A list,
VAR: Vector autoregressive model. The results are statistically significant only
SVAR: Structure vector autoregression model
Tvp-var: Time Varying parameters-stochastic Volatility-Vector Auto Regression. The time-varying parameter random volatility vector autoregressive model is different from VAR in that the model does not have the assumption of homoscedasticity and is more consistent with reality. Moreover, the time-varying parameter assumes random volatility, which can better capture the relationship and characteristics of economic variables in different historical backgrounds (time-varying influence). Incorporating random volatility into TVP estimation can significantly improve the estimation performance. In the VAR model, all parameters follow the first-order wandering process.
The concept of random fluctuation is very important in TVP-VAR. Random fluctuation was put forward by Black in 1976, and then has great development in economic measurement. In recent years, random fluctuations are often used in macroeconomic empirical analysis. In many cases, the generation of economic data has the drift coefficient and random fluctuations in the process of impact, if this is the case, then using model with time-varying coefficients but with constant volatility will cause a problem, that is due to ignore the possibility of disturbance in the volatility changes, estimated the time-varying coefficient of deviation may occur. To avoid this problem, the TVP-VAR model assumes random volatility. Although likelihood functions are difficult to handle, so random volatility makes estimation difficult, models can be estimated using markov chain Monte Carlo (MCMC) methods in Bayesian inference.
Starting from the estimation algorithm of TVP regression model with random fluctuation (the univariate case of TVP-VAR model).
For MCMC algorithm, PI (theta) is a prior density, posterior distribution of PI (theta and alpha, h | y) ^ 4.
Iv. Data stability is also required for TVP-VAR modeling. ADF unit root test can be used to test the stationarity of data. The uneven data can be differentiated until it is stable, and the data after difference can be used for modeling.
It should be noted that in tVP-VAR modeling, the order between variables will affect the final empirical results. The order of variables may be one reason why the empirical results do not conform to expectations. It is best to focus on the variables in the first place so that they will be shown better in the time-varying diagram.
Ii. Source code
% = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = = =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) * * VB_OLS tau. 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) == 0disp(irep); toc; end %% Draw1: 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')
Copy the code3. Operation results
Fourth, note
Version: 2014 a