cointegrationTests.m

clear;

% 1 minute data on EWA-EWC
load('inputData_ETF', 'tday', 'syms', 'cl');
idxA=find(strcmp('EWA', syms));
idxC=find(strcmp('EWC', syms));
disp(idxA); disp(idxC);

x=cl(:, idxA);
y=cl(:, idxC);

plot(x);
hold on;
plot(y, 'g');

legend('EWA', 'EWC');
figure;

scatter(x, y);

figure;

regression_result=ols(y, [x ones(size(x))]);
hedgeRatio=regression_result.beta(1);
disp(hedgeRatio);

plot(y-hedgeRatio*x);

% Assume a non-zero offset but no drift, with lag=1.
results=cadf(y, x, 0, 1); % cadf is a function in the jplv7 (spatial-econometrics.com) package. We pick y to be the dependent variable again.

% Print out results
prt(results);

% Output:
%  Augmented DF test for co-integration variables:      variable   1,variable   2  
% CADF t-statistic        # of lags   AR(1) estimate 
%      -3.64346635                1        -0.020411 
% 
%    1% Crit Value    5% Crit Value   10% Crit Value 
%           -3.880           -3.359           -3.038 

% Combine the two time series into a matrix y2 for input into Johansen test
y2=[y, x];
results=johansen(y2, 0, 1); % johansen test with non-zero offset but zero drift, and with the lag k=1.

% Print out results
prt(results);

% Output:
%  Johansen MLE estimates 
% NULL:                  Trace Statistic         Crit 90%         Crit 95%         Crit 99% 
% r <= 0   variable   1           19.983           13.429           15.494           19.935 
% r <= 1   variable   2            3.983            2.705            3.841            6.635 
% 
% NULL:                  Eigen Statistic         Crit 90%         Crit 95%         Crit 99% 
% r <= 0   variable   1           16.000           12.297           14.264           18.520 
% r <= 1   variable   2            3.983            2.705            3.841            6.635 


% Adding IGE to the portfolio

idxI=find(strcmp('IGE', syms));
z=cl(:, idxI);
y3=[y2, z];

results=johansen(y3, 0, 1); % johansen test with non-zero offset but zero drift, and with the lag k=1.

% Print out results
prt(results);

% Output:
%  Johansen MLE estimates 
% NULL:                  Trace Statistic         Crit 90%         Crit 95%         Crit 99% 
% r <= 0   variable   1           34.429           27.067           29.796           35.463 
% r <= 1   variable   2           17.532           13.429           15.494           19.935 
% r <= 2   variable   3            4.471            2.705            3.841            6.635 
% 
% NULL:                  Eigen Statistic         Crit 90%         Crit 95%         Crit 99% 
% r <= 0   variable   1           16.897           18.893           21.131           25.865 
% r <= 1   variable   2           13.061           12.297           14.264           18.520 
% r <= 2   variable   3            4.471            2.705            3.841            6.635 

results.eig % Display the eigenvalues

% ans =
% 
%     0.0112
%     0.0087
%     0.0030
    
results.evec % Display the eigenvectors

% ans =
% 
%    -1.0460   -0.5797   -0.2647
%     0.7600   -0.1120   -0.0790
%     0.2233    0.5316    0.0952
    
yport=sum(repmat(results.evec(:, 1)', [size(y3, 1) 1]).*y3, 2); % (net) market value of portfolio

% Find value of lambda and thus the halflife of mean reversion by linear regression fit
ylag=lag(yport, 1);  % lag is a function in the jplv7 (spatial-econometrics.com) package.
deltaY=yport-ylag;
deltaY(1)=[]; % Regression functions cannot handle the NaN in the first bar of the time series.
ylag(1)=[];
regress_results=ols(deltaY, [ylag ones(size(ylag))]); % ols is a function in the jplv7 (spatial-econometrics.com) package.
halflife=-log(2)/regress_results.beta(1);

fprintf(1, 'halflife=%f days\n', halflife);

% halflife=22.662578  days
% 
%  Apply a simple linear mean reversion strategy to EWA-EWC-IGE
lookback=round(halflife); % setting lookback to the halflife found above

numUnits =-(yport-movingAvg(yport, lookback))./movingStd(yport, lookback); % capital invested in portfolio in dollars.  movingAvg and movingStd are functions from epchan.com/book2
positions=repmat(numUnits, [1 size(y3, 2)]).*repmat(results.evec(:, 1)', [size(y3, 1) 1]).*y3; % results.evec(:, 1)' can be viewed as the capital allocation, while positions is the dollar capital in each ETF.
pnl=sum(lag(positions, 1).*(y3-lag(y3, 1))./lag(y3, 1), 2); % daily P&L of the strategy
ret=pnl./sum(abs(lag(positions, 1)), 2); % return is P&L divided by gross market value of portfolio
ret(isnan(ret))=0;

figure;
plot(cumprod(1+ret)-1); % Cumulative compounded return

fprintf(1, 'APR=%f Sharpe=%f\n', prod(1+ret).^(252/length(ret))-1, sqrt(252)*mean(ret)/std(ret));
% APR=0.125739 Sharpe=1.391310