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    %% Infer Conditional Variances  
% Infer the conditional variances from an AR(1) and GARCH(1,1) composite model.   

% Copyright 2015 The MathWorks, Inc.


%% 
% Specify an AR(1) model using known parameters. Set the variance equal
% to a |garch| model. 
Mdl = arima('AR',{0.8,-0.3},'Constant',0);
MdlVar = garch('Constant',0.0002,'GARCH',0.6,...
	'ARCH',0.2);
Mdl.Variance = MdlVar;  

%% 
% Simulate response data with 102 observations. 
rng 'default';
Y = simulate(Mdl,102);  

%% 
% Infer conditional variances for the last 100 observations without using
% presample data. 
[Ew,Vw] = infer(Mdl,Y(3:end));  

%% 
% Infer conditional variances for the last 100 observations using the first
% two observations as presample data. 
[E,V] = infer(Mdl,Y(3:end),'Y0',Y(1:2));  

%% 
% Plot the two sets of conditional variances for comparison. Examine the
% first few observations to see the slight difference between the series
% at the beginning. 
figure;
subplot(2,1,1);
plot(Vw,'r','LineWidth',2);
hold on;
plot(V);
legend('Without Presample','With Presample');
title 'Inferred Conditional Variances';
hold off

subplot(2,1,2);
plot(Vw(1:5),'r','LineWidth',2);
hold on;
plot(V(1:5));
legend('Without Presample','With Presample');
title 'Beginning of Series';
hold off