sar_g.m

计量工具箱

% compute posterior means
beta = mean(bsave)';
rho = mean(psave);
sige = mean(ssave);
results.sige = sige;
vmean = in;
[nobs,nvar] = size(x);
if logmflag == 1
          AI = inv(x'*x + sige*TI);
          b0 = AI*(x'*y + sige*TIc);
          bd = AI*(x'*Wy + sige*TIc);
          e0 = y - x*b0;
          ed = Wy - x*bd;
          epe0 = e0'*e0;
          eped = ed'*ed;
          epe0d = ed'*e0;
logdetx = log(det(x'*x + sige*TI));
mlike = sar_marginal(detval,e0,ed,epe0,eped,epe0d,nobs,nvar,logdetx,a1,a2);
yhat = (speye(nobs) - rho*W)\(x*beta);
e = y - yhat;
elseif logmflag == 0
yhat = (speye(nobs) - rho*W)\(x*beta);
e = y - yhat;
end;

    
% compute R-squared
epe = e'*e;
sige = epe/(n-k);
results.sigma = sige;
ym = y - mean(y);
rsqr1 = epe;
rsqr2 = ym'*ym;
results.rsqr = 1.0 - rsqr1/rsqr2; % r-squared
rsqr1 = rsqr1/(nobs-nvar);
rsqr2 = rsqr2/(nobs-1.0);
results.rbar = 1 - (rsqr1/rsqr2); % rbar-squared

time = etime(clock,timet);

results.meth  = 'sar_g';
results.beta = beta;
results.rho = rho;
results.bdraw = bsave;
results.pdraw = psave;
results.sdraw = ssave;
if logmflag ==1
results.mlike = mlike;
end;
results.vmean = vmean;
results.yhat  = yhat;
results.resid = e;
results.bmean = c;
results.bstd  = sqrt(diag(T));
results.ndraw = ndraw;
results.nomit = nomit;
results.time  = time;
results.time1 = time1;
results.time2 = time2;
results.time3 = time3;
results.nu = nu;
results.d0 = d0;
results.a1 = a1;
results.a2 = a2;
results.tflag = 'plevel';
results.rmax = rmax; 
results.rmin = rmin;
results.lflag = ldetflag;
results.lndet = detval;
results.novi  = novi_flag;
results.priorb = inform_flag;

if mm~= 0
results.rdraw = rsave;
results.m     = mm;
results.k     = kk;
else
results.r     = rval;
results.rdraw = 0;
end;


otherwise
error('sar_g: unrecognized prior.novi_flag value on input');
% we should never get here

end; % end of homoscedastic vs. heteroscedastic vs. log-marginal options

% =========================================================================
% support functions below
% =========================================================================

function rho = draw_rho(detval,epe0,eped,epe0d,n,k,rho,a1,a2,logdetx)

nmk = (n-k)/2;
nrho = length(detval(:,1));
iota = ones(nrho,1);

z = epe0*iota - 2*detval(:,1)*epe0d + detval(:,1).*detval(:,1)*eped;
z = -nmk*log(z);
%C = gammaln(nmk)*iota -nmk*log(2*pi)*iota - 0.5*logdetx*iota;
den =  detval(:,2) + z;

bprior = beta_prior(detval(:,1),a1,a2);
den = den + log(bprior);
n = length(den);
y = detval(:,1);
adj = max(den);
den = den - adj;
x = exp(den);
% trapezoid rule
isum = sum((y(2:n,1) + y(1:n-1,1)).*(x(2:n,1) - x(1:n-1,1))/2);
z = abs(x/isum);
den = cumsum(z);

rnd = unif_rnd(1,0,1)*sum(z);
ind = find(den <= rnd);
idraw = max(ind);
if (idraw > 0 & idraw < nrho)
rho = detval(idraw,1);
end;

% To see how this works, uncomment the following lines
% plot(detval(:,1),den/1000,'-');
% line([detval(idraw,1) detval(idraw,1)],[0 den(idraw,1)/1000]);
% hold on;
% line([detval(idraw,1) 0],[den(idraw,1)/1000 den(idraw,1)/1000]);
% drawnow;
% pause;


function [nu,d0,rval,mm,kk,rho,sige,rmin,rmax,detval,ldetflag,eflag,order,iter,novi_flag,c,T,inform_flag,a1,a2,logmflag] = sar_parse(prior,k)
% PURPOSE: parses input arguments for sar_g models
% ---------------------------------------------------
%  USAGE: [nu,d0,rval,mm,kk,rho,sige,rmin,rmax,detval, ...
%         ldetflag,eflag,mflag,order,iter,novi_flag,c,T,inform_flag,a1,a2,logmflag = 
%                           sar_parse(prior,k)
% where info contains the structure variable with inputs 
% and the outputs are either user-inputs or default values
% ---------------------------------------------------

% set defaults

logmflag = 1;  % default to compute log-marginal
eflag = 0;     % default to not computing eigenvalues
ldetflag = 1;  % default to 1999 Pace and Barry MC determinant approx
mflag = 1;     % default to compute log marginal likelihood
order = 50;    % there are parameters used by the MC det approx
iter = 30;     % defaults based on Pace and Barry recommendation
rmin = -1;     % use -1,1 rho interval as default
rmax = 1;
detval = 0;    % just a flag
rho = 0.5;
sige = 1.0;
rval = 4;
mm = 0;
kk = 0;
nu = 0;
d0 = 0;
a1 = 1.0;
a2 = 1.0;
c = zeros(k,1);   % diffuse prior for beta
T = eye(k)*1e+12;
prior_beta = 0;   % flag for diffuse prior on beta
novi_flag = 0; % do vi-estimates
inform_flag = 0;

fields = fieldnames(prior);
nf = length(fields);
if nf > 0
 for i=1:nf
    if strcmp(fields{i},'nu')
        nu = prior.nu;
    elseif strcmp(fields{i},'d0')
        d0 = prior.d0;  
    elseif strcmp(fields{i},'rval')
        rval = prior.rval; 
    elseif strcmp(fields{i},'logm')
       logmflag = prior.logm; 
    elseif strcmp(fields{i},'a1')
       a1 = prior.a1; 
    elseif strcmp(fields{i},'a2')
       a2 = prior.a2; 
    elseif strcmp(fields{i},'m')
        mm = prior.m;
        kk = prior.k;
        rval = gamm_rnd(1,1,mm,kk);    % initial value for rval   
    elseif strcmp(fields{i},'beta')
        c = prior.beta; inform_flag = 1; % flag for informative prior on beta
    elseif strcmp(fields{i},'bcov')
        T = prior.bcov; inform_flag = 1;
    elseif strcmp(fields{i},'rmin')
        rmin = prior.rmin; eflag = 0;
    elseif strcmp(fields{i},'rmax')
        rmax = prior.rmax;  eflag = 0;
    elseif strcmp(fields{i},'lndet')
    detval = prior.lndet;
    ldetflag = -1;
    eflag = 0;
    rmin = detval(1,1);
    nr = length(detval);
    rmax = detval(nr,1);
    elseif strcmp(fields{i},'lflag')
        tst = prior.lflag;
        if tst == 0,
        ldetflag = 0; 
        elseif tst == 1,
        ldetflag = 1; 
        elseif tst == 2,
        ldetflag = 2; 
        else
        error('sar_g: unrecognizable lflag value on input');
        end;
    elseif strcmp(fields{i},'order')
        order = prior.order;  
    elseif strcmp(fields{i},'iter')
        iter = prior.iter; 
    elseif strcmp(fields{i},'novi')
        novi_flag = prior.novi;
    elseif strcmp(fields{i},'dflag')
        metflag = prior.dflag;
    elseif strcmp(fields{i},'eig')
        eflag = prior.eig;
    end;
 end;

 
else, % the user has input a blank info structure
      % so we use the defaults
end; 

function [rmin,rmax,time2] = sar_eigs(eflag,W,rmin,rmax,n);
% PURPOSE: compute the eigenvalues for the weight matrix
% ---------------------------------------------------
%  USAGE: [rmin,rmax,time2] = far_eigs(eflag,W,rmin,rmax,W)
% where eflag is an input flag, W is the weight matrix
%       rmin,rmax may be used as default outputs
% and the outputs are either user-inputs or default values
% ---------------------------------------------------


if eflag == 1 % compute eigenvalues
t0 = clock;
opt.tol = 1e-3; opt.disp = 0;
lambda = eigs(sparse(W),speye(n),1,'SR',opt);  
rmin = 1/real(lambda);   
rmax = 1;
time2 = etime(clock,t0);
else
time2 = 0;
end;


function [detval,time1] = sar_lndet(ldetflag,W,rmin,rmax,detval,order,iter);
% PURPOSE: compute the log determinant |I_n - rho*W|
% using the user-selected (or default) method
% ---------------------------------------------------
%  USAGE: detval = far_lndet(lflag,W,rmin,rmax)
% where eflag,rmin,rmax,W contains input flags 
% and the outputs are either user-inputs or default values
% ---------------------------------------------------


% do lndet approximation calculations if needed
if ldetflag == 0 % no approximation
t0 = clock;    
out = lndetfull(W,rmin,rmax);
time1 = etime(clock,t0);
tt=rmin:.001:rmax; % interpolate a finer grid
outi = interp1(out.rho,out.lndet,tt','spline');
detval = [tt' outi];
    
elseif ldetflag == 1 % use Pace and Barry, 1999 MC approximation

t0 = clock;    
out = lndetmc(order,iter,W,rmin,rmax);
time1 = etime(clock,t0);
results.limit = [out.rho out.lo95 out.lndet out.up95];
tt=rmin:.001:rmax; % interpolate a finer grid
outi = interp1(out.rho,out.lndet,tt','spline');
detval = [tt' outi];

elseif ldetflag == 2 % use Pace and Barry, 1998 spline interpolation

t0 = clock;
out = lndetint(W,rmin,rmax);
time1 = etime(clock,t0);
tt=rmin:.001:rmax; % interpolate a finer grid
outi = interp1(out.rho,out.lndet,tt','spline');
detval = [tt' outi];

elseif ldetflag == -1 % the user fed down a detval matrix
    time1 = 0;
        % check to see if this is right
        if detval == 0
            error('sar_g: wrong lndet input argument');
        end;
        [n1,n2] = size(detval);
        if n2 ~= 2
            error('sar_g: wrong sized lndet input argument');
        elseif n1 == 1
            error('sar_g: wrong sized lndet input argument');
        end;          
end;

function  out = sar_marginal(detval,e0,ed,epe0,eped,epe0d,nobs,nvar,logdetx,a1,a2)
% PURPOSE: returns a vector of the log-marginal over a grid of rho-values
% -------------------------------------------------------------------------
% USAGE: out = sar_marginal(detval,e0,ed,epe0,eped,epe0d,nobs,nvar,logdetx,a1,a2)
% where:       detval = an ngrid x 2 matrix with rho-values and lndet values
%                  e0 = y - x*b0;
%                 ed = Wy - x*bd;
%               epe0 = e0'*e0;
%               eped = ed'*ed;
%              epe0d = ed'*e0;
%               nobs = # of observations
%               nvar = # of explanatory variables
%            logdetx = log(det(x'*x))
%                 a1 = parameter for beta prior on rho
%                 a2 = parameter for beta prior on rho
% -------------------------------------------------------------------------
% RETURNS: out = a structure variable
%        out = log marginal, a vector the length of detval
% -------------------------------------------------------------------------

% written by:
% James P. LeSage, 7/2003
% Dept of Economics
% University of Toledo
% 2801 W. Bancroft St,
% Toledo, OH 43606
% [e-mail]

n = length(detval);
nmk = (nobs-nvar)/2;
% C is a constant of integration that can vary with nvars, so for model
% comparisions involving different nvars we need to include this
bprior = beta_prior(detval(:,1),a1,a2);
C = log(bprior) + gammaln(nmk) - nmk*log(2*pi) - 0.5*logdetx;
iota = ones(n,1);
z = epe0*iota - 2*detval(:,1)*epe0d + detval(:,1).*detval(:,1)*eped;
den = C + detval(:,2) - nmk*log(z);
out = real(den);

function  out = sar_marginal2(detval,e0,ed,epe0,eped,epe0d,nobs,nvar,a1,a2,c,TI,xs,ys,sige,W)
% PURPOSE: returns a vector of the log-marginal over a grid of rho-values
%          for the case of an informative prior on beta
% -------------------------------------------------------------------------
% USAGE: out = sar_marginal(detval,e0,ed,epe0,eped,epe0d,nobs,nvar,logdetx,a1,a2)
% where:       detval = an ngrid x 2 matrix with rho-values and lndet values
%                  e0 = y - x*b0;
%                 ed = Wy - x*bd;
%               epe0 = e0'*e0;
%               eped = ed'*ed;
%              epe0d = ed'*e0;
%               nobs = # of observations
%               nvar = # of explanatory variables
%            logdetx = log(det(x'*x))
%                 a1 = parameter for beta prior on rho
%                 a2 = parameter for beta prior on rho
%                 c = prior mean for beta
%                TI = prior var-cov for beta
%                xs = x*sqrt(V) or x if homoscedastic model
%                ys = y*sqrt(V) or y is homoscedastic model
% -------------------------------------------------------------------------
% RETURNS: out = a structure variable
%        out = log marginal, a vector the length of detval
% -------------------------------------------------------------------------

% written by:
% James P. LeSage, 7/2003
% Dept of Economics
% University of Toledo
% 2801 W. Bancroft St,
% Toledo, OH 43606
% [e-mail]

n = length(detval);
nmk = (nobs-nvar)/2;
% C is a constant of integration that can vary with nvars, so for model
% comparisions involving different nvars we need to include this
bprior = beta_prior(detval(:,1),a1,a2);
C = log(bprior) + gammaln(nmk) - nmk*log(2*pi);
iota = ones(n,1);
z = epe0*iota - 2*detval(:,1)*epe0d + detval(:,1).*detval(:,1)*eped;
% add quadratic terms based on prior for beta
Q1 = zeros(n,1);
Q2 = zeros(n,1);
xpxi = inv(xs'*xs);
sTI = sige*TI;
xpxis = inv(xs'*xs + sTI);
logdetx = log(det(xpxis));
C = C - 0.5*logdetx;
          for i=1:n;
           rho = detval(i,1);
           D = speye(nobs) - rho*W;
           bhat = xpxi*(xs'*D*ys);
           beta = xpxis*(xs'*D*ys + sTI*c); 
           Q1(i,1) = (c - beta)'*sTI*(c - beta);
           Q2(i,1) = (bhat - beta)'*(xs'*xs)*(bhat - beta);
          end;

den = C + detval(:,2) - nmk*log(z + Q1 + Q2);
out = real(den);