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. 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 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 z t {displaystyle z_{t}} , the partial autocorrelation of lag k, denoted α ( k ) {displaystyle alpha (k)} , is the autocorrelation between z t {displaystyle z_{t}} and z t + k {displaystyle z_{t+k}} with the linear dependence of z t {displaystyle z_{t}} on z t + 1 {displaystyle z_{t+1}} through z t + k − 1 {displaystyle z_{t+k-1}} removed; equivalently, it is the autocorrelation between z t + 1 {displaystyle z_{t+1}} and z t + k + 1 {displaystyle z_{t+k+1}} that is not accounted for by lags 1 to k , inclusive. where P t , k ( x ) {displaystyle P_{t,k}(x)} is surjective operator of orthogonal projection of x {displaystyle x} onto the linear subspace of Hilbert space spanned by x t + 1 , … , x t + k {displaystyle x_{t+1},dots ,x_{t+k}} .