Autocovariance

Definition

$$\begin{array}{rl}{\mu }_t& =E[Y_t]\\ {\gamma }_{t,s}& =\mathrm{Cov}(Y_t,Y_s)=E[(Y_t-{\mu }_t)(Y_s-{\mu }_s)]\\ {\rho }_{t,s}& =\mathrm{Corr}(Y_t,Y_s)=\frac{{\gamma }_{t,s}}{\sqrt{{\gamma }_{t,t}{\gamma }_{s,s}}}\end{array}$$

Tip

Respectively, cov and corr of 2 random variables from same sequence at two time points

Properties

• ${\gamma }_{t,t}=\mathrm{Var}(Y_t)$
• ${\rho }_{t,t}=1$
• ${\rho }_{t,s}={\rho }_{s,t}$
• ${\gamma }_{t,s}={\gamma }_{s,t}$
• $|{\gamma }_{t,s}|\le \sqrt{{\gamma }_{t,t}{\gamma }_{s,s}}$
• $|{\rho }_{t,s}|\le 1$