MATLAB Examples

Estimate Random Parameter of State-Space Model

This example shows how to estimate a random, autoregressive coefficient of a state in a state-space model. That is, this example takes a Bayesian view of state-space model parameter estimation by using the "zig-zag" estimation method.

Contents

Suppose that two states ($x_{1,t}$ and $x_{2,t}$) represent the net exports of two countries at the end of the year.

  • $x_{1,t}$ is a unit root process with a disturbance variance of $\sigma_1^2$.
  • $x_{2,t}$ is an AR(1) process with an unknown, random coefficient and a disturbance variance of $\sigma_2^2$.
  • An observation ($y_t$) is the exact sum of the two net exports. That is, the net exports of the individual states are unknown.

Symbolically, the true state-space model is

$$\begin{array}{l}
{x_{1,t}} = {x_{1,t - 1}} + \sigma_1{u_{1,t}}\\
{x_{2,t}} = \phi {x_{2,t - 1}} + \sigma_2{u_{2,t}}\\
{y_t} = {x_{1,t}} + {x_{2,t}}
\end{array}$$

Simulate Data

Simulate 100 years of net exports from:

  • A unit root process with a mean zero, Gaussian noise series that has variance $0.1^2$.
  • An AR(1) process with an autoregressive coefficient of 0.6 and a mean zero, Gaussian noise series that has variance $0.2^2$.
  • $x_{1,0} = x_{2,0} = 0$.
  • Create an observation series by summing the two net exports per year.
rng(100); % For reproducibility
T = 150;
sigma1 = 0.1;
sigma2 = 0.2;
phi = 0.6;

u1 = randn(T,1)*sigma1;
x1 = cumsum(u1);
Mdl2 = arima('AR',phi,'Variance',sigma2^2,'Constant',0);
x2 = simulate(Mdl2,T,'Y0',0);
y = x1 + x2;

figure;
plot([x1 x2 y])
legend('x_1','x_2','y','Location','Best');
ylabel('Net exports');
xlabel('Period');

The Zig-Zag Estimation Method

Treat $\phi$ as if it is unknown and random, and use the zig-zag method to recover its distribution. To implement the zig-zag method:

1. Choose an initial value for $\phi$ in the interval (-1,1), and denote it $\phi_z$.

2. Create the true state-space model, that is, an ssm model object that represents the data-generating process.

3. Use the simulation smoother (simsmooth) to draw a random path from the distribution of the second smoothed states. Symbolically, $x_{2,z,t} \sim P(x_{2,t}|y_t,x_{1,t},\phi = \phi_z)$.

4. Create another state-space model that has this form

$$\begin{array}{l}
{\phi_{z,t}} = {\phi_{z,t - 1}}\\
{x_{2,z,t}} = {x_{2,z,t - 1}} \phi_{z,t}  + 0.8{u_{2,t}}\\
\end{array}$$

In words, $\phi_{z,t}$ is a static state and $x_{2,z,t}$ is an "observed" series with time varying coefficient $C_t = x_{2,z,t-1}$.

5. Use the simulation smoother to draw a random path from the distribution of the smoothed $\phi_{z,t}$ series. Symbolically, $\phi_{z,t} \sim P(\phi|x_{2,z,t})$, where $x_{2,z,t}$ encompasses the structure of the true state-space model and the observations. $\phi_{z,t}$ is static, so you can just reserve one value ($\phi_z$).

6. Repeat steps 2 - 5 many times and store $\phi_{z}$ each iteration.

7. Perform diagnostic checks on the simulation series. That is, construct:

  • Trace plots to determine the burn in period and whether the Markov chain is mixing well.
  • Autocorrelation plots to determine how many draws need removing to obtain a well-mixed Markov chain.

8. The remaining series represents draws from the posterior distribution of $\phi$. You can compute descriptive statistics, or plot a histogram to determine the qualities of the distribution.

Estimate Random Coefficient Using Zig-Zag Method

Specify initial values, preallocate, and create the true state-space model.

phi0 = -0.3;             % Initial value of phi
Z = 1000;                % Number of times to iterate the zig-zag method
phiz = [phi0; nan(Z,1)]; % Preallocate

A = [1 0; 0 NaN];
B = [sigma1; sigma2];
C = [1 1];
Mdl = ssm(A,B,C,'StateType',[2; 0]);

Mdl is an ssm model object. The NaN acts as a placeholder for $\phi$.

Iterate steps 2 - 5 of the zig-zag method.

for j = 2:(Z + 1);
    % Draw a random path from smoothed x_2 series.
    xz = simsmooth(Mdl,y,'Params',phiz(j-1));
    % The second column of xz is a draw from the posterior distribution of x_2.

    % Create the intermediate state-space model.
    Az = 1;
    Bz = 0;
    Cz = num2cell(xz((1:(end - 1)),2));
    Dz = sigma2;
    Mdlz = ssm(Az,Bz,Cz,Dz,'StateType',2);

    % Draw a random path from the smoothed phiz series.
    phizvec = simsmooth(Mdlz,xz(2:end,2));
    phiz(j) = phizvec(1);
    % phiz(j) is a draw from the posterior distribution of phi
end

phiz is a Markov chain. Before analyzing the posterior distribution of $\phi$, you should assess whether to impose a burn-in period, or the severity of the autocorrelation in the chain.

Determine Quality of Simulation

Draw a trace plot for the first 100, 500, and all of the random draws.

vec = [100 500 Z];
figure;
for j = 1:3;
    subplot(3,1,j)
    plot(phiz(1:vec(j)));
    title('Trace Plot for \phi');
    xlabel('Simulation number');
    axis tight;
end

According to the first plot, transient effects die down after about 20 draws. Therefore, a short burn-in period should suffice. The plot of the entire simulation shows that the series settles around a center.

Plot the autocorrelation function of the series after removing the first 20 draws.

burnOut = 21:Z;

figure;
autocorr(phiz(burnOut));

The autocorrelation function dies out rather quickly. It doesn't seem like autocorrelation in the chain is an issue.

Determine qualities of the posterior distribution of $\phi$ by computing simulation statistics and by plotting a histogram of the reduced set of random draws.

xbar = mean(phiz(burnOut))
xstd = std(phiz(burnOut))
ci = norminv([0.025,0.975],xbar,xstd); % 95% confidence interval

figure;
histogram(phiz(burnOut),'Normalization','pdf');
h = gca;
hold on;
simX = linspace(h.XLim(1),h.XLim(2),100);
simPDF = normpdf(simX,xbar,xstd);
plot(simX,simPDF,'k','LineWidth',2);
h1 = plot([xbar xbar],h.YLim,'r','LineWidth',2);
h2 = plot([0.6 0.6],h.YLim,'g','LineWidth',2);
h3 = plot([ci(1) ci(1)],h.YLim,'--r',...
    [ci(2) ci(2)],h.YLim,'--r','LineWidth',2);
legend([h1 h2 h3(1)],{'Simulation Mean','True Mean','95% CI'});
h.XTick = sort([h.XTick xbar]);
h.XTickLabel{h.XTick == xbar} = xbar;
h.XTickLabelRotation = 90;
xbar =

    0.5104


xstd =

    0.0988

The posterior distribution of $\phi$ is roughly normal with mean and standard deviation approximately 0.51 and 0.1, respectively. The true mean of $\phi$ is 0.6, and it is less than one standard deviation to the right of the simulation mean.

Compute the maximum likelihood estimate of $\phi$. That is, treat $\phi$ as a fixed, but unknown parameter, and then estimate Mdl using the Kalman filter and maximum likelihood.

[~,estParams] = estimate(Mdl,y,phi0)
Method: Maximum likelihood (fminunc)
Sample size: 150
Logarithmic  likelihood:     -10.1434
Akaike   info criterion:      22.2868
Bayesian info criterion:      25.2974
      |     Coeff       Std Err     t Stat       Prob  
-------------------------------------------------------
 c(1) |  0.53590       0.19183    2.79360      0.00521 
      |                                                
      |   Final State   Std Dev    t Stat        Prob  
 x(1) | -0.85059       0.00000   -6.45811e+08   0      
 x(2) |  0.00454        0          Inf          0      

estParams =

    0.5359

The MLE of $\phi$ is 0.54. Both estimates are within one standard deviation or standard error from the true value of $\phi$.