www.gusucode.com > econ 案例源码程序 matlab代码 > econ/SimulateanARMAErrorModelExample.m

    %% Simulate an ARMA Error Model  
% This example shows how to simulate responses from a regression model with
% ARMA errors without specifying a presample.   

% Copyright 2015 The MathWorks, Inc.


%% 
% Specify the regression model with ARMA(2,1) errors:
%
% $$\begin{array}{l}{y_t} = 2 + {X_t}\left[ \begin{array}{l} - 2\\1.5\end{array} \right] + {u_t}\\{u_t} = 0.9{u_{t - 1}} - 0.1{u_{t - 2}} + {\varepsilon _t} + 0.5{\varepsilon _{t - 1}},\end{array}$$
%
% where $\varepsilon_t$ is distributed with 15 degrees of freedom and variance 1. 
Beta = [-2; 1.5];
Intercept = 2;
a1 = 0.9;
a2 = -0.1;
b1 = 0.5;
Variance = 1;
Distribution = struct('Name','t','DoF',15);
Mdl = regARIMA('AR',{a1, a2},'MA',b1,...
    'Distribution',Distribution,'Intercept',Intercept,...
    'Beta',Beta,'Variance',Variance);  

%% 
% Generate two length _T_ = 50 predictor series by random selection from
% the standard Gaussian distribution. 
T = 50;
rng(6); % For reproducibility
X = randn(T,2); 

%%
% The software treats the predictors as nonstochastic series.  

%% 
% Generate and plot one sample path of responses from |Mdl|. 
rng(7);
ySim = simulate(Mdl,T,'X',X);

figure
plot(ySim)
title('{\bf Simulated Response Series}')    

%%
% |simulate| requires:
%
% * |P = 2| presample unconditional disturbances to initialize the autoregressive
% component of the error series.  
% * |Q = 1| presample innovations to initialize the moving average component
% of the error series.   
%
%%
% Without them, as in this case, |simulate| sets the necessary presample
% errors to 0.  

%% 
% Alternatively, use |filter| to filter a random innovation series through
% |Mdl|. 
rng(7);
e = randn(T,1);
yFilter = filter(Mdl,e,'X',X);

figure
plot(yFilter)
title('{\bf Simulated Response Series Using Filtered Innovations}')    

%%
% The plots suggest that the simulated responses and the responses generated
% from the filtered innovations are equivalent.  

%% 
% Simulate 1000 response paths from |Mdl|. Assess transient effects by plotting
% the unconditional disturbance (|U|) variances across the simulated paths
% at each period. 
numPaths = 1000;
[Y,~,U] = simulate(Mdl,T,'NumPaths',numPaths,'X',X);

figure
h1 = plot(Y,'Color',[.85,.85,.85]);
title('{\bf 1000 Simulated Response Paths}')
hold on
h2 = plot(1:T,Intercept+X*Beta,'k--','LineWidth',2);
legend([h1(1),h2],'Simulated Path','Mean')
hold off

figure
h1 = plot(var(U,0,2),'r','LineWidth',2);
hold on
theoVarFix = Variance*(a1*b1*(1+a2)+(1-a2)*(1+a1*b1+b1^2))/...
    ((1+a2)*((1-a2)^2-a1^2));
h2 = plot([1 T],[theoVarFix theoVarFix],'k--','LineWidth',2);
title('{\bf Unconditional Disturbance Variance}')
legend([h1,h2],'Simulation Variance','Theoretical Variance',...
    'Location','Best')
hold off    

%%
% The simulated paths follow their theoretical mean, $c + X\beta$, which is
% not constant over time (and might look nonstationary).    

%%
% The variance of the process is not constant, but levels off at the theoretical
% variance by the 10th period. The theoretical variance of the ARMA(2,1)
% error model is: 
%
% $$\frac{{\sigma _\varepsilon ^2\left[ {{a_1}{b_1}\left( {1 + {a_2}} \right) + \left( {1 - {a_2}} \right)\left( {1 + {a_1}{b_1} + b_1^2} \right)} \right]}}{{{{\left( {1 + {a_2}} \right)}^2}\left[ {{{\left( {1 - {a_2}} \right)}^2} - a_1^2} \right]}} = \frac{{\left[ {0.9(0.5)\left( {1 - 0.1} \right) + \left( {1 + 0.1} \right)\left( {1 + 0.9(0.5) + {{0.5}^2}} \right)} \right]}}{{{{\left( {1 - 0.1} \right)}^2}\left[ {{{\left( {1 + 0.1} \right)}^2} - {{0.9}^2}} \right]}} = {\rm{6}}{\rm{.32}}.$$
%

%% 
% You can reduce transient effects by partitioning the simulated data into
% a burn-in portion and a portion for analysis. Do not use the burn-in portion
% for analysis. Include enough periods in the burn-in portion to overcome
% the transient effects. 
burnIn = 1:10;
notBurnIn = burnIn(end)+1:T;
Y = Y(notBurnIn,:);
X = X(notBurnIn,:);
U = U(notBurnIn,:);
figure
h1 = plot(notBurnIn,Y,'Color',[.85,.85,.85]);
hold on
h2 = plot(notBurnIn,Intercept+X*Beta,'k--','LineWidth',2);
title('{\bf 1000 Simulated Response Paths for Analysis}')
legend([h1(1),h2],'Simulated Path','Mean')
axis tight
hold off

figure
h1 = plot(notBurnIn,var(U,0,2),'r','LineWidth',2);
hold on
h2 = plot([notBurnIn(1) notBurnIn(end)],...
    [theoVarFix theoVarFix],'k--','LineWidth',2);
title('{\bf Converged Unconditional Disturbance Variance}')
legend([h1,h2],'Simulation Variance','Theoretical Variance')
axis tight
hold off       

%%
% Unconditional disturbance simulation variances fluctuate around the theoretical
% variance due to Monte Carlo sampling error. Be aware that the exclusion
% of the burn-in sample from analysis reduces the effective sample size.