 # Chapter 3. Multivariate Volatility Models (in MATLAB/Julia)

Copyright 2011 - 2019 Jon Danielsson. This code is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version. This code is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. The GNU General Public License is available at: https://www.gnu.org/licenses/.

The GARCH functionality in the econometric toolbox in Matlab cannot do univariate GARCH.

Kevin Sheppard's MFE toolbox is much better, while not as user friendly, it is much better written and is certainly more comprehensive. It can be downloaded here and the documentation here is quite detailed.

##### Listing 3.1/3.2: Download stock prices in MATLAB Last updated August 2016

p = p(:,[1,2]);                % consider first two stocks
y = diff(log(p))*100;          % convert prices to returns
y(:,1)=y(:,1)-mean(y(:,1));    % subtract mean
y(:,2)=y(:,2)-mean(y(:,2));
T = length(y);

##### Listing 3.1/3.2: Download stock prices in Julia Last updated June 2018

using CSV;
y1 = diff(log.(p[:,1])).*100;              # consider first two stocks
y2 = diff(log.(p[:,2])).*100;              # convert prices to returns
y1 = y1-mean(y1);                          # subtract mean
y2 = y2-mean(y2);
y = hcat(y1,y2);                           # combine both series horizontally
T = size(y,1);                             # get the height of timeseries


##### Listing 3.3/3.4: EWMA in MATLAB Last updated June 2018

%% create a matrix to hold covariance matrix for each t
EWMA = nan(T,3);
lambda = 0.94;
S = cov(y);                                      % initial (t=1) covar matrix
EWMA(1,:) = S([1,4,2]);                          % extract var and covar
for i = 2:T                                      % loop though the sample
S = lambda*S+(1-lambda)* y(i-1,:)'*y(i-1,:);
EWMA(i,:) = S([1,4,2]);                      % convert matrix to vector
end
EWMArho = EWMA(:,3)./sqrt(EWMA(:,1).*EWMA(:,2)); % calculate correlations

##### Listing 3.3/3.4: EWMA in Julia Last updated June 2018

## create a matrix to hold covariance matrix for each t
EWMA = fill!(Array{Float64}(T,3), NaN)
lambda = 0.94
S = cov(y)                                       # initial (t=1) covar matrix
EWMA[1,:] = [S, S, S]                   # extract var and covar
for i in range(2,T-1)                            # loop though the sample
S = lambda*S+(1-lambda)*y[i-1,:]*(y[i-1,:])'
EWMA[i,:] = [S, S, S]               # convert matrix to vector
end
EWMArho = EWMA[:,3]./sqrt.(EWMA[:,1].*EWMA[:,2]) # calculate correlations


##### Listing 3.5/3.6: OGARCH in MATLAB Last updated August 2016

[par, Ht] = o_mvgarch(y,2, 1,1,1);
Ht = reshape(Ht,4,T)';
%% Ht comes from o_mvgarch as a 3D matrix, this transforms it into a 2D matrix
OOrho = Ht(:,3) ./ sqrt(Ht(:,1) .* Ht(:,4));
%% OOrho is a vector of correlations

##### Listing 3.5/3.6: OGARCH in Julia Last updated June 2018

## No OGARCH code available in Julia at present


##### Listing 3.7/3.8: DCC in MATLAB Last updated August 2016

[p, lik, Ht] = dcc(y,1,1,1,1);
Ht = reshape(Ht,4,T)';
DCCrho = Ht(:,3) ./ sqrt(Ht(:,1) .* Ht(:,4));
%% DCCrho is a vector of correlations

##### Listing 3.7/3.8: DCC in Julia Last updated June 2018

## No DCC code available in Julia at present


##### Listing 3.9/3.10: Correlation comparison in MATLAB Last updated June 2018

plot([EWMArho,OOrho,DCCrho])
legend('EWMA','DCC','OGARCH','Location','SouthWest')

##### Listing 3.9/3.10: Correlation comparison in Julia Last updated June 2018

## No OGARCH/DCC code available in Julia at present