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    %% Specify AR Lags When Estimating FGLS Coefficients and Standard Errors
% Suppose the sensitivity of the U.S. Consumer Price Index (CPI) to changes
% in the paid compensation of employees (COE) is of interest.  This example
% enhances the analysis outlined in the example <docid:econ_ug.buicqm4-12>.

% Copyright 2015 The MathWorks, Inc.


%%
% Load the U.S. macroeconomic data set. 
load Data_USEconModel

%%
% The series are nonstationary.  Stabilize them by applying the log, and
% then the first difference.
CPI = diff(log(DataTable.CPIAUCSL));
COE = diff(log(DataTable.COE));

%%
% Regress |CPI| onto |COE| including an intercept to obtain OLS estimates.
% Plot correlograms for the residuals.
Mdl = fitlm(COE,CPI);
u = Mdl.Residuals.Raw;

figure;
subplot(2,1,1)
autocorr(u);
subplot(2,1,2);
parcorr(u);

%%
% The correlograms suggest that the innovations have significant AR
% effects. According to <docid:econ_ug.bs7alhb>, the innovations seem to
% comprise an AR(3) series.

%%
% Estimate the regression coefficients using FGLS.  By default, |fgls|
% assumes that the innovations are autoregressive.  Specify that the
% innovations are AR(3) using the |'arLags'| name-value pair argument.
[coeff,se] = fgls(CPI,COE,'arLags',3,'display','final');

%%
% If the COE series is exogenous with respect to the CPI, then the FGLS
% estimates (|coeff|) are consistent and asymptotically more efficient than 
% the OLS estimates.