# Create State-Space Model Containing ARMA State

This example shows how to create a stationary ARMA model subject to measurement error using `ssm`.

To explicitly create a state-space model, it is helpful to write the state and observation equations in matrix form. In this example, the state of interest is the ARMA(2,1) process

`${x}_{t}=c+{\varphi }_{1}{x}_{t-1}+{\varphi }_{2}{x}_{t-2}+{u}_{t}+{\theta }_{1}{u}_{t-1},$`

where ${u}_{t}$ is Gaussian with mean 0 and known standard deviation 0.5.

The variables ${x}_{t}$, ${x}_{t-1}$, and ${u}_{t}$ are in the state-space model framework. Therefore, the terms $c$, ${\varphi }_{2}{x}_{t-2}$, and ${\theta }_{1}{u}_{t-1}$ require "dummy states" to be included in the model.

The state equation is

`$\left[\begin{array}{c}{x}_{1,t}\\ {x}_{2,t}\\ {x}_{3,t}\\ {x}_{4,t}\end{array}\right]=\left[\begin{array}{cccc}{\varphi }_{1}& c& {\varphi }_{2}& {\theta }_{1}\\ 0& 1& 0& 0\\ 1& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_{1,t-1}\\ {x}_{2,t-1}\\ {x}_{3,t-1}\\ {x}_{4,t-1}\end{array}\right]+\left[\begin{array}{c}0.5\\ 0\\ 0\\ 1\end{array}\right]{u}_{1,t}$`

Note that:

• c corresponds to a state (${x}_{2,t}$) that is always `1`.

• ${x}_{3,t}={x}_{1,t-1}$, and ${x}_{1,t}$ has the term ${\varphi }_{2}{x}_{3,t-1}={\varphi }_{2}{x}_{1,t-2}$.

• ${x}_{1,t}$ has the term $0.5{u}_{1,t}$. `ssm` puts state disturbances as Gaussian random variables with mean 0 and variance 1. Therefore, the factor `0.5` is the standard deviation of the state disturbance.

• ${x}_{4,t}={u}_{1,t}$, and ${x}_{1,t}$ has the term ${\theta }_{1}{x}_{4,t}={\theta }_{1}{u}_{1,t-1}$.

The observation equation is unbiased for the ARMA(2,1) state process. The observation innovations are Gaussian with mean 0 and known standard deviation 0.1. Symbolically, the observation equation is

`${y}_{t}=\left[\begin{array}{cccc}1& 0& 0& 0\end{array}\right]\left[\begin{array}{c}{x}_{1,t}\\ {x}_{2,t}\\ {x}_{3,t}\\ {x}_{4,t}\end{array}\right]+0.1{\epsilon }_{t}.$`

You can include a measurement-sensitivity factor (a bias) by replacing `1` in the row vector by a scalar or unknown parameter.

Define the state-transition coefficient matrix. Use `NaN` values to indicate unknown parameters.

`A = [NaN NaN NaN NaN; 0 1 0 0; 1 0 0 0; 0 0 0 0];`

`B = [0.5; 0; 0; 1];`

Define the measurement-sensitivity coefficient matrix.

`C = [1 0 0 0];`

Define the observation-innovation coefficient matrix.

`D = 0.1;`

Use `ssm` to create the state-space model. Set the initial-state mean (`Mean0`) to a vector of zeros and covariance matrix (`Cov0`) to the identity matrix, except set the mean and variance of the constant state to `1` and `0`, respectively. Specify the type of initial state distributions (`StateType`) by noting that:

• ${x}_{1,t}$ is a stationary, ARMA(2,1) process.

• ${x}_{2,t}$ is the constant 1 for all periods.

• ${x}_{3,t}$ is the lagged ARMA process, so it is stationary.

• ${x}_{4,t}$ is a white-noise process, so it is stationary.

```Mean0 = [0; 1; 0; 0]; Cov0 = eye(4); Cov0(2,2) = 0; StateType = [0; 1; 0; 0]; Mdl = ssm(A,B,C,D,'Mean0',Mean0,'Cov0',Cov0,'StateType',StateType);```

`Mdl` is an `ssm` model. You can use dot notation to access its properties. For example, print `A` by entering `Mdl.A`.

Use `disp` to verify the state-space model.

`disp(Mdl)`
```State-space model type: ssm State vector length: 4 Observation vector length: 1 State disturbance vector length: 1 Observation innovation vector length: 1 Sample size supported by model: Unlimited Unknown parameters for estimation: 4 State variables: x1, x2,... State disturbances: u1, u2,... Observation series: y1, y2,... Observation innovations: e1, e2,... Unknown parameters: c1, c2,... State equations: x1(t) = (c1)x1(t-1) + (c2)x2(t-1) + (c3)x3(t-1) + (c4)x4(t-1) + (0.50)u1(t) x2(t) = x2(t-1) x3(t) = x1(t-1) x4(t) = u1(t) Observation equation: y1(t) = x1(t) + (0.10)e1(t) Initial state distribution: Initial state means x1 x2 x3 x4 0 1 0 0 Initial state covariance matrix x1 x2 x3 x4 x1 1 0 0 0 x2 0 0 0 0 x3 0 0 1 0 x4 0 0 0 1 State types x1 x2 x3 x4 Stationary Constant Stationary Stationary ```

If you have a set of responses, you can pass them and `Mdl` to `estimate` to estimate the parameters.