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 $\mu _{i}$ and variance $\sigma _{i}^{2}$ for all times i. Then the definition of the serial correlation between any two time s and t is

$R(s,t)={\frac {E[(X_{t}-\mu _{t})(X_{s}-\mu _{s})]}{\sigma _{s}\sigma _{t}}}$

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 X_t 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

$R(\tau )={\frac {E[(X_{t}-\mu )(X_{t+\tau }-\mu )]}{\sigma ^{2}}}$

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

$R(\tau )=R(-\tau )$ 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.