# Compare Simulation Smoother to Smoothed States

This example shows how the results of the state-space model simulation smoother (simsmooth) compare to the smoothed states (smooth).

Suppose that the relationship between the change in the unemployment rate (${x}_{1,t}$) and the nominal gross national product (nGNP) growth rate (${x}_{3,t}$) can be expressed in the following, state-space model form.

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

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

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 on ${x}_{1,t}$.

• ${x}_{3,t}$ is the nGNP growth rate at time t.

• ${x}_{4,t}$ is a dummy state for the MA(1) effect on ${x}_{3,t}$.

• ${y}_{1,t}$ is the observed change in the unemployment rate.

• ${y}_{2,t}$ is the observed nGNP growth rate.

• ${u}_{1,t}$ and ${u}_{2,t}$ are Gaussian series of state disturbances having mean 0 and standard deviation 1.

• ${\epsilon }_{1,t}$ is the Gaussian series of observation innovations having mean 0 and standard deviation ${\sigma }_{1}$.

• ${\epsilon }_{2,t}$ is the Gaussian series of observation innovations having mean 0 and standard deviation ${\sigma }_{2}$.

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

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
y = zeros(T-1,2);                          % Preallocate
y(:,1) = diff(u);
y(:,2) = diff(log(gnpn));

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.

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

Specify the state-space model using ssm. Verify that the model specification is consistent with the state-space model.

Mdl = ssm(A,B,C,D)
Mdl =
State-space model type: ssm

State vector length: 4
Observation vector length: 2
State disturbance vector length: 2
Observation innovation vector length: 2
Sample size supported by model: Unlimited
Unknown parameters for estimation: 8

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) + (c3)x2(t-1) + (c4)x3(t-1) + u1(t)
x2(t) = u1(t)
x3(t) = (c2)x1(t-1) + (c5)x3(t-1) + (c6)x4(t-1) + u2(t)
x4(t) = u2(t)

Observation equations:
y1(t) = x1(t) + (c7)e1(t)
y2(t) = x3(t) + (c8)e2(t)

Initial state distribution:

Initial state means are not specified.
Initial state covariance matrix is not specified.
State types are not specified.

Estimate the model parameters, and use a random set of initial parameter values for optimization. Restrict the estimate of ${\sigma }_{1}$ and ${\sigma }_{2}$ to all positive, real numbers using the 'lb' name-value pair argument. For numerical stability, specify the Hessian when the software computes the parameter covariance matrix, using the 'CovMethod' name-value pair argument.

rng(1);
params0 = rand(8,1);
[EstMdl,estParams] = estimate(Mdl,y,params0,...
'lb',[-Inf -Inf -Inf -Inf -Inf -Inf 0 0],'CovMethod','hessian');
Method: Maximum likelihood (fmincon)
Sample size: 61
Logarithmic  likelihood:     -199.397
Akaike   info criterion:      414.793
Bayesian info criterion:       431.68
|     Coeff       Std Err    t Stat     Prob
----------------------------------------------------
c(1) |  0.03387       0.15213     0.22263  0.82383
c(2) | -0.01258       0.05749    -0.21876  0.82684
c(3) |  2.49855       0.22764    10.97570   0
c(4) |  0.77438       2.58647     0.29939  0.76464
c(5) |  0.13994       2.64359     0.05294  0.95778
c(6) |  0.00367       2.45472     0.00150  0.99881
c(7) |  0.00535       2.11315     0.00253  0.99798
c(8) |  0.00032       0.12685     0.00253  0.99798
|
|   Final State   Std Dev     t Stat    Prob
x(1) |  1.39999       0.00535   261.57973   0
x(2) |  0.21778       0.91641     0.23765  0.81216
x(3) |  0.04730       0.00032   147.40073   0
x(4) |  0.03568       0.00033   107.76114   0

EstMdl is an ssm model, and you can access its properties using dot notation.

Simulate 1e4 paths of observations from the fitted, state-space model EstMdl using the simulation smoother. Specify to simulate observations for each period.

numPaths = 1e4;
SimX = simsmooth(EstMdl,y,'NumPaths',numPaths);

SimX is a T - 1-by- 4-by- numPaths matrix containing the simulated states. The rows of SimX correspond to periods, the columns correspond to a state in the model, and the pages correspond to paths.

Estimate the smoothed state means, standard deviations, and 95% confidence intervals.

SmoothBar = mean(SimX,3);
SmoothSTD = std(SimX,0,3);
SmoothCIL = SmoothBar - 1.96*SmoothSTD;
SmoothCIU = SmoothBar + 1.96*SmoothSTD;

Estimate smooth states using smooth.

SmoothX = smooth(EstMdl,y);

Plot the smoothed states, and the means of the simulated states and their 95% confidence intervals.

figure
h = plot(dates(2:T),SmoothBar(:,1),'-r',...
dates(2:T),SmoothCIL(:,1),':b',...
dates(2:T),SmoothCIU(:,1),':b',...
dates(2:T),SmoothX(:,1),':k',...
'LineWidth',3);
xlabel 'Period';
ylabel 'Unemployment rate';
legend(h([1,2,4]),{'Simulated, smoothed state mean','95% confidence interval',...
'Smoothed states'},'Location','Best');
title 'Smoothed Unemployment Rate';
axis tight

figure
h = plot(dates(2:T),SmoothBar(:,3),'-r',...
dates(2:T),SmoothCIL(:,3),':b',...
dates(2:T),SmoothCIU(:,3),':b',...
dates(2:T),SmoothX(:,3),':k',...
'LineWidth',3);
xlabel 'Period';
ylabel 'nGNP';
legend(h([1,2,4]),{'Simulated, smoothed state mean','95% confidence interval',...
'Smoothed states'},'Location','Best');
title 'Smoothed nGNP';
axis tight

The simulated state means are practically identical to the smoothed states.