Autocorrelation

The autocorrelation coefficient

We say that a random variable $y_t$ is autocorrelated if its value at a period $t$ depends to some degree on previous realizations. In other words, a variable is autocorrelated if we can explain its value today using its lagged values from the past. This is equivalent to saying that the variable is serially correlated or that it presents a serial correlation.

The concept of serial correlation is intimately connected to the concept of weak stationarity. The autocorrelation coefficient, $\rho_j$, is, in fact, a function of the autocovariance coefficient, $\gamma_j$.

This definition means that the correlation between $y_t$ and $y_{t-j}$, $\rho_j$ is the covariance between the two points over the unconditional variance, $\gamma_0$, of the process. Weak stationarity ensures that $\rho_j$ ...