Serial Correlation in Time Series

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In statistics, the serial correlation of a random process describes the correlation between values of the process at different points in time, as a function of the two times or of the time difference. Let X be some repeatable process, and i be some point in time after the start of that process. (i may be an integer for a discrete-time process or a real number for a continuous-time process.) Then Xi is the value (or realization) produced by a given run of the process at time i. Suppose that the process is further known to have defined values for mean and variance for all times i. Then the definition of the serial correlation between any two time s and t is

where "E" is the expected value operator. Note that this expression is not well-defined for all time series or processes, because the variance may be zero (for a constant process) or infinite. If the function R is well-defined, its value must lie in the range [−1, 1], with 1 indicating perfect correlation and −1 indicating perfect anti-correlation. If Xt is a second-order stationary process then the mean μ and the variance σ2 are time-independent, and further the autocorrelation depends only on the difference between t and s: the correlation depends only on the time-distance between the pair of values but not on their position in time. This further implies that the autocorrelation can be expressed as a function of the time-lag, and that this would be an even function of the lag τ = s − t. This gives the more familiar form

and the fact that this is an even function can be stated as

It is common practice in some disciplines, other than statistics and time series analysis, to drop the normalization by σ2 and use the term "serial correlation" interchangeably with "serial covariance". However, the normalization is important both because the interpretation of the serial correlation as a correlation provides a scale-free measure of the strength of statistical dependence, and because the normalization has an effect on the statistical properties of the estimated serial correlations.