**Class: **ssm

Forecast states and observations of state-space models

`[`

returns forecasted observations (`Y`

,`YMSE`

]
= forecast(`Mdl`

,`numPeriods`

,`Y0`

)`Y`

)
and their corresponding variances (`YMSE`

) from forecasting
the state-space model `Mdl`

using
a `numPeriods`

forecast horizon and in-sample observations `Y0`

.

`[`

uses
additional options specified by one or more `Y`

,`YMSE`

]
= forecast(`Mdl`

,`numPeriods`

,`Y0`

,`Name,Value`

)`Name,Value`

pair
arguments. For example, for state-space models that include a linear
regression component in the observation model, include in-sample predictor
data, predictor data for the forecast horizon, and the regression
coefficient.

`Mdl`

— Standard state-space model`ssm`

model objectStandard state-space model, specified as an `ssm`

model
object returned by `ssm`

or `estimate`

.

If `Mdl`

is not fully specified (that is, `Mdl`

contains
unknown parameters), then specify values for the unknown parameters
using the `'`

`Params`

`'`

name-value
pair argument. Otherwise, the software issues an error. `estimate`

returns
fully-specified state-space models.

`Mdl`

does not store observed responses or
predictor data. Supply the data wherever necessary using the appropriate
input or name-value pair arguments.

`numPeriods`

— Forecast horizonpositive integer

Forecast horizon, specified as a positive integer. That is,
the software returns 1,..,`numPeriods`

forecasts.

**Data Types: **`double`

`Y0`

— In-sample, observed responsescell vector of numeric vectors | numeric matrix

In-sample, observed responses, specified as a cell vector of numeric vectors or a matrix.

If

`Mdl`

is time invariant, then`Y0`

is a*T*-by-*n*numeric matrix, where each row corresponds to a period and each column corresponds to a particular observation in the model. Therefore,*T*is the sample size and*m*is the number of observations per period. The last row of`Y`

contains the latest observations.If

`Mdl`

is time varying with respect to the observation equation, then`Y`

is a*T*-by-1 cell vector. Each element of the cell vector corresponds to a period and contains an*n*-dimensional vector of observations for that period. The corresponding dimensions of the coefficient matrices in_{t}`Mdl.C{t}`

and`Mdl.D{t}`

must be consistent with the matrix in`Y{t}`

for all periods. The last cell of`Y`

contains the latest observations.

If `Mdl`

is an estimated state-space
model (that is, returned by `estimate`

), then it is best practice to set `Y0`

to
the same data set that you used to fit `Mdl`

.

`NaN`

elements indicate missing
observations. For details on how the Kalman filter accommodates missing
observations, see Algorithms.

**Data Types: **`double`

| `cell`

Specify optional
comma-separated pairs of `Name,Value`

arguments. `Name`

is
the argument name and `Value`

is the corresponding value.
`Name`

must appear inside quotes. You can specify several name and value
pair arguments in any order as
`Name1,Value1,...,NameN,ValueN`

.

`'Beta',beta,'Predictors',Z`

specifies
to deflate the observations by the regression component composed of
the predictor data `Z`

and the coefficient matrix `beta`

.`'A'`

— Forecast-horizon, state-transition, coefficient matricescell vector of numeric matrices

Forecast-horizon, state-transition, coefficient matrices, specified
as the comma-separated pair consisting of `'A'`

and
a cell vector of numeric matrices.

`A`

must contain at least`numPeriods`

cells. Each cell must contain a matrix specifying how the states transition in the forecast horizon. If the length of`A`

is greater than`numPeriods`

, then the software uses the first`numPeriods`

cells. The last cell indicates the latest period in the forecast horizon.If

`Mdl`

is time invariant with respect to the states, then each cell of`A`

must contain an*m*-by-*m*matrix, where*m*is the number of the in-sample states per period. By default, the software uses`Mdl.A`

throughout the forecast horizon.If

`Mdl`

is time varying with respect to the states, then the dimensions of the matrices in the cells of`A`

can vary, but the dimensions of each matrix must be consistent with the matrices in`B`

and`C`

in the corresponding periods. By default, the software uses`Mdl.A{end}`

throughout the forecast horizon.

The matrices in `A`

cannot contain `NaN`

values.

**Data Types: **`cell`

`'B'`

— Forecast-horizon, state-disturbance-loading, coefficient matricescell vector of matrices

Forecast-horizon, state-disturbance-loading, coefficient matrices,
specified as the comma-separated pair consisting of `'B'`

and
a cell vector of matrices.

`B`

must contain at least`numPeriods`

cells. Each cell must contain a matrix specifying how the states transition in the forecast horizon. If the length of`B`

is greater than`numPeriods`

, then the software uses the first`numPeriods`

cells. The last cell indicates the latest period in the forecast horizon.If

`Mdl`

is time invariant with respect to the states and state disturbances, then each cell of`B`

must contain an*m*-by-*k*matrix, where*m*is the number of the in-sample states per period, and*k*is the number of in-sample, state disturbances per period. By default, the software uses`Mdl.B`

throughout the forecast horizon.If

`Mdl`

is time varying, then the dimensions of the matrices in the cells of`B`

can vary, but the dimensions of each matrix must be consistent with the matrices in`A`

in the corresponding periods. By default, the software uses`Mdl.B{end}`

throughout the forecast horizon.

The matrices in `B`

cannot contain `NaN`

values.

**Data Types: **`cell`

`'C'`

— Forecast-horizon, measurement-sensitivity, coefficient matricescell vector of matrices

Forecast-horizon, measurement-sensitivity, coefficient matrices,
specified as the comma-separated pair consisting of `'C'`

and
a cell vector of matrices.

`C`

must contain at least`numPeriods`

cells. Each cell must contain a matrix specifying how the states transition in the forecast horizon. If the length of`C`

is greater than`numPeriods`

, then the software uses the first`numPeriods`

cells. The last cell indicates the latest period in the forecast horizon.If

`Mdl`

is time invariant with respect to the states and the observations, then each cell of`C`

must contain an*n*-by-*m*matrix, where*n*is the number of the in-sample observations per period, and*m*is the number of in-sample states per period. By default, the software uses`Mdl.C`

throughout the forecast horizon.If

`Mdl`

is time varying with respect to the states or the observations, then the dimensions of the matrices in the cells of`C`

can vary, but the dimensions of each matrix must be consistent with the matrices in`A`

and`D`

in the corresponding periods. By default, the software uses`Mdl.C{end}`

throughout the forecast horizon.

The matrices in `C`

cannot contain `NaN`

values.

**Data Types: **`cell`

`'D'`

— Forecast-horizon, observation-innovation, coefficient matricescell vector of matrices

Forecast-horizon, observation-innovation, coefficient matrices,
specified as the comma-separated pair consisting of `'D'`

and
a cell vector of matrices.

`D`

must contain at least`numPeriods`

cells. Each cell must contain a matrix specifying how the states transition in the forecast horizon. If the length of`D`

is greater than`numPeriods`

, then the software uses the first`numPeriods`

cells. The last cell indicates the latest period in the forecast horizon.If

`Mdl`

is time invariant with respect to the observations and the observation innovations, then each cell of`D`

must contain an*n*-by-*h*matrix, where*n*is the number of the in-sample observations per period, and*h*is the number of in-sample, observation innovations per period. By default, the software uses`Mdl.D`

throughout the forecast horizon.If

`Mdl`

is time varying with respect to the observations or the observation innovations, then the dimensions of the matrices in the cells of`D`

can vary, but the dimensions of each matrix must be consistent with the matrices in`C`

in the corresponding periods. By default, the software uses`Mdl.D{end}`

throughout the forecast horizon.

The matrices in `D`

cannot contain `NaN`

values.

**Data Types: **`cell`

`'Beta'`

— Regression coefficients`[]`

(default) | numeric matrixRegression coefficients corresponding to predictor variables,
specified as the comma-separated pair consisting of `'Beta'`

and
a *d*-by-*n* numeric matrix. *d* is
the number of predictor variables (see `Predictors0`

and `PredictorsF`

)
and *n* is the number of observed response series
(see `Y0`

).

If you specify

`Beta`

, then you must also specify`Predictors0`

and`PredictorsF`

.If

`Mdl`

is an estimated state-space model, then specify the estimated regression coefficients stored in`Mdl.estParams`

.

By default, the software excludes a regression component from the state-space model.

`'Predictors0'`

— In-sample, predictor variables in state-space model observation equation`[]`

(default) | matrixIn-sample, predictor variables in the state-space model observation
equation, specified as the comma-separated pair consisting of `'Predictors0'`

and
a matrix. The columns of `Predictors0`

correspond
to individual predictor variables. `Predictors0`

must
have *T* rows, where row *t* corresponds
to the observed predictors at period *t* (*Z _{t}*).
The expanded observation equation is

$${y}_{t}-{Z}_{t}\beta =C{x}_{t}+D{u}_{t}.$$

In other words, the
software deflates the observations using the regression component. *β* is
the time-invariant vector of regression coefficients that the software
estimates with all other parameters.

If there are

*n*observations per period, then the software regresses all predictor series onto each observation.If you specify

`Predictors0`

, then`Mdl`

must be time invariant. Otherwise, the software returns an error.If you specify

`Predictors0`

, then you must also specify`Beta`

and`PredictorsF`

.If

`Mdl`

is an estimated state-space model (that is, returned by`estimate`

), then it is best practice to set`Predictors0`

to the same predictor data set that you used to fit`Mdl`

.

By default, the software excludes a regression component from the state-space model.

**Data Types: **`double`

`'PredictorsF'`

— Forecast-horizon, predictor variables in state-space model observation equation`[]`

(default) | numeric matrixIn-sample, predictor variables in the state-space model observation
equation, specified as the comma-separated pair consisting of `'Predictors0'`

and
a *T*-by-*d* numeric matrix. *T* is
the number of in-sample periods and *d* is the number
of predictor variables. Row *t* corresponds to the
observed predictors at period *t* (*Z _{t}*).
The expanded observation equation is

$${y}_{t}-{Z}_{t}\beta =C{x}_{t}+D{u}_{t}.$$

In other words, the
software deflates the observations using the regression component. *β* is
the time-invariant vector of regression coefficients that the software
estimates with all other parameters.

If there are

*n*observations per period, then the software regresses all predictor series onto each observation.If you specify

`Predictors0`

, then`Mdl`

must be time invariant. Otherwise, the software returns an error.If you specify

`Predictors0`

, then you must also specify`Beta`

and`PredictorsF`

.If

`Mdl`

is an estimated state-space model (that is, returned by`estimate`

), then it is best practice to set`Predictors0`

to the same predictor data set that you used to fit`Mdl`

.

By default, the software excludes a regression component from the state-space model.

**Data Types: **`double`

`Y`

— Forecasted observationsmatrix | cell vector of numeric vectors

Forecasted observations, returned as a matrix or a cell vector of numeric vectors.

If `Mdl`

is a time-invariant, state-space model
with respect to the observations, then `Y`

is a `numPeriods`

-by-*n* matrix.

If `Mdl`

is a time-varying, state-space model
with respect to the observations, then `Y`

is a `numPeriods`

-by-1
cell vector of numeric vectors. Cell *t* of `Y`

contains
an *n _{t}*-by-1 numeric vector
of forecasted observations for period

`YMSE`

— Error variances of forecasted observationsmatrix | cell vector of numeric vectors

Error variances of forecasted observations, returned as a matrix or a cell vector of numeric vectors.

If `Mdl`

is a time-invariant, state-space model
with respect to the observations, then `YMSE`

is
a `numPeriods`

-by-*n* matrix.

If `Mdl`

is a time-varying, state-space model
with respect to the observations, then `YMSE`

is
a `numPeriods`

-by-1 cell vector of numeric vectors.
Cell *t* of `YMSE`

contains an *n _{t}*-by-1
numeric vector of error variances for the corresponding forecasted
observations for period

`X`

— State forecastsmatrix | cell vector of numeric vectors

State forecasts, returned as a matrix or a cell vector of numeric vectors.

If `Mdl`

is a time-invariant, state-space model
with respect to the states, then `X`

is a `numPeriods`

-by-*m* matrix.

If `Mdl`

is a time-varying, state-space model
with respect to the states, then `X`

is a `numPeriods`

-by-1
cell vector of numeric vectors. Cell *t* of `X`

contains
an *m _{t}*-by-1 numeric vector
of forecasted observations for period

`XMSE`

— Error variances of state forecastsmatrix | cell vector of numeric vectors

Error variances of state forecasts, returned as a matrix or a cell vector of numeric vectors.

If `Mdl`

is a time-invariant, state-space model
with respect to the states, then `XMSE`

is a `numPeriods`

-by-*m* matrix.

If `Mdl`

is a time-varying, state-space model
with respect to the states, then `XMSE`

is a `numPeriods`

-by-1
cell vector of numeric vectors. Cell *t* of `XMSE`

contains
an *m _{t}*-by-1 numeric vector
of error variances for the corresponding forecasted observations for
period

Suppose that a latent process is an AR(1). Subsequently, the state equation is

$${x}_{t}=0.5{x}_{t-1}+{u}_{t},$$

where $${u}_{t}$$ is Gaussian with mean 0 and standard deviation 1.

Generate a random series of 100 observations from $${x}_{t}$$, assuming that the series starts at 1.5.

T = 100; ARMdl = arima('AR',0.5,'Constant',0,'Variance',1); x0 = 1.5; rng(1); % For reproducibility x = simulate(ARMdl,T,'Y0',x0);

Suppose further that the latent process is subject to additive measurement error. Subsequently, the observation equation is

$${y}_{t}={x}_{t}+{\epsilon}_{t},$$

where $${\epsilon}_{t}$$ is Gaussian with mean 0 and standard deviation 0.75. Together, the latent process and observation equations compose a state-space model.

Use the random latent state process (`x`

) and the observation equation to generate observations.

y = x + 0.75*randn(T,1);

Specify the four coefficient matrices.

A = 0.5; B = 1; C = 1; D = 0.75;

Specify the state-space model using the coefficient matrices.

Mdl = ssm(A,B,C,D)

Mdl = State-space model type: ssm State vector length: 1 Observation vector length: 1 State disturbance vector length: 1 Observation innovation vector length: 1 Sample size supported by model: Unlimited State variables: x1, x2,... State disturbances: u1, u2,... Observation series: y1, y2,... Observation innovations: e1, e2,... State equation: x1(t) = (0.50)x1(t-1) + u1(t) Observation equation: y1(t) = x1(t) + (0.75)e1(t) Initial state distribution: Initial state means x1 0 Initial state covariance matrix x1 x1 1.33 State types x1 Stationary

`Mdl`

is an `ssm`

model. Verify that the model is correctly specified using the display in the Command Window. The software infers that the state process is stationary. Subsequently, the software sets the initial state mean and covariance to the mean and variance of the stationary distribution of an AR(1) model.

Forecast the observations 10 periods into the future, and estimate their variances.

numPeriods = 10; [ForecastedY,YMSE] = forecast(Mdl,numPeriods,y);

Plot the forecasts with the in-sample responses, and 95% Wald-type forecast intervals.

ForecastIntervals(:,1) = ForecastedY - 1.96*sqrt(YMSE); ForecastIntervals(:,2) = ForecastedY + 1.96*sqrt(YMSE); figure plot(T-20:T,y(T-20:T),'-k',T+1:T+numPeriods,ForecastedY,'-.r',... T+1:T+numPeriods,ForecastIntervals,'-.b',... T:T+1,[y(end)*ones(3,1),[ForecastedY(1);ForecastIntervals(1,:)']],':k',... 'LineWidth',2) hold on title({'Observed Responses and Their Forecasts'}) xlabel('Period') ylabel('Responses') legend({'Observations','Forecasted observations','95% forecast intervals'},... 'Location','Best') hold off

Suppose that the linear relationship between the change in the unemployment rate and the nominal gross national product (nGNP) growth rate is of interest. Suppose further that the first difference of the unemployment rate is an ARMA(1,1) series. Symbolically, and in state-space form, the model is

$$\begin{array}{l}\left[\begin{array}{c}{x}_{1,t}\\ {x}_{2,t}\end{array}\right]=\left[\begin{array}{cc}\varphi & \theta \\ 0& 0\end{array}\right]\left[\begin{array}{c}{x}_{1,t-1}\\ {x}_{2,t-1}\end{array}\right]+\left[\begin{array}{c}1\\ 1\end{array}\right]{u}_{1,t}\\ {y}_{t}-\beta {Z}_{t}={x}_{1,t}+\sigma {\epsilon}_{t},\end{array}$$

where:

$${x}_{1,t}$$ is the change in the unemployment rate at time

*t*.$${x}_{2,t}$$ is a dummy state for the MA(1) effect.

$${y}_{1,t}$$ is the observed change in the unemployment rate being deflated by the growth rate of nGNP ($${Z}_{t}$$).

$${u}_{1,t}$$ is the Gaussian series of state disturbances having mean 0 and standard deviation 1.

$${\epsilon}_{t}$$ is the Gaussian series of observation innovations having mean 0 and standard deviation $$\sigma $$.

Load the Nelson-Plosser data set, which contains the unemployment rate and nGNP series, among other things.

`load Data_NelsonPlosser`

Preprocess the data by taking the natural logarithm of the nGNP series, and the first difference of each series. Also, remove the starting `NaN`

values from each series.

isNaN = any(ismissing(DataTable),2); % Flag periods containing NaNs gnpn = DataTable.GNPN(~isNaN); u = DataTable.UR(~isNaN); T = size(gnpn,1); % Sample size Z = [ones(T-1,1) diff(log(gnpn))]; y = diff(u);

Though this example removes missing values, the software can accommodate series containing missing values in the Kalman filter framework.

To determine how well the model forecasts observations, remove the last 10 observations for comparison.

numPeriods = 10; % Forecast horizon isY = y(1:end-numPeriods); % In-sample observations oosY = y(end-numPeriods+1:end); % Out-of-sample observations ISZ = Z(1:end-numPeriods,:); % In-sample predictors OOSZ = Z(end-numPeriods+1:end,:); % Out-of-sample predictors

Specify the coefficient matrices.

A = [NaN NaN; 0 0]; B = [1; 1]; C = [1 0]; D = NaN;

Specify the state-space model using `ssm`

.

Mdl = ssm(A,B,C,D);

Estimate the model parameters, and use a random set of initial parameter values for optimization. Specify the regression component and its initial value for optimization using the `'Predictors'`

and `'Beta0'`

name-value pair arguments, respectively. Restrict the estimate of $$\sigma $$ to all positive, real numbers. For numerical stability, specify the Hessian when the software computes the parameter covariance matrix, using the `'CovMethod'`

name-value pair argument.

params0 = [0.3 0.2 0.1]; % Chosen arbitrarily [EstMdl,estParams] = estimate(Mdl,isY,params0,'Predictors',ISZ,... 'Beta0',[0.1 0.2],'lb',[-Inf,-Inf,0,-Inf,-Inf],'CovMethod','hessian');

Method: Maximum likelihood (fmincon) Sample size: 51 Logarithmic likelihood: -87.2409 Akaike info criterion: 184.482 Bayesian info criterion: 194.141 | Coeff Std Err t Stat Prob ---------------------------------------------------------- c(1) | -0.31780 0.19429 -1.63572 0.10190 c(2) | 1.21242 0.48882 2.48032 0.01313 c(3) | 0.45583 0.63930 0.71302 0.47584 y <- z(1) | 1.32407 0.26313 5.03201 0 y <- z(2) | -24.48733 1.90115 -12.88024 0 | | Final State Std Dev t Stat Prob x(1) | -0.38117 0.42842 -0.88971 0.37363 x(2) | 0.23402 0.66222 0.35339 0.72380

`EstMdl`

is an `ssm`

model, and you can access its properties using dot notation.

Forecast observations over the forecast horizon. `EstMdl`

does not store the data set, so you must pass it in appropriate name-value pair arguments.

[fY,yMSE] = forecast(EstMdl,numPeriods,isY,'Predictors0',ISZ,... 'PredictorsF',OOSZ,'Beta',estParams(end-1:end));

`fY`

is a 10-by-1 vector containing the forecasted observations, and `yMSE`

is a 10-by-1 vector containing the variances of the forecasted observations.

Obtain 95% Wald-type forecast intervals. Plot the forecasted observations with their true values and the forecast intervals.

ForecastIntervals(:,1) = fY - 1.96*sqrt(yMSE); ForecastIntervals(:,2) = fY + 1.96*sqrt(yMSE); figure h = plot(dates(end-numPeriods-9:end-numPeriods),isY(end-9:end),'-k',... dates(end-numPeriods+1:end),oosY,'-k',... dates(end-numPeriods+1:end),fY,'--r',... dates(end-numPeriods+1:end),ForecastIntervals,':b',... dates(end-numPeriods:end-numPeriods+1),... [isY(end)*ones(3,1),[oosY(1);ForecastIntervals(1,:)']],':k',... 'LineWidth',2); xlabel('Period') ylabel('Change in the unemployment rate') legend(h([1,3,4]),{'Observations','Forecasted responses',... '95% forecast intervals'}) title('Observed and Forecasted Changes in the Unemployment Rate')

This model seems to forecast the changes in the unemployment rate well.

Suppose that a latent process is an AR(1). Subsequently, the state equation is

$${x}_{t}=0.5{x}_{t-1}+{u}_{t},$$

where $${u}_{t}$$ is Gaussian with mean 0 and standard deviation 1.

Generate a random series of 100 observations from $${x}_{t}$$, assuming that the series starts at 1.5.

T = 100; ARMdl = arima('AR',0.5,'Constant',0,'Variance',1); x0 = 1.5; rng(1); % For reproducibility x = simulate(ARMdl,T,'Y0',x0);

Suppose further that the latent process is subject to additive measurement error. Subsequently, the observation equation is

$${y}_{t}={x}_{t}+{\epsilon}_{t},$$

where $${\epsilon}_{t}$$ is Gaussian with mean 0 and standard deviation 0.75. Together, the latent process and observation equations compose a state-space model.

Use the random latent state process (`x`

) and the observation equation to generate observations.

y = x + 0.75*randn(T,1);

Specify the four coefficient matrices.

A = 0.5; B = 1; C = 1; D = 0.75;

Specify the state-space model using the coefficient matrices.

Mdl = ssm(A,B,C,D);

`Mdl`

is an `ssm`

model.

Forecast the states 10 periods into the future, and estimate their variances.

numPeriods = 10; [~,~,ForecastedX,XMSE] = forecast(Mdl,numPeriods,y);

Plot the forecasts with the smoothed states, and 95% Wald-type forecast intervals.

smoothX = smooth(Mdl,y); ForecastIntervals(:,1) = ForecastedX - 1.96*sqrt(XMSE); ForecastIntervals(:,2) = ForecastedX + 1.96*sqrt(XMSE); figure plot(T-20:T,smoothX(T-20:T),'-k',T+1:T+numPeriods,ForecastedX,'-.r',... T+1:T+numPeriods,ForecastIntervals,'-.b',... T:T+1,[smoothX(end)*ones(3,1),[ForecastedX(1);ForecastIntervals(1,:)']],... ':k','LineWidth',2) hold on title({'Smoothed and Forecasted States'}) xlabel('Period') ylabel('States') legend({'Smoothed states','Forecasted states','95% forecast intervals'},... 'Location','Best') hold off

`Mdl`

does not store the response data, predictor
data, and the regression coefficients. Supply them whenever necessary
using the appropriate input or name-value pair arguments.

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.

[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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