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 $$0.5u_{1,t}$$. 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];

Define the state-disturbance-loading coefficient matrix.

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.