Partial autocorrelation function

In time series analysis, the partial autocorrelation function (PACF) gives the partial correlation of a stationary time series with its own lagged values, regressed the values of the time series at all shorter lags. It contrasts with the autocorrelation function, which does not control for other lags. This function plays an important role in data analysis aimed at identifying the extent of the lag in an autoregressive (AR) model. The use of this function was introduced as part of the Box–Jenkins approach to time series modelling, whereby plotting the partial autocorrelative functions one could determine the appropriate lags p in an AR (p) model or in an extended ARIMA (p,d,q) model. Given a time series , the partial autocorrelation of lag , denoted , is the autocorrelation between and with the linear dependence of on through removed. Equivalently, it is the autocorrelation between and that is not accounted for by lags through , inclusive.where and are linear combinations of that minimize the mean squared error of and respectively. For stationary processes, the coefficients in and are the same, but reversed: The theoretical partial autocorrelation function of a stationary time series can be calculated by using the Durbin–Levinson Algorithm:where for and is the autocorrelation function. The formula above can be used with sample autocorrelations to find the sample partial autocorrelation function of any given time series. The following table summarizes the partial autocorrelation function of different models: The behavior of the partial autocorrelation function mirrors that of the autocorrelation function for autoregressive and moving-average models. For example, the partial autocorrelation function of an AR(p) series cuts off after lag p similar to the autocorrelation function of an MA(q) series with lag q. In addition, the autocorrelation function of an AR(p) process tails off just like the partial autocorrelation function of an MA(q) process. Partial autocorrelation is a commonly used tool for identifying the order of an autoregressive model.