Forecast State-Space Model Containing Regime Change in the Forecast Horizon

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This example shows how to forecast a time-varying, state-space model, in which there is a regime change in the forecast horizon.

Suppose that you observed a multivariate process for 75 periods, and you want to forecast the process 25 periods into the future. Also, suppose that you can specify the process as a state-space model. For periods 1 through 50, the state-space model has one state: a stationary AR(2) model with a constant term. At period 51, the state-space model includes a random walk. The states are observed unbiasedly, but with additive measurement error. Symbolically, the model is

$\begin{array}{l} [\begin{array}{cccccccccccccccccccc} x_{1, t} \\ x_{2, t} \\ x_{3, t} \\ x_{4, t} \end{array}] = [\begin{array}{cccccccccccccccccccc} 0.5 & - 0.2 & 1 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccccccccccccccccccc} x_{1, t - 1} \\ x_{2, t - 1} \\ x_{3, t - 1} \\ x_{4, t - 1} \end{array}] + [\begin{array}{cccccccccccccccccccc} 0.1 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0.5 \end{array}] [\begin{array}{cccccccccccccccccccc} u_{1, t} \\ u_{2, t} \end{array}] \\ y_{t} = [\begin{array}{cccccccccccccccccccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}] [\begin{array}{cccccccccccccccccccc} x_{1, t} \\ x_{2, t} \\ x_{3, t} \\ x_{4, t} \end{array}] + [\begin{array}{cccccccccccccccccccc} 0.3 & 0 \\ 0 & 0.2 \end{array}] [\begin{array}{cccccccccccccccccccc} ε_{1, t} \\ ε_{2, t} \end{array}] \end{array} .$

For periods 1 through 50, the random walk process is not in the model.

Specify the in-sample, coefficient matrices.

A1 = {[0.5 0.2 1; 1 0 0; 0 0 1]};                % A for periods 1 - 50
A2 = {[0.5 0.2 1; 1 0 0; 0 0 1; 0 0 0]};         % A for period 51
A3 = {[0.5 0.2 1 0; 1 0 0 0; 0 0 1 0; 0 0 0 1]}; % A for periods 51 - 75
A = [repmat(A1,50,1); A2; repmat(A3,24,1)];

B1 = {[0.1; 0; 0]};              % B for periods 1 - 50
B3 = {[0.1 0; 0 0; 0 0; 0 0.5]}; % B for periods 51 - 75
B = [repmat(B1,50,1); repmat(B3,25,1)];

C1 = {[1 0 0]};            % C for periods 1 - 50
C3 = {[1 0 0 0; 0 0 0 1]}; % C for periods 51 - 75
C = [repmat(C1,50,1); repmat(C3,25,1)];

D1 = {0.3};            % D for periods 1 - 50
D3 = {[0.3 0; 0 0.2]}; % D for periods 51 - 75
D = [repmat(D1,50,1); repmat(D3,25,1)];

Specify the state space model, and the initial state means and covariance matrix. It is best practice to specify the types of each state using the 'StateType' name-value pair argument. Only specify the initial state parameters for the three states that start the state-space model.

Mean0 = [1/(1-0.5-0.2); 1/(1-0.5-0.2); 1];
Cov0 = [0.02 0.01 0; 0.01 0.02 0; 0 0 0];
StateType = [0; 0; 1];
Mdl = ssm(A,B,C,D,'Mean0',Mean0,'Cov0',Cov0,'StateType',StateType)

Mdl = 
State-space model type: ssm

State vector length: Time-varying
Observation vector length: Time-varying
State disturbance vector length: Time-varying
Observation innovation vector length: Time-varying
Sample size supported by model: 75

State variables: x1, x2,...
State disturbances: u1, u2,...
Observation series: y1, y2,...
Observation innovations: e1, e2,...

State equations of period 1, 2, 3,..., 50:
x1(t) = (0.50)x1(t-1) + (0.20)x2(t-1) + x3(t-1) + (0.10)u1(t)
x2(t) = x1(t-1)
x3(t) = x3(t-1)

State equations of period 51:
x1(t) = (0.50)x1(t-1) + (0.20)x2(t-1) + x3(t-1) + (0.10)u1(t)
x2(t) = x1(t-1)
x3(t) = x3(t-1)
x4(t) = (0.50)u2(t)

State equations of period 52, 53, 54,..., 75:
x1(t) = (0.50)x1(t-1) + (0.20)x2(t-1) + x3(t-1) + (0.10)u1(t)
x2(t) = x1(t-1)
x3(t) = x3(t-1)
x4(t) = x4(t-1) + (0.50)u2(t)


Observation equation of period 1, 2, 3,..., 50:
y1(t) = x1(t) + (0.30)e1(t)

Observation equations of period 51, 52, 53,..., 75:
y1(t) = x1(t) + (0.30)e1(t)
y2(t) = x4(t) + (0.20)e2(t)


Initial state distribution:

Initial state means
  x1    x2   x3 
 3.33  3.33   1 

Initial state covariance matrix
     x1    x2    x3 
 x1  0.02  0.01   0 
 x2  0.01  0.02   0 
 x3   0     0     0 

State types
     x1          x2         x3    
 Stationary  Stationary  Constant

Mdl is a time-varying, ssm model without unknown parameters. The software sets initial state means and covariance values based on the type of state.

Simulate 75 observations from Mdl.

rng(1); % For reproducibility
Y = simulate(Mdl,75);

y is a 75-by-1 cell vector. Cells 1 through 50 contain scalars, and cells 51 through 75 contain 2-by-1 numeric vectors. Cell j corresponds to the observations of period j, specified by the observation model.

Plot the simulated responses.

y1 = cell2mat(Y(51:75));              % Observations for periods 1 - 50
d1 = cell2mat(Y(51:75));
y2 = [d1(((1:25)*2)-1) d1((1:25)*2)]; % Observations for periods 51 - 75

figure
plot(1:75,[y1;y2(:,1)],'-k',1:75,[nan(50,1);y2(:,2)],'-r','LineWidth',2')
title('In-sample Observations')
ylabel('Observations')
xlabel('Period')
legend({'AR(2)','Random walk'})

Figure contains an axes object. The axes object with title In-sample Observations, xlabel Period, ylabel Observations contains 2 objects of type line. These objects represent AR(2), Random walk.

Suppose that the random walk process drops out of the state space in the 20th period of the forecast horizon.

Specify the coefficient matrices for the forecast period.

A4 = {[0.5 0.2 1 0; 1 0 0 0; 0 0 1 0; 0 0 0 1]};  % A for periods 76 - 95
A5 = {[0.5 0.2 1 0; 1 0 0 0; 0 0 1 0]};           % A for period 96
A6 = {[0.5 0.2 1; 1 0 0; 0 0 1]};                 % A for periods 97 - 100
fhA = [repmat(A4,20,1); A5; repmat(A6,4,1)];

B4 = {[0.1 0; 0 0; 0 0; 0 0.5]}; % B for periods 76 - 95
B6 = {[0.1; 0; 0]};              % B for periods 96 - 100
fhB = [repmat(B4,20,1); repmat(B6,5,1)];

C4 = {[1 0 0 0; 0 0 0 1]}; % C for periods 76 - 95
C6 = {[1 0 0]};            % C for periods 96 - 100
fhC = [repmat(C4,20,1); repmat(C6,5,1)];

D4 = {[0.3 0; 0 0.2]}; % D for periods 76 - 95
D6 = {0.3};            % D for periods 96 - 100
fhD = [repmat(D4,20,1); repmat(D6,5,1)];

Forecast observations over the forecast horizon.

FY = forecast(Mdl,25,Y,'A',fhA,'B',fhB,'C',fhC,'D',fhD);

FY is a 25-by-1 cell vector. Cells 1 through 20 contain 2-by-1 numeric vectors, and cells 51 through 75 contain scalars. Cell j corresponds to the observations of period j, specified by the forecast-horizon, observation model.

Plot the forecasts with the in-sample observations.

d2 = cell2mat(FY(1:20));
FY1 = [d2(((1:20)*2)-1) d2((1:20)*2)]; % Forecasts for periods 76 - 95
FY2 = cell2mat(FY(21:25));             % Forecasts for periods 96 - 100

figure
plot(1:75,[y1;y2(:,1)],'-k',1:75,[nan(50,1);y2(:,2)],'-r',...
    76:100,[FY1(:,1); FY2],'.-k',76:100,[FY1(:,2); nan(5,1)],'.-r',...
    75:76,[y2(end,1) FY1(1,1)],':k',75:76,[y2(end,2) FY1(1,2)],':r',...
    'LineWidth',2')
title('In-sample and Forecasted Observations')
ylabel('Observations')
xlabel('Period')
xlim([50,100])
legend({'In-sample AR(2)','In-sample random walk',...
    'AR(2), forecasted observations',...
    'Random walk, forecasted observations'},'Location','Best')%% Title

Figure contains an axes object. The axes object with title In-sample and Forecasted Observations, xlabel Period, ylabel Observations contains 6 objects of type line. These objects represent In-sample AR(2), In-sample random walk, AR(2), forecasted observations, Random walk, forecasted observations.

Forecast State-Space Model Containing Regime Change in the Forecast Horizon

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