at period *t*, given all information up to period
*t* – 1.

### Smoothed States

*Smoothed states* are estimated states at period
*t*, which are updated using all available information (for
example, all of the observed responses).

The *m _{t}*-by-1 vector of smoothed states at
period

*t*is $${x}_{t|T}=E({x}_{t}|{y}_{T},\mathrm{...},{y}_{1})$$. The estimated vector of smoothed states is

$${\widehat{x}}_{t|T}={\widehat{x}}_{t|t-1}+{P}_{t|t-1}{r}_{t},$$

where:

$${\widehat{x}}_{t|t-1}$$ are the state forecasts at period

*t*using the observed responses from periods 1 to*t*– 1.$${P}_{t|t-1}$$ is the estimated variance-covariance matrix of the state forecasts, given all information up to period

*t*– 1.$${r}_{t}={\displaystyle \sum}_{s=t}^{T}\left\{\left[{\displaystyle \prod}_{j=t}^{s-1}\left({A}_{t}-{K}_{t}{C}_{t}\right)\right]{C}_{s}^{\prime}{V}_{s|s-1}^{-1}{\nu}_{s}\right\},$$ where,

*K*is the_{t}*m*-by-_{t}*h*raw Kalman gain matrix for period_{t}*t*.$${V}_{t|t-1}={C}_{t}{P}_{t|t-1}{C}_{t}^{\prime}+{D}_{t}{D}_{t}^{\prime}$$, which is the estimated variance-covariance matrix of the forecasted observations.

$${\nu}_{t}={y}_{t}-{\widehat{y}}_{t|t-1}$$, which is the difference between the observation and its forecast at period

*t*.

### Smoothed State Disturbances

*Smoothed state disturbances* are estimated,
state disturbances at period *t*, which are updated using all
available information (for example, all of the observed responses).

The *k _{t}*-by-1 vector of smoothed, state
disturbances at period

*t*is $${u}_{t|T}=E\left({u}_{t}|{y}_{T},\mathrm{...},{y}_{1}\right)$$. The estimated vector of smoothed, state disturbances is

$${\widehat{u}}_{t|T}={B}_{t}^{\prime}{r}_{t},$$

where *r _{t}* is the
variable in the formula to estimate the smoothed
states.

At period *t*, the smoothed state disturbances have
variance-covariance matrix

$${U}_{t|T}=I-{B}_{t}^{\prime}{N}_{t}{B}_{t},$$

where *N _{t}* is the
variable in the formula to estimate the variance-covariance matrix of the smoothed
states.

The software computes smoothed estimates using backward recursion of the Kalman filter.

### Forecasted Observations

*s*-step-ahead, forecasted observations are estimates of the
observations at period *t* using all information (for example, the
observed responses) up to period *t* – *s*.

The *n _{t}*-by-1 vector of 1-step-ahead,
forecasted observations at period

*t*is $${y}_{t|t-1}=E\left({y}_{t}|{y}_{t-1},\mathrm{...},{y}_{1}\right)$$. The estimated vector of forecasted observations is

$${\widehat{y}}_{t|t-1}={C}_{t}{\widehat{x}}_{t|t-1},$$

where $${\widehat{x}}_{t|t-1}$$ is the *m _{t}*-by-1 estimated
vector of state forecasts at
period

*t*.

At period *t*, the 1-step-ahead, forecasted observations have
variance-covariance matrix

$${V}_{t|t-1}=Var\left({y}_{t}|{y}_{t-1},\mathrm{...},{y}_{1}\right)={C}_{t}{P}_{t|t-1}{C}_{t}^{\prime}+{D}_{t}{D}_{t}^{\prime}.$$

where $${P}_{t|t-1}$$ is the estimated variance-covariance matrix of the state forecasts
at period *t*, given all information up to period
*t* – 1.

In general, the *s*-step-ahead, vector of state forecasts is $${x}_{t|t-s}=E\left({x}_{t}|{y}_{t-s},\mathrm{...},{y}_{1}\right)$$. The *s*-step-ahead, forecasted observation
vector is

$${\widehat{y}}_{t+s|t}={C}_{t+s}{\widehat{x}}_{t+s|t}.$$

### Smoothed Observation Innovations

*Smoothed observation innovations* are estimated, observation
innovations at period *t*, which are updated using all available
information (for example, all of the observed responses).

The *h _{t}*-by-1 vector of smoothed,
observation innovations at period

*t*is $${\epsilon}_{t|T}=E\left({\epsilon}_{t}|{y}_{T},\mathrm{...},{y}_{1}\right)$$. The estimated vector of smoothed, observation innovations is

$${\widehat{\epsilon}}_{t}={D}_{t}^{\prime}{V}_{t|t-1}^{-1}{\nu}_{t}-{D}_{t}^{\prime}{K}_{t}^{\prime}{r}_{t+1},$$

where:

*r*and_{t}*ν*are the variables in the formula to estimate the smoothed states._{t}*K*is the_{t}*m*-by-_{t}*h*raw Kalman gain matrix for period_{t}*t*.$${V}_{t|t-1}={C}_{t}{P}_{t|t-1}{C}_{t}^{\prime}+{D}_{t}{D}_{t}^{\prime}$$, which is the estimated variance-covariance matrix of the forecasted observations.

At period *t*, the smoothed observation innovations have
variance-covariance matrix

$${E}_{t|T}=I-{D}_{t}^{\prime}\left({V}_{t|t-1}^{-1}-{K}_{t}^{\prime}{N}_{t+1}{K}_{t}\right){D}_{t}.$$

The software computes smoothed estimates using backward recursion of the Kalman filter.

### Kalman Gain

The

*raw Kalman gain*is a matrix that indicates how much to weigh the observations during recursions of the Kalman filter.The raw Kalman gain is an

*m*-by-_{t }*h*matrix computed using_{t}$${K}_{t}={P}_{t|t-1}{C}_{t}^{\prime}{\left({C}_{t}{P}_{t|t-1}{C}_{t}^{\prime}+{D}_{t}{D}_{t}^{\prime}\right)}^{-1},$$

where $${P}_{t|t-1}$$ is the estimated variance-covariance matrix of the state forecasts, given all information up to period

*t*– 1.The value of the raw Kalman gain determines how much weight to put on the observations. For a given estimated observation innovation, if the maximum eigenvalue of

*D*is relatively small, then the raw Kalman gain imparts a relatively large weight on the observations. If the maximum eigenvalue of_{t}D_{t}′*D*is relatively large, then the raw Kalman gain imparts a relatively small weight on the observations. Consequently, the filtered states at period_{t}D_{t}′*t*are close to the corresponding state forecasts.Consider obtaining the 1-step-ahead state forecasts for period

*t*+ 1 using all information up to period*t*. The*adjusted Kalman gain*($${K}_{adj,t}$$) is the amount of weight put on the estimated observation innovation for period*t*($${\widehat{\epsilon}}_{t}$$) as compared to the 2-step-ahead state forecast ($${\widehat{x}}_{t+1|t-1}$$).That is,

$${\widehat{x}}_{t+1|t}={A}_{t}{\widehat{x}}_{t|t}={A}_{t}{\widehat{x}}_{t|t-1}+{A}_{t}{K}_{t}{\widehat{\epsilon}}_{t}={\widehat{x}}_{t+1|t-1}+{K}_{adj,t}{\widehat{\epsilon}}_{t}.$$

### Backward Recursion of the Kalman Filter

*Backward recursion of the Kalman filter* estimates smoothed
states, state disturbances, and observation innovations.

The software estimates the smoothed values by:

Setting

*r*_{T + 1}= 0, and*N*_{T + 1}to an*m*-by-_{T}*m*matrix of 0s_{T}For

*t*=*T*,...,1, it recursively computes:*r*(see Smoothed States)_{t}$${\widehat{x}}_{t|T}$$, which is the matrix of smoothed states

*N*(see Smoothed States)_{t}$${P}_{t|T}$$, which is the estimated variance-covariance matrix of the smoothed states

$${\widehat{u}}_{t|T}$$, which is the matrix of smoothed state disturbances

$${U}_{t|T}$$, which is the estimated variance-covariance matrix of the smoothed state disturbances

$${\widehat{\epsilon}}_{t|T}$$, which is the matrix of smoothed observation innovations

$${E}_{t|T}$$, which is the estimated variance-covariance matrix of the smoothed observation innovations

### Diffuse Kalman Filter

Consider a state-space model written so that the *m* diffuse
states (*x _{d}*) are segregated from the

*n*stationary states (

*x*). That is, the moments of the initial distributions are

_{s}$${\mu}_{0}=\left[\begin{array}{c}{\mu}_{d0}\\ {\mu}_{s0}\end{array}\right]\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\Sigma}_{0}=\left[\begin{array}{cc}{\Sigma}_{d0}& 0\\ 0& {\Sigma}_{s0}\end{array}\right].$$

*μ*_{d0}is an*m*-vector of zeros*μ*_{s0}is an*n*-vector of real numbers*Σ*_{d0}=*κI*, where_{m}*I*is the_{m}*m*-by-*m*identity matrix and*κ*is a positive real number.*Σ*_{s0}is an*n*-by-*n*positive definite matrix.The diffuse states are uncorrelated with each other and the stationary states.

One way to analyze such a model is to set *κ* to a relatively
large, positive real number, and then implement the standard Kalman filter (see `ssm`

). This treatment is an approximation to an analysis that treats the
diffuse states as if their initial state covariance approaches infinity.

The *diffuse Kalman filter* or *exact-initial
Kalman filter*
[68]
treats the diffuse states by taking *κ* to ∞. The diffuse
Kalman filter filters in two stages: the first stage initializes the model so that
it can subsequently be filtered using the standard Kalman filter, which is the
second stage. The initialization stage mirrors the standard Kalman filter. It sets
all initial filtered states to zero, and then augments that vector of initial
filtered states with the identity matrix, which composes an (*m* +
*n*)-by-(*m* + *n* + 1)
matrix. After a sufficient number of periods, the precision matrices become
nonsingular. That is, the diffuse Kalman filter uses enough periods at the beginning
of the series to initialize the model. You can consider this period as presample
data.

The second stage commences when the precision matrices are nonsingular.
Specifically, the initialization stage returns a vector of filtered states and their
precision matrix. Then, the standard Kalman filter uses those estimates and the
remaining data to filter, smooth, and estimate parameters. For more details, see
`dssm`

and [68], Sec. 5.2.

## References

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

## See Also

### Objects

### Functions

`estimate`

|`estimate`

|`filter`

|`filter`

|`update`

|`smooth`

|`smooth`

|`forecast`

|`forecast`

|`irf`

|`irfplot`