Explicitly Specify an ARMA Model with Known and Unknown Parameter Values

This example shows how to specify an stationary ARMA model subject to measurement error using ssm.

To explicitly specify 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

$$x_{t} = c + \phi_1 x_{t-1} + \phi_2 x_{t-2} + u_t + \theta_1 u_{t-1},$$

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 $c$, $\phi_2x_{t-2}$, and $\theta_1 u_{t-1}$ require "dummy states" to be included in the model.

Subsequently, the state equation is

$$\left[ {\begin{array}{*{20}{c}}{{x_{1,t}}}\\{{x_{2,t}}}\\{{x_{3,t}}}\\{{x_{4,t}}}\end{array}}
\right] = \left[
{\begin{array}{*{20}{c}}\phi_1&c&\phi_2&\theta_1\\0&1&0&0\\1&0&0&0\\0&0&0&0\end{array}} \right]\left[
{\begin{array}{*{20}{c}}{{x_{1,t - 1}}}\\{{x_{2,t - 1}}}\\{{x_{3,t -
1}}}\\{{x_{4,t -
1}}}\end{array}} \right] + \left[ {\begin{array}{*{20}{c}}0.5\\0\\0\\1\end{array}} \right]u_{1,t}$$

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 $\phi_2x_{3,t - 1} = \phi_2x_{1,t - 2}$.

  • $x_{1,t}$ 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 $\theta_1x_{4,t} = \theta_1u_{1,t-1}$.

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

$${y_t} = \left[ {\begin{array}{*{20}{c}}1&0&0&0\end{array}}
\right]\left[
{\begin{array}{*{20}{c}}{{x_{1,t}}}\\{{x_{2,t}}}\\{{x_{3,t}}}\\{{x_{4,t}}}\end{array}}
\right]+0.1\varepsilon_t.$$

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 specify 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 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.

See Also

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