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Simulate Conditional Variance Model

This example shows how to simulate a conditional variance model using simulate.

Step 1. Load the data and specify the model.

Load the Deutschmark/British pound foreign exchange rate data included with the toolbox, and convert to returns. Specify and fit a GARCH(1,1) model.

r = price2ret(Data);
T = length(r);
Mdl = garch(1,1);
EstMdl = estimate(Mdl,r);

GARCH(1,1) Conditional Variance Model (Gaussian Distribution):

Value       StandardError    TStatistic      PValue
__________    _____________    __________    __________

Constant    1.0535e-06      3.505e-07        3.0059       0.0026482
GARCH{1}       0.80657        0.01291        62.474               0
ARCH{1}        0.15436       0.011575        13.336      1.4363e-40
v0 = infer(EstMdl,r);

Step 2. Simulate foreign exchange rate returns.

Use the fitted model to simulate 25 realizations of foreign exchange rate returns and conditional variances over a 1000-period forecast horizon. Use the observed returns and inferred conditional variances as presample innovations and variances, respectively.

rng default; % For reproducibility
[V,Y] = simulate(EstMdl,1000,'NumPaths',25,...
'E0',r,'V0',v0);

figure
subplot(2,1,1)
plot(v0)
hold on
plot(T+1:T+1000,V)
xlim([0,T+1000])
title('Conditional Variances')
hold off

subplot(2,1,2)
plot(r)
hold on
plot(T+1:T+1000,Y)
xlim([0,T+1000])
title('Returns')
hold off Step 3. Plot the returns distribution at a future time.

Use simulations to generate a forecast distribution of foreign exchange returns 500 days into the future. Generate 1000 sample paths to estimate the distribution.

rng default; % For reproducibility
[V,Y] = simulate(EstMdl,500,'NumPaths',1000,...
'E0',r-EstMdl.Offset,'V0',v0);

figure
histogram(Y(500,:),10)
title('Return Distribution in 500 Days') 