Regression

Regression measures the linear relation between two timeseries.

$Y=\alpha+\beta*X+\epsilon$ $\left[ \begin{array} {cccc} r_{11} \: r_{12} \ldots r_{1n} \\ r_{21} \: r_{22} \ldots r_{2n} \\ \vdots \\ r_{k1} \: r_{k2} \ldots r_{kn} \end{array}\right] = \left[ \begin{array}{c} \alpha_1 \\ \alpha_2 \\ \vdots \\ \alpha_n \end{array} \right]^T + \left[ \begin{array}{c} \beta_1 \\ \beta_2 \\ \vdots \\ \beta_n \end{array} \right]^T \left[ \begin{array}{cc} 1 \: r_{X1} \\ 1 \: r_{X2} \\ \vdots \\ 1 \: r_{Xk} \end{array} \right]$

In Matlab/Octave:

[alpha;beta]=[1, r_X]  \ r_kn;
residuals = r_kn - ([1,r_X]*[alpha;beta]);
sigma_res = std(residuals);
alpha_adj = alpha ./ sigma_res;

Some wordings:

$ESS = \sum_t{(\hat{y}_t - \bar{y})^2}$ explained sum of squares

$RSS = \sum_t{\epsilon_t^2}$ residual sum of squares

$TSS = \sum_t{(y_t - \bar{y})^2}$ total sum of squares

$TSS = ESS + RSS$

Goodness of fit (R²):

$R^2 = 1 - \frac{ESS}{TSS}$

adjusted R² by loss of degree of freedom (k variables, T observations): ${R_{adj}}^2 = 1 - [ \frac{T - 1}{T - k} (1 - R^2) ]$