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 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.

Definition

Given a time series $z_{t}$ , the partial autocorrelation of lag $k$ , denoted $\phi _{k,k}$ , is the autocorrelation between $z_{t}$ and $z_{t+k}$ with the linear dependence of $z_{t}$ on $z_{t+1}$ through $z_{t+k-1}$ removed. Equivalently, it is the autocorrelation between $z_{t}$ and $z_{t+k}$ that is not accounted for by lags $1$ through $k-1$ , inclusive. $\phi _{1,1}=\operatorname {corr} (z_{t+1},z_{t}),{\text{ for }}k=1,$ $\phi _{k,k}=\operatorname {corr} (z_{t+k}-{\hat {z}}_{t+k},\,z_{t}-{\hat {z}}_{t}),{\text{ for }}k\geq 2,$ where ${\hat {z}}_{t+k}=\beta _{1}z_{t+k-1}+\beta _{2}z_{t+k-2}+...+\beta _{k-1}z_{t+1}$ is the linear combination of $\{z_{t+k-1},z_{t+k-2},...,z_{t+1}\}$ that minimizes the mean squared error, $\mathrm {E} [z_{t+k}-{\hat {z}}_{t+k}]^{2}$ . Similarly, ${\hat {z}}_{t}=\beta _{1}z_{t+1}+\beta _{2}z_{t+2}+...+\beta _{k-1}z_{t+k-1}$ is a linear combination minimizing $\mathrm {E} [z_{t}-{\hat {z}}_{t}]^{2}$ . For stationary processes, the coefficients $\beta _{1},\beta _{2},...,\beta _{k-1}$ are the same.^[2]

Calculation

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

There are algorithms for estimating the partial autocorrelation based on the sample autocorrelations. The formula above can be used with sample autocorrelations to find the sample partial autocorrelation function of any given time series.^[6]^[7]

Autoregressive Model Identification

Partial autocorrelation plots are a commonly used tool for identifying the order of an autoregressive model.^[6] The partial autocorrelation of an AR(p) process is zero at lag $p+1$ and greater. If the sample autocorrelation plot indicates that an AR model may be appropriate, then the sample partial autocorrelation plot is examined to help identify the order. One looks for the point on the plot where the partial autocorrelations for all higher lags are essentially zero. Placing on the plot an indication of the sampling uncertainty of the sample PACF is helpful for this purpose: this is usually constructed on the basis that the true value of the PACF, at any given positive lag, is zero. This can be formalised as described below.

An approximate test that a given partial correlation is zero (at a 5% significance level) is given by comparing the sample partial autocorrelations against the critical region with upper and lower limits given by $\pm 1.96/{\sqrt {n}}$ , where n is the record length (number of points) of the time-series being analysed. This approximation relies on the assumption that the record length is at least moderately large (say $n>30$ ) and that the underlying process has finite second moment.

References

^ Brockwell, Peter J.; Davis, Richard A. (2016). "Modeling and Forecasting with ARMA Processes". Introduction to Time Series and Forecasting (Third ed.). Springer International Publishing. p. 132. ISBN 978-3319298528.
^ Shumway, Robert H.; Stoffer, David S. (2017). Time Series Analysis and Its Applications: With R Examples. Springer Texts in Statistics. Cham: Springer International Publishing. pp. 97–98. doi:10.1007/978-3-319-52452-8. ISBN 978-3-319-52451-1.
^ Durbin, J. (1960). "The Fitting of Time-Series Models". Revue de l'Institut International de Statistique / Review of the International Statistical Institute. 28 (3): 233–244. doi:10.2307/1401322. ISSN 0373-1138.
^ Shumway, Robert H.; Stoffer, David S. (2017). Time Series Analysis and Its Applications: With R Examples. Springer Texts in Statistics. Cham: Springer International Publishing. pp. 103–104. doi:10.1007/978-3-319-52452-8. ISBN 978-3-319-52451-1.
^ Enders, Walter (2004). Applied econometric time series (2nd ed.). Hoboken, NJ: J. Wiley. pp. 65–67. ISBN 0-471-23065-0. OCLC 52387978.
^ ^a ^b Box, George E. P.; Reinsel, Gregory C.; Jenkins, Gwilym M. (2008). Time Series Analysis: Forecasting and Control (4th ed.). Hoboken, New Jersey: John Wiley. ISBN 9780470272848.
^ Brockwell, Peter J.; Davis, Richard A. (1991). Time Series: Theory and Methods (2nd ed.). New York, NY: Springer. pp. 102, 243–245. ISBN 9781441903198.

This article incorporates public domain material from http://www.itl.nist.gov/div898/handbook/pmc/section4/pmc4463.htm. National Institute of Standards and Technology. {{citation}}: External link in |title= (help)

[1] Brockwell, Peter J.; Davis, Richard A. (2016). "Modeling and Forecasting with ARMA Processes". Introduction to Time Series and Forecasting (Third ed.). Springer International Publishing. p. 132. ISBN 978-3319298528.

[2] Shumway, Robert H.; Stoffer, David S. (2017). Time Series Analysis and Its Applications: With R Examples. Springer Texts in Statistics. Cham: Springer International Publishing. pp. 97–98. doi:10.1007/978-3-319-52452-8. ISBN 978-3-319-52451-1.

[3] Durbin, J. (1960). "The Fitting of Time-Series Models". Revue de l'Institut International de Statistique / Review of the International Statistical Institute. 28 (3): 233–244. doi:10.2307/1401322. ISSN 0373-1138.

[4] Shumway, Robert H.; Stoffer, David S. (2017). Time Series Analysis and Its Applications: With R Examples. Springer Texts in Statistics. Cham: Springer International Publishing. pp. 103–104. doi:10.1007/978-3-319-52452-8. ISBN 978-3-319-52451-1.

[5] Enders, Walter (2004). Applied econometric time series (2nd ed.). Hoboken, NJ: J. Wiley. pp. 65–67. ISBN 0-471-23065-0. OCLC 52387978.

[:0-6] Box, George E. P.; Reinsel, Gregory C.; Jenkins, Gwilym M. (2008). Time Series Analysis: Forecasting and Control (4th ed.). Hoboken, New Jersey: John Wiley. ISBN 9780470272848.

[7] Brockwell, Peter J.; Davis, Richard A. (1991). Time Series: Theory and Methods (2nd ed.). New York, NY: Springer. pp. 102, 243–245. ISBN 9781441903198.

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