Estimate State-Space Model Containing Regression Component

This example shows how to fit a state-space model that has an observation-equation regression component.

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. 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);

This example proceeds using series without NaN values. However, using the Kalman filter framework, the software can accommodate series containing missing values.

Specify the coefficient matrices. Use NaN values to indicate unknown parameters.

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

Specify the state-space model using ssm. Since $$x_{1,t}$$ is an ARMA(1,1) process, and $$x_{2,t}$$ is white noise, specify that they are stationary processes.

StateType = [0; 0];
Mdl = ssm(A,B,C,D,'StateType',StateType);

Estimate the model parameters. 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, but allow all other parameters to be unbounded.

params0 = [0.3 0.2 0.1]; % Chosen arbitrarily
EstMdl = estimate(Mdl,y,params0,'Predictors',Z,'Beta0',[0.1 0.1],...
    'lb',[-Inf,-Inf,0,-Inf,-Inf]);

Method: Maximum likelihood (fmincon)
Sample size: 61
Logarithmic  likelihood:     -99.7245
Akaike   info criterion:      209.449
Bayesian info criterion:      220.003
           |      Coeff       Std Err    t Stat     Prob  
----------------------------------------------------------
 c(1)      |  -0.34098       0.29608    -1.15164  0.24948 
 c(2)      |   1.05003       0.41377     2.53771  0.01116 
 c(3)      |   0.48592       0.36790     1.32079  0.18657 
 y <- z(1) |   1.36121       0.22338     6.09358   0      
 y <- z(2) | -24.46711       1.60018   -15.29024   0      
           |                                              
           |    Final State   Std Dev     t Stat    Prob  
 x(1)      |   1.01264       0.44690     2.26592  0.02346 
 x(2)      |   0.77718       0.58917     1.31912  0.18713 

A table of estimates and statistics output to the Command Window. EstMdl is an ssm model, and you can access its properties using dot notation.