Forecast State-Space Model Using Monte-Carlo Methods

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This example shows how to forecast a state-space model using Monte-Carlo methods, and to compare the Monte-Carlo forecasts to the theoretical forecasts.

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.

$[\begin{array}{cccccccccccccccccccc} x_{1, t} \\ x_{2, t} \\ x_{3, t} \\ x_{4, t} \end{array}] = [\begin{array}{cccccccccccccccccccc} ϕ_{1} & θ_{1} & γ_{1} & 0 \\ 0 & 0 & 0 & 0 \\ γ_{2} & 0 & ϕ_{2} & θ_{2} \\ 0 & 0 & 0 & 0 \end{array}] [\begin{array}{cccccccccccccccccccc} x_{1, t - 1} \\ x_{2, t - 1} \\ x_{3, t - 1} \\ x_{4, t - 1} \end{array}] + [\begin{array}{cccccccccccccccccccc} 1 & 0 \\ 1 & 0 \\ 0 & 1 \\ 0 & 1 \end{array}] [\begin{array}{cccccccccccccccccccc} u_{1, t} \\ u_{2, t} \end{array}]$

$[\begin{array}{cccccccccccccccccccc} y_{1, t} \\ y_{2, t} \end{array}] = [\begin{array}{cccccccccccccccccccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array}] [\begin{array}{cccccccccccccccccccc} x_{1, t} \\ x_{2, t} \\ x_{3, t} \\ x_{4, t} \end{array}] + [\begin{array}{cccccccccccccccccccc} σ_{1} & 0 \\ 0 & σ_{2} \end{array}] [\begin{array}{cccccccccccccccccccc} ε_{1, t} \\ ε_{2, t} \end{array}],$

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 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.
$ε_{1, t}$ is the Gaussian series of observation innovations having mean 0 and standard deviation $σ_{1}$ .
$ε_{2, t}$ is the Gaussian series of observation innovations having mean 0 and standard deviation $σ_{2}$ .

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
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.

To determine how well the model forecasts observations, remove the last 10 observations for comparison.

numPeriods = 10;                   % Forecast horizon
isY = y(1:end-numPeriods,:);       % In-sample observations
oosY = y(end-numPeriods+1:end,:);  % Out-of-sample observations

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 $σ_{1}$ and $σ_{2}$ 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,isY,params0,...
    'lb',[-Inf -Inf -Inf -Inf -Inf -Inf 0 0],'CovMethod','hessian');

Method: Maximum likelihood (fmincon)
Sample size: 51
Logarithmic  likelihood:      -170.92
Akaike   info criterion:       357.84
Bayesian info criterion:      373.295
      |     Coeff       Std Err    t Stat     Prob  
----------------------------------------------------
 c(1) |  0.06750       0.16548     0.40791  0.68334 
 c(2) | -0.01372       0.05887    -0.23302  0.81575 
 c(3) |  2.71201       0.27039    10.03006   0      
 c(4) |  0.83816       2.84586     0.29452  0.76836 
 c(5) |  0.06274       2.83470     0.02213  0.98234 
 c(6) |  0.05196       2.56873     0.02023  0.98386 
 c(7) |  0.00272       2.40771     0.00113  0.99910 
 c(8) |  0.00016       0.13942     0.00113  0.99910 
      |                                             
      |   Final State   Std Dev     t Stat    Prob  
 x(1) | -0.00000       0.00272    -0.00033  0.99973 
 x(2) |  0.12237       0.92954     0.13164  0.89527 
 x(3) |  0.04049       0.00016   256.77783   0      
 x(4) |  0.01183       0.00016    72.52162   0

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

Filter the estimated, state-space model, and extract the filtered states and their variances from the final period.

[~,~,Output] = filter(EstMdl,isY);

Modify the estimated, state-space model so that the initial state means and covariances are the filtered states and their covariances of the final period. This sets up simulation over the forecast horizon.

EstMdl1 = EstMdl;
EstMdl1.Mean0 = Output(end).FilteredStates;
EstMdl1.Cov0 = Output(end).FilteredStatesCov;

Simulate 5e5 paths of observations from the fitted, state-space model EstMdl. Specify to simulate observations for each period.

numPaths = 5e5;
SimY = simulate(EstMdl1,10,'NumPaths',numPaths);

SimY is a 10-by- 2-by- numPaths array containing the simulated observations. The rows of SimY correspond to periods, the columns correspond to an observation in the model, and the pages correspond to paths.

Estimate the forecasted observations and their 95% confidence intervals in the forecast horizon.

MCFY = mean(SimY,3);
CIFY = quantile(SimY,[0.025 0.975],3);

Estimate the theoretical forecast bands.

[Y,YMSE] = forecast(EstMdl,10,isY);
Lb = Y - sqrt(YMSE)*1.96;
Ub = Y + sqrt(YMSE)*1.96;

Plot the forecasted observations with their true values and the forecast intervals.

figure
h = plot(dates(end-numPeriods-9:end),[isY(end-9:end,1);oosY(:,1)],'-k',...
    dates(end-numPeriods+1:end),MCFY(end-numPeriods+1:end,1),'.-r',...
    dates(end-numPeriods+1:end),CIFY(end-numPeriods+1:end,1,1),'-b',...
    dates(end-numPeriods+1:end),CIFY(end-numPeriods+1:end,1,2),'-b',...
    dates(end-numPeriods+1:end),Y(:,1),':c',...
    dates(end-numPeriods+1:end),Lb(:,1),':m',...
    dates(end-numPeriods+1:end),Ub(:,1),':m',...
    'LineWidth',3);
xlabel('Period')
ylabel('Change in the unemployment rate')
legend(h([1,2,4:6]),{'Observations','MC forecasts',...
    '95% forecast intervals','Theoretical forecasts',...
    '95% theoretical intervals'},'Location','Best')
title('Observed and Forecasted Changes in the Unemployment Rate')

Figure contains an axes object. The axes object with title Observed and Forecasted Changes in the Unemployment Rate, xlabel Period, ylabel Change in the unemployment rate contains 7 objects of type line. These objects represent Observations, MC forecasts, 95% forecast intervals, Theoretical forecasts, 95% theoretical intervals.

figure
h = plot(dates(end-numPeriods-9:end),[isY(end-9:end,2);oosY(:,2)],'-k',...
    dates(end-numPeriods+1:end),MCFY(end-numPeriods+1:end,2),'.-r',...
    dates(end-numPeriods+1:end),CIFY(end-numPeriods+1:end,2,1),'-b',...
    dates(end-numPeriods+1:end),CIFY(end-numPeriods+1:end,2,2),'-b',...
    dates(end-numPeriods+1:end),Y(:,2),':c',...
    dates(end-numPeriods+1:end),Lb(:,2),':m',...
    dates(end-numPeriods+1:end),Ub(:,2),':m',...
    'LineWidth',3);
xlabel('Period')
ylabel('nGNP growth rate')
legend(h([1,2,4:6]),{'Observations','MC forecasts',...
    '95% MC intervals','Theoretical forecasts','95% theoretical intervals'},...
    'Location','Best')
title('Observed and Forecasted nGNP Growth Rates')

Figure contains an axes object. The axes object with title Observed and Forecasted nGNP Growth Rates, xlabel Period, ylabel nGNP growth rate contains 7 objects of type line. These objects represent Observations, MC forecasts, 95% MC intervals, Theoretical forecasts, 95% theoretical intervals.

Forecast State-Space Model Using Monte-Carlo Methods

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