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

    %% Filter States of State-Space Model Containing Regression Component
% Suppose that the linear relationship between the change in the
% unemployment rate and the nominal gross national product (nGNP) growth 
% rate is of interest. Suppose further that the first difference of the 
% unemployment rate is an ARMA(1,1) series. Symbolically, and in
% state-space form, the model is
%
% $$\begin{array}{l}
% \left[ {\begin{array}{*{20}{c}}
% {{x_{1,t}}}\\
% {{x_{2,t}}}
% \end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
% \phi &\theta \\
% 0&0
% \end{array}} \right]\left[ {\begin{array}{*{20}{c}}
% {{x_{1,t - 1}}}\\
% {{x_{2,t - 1}}}
% \end{array}} \right] + \left[ {\begin{array}{*{20}{c}}
% 1\\
% 1
% \end{array}} \right]
% {{u_{1,t}}}\\
% {y_t} - \beta {Z_t} = {x_{1,t}} + \sigma\varepsilon_t,
% \end{array}$$
%
% 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.
% * $y_{1,t}$ is the observed change in the unemployment rate being
% deflated by the growth rate of nGNP ($Z_t$).
% * $u_{1,t}$ is the Gaussian series of state disturbances having mean 0 and
% standard deviation 1.
% *  $\varepsilon_t$ is the Gaussian series of observation innovations having
% mean 0 and standard deviation $\sigma$.
%
%%
% Load the Nelson-Plosser data set, which contains the unemployment rate and
% nGNP series, among other things.

% Copyright 2015 The MathWorks, Inc.

load Data_NelsonPlosser
%%
% Preprocess the data by taking the natural logarithm of the nGNP series,
% and the first difference of each series.   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
Z = [ones(T-1,1) diff(log(gnpn))];
y = diff(u);
%%
% Though this example removes missing values, the software can accommodate series
% containing missing values in the Kalman filter framework.
%%
% Specify the coefficient matrices.
A = [NaN NaN; 0 0];
B = [1; 1];
C = [1 0];
D = NaN;
%%
% Specify the state-space model using |ssm|.
Mdl = ssm(A,B,C,D);
%%
% Estimate the model parameters, and use a random set of initial parameter
% values for optimization. Specify the regression component and its initial
% value for optimization using the |'Predictors'| and |'Beta0'| name-value
% pair arguments, respectively. Restrict the estimate of $\sigma$ to all
% positive, real numbers.
params0 = [0.3 0.2 0.2];
[EstMdl,estParams] = estimate(Mdl,y,params0,'Predictors',Z,...
    'Beta0',[0.1 0.2],'lb',[-Inf,-Inf,0,-Inf,-Inf]);
%%
% |EstMdl| is an |ssm| model, and you can access its properties using dot
% notation.
%%
% Filter the estimated state-space model. |EstMdl| does not store the data
% or the regression coefficients, so you must pass in them in using the
% name-value pair arguments |'Predictors'| and |'Beta'|, respectively. Plot
% the estimated, filtered states.  Recall that the first state is the
% change in the unemployment rate, and the second state helps build the first.
filteredX = filter(EstMdl,y,'Predictors',Z,'Beta',estParams(end-1:end));

figure
plot(dates(end-(T-1)+1:end),filteredX(:,1));
xlabel('Period')
ylabel('Change in the unemployment rate')
title('Filtered Change in the Unemployment Rate')