Intime series analysis, thepartial autocorrelation function (PACF) gives thepartial 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 theautocorrelation 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 anautoregressive (AR) model. The use of this function was introduced as part of theBox–Jenkins approach to time series modelling, whereby plotting the partial autocorrelative functions one could determine the appropriate lagsp in an AR (p)model or in an extendedARIMA (p,d,q) model.

Given a time series${\displaystyle z_t}$, the partial autocorrelation of lag${\displaystyle k}$, denoted${\displaystyle {\varphi }_{k,k}}$, is theautocorrelation between${\displaystyle z_t}$ and${\displaystyle z_{t+k}}$ with the linear dependence of${\displaystyle z_t}$ on${\displaystyle z_{t+1}}$ through${\displaystyle z_{t+k-1}}$ removed. Equivalently, it is the autocorrelation between${\displaystyle z_t}$ and${\displaystyle z_{t+k}}$ that is not accounted for by lags${\displaystyle 1}$ through${\displaystyle k-1}$, inclusive.^[1]$${\displaystyle {\varphi }_{1,1}=\mathrm{corr}(z_{t+1},z_t),\text{ for }k=1,}$$$${\displaystyle {\varphi }_{k,k}=\mathrm{corr}(z_{t+k}-{\hat{z}}_{t+k},z_t-{\hat{z}}_t),\text{ for }k\ge 2,}$$where${\displaystyle {\hat{z}}_{t+k}}$ and${\displaystyle {\hat{z}}_t}$ arelinear combinations of${\displaystyle \{z_{t+1},z_{t+2},...,z_{t+k-1}\}}$ that minimize themean squared error of${\displaystyle z_{t+k}}$ and${\displaystyle z_t}$ respectively. Forstationary processes, the coefficients in${\displaystyle {\hat{z}}_{t+k}}$ and${\displaystyle {\hat{z}}_t}$ are the same, but reversed:^[2]$${\displaystyle {\hat{z}}_{t+k}={\beta }_1{z}_{t+k-1}+\cdots +{\beta }_{k-1}{z}_{t+1}\text{and}{\hat{z}}_t={\beta }_1{z}_{t+1}+\cdots +{\beta }_{k-1}{z}_{t+k-1}.}$$

The theoretical partial autocorrelation function of a stationary time series can be calculated by using the Durbin–Levinson Algorithm:$${\displaystyle {\varphi }_{n,n}=\frac{\rho (n)-\sum _{k=1}^{n-1}{\varphi }_{n-1,k}\rho (n-k)}{1-\sum _{k=1}^{n-1}{\varphi }_{n-1,k}\rho (k)}}$$where${\displaystyle {\varphi }_{n,k}={\varphi }_{n-1,k}-{\varphi }_{n,n}{\varphi }_{n-1,n-k}}$ for${\displaystyle 1\le k\le n-1}$ and${\displaystyle \rho (n)}$ is the autocorrelation function.^[3][4][5]

The formula above can be used with sample autocorrelations to find the sample partial autocorrelation function of any given time series.^[6][7]

The following table summarizes the partial autocorrelation function of different models:^[5][8]

Model	PACF
White noise	The partial autocorrelation is 0 for all lags.
Autoregressive model	The partial autocorrelation for an AR(p) model is nonzero for lags less than or equal top and 0 for lags greater thanp.
Moving-average model	If${\displaystyle {\varphi }_{1,1}>0}$, the partial autocorrelationoscillates to 0.
	If, the partial autocorrelationgeometrically decays to 0.
Autoregressive–moving-average model	An ARMA(p,q) model's partial autocorrelation geometrically decays to 0 but only after lags greater thanp.

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 lagp similar to the autocorrelation function of an MA(q) series with lagq. In addition, the autocorrelation function of an AR(p) process tails off just like the partial autocorrelation function of an MA(q) process.^[2]

Partial autocorrelation is a commonly used tool for identifying the order of an autoregressive model.^[6] As previously mentioned, the partial autocorrelation of an AR(p) process is zero at lags greater thanp.^[5][8] If an AR model is determined to be appropriate, then the sample partial autocorrelation plot is examined to help identify the order.

The partial autocorrelation of lags greater thanp for an AR(p) time series are approximately independent andnormal with amean of 0.^[9] Therefore, aconfidence interval can be constructed by dividing a selectedz-score by${\displaystyle \sqrt{n}}$. Lags with partial autocorrelations outside of the confidence interval indicate that the AR model's order is likely greater than or equal to the lag. Plotting the partial autocorrelation function and drawing the lines of the confidence interval is a common way to analyze the order of an AR model. To evaluate the order, one examines the plot to find the lag after which the partial autocorrelations are all within the confidence interval. This lag is determined to likely be the AR model's order.^[1]

• Time domain analysis
• Covariance and correlation
• Time series

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