一、简介
VAR:向量自回归模型，结果仅具有统计上的 意义
SVAR：结构向量自回归模型
TVP-VAR：Time Varying Parameter-Stochastic Volatility-Vector Auto Regression。时变参数随机波动率向量自回归模型，与VAR 不同的是，模型没有同方差的假定，更符合实际。并且时变参数假定随机波动率，更能捕捉到经济变量在不同时代背景下所具有的关系和特征（时变影响）。将随机波动性纳入TVP估算中可以显着提高估算性能。在VAR模型中，所有的参数遵循一阶游走过程。
随机波动的概念在TVP-VAR中很重要，随机波动是1976年由Black提出，随后在经济计量中有很大的发展。近几年，随机波动也经常被用在宏观经济的经验分析中。很多情况下，经济数据的产生过程中具有漂移系数和随机波动的冲击，如果是这种情况，那么使用具有时变系数但具有恒定波动性的模型会引起一个问题，即由于忽略了扰动中波动性的可能变化，估计的时变系数可能会出现偏差。为了避免这种问题，TVP-VAR模型假设了随机波动性。尽管似然函数难以处理，所以随机波动率使估计变得困难，但是可以在贝叶斯推断中使用马尔可夫链蒙特卡罗（MCMC）方法来估计模型。
从 具有随机波动性的时变参数（TVP）回归模型的估计算法(TVP-VAR模型的单变量情况)说起。
对于MCMC算法，π(θ)为先验密度，后验分布 π(θ,α,h|y）^4。
四、进行TVP-VAR建模时，也需要数据平稳。可以用ADF单位根检验法检验数据的平稳性，不平稳的数据可以做差分，直至平稳，用差分后的数据进行建模。
需要注意的是：TVP-VAR建模时，变量之间的顺序会影响到最后的实证结果。变量的顺序问题可能是实证结果不符合预期的一个原因。最好把关注的变量放在首位，这样在时变关系图中就能得到较好的展示。
二、源代码
%==================================4.1 Main================================
clear;
clc;
% Load Korobilis (2008) quarterly data
load ydata.dat; % data
load yearlab.dat; % data labels
%%
%----------------------------------BASICS----------------------------------
Y=ydata;
t=size(Y,1); % t - The total number of periods in the raw data (t=215)
M=size(Y,2); % M - The dimensionality of Y (i.e. the number of variables)(M=3)
tau = 40; % tau - the size of the training sample (the first forty quarters)
p = 2; % p - number of lags in the VAR model
%% Generate the Z_t matrix, i.e. the regressors in the model.
ylag = mlag2(Y,p); % This function generates a 215x6 matrix with p lags of variable Y.
ylag = ylag(p+tau+1:t,:); % Then remove our training sample, so now a 173x6 matrix.
K = M + p*(M^2); % K is the number of elements in the state vector
% Here we distribute the lagged y data into the Z matrix so it is
% conformable with a beta_t matrix of coefficients.
Z = zeros((t-tau-p)*M,K);
for i = 1:t-tau-p
ztemp = eye(M);
for j = 1:p
xtemp = ylag(i,(j-1)*M+1:j*M);
xtemp = kron(eye(M),xtemp);
ztemp = [ztemp xtemp];
end
Z((i-1)*M+1:i*M,:) = ztemp;
end
% Redefine our variables to exclude the training sample and the first two
% lags that we take as given, taking total number of periods (t) from 215
% to 173.
y = Y(tau+p+1:t,:)';
yearlab = yearlab(tau+p+1:t);
t=size(y,2); % t now equals 173
%% --------------------MODEL AND GIBBS PRELIMINARIES-----------------------
nrep = 5000; % Number of sample draws
nburn = 2000; % Number of burn-in-draws
it_print = 100; % Print in the screen every "it_print"-th iteration
%% INITIAL STATE VECTOR PRIOR
% We use the first 40 observations (tau) to run a standard OLS of the
% measurement equation, using the function ts_prior. The result is
% estimates for priors for B_0 and Var(B_0).
[B_OLS,VB_OLS]= ts_prior(Y,tau,M,p);
% Given the distributions we have, we now have to define our priors for B,
% Q and Sigma. These are set in accordance with how they are set in
% Primiceri (2005). These are the hyperparameters of the beta, Q and Sigma
% initial priors.
B_0_prmean = B_OLS;
B_0_prvar = 4*VB_OLS;
Q_prmean = ((0.01).^2)*tau*VB_OLS;
Q_prvar = tau;
Sigma_prmean = eye(M);
Sigma_prvar = M+1;
% To start the Kalman filtering assign arbitrary values that are in support
% of their priors, Q and Sigma.
consQ = 0.0001;
Qdraw = consQ*eye(K);
Sigmadraw = 0.1*eye(M);
% Create some matrices for storage that will be filled in once we
% start the Gibbs sampling.
Btdraw = zeros(K,t);
Bt_postmean = zeros(K,t);
Qmean = zeros(K,K);
Sigmamean = zeros(M,M);
%% -------------------------IRF-PRELIMINARIES------------------------------
nhor = 21; % The number of periods in the impulse response function.
% Matricies to be filled containing IRFs for 1975q1, 1981q3, 1996q1. The
% dimensions correspond to the iterations of the gibbs sample, each of the
% variables, and each of the 21 periods of the IRF analysis.
imp75 = zeros(nrep,M,nhor);
imp81 = zeros(nrep,M,nhor);
imp96 = zeros(nrep,M,nhor);
% This corresponds to variable J introduced in equation (14) in the report
bigj = zeros(M,M*p);
bigj(1:M,1:M) = eye(M);
%% ================ START GIBBS SAMPLING ==================================
tic; % This is just a timer
disp('Number of iterations');
for irep = 1:nrep + nburn % 7000 gibbs iterations starts here
% Print iterations - this just updates on the progress of the sampling
if mod(irep,it_print) == 0
disp(irep);toc;
end
%% Draw 1: B_t from p(B_t|y,Sigma)
% We use the function 'carter_kohn_hom' to to run the FFBS algorithm.
% This results in a 21x173 matrix, corresponding to one Gibbs sample
% draw of each of the coefficients in each time period. The inputs
% Sigmadraw and Qdraw are updated for each Gibbs sample repetition.
[Btdraw] = carter_kohn_hom(y,Z,Sigmadraw,Qdraw,K,M,t,B_0_prmean,B_0_prvar);
%% Draw 2: Q from p(Q^{
-1}|y,B_t) which is i-Wishart
% We draw Q from an Inverse Wishart distribution. The parameters
% of the distribution are derived as equation (11) in the main report.
% The mean is taken as the inverse of the accumulated sum of squared
% errors added to the prior mean, and the variance is simply t.
% Differencing Btdraw to create the sum of squared errors
Btemp = Btdraw(:,2:t)' - Btdraw(:,1:t-1)';
sse_2Q = zeros(K,K);
for i = 1:t-1
sse_2Q = sse_2Q + Btemp(i,:)'*Btemp(i,:);
end
Qinv = inv(sse_2Q + Q_prmean); % compute mean to use for Wishart draw
Qinvdraw = wish(Qinv,t+Q_prvar); % draw inv q from the wishart distribution
Qdraw = inv(Qinvdraw); % find non-inverse q
%% Draw 3: Sigma from p(Sigma|y,B_t) which is i-Wishart
% We draw Sigma from an Inverse Wishart distribution. The parameters
% of the distirbution are derived as equation (10) in the main report.
% The mean is taken as the inverse of the sum of squared residuals
% added to the prior mean. The variance is simply t.
% Find residuals using data and the current draw of coefficients
resids = zeros(M,t);
for i = 1:t
resids(:,i) = y(:,i) - Z((i-1)*M+1:i*M,:)*Btdraw(:,i);
end
% Create a matrix for the accumulated sum of squared residuals, to
% be used as the mean parameter in the i-Wishart draw below.
sse_2S = zeros(M,M);
for i = 1:t
sse_2S = sse_2S + resids(:,i)*resids(:,i)';
end
Sigmainv = inv(sse_2S + Sigma_prmean); % compute mean to use for the Wishart
Sigmainvdraw = wish(Sigmainv,t+Sigma_prvar); % draw from the Wishsart distribution
Sigmadraw = inv(Sigmainvdraw); % turn into non-inverse Sigma
%% IRF
% We only apply IRF analysis once we have exceeded the burn-in draws phase.
if irep > nburn;
% Create matrix that is going to contain all beta draws over
% which we will take the mean after the Gibbs sampler as our moment
% estimate:
Bt_postmean = Bt_postmean + Btdraw;
% biga is the A matrix of the VAR(1) version of our VAR(2) model,
% found in equation (12. biga changes in every period of the
% analysis, because the coefficients are time varying, so we
% apply the analysis below in every time period.
biga = zeros(M*p,M*p);
for j = 1:p-1
biga(j*M+1:M*(j+1),M*(j-1)+1:j*M) = eye(M); % fill the A matrix with identity matrix (3) in bottom left corner
end
% The following procedure is applied separately in each time period.
% This loop takes coefficients of the relevant time period from
% Bt_draw (which contains all coefficients for all t) and uses
% them to update the biga matrix, so that it can change for
% every t.
for i = 1:t
bbtemp = Btdraw(M+1:K,i); % get the draw of B(t) at time i=1,...,T (exclude intercept)
splace = 0;
for ii = 1:p
for iii = 1:M
biga(iii,(ii-1)*M+1:ii*M) = bbtemp(splace+1:splace+M,1)'; % Load non-intercept coefficient draws
splace = splace + M;
end
end
% Next we want to create a shock matrix in which the third
% column is [0 0 1]', therefore implementing a unit shock
% in the interest rate.
shock = eye(3);
% Now get impulse responses for 1 through nhor future
% periods. impresp is a 3x63 matrix which contains 9
% response values in total for each period, 3 for each
% variable. These three responses correspond to the 3
% possible shocks that are contained in the schock
% matrix
% bigai is updated through mulitiplication with the
% coefficient matrix after each time period.
% Create a results matrix to store impulse responses in all periods
impresp = zeros(M,M*nhor);
% Fill in the first period of the results matrix with the shock (as defined above)
impresp(1:M,1:M) = shock;
% Create a separate variable for the a matrix so that we
% can update it for each period of the IRF analysis.
bigai = biga;
% This follows the impulse response function as in equation 15.
% Fill in each period of the results matrix according to
% the impulse response function formula.
for j = 1:nhor-1
impresp(:,j*M+1:(j+1)*M) = bigj*bigai*bigj'*shock;
bigai = bigai*biga; % update the coefficient matrix for next period
end
% The section below keeps only the responses that we are interested in:
% - those from the periods 1975q1, 1981q3, and 1996q1
% - those that correspond to the shock in the interest
% rate (i.e. those caused by the third column of our shock
% matrix).
if yearlab(i,1) == 1975.00; % store only IRF from 1975:Q1
impf_m = zeros(M,nhor);
jj=0;
for ij = 1:nhor
jj = jj + M; % select only the third column for each time period of the IRF
impf_m(:,ij) = impresp(:,jj);
end
% For each iteration of the Gibbs sampler, fill in the
% results along the first dimension
imp75(irep-nburn,:,:) = impf_m;
end
if yearlab(i,1) == 1981.50; % store only IRF from 1981:Q3
impf_m = zeros(M,nhor);
jj=0;
for ij = 1:nhor
jj = jj + M; % select only the third column for each time period of the IRF
impf_m(:,ij) = impresp(:,jj);
end
% For each iteration of the Gibbs sample, fill in the
% results along the first dimension
imp81(irep-nburn,:,:) = impf_m;
end
if yearlab(i,1) == 1996.00; % store only IRF from 1996:Q1
impf_m = zeros(M,nhor);
jj=0;
for ij = 1:nhor
jj = jj + M; % select only the third column for each time period of the IRF
impf_m(:,ij) = impresp(:,jj);
end
% For each iteration of the Gibbs sample, fill in the
% results along the first dimension
imp96(irep-nburn,:,:) = impf_m;
end
end % End getting impulses for each time period
end % End the impulse response calculation section
end % End main Gibbs loop (for irep = 1:nrep+nburn)
clc;
toc; % Stop timer and print total time
%% ================ END GIBBS SAMPLING ==================================
% Even though it is not used in our IRF analysis since we are integrating
% that into the Gibbs sampling loop, here is how to take the mean of the
% draw of the betas as moment estimate:
Bt_postmean = Bt_postmean./nrep;
%% Graphs and tables
% This works out the percentage range of that each variable's coefficient
% spans across time. I.e. to what extent was the TVP facility used by each
% variable in the model? This is calculated by finding the range for each
% variable as a percentage of the mean coefficient size for that variable.
% The result is a 21x1 vector, and it is reshaped into a 3x7 matrix in order
% to map onto the system of equations (2,3,and 4) set out in the report.
Bt_range = ones(21,1)
for i = 1:21
Bt_range(i) = abs((max(Bt_postmean(i,:))-min(Bt_postmean(i,:)))/mean(Bt_postmean(i,:)))
end
Bt_range = reshape(Bt_range,3,7)
% Create a table of coefficient ranges for export to the report
rowNames = {
'Inflation','Unemployment','Interest Rate'};
colNames = {
'Intercept','Inf_1','Unemp_1', 'IR_1','Inf_2','Unemp_2', 'IR_2'};
pc_change_table = array2table(Bt_range,'RowNames',rowNames,'VariableNames',colNames)
writetable(pc_change_table,'pc_change.csv')
% Now plot a separate chart for each of the coefficients
figure
for i = 1:21
subplot(7,3,i)
plot(1:t,Bt_postmean(i,:))
end
% Now we move to plotting the IRF. This section takes moments along the
% first dimension, i.e. across the Gibbs sample iterations. The moments
% are for the 16th, 50th and 84th percentile.
qus = [.16, .5, .84];
imp75XY=squeeze(quantile(imp75,qus));
imp81XY=squeeze(quantile(imp81,qus));
imp96XY=squeeze(quantile(imp96,qus));
% Plot impulse responses
figure
set(0,'DefaultAxesColorOrder',[0 0 0],...
'DefaultAxesLineStyleOrder','--|-|--')
subplot(3,3,1)
plot(1:nhor,squeeze(imp75XY(:,1,:)))
title('Impulse response of inflation, 1975:Q1')
xlim([1 nhor])
ylim([-0.2 0.1])
% % yline(0)
set(gca,'XTick',0:3:nhor)
subplot(3,3,2)
plot(1:nhor,squeeze(imp75XY(:,2,:)))
title('Impulse response of unemployment, 1975:Q1')
xlim([1 nhor])
ylim([-0.2 0.2])
% yline(0)
set(gca,'XTick',0:3:nhor)
subplot(3,3,3)
%ylim([0 1])
% % yline(0)
plot(1:nhor,squeeze(imp75XY(:,3,:)))
title('Impulse response of interest rate, 1975:Q1')
xlim([1 nhor])
%ylim([-0.3 0.1])
% yline(0)
set(gca,'XTick',0:3:nhor)
subplot(3,3,4)
plot(1:nhor,squeeze(imp81XY(:,1,:)))
title('Impulse response of inflation, 1981:Q3')
xlim([1 nhor])
ylim([-0.2 0.1])
% yline(0)
set(gca,'XTick',0:3:nhor)
subplot(3,3,5)
plot(1:nhor,squeeze(imp81XY(:,2,:)))
title('Impulse response of unemployment, 1981:Q3')
xlim([1 nhor])
ylim([-0.2 0.2])
% yline(0)
set(gca,'XTick',0:3:nhor)
subplot(3,3,6)
plot(1:nhor,squeeze(imp81XY(:,3,:)))
title('Impulse response of interest rate, 1981:Q3')
xlim([1 nhor])
%ylim([-0.4 0.1])
% yline(0)
set(gca,'XTick',0:3:nhor)
subplot(3,3,7)
plot(1:nhor,squeeze(imp96XY(:,1,:)))
title('Impulse response of inflation, 1996:Q1')
xlim([1 nhor])
ylim([-0.2 0.1])
% yline(0)
set(gca,'XTick',0:3:nhor)
subplot(3,3,8)
plot(1:nhor,squeeze(imp96XY(:,2,:)))
title('Impulse response of unemployment, 1996:Q1')
xlim([1 nhor])
ylim([-0.2 0.2])
% yline(0)
set(gca,'XTick',0:3:nhor)
subplot(3,3,9)
plot(1:nhor,squeeze(imp96XY(:,3,:)))
title('Impulse response of interest rate, 1996:Q1')
xlim([1 nhor])
%ylim([0 1])
% yline(0)
set(gca,'XTick',0:3:nhor)
disp(' ')
disp('To plot impulse responses, use: plot(1:nhor,squeeze(imp75XY(:,VAR,:))) ')
disp(' ')
disp('where VAR=1 for impulses of inflation, VAR=2 for unemployment and VAR=3 for interest rate')
三、运行结果
四、备注
完整代码或者代写添加QQ2449341593
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【数据分析】基于matlab时变参数随机波动率向量自回归模型（TVP-VAR）【含Matlab源码 099期】