2.2 Correlation and Autocorrelation Function

The correlation coefficient between two random variables X and Y is defined as

Inline

where μx and μy are the mean of X and Y, respectively, and it is assumed that the variances exist. This coefficient measures the strength of linear dependence between X and Y, and it can be shown that − 1 ≤ ρx, y ≤ 1 and ρx, y = ρy, x. The two random variables are uncorrelated if ρx, y = 0. In addition, if both X and Y are normal random variables, then ρx, y = 0 if and only if X and Y are independent. When the sample Inline is available, the correlation can be consistently estimated by its sample counterpart

Inline

where Inline and Inline are the sample mean of X and Y, respectively.

Autocorrelation Function (ACF)

Consider a weakly stationary return series rt. When the linear dependence between rt and its past values rti is of interest, the concept of correlation is generalized to autocorrelation. The correlation coefficient between rt and rt−ℓ is called the lag- autocorrelation of rt

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