Nonlinear Granger Causality Test Using Markov Chain in MATLAB: Code Example

%25%20%E7%94%9F%E6%88%90%E6%A8%A1%E6%95%B0%E6%95%B0%E6%8D%AE%0A%25T%20%3D%201000%3B%20%25%20%E6%95%B0%E6%8D%AE%E9%95%BF%E5%BA%A6%0A%25x%20%3D%20zeros(T%2C%201)%3B%0A%25y%20%3D%20zeros(T%2C%201)%3B%0A%25epsilon%20%3D%20randn(T%2C%201)%3B%0A%0A%25for%20t%20%3D%203%3AT%0A%25%20%20x(t)%20%3D%200.5x(t-1)%20%2B%200.25x(t-2)%20%2B%20epsilon(t)%3B%0A%25%20%20y(t)%20%3D%200.5y(t-1)%20%2B%200.25y(t-2)%20%2B%20epsilon(t)%3B%0A%25end%0A%0A%25%20%E8%AE%BE%E7%BD%AE%E6%A8%A1%E5%9E%8B%E5%8F%82%E6%95%B0%0A%25p%20%3D%202%3B%20%25%20%E6%97%A7%E5%90%8E%E9%98%B6%E6%95%B0%0A%25q%20%3D%202%3B%20%25%20%E7%8A%B6%E6%80%81%E9%98%B6%E6%95%B0%0A%0A%25%20%E6%9E%84%E5%BB%BA%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%E7%9F%A9%E9%98%B6%0A%25A%20%3D%20zeros(2^q%2C%202^q)%3B%0A%25A(1%3A2^(q-1)%2C%201%3A2^(q-1))%20%3D%20eye(2^(q-1))%3B%0A%25A(2^(q-1)%2B1%3Aend%2C%201%3A2^(q-1))%20%3D%20eye(2^(q-1))%3B%0A%25A(2^(q-1)%2B1%3Aend%2C%202^(q-1)%2B1%3Aend)%20%3D%20eye(2^(q-1))%3B%0A%0A%25%20%E6%9E%84%E5%BB%BA%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%E6%A6%82%E7%8E%87%E7%9F%A9%E9%98%B6%0A%25P%20%3D%20ones(2^q%2C%202^q)%20/%202^q%3B%0A%0A%25%20%E4%BC%8F%E8%AE%A1%E6%A8%A1%E5%9E%8B%E5%8F%82%E6%95%B0%0A%25theta_x%20%3D%20zeros(p%2C%202^q)%3B%20%25%20x%E7%9A%84%E6%A8%A1%E5%9E%8B%E5%8F%82%E6%95%B0%0A%25theta_y%20%3D%20zeros(p%2C%202^q)%3B%20%25%20y%E7%9A%84%E6%A8%A1%E5%9E%8B%E5%8F%82%E6%95%B0%0A%0A%25for%20s%20%3D%201%3A2^q%0A%25%20%20X%20%3D%20zeros(T-p%2C%20p)%3B%0A%25%20%20Y%20%3D%20zeros(T-p%2C%20p)%3B%0A%25%20%20for%20t%20%3D%20p%2B1%3AT%0A%25%20%20%20%20X(t-p%2C%20:)%20%3D%20x(t-1%3A-1%3At-p)'%3B%0A%25%20%20%20%20Y(t-p%2C%20:)%20%3D%20y(t-1%3A-1%3At-p)'%3B%0A%25%20%20end%0A%0A%25%20%20%25%20%E4%BD%BF%E7%94%A8OLS%E4%BC%8F%E8%AE%A1%E6%A8%A1%E5%9E%8B%E5%8F%82%E6%95%B0%0A%25%20%20theta_x(:,%20s)%20%3D%20(X'%20*%20X)%20(%20(X'%20*%20x(p%2B1%3AT)%20-%20X'%20*%20A(s%2C%201%3A2^(q-1))'%20*%20x(p%2B1%3AT))%3B%0A%25%20%20theta_y(:,%20s)%20%3D%20(Y'%20*%20Y)%20(%20(Y'%20*%20y(p%2B1%3AT)%20-%20Y'%20*%20A(s%2C%202^(q-1)%2B1%3Aend)'%20*%20y(p%2B1%3AT))%3B%0A%25end%0A%0A%25%20%E8%AE%A1%E7%AE%97%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%E6%A6%82%E7%8E%87%0A%25pi_x%20%3D%20zeros(2^q%2C%20T)%3B%0A%25pi_y%20%3D%20zeros(2^q%2C%20T)%3B%0A%25pi_x(:,%201)%20%3D%20P(:,%201)%3B%0A%25pi_y(:,%201)%20%3D%20P(:,%201)%3B%0A%0A%25for%20t%20%3D%202%3AT%0A%25%20%20pi_x(:,%20t)%20%3D%20P'%20*%20pi_x(:,%20t-1)%3B%0A%25%20%20pi_y(:,%20t)%20%3D%20P'%20*%20pi_y(:,%20t-1)%3B%0A%25end%0A%0A%25%20%E8%AE%A1%E7%AE%97Granger%E5%9B%A0%E6%9E%9C%E6%80%A7%E6%8C%87%E6%95%B0%0A%25GC_x_to_y%20%3D%20zeros(2^q%2C%202^q)%3B%20%25%20x%20-%3E%20y%E7%9A%84Granger%E5%9B%A0%E6%9E%9C%E6%80%A7%E6%8C%87%E6%95%B0%E7%9F%A9%E9%98%B6%0A%25GC_y_to_x%20%3D%20zeros(2^q%2C%202^q)%3B%20%25%20y%20-%3E%20x%E7%9A%84Granger%E5%9B%A0%E6%9E%9C%E6%80%A7%E6%8C%87%E6%95%B0%E7%9F%A9%E9%98%B6%0A%0A%25for%20s1%20%3D%201%3A2^q%0A%25%20%20for%20s2%20%3D%201%3A2^q%0A%25%20%20%20%20GC_x_to_y(s1%2C%20s2)%20%3D%20log(pi_y(s2%2C%20end)%20/%20pi_y(s2%2C%201))%20-%20log(pi_y(s2%2C%20end)%20/%20pi_y(s2%2C%201))%3B%0A%25%20%20%20%20GC_y_to_x(s1%2C%20s2)%20%3D%20log(pi_x(s2%2C%20end)%20/%20pi_x(s2%2C%201))%20-%20log(pi_x(s2%2C%20end)%20/%20pi_x(s2%2C%201))%3B%0A%25%20%20end%0A%25end%0A%0A%25%20%E8%BE%93%E5%87%BA%E7%BB%93%E6%9E%9C%0A%25disp('Granger%20causality%20from%20x%20to%20y:')%0A%25disp(GC_x_to_y)%0A%0A%25disp('Granger%20causality%20from%20y%20to%20x:')%0A%25disp(GC_y_to_x)%0A%0A%25%E5%9C%A8%E8FD9%E4%E2%88%8F%E7%A4%BAE%E4%BE%8B%E4%B8%AD%EF%BC%8C%E6%88%91%E4%BB%AC%E9996%E7%88%B7%E7%94%9F%E6%88%90%E4%BA%86%E4%B8%80%E4%B8%AA%E6%A8%A1%E6%95%B0%E6%95%B0%E6%8D%AE%EF%BC%8C%E7%8C%AB0%E5%90%8E%E8BBE%E7%BD%AE%E4%BA%86%E6%A8%A1%E5%9E%8B%E5%8F%82%E6%95%B0%E548C%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%E7%9F%A9%E9%98%B6%E3%80%82%E9%9A%8F%E7%9F%A5%EF%BC%8C%E6%88%91%E4%BB%AC%E4%BD%BF%E7%94%A8OLS%E4%BC%8F%E8%AE%A1%E6%A8%A1%E5%9E%8B%E5%8F%82%E6%95%B0%EF%BC%8C%E5E76E%E8%AE%A1%E7%AE%97%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%E6%A6%82%E7%8E%87%E3%80%82%E6%9C%80%E5%90%8E%EF%BC%8C%E6%8C%82%E6%9E%9C%E7%8A%B6%E6%80%81%E8%BD%AC%E7%A7%BB%E6%A6%82%E7%8E%87%E8BA%8F%E8A8%88%E7%AE%97%E4%BA%86Granger%E5%9B%A0%E6%9E%9C%E6%80%A7%E6%8C%87%E6%95%B0%E3%80%82%0A%0A%25%E8BF%83%E8BF%9C%E6%B3%A8%E6%84%8F%EF%BC%8C%E8FD9%E5%8F%AA%E6%98%AF%E4%B8%80%E4%B8%AA%E7%AE%80%E5%8D%95%E7%9A%84%E7%A4%BAE%E4%BE%8B%E7%A0%81%E7%A2%BC%EF%BC%8C%E5B9%B4%E5BA%95%E5BA%94%E7%94%A8%E4%B8%AD%E58F%AF%E80FD%E9700%E8981%E6%8C%82%E6%9E%9C%E58F%8A%E980%81%E95EE%E9898%E9996%E884C%E694%B9%E695%B4%E548C%E689%A9%E5BC%A0%E3%80%82