Partial autocorrelation function: Difference between revisions

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==Definition==

Given a time series <math>z_t</math>, the partial autocorrelation of lag <math>k</math>, denoted <math>\phi_{k,k}</math>, is the [[autocorrelation]] between <math>z_t</math> and <math>z_{t+k}</math> with the linear dependence of <math>z_t</math> on <math>z_{t+1}</math> through <math>z_{t+k-1}</math> removed. Equivalently, it is the autocorrelation between <math>z_t</math> and <math>z_{t+k}</math> that is not accounted for by lags <math>1</math> through <math>k-1</math>, inclusive.<ref name=":3">{{Cite web |title=6.4.4.6.3. Partial Autocorrelation Plot |url=https://www.itl.nist.gov/div898/handbook/pmc/section4/pmc4463.htm |access-date=2022-07-14 |website=www.itl.nist.gov}}</ref><math display="block">\phi_{1,1} = \operatorname{corr}(z_{t+1}, z_{t}),\text{ for }k= 1,</math><math display="block">\phi_{k,k} = \operatorname{corr}(z_{t+k} - \hat{z}_{t+k},\, z_{t} - \hat{z}_{t}),\text{ for }k\geq 2,</math>where <math>\hat{z}_{t+k}</math> and <math>\hat{z}_t</math> are [[Linearlinear combination|linear combinations]]s of <math>\{z_{t+1}, z_{t+2}, ..., z_{t+k-1}\}</math> that minimize the [[mean squared error]] of <math>z_{t+k}</math> and <math>z_t</math> respectively. For [[Stationarystationary process|stationary processes]]es, <math>\hat{z}_{t+k}</math> and <math>\hat{z}_t</math> are the same.<ref name=":4">{{Cite book |last=Shumway |first=Robert H. |url=http://link.springer.com/10.1007/978-3-319-52452-8 |title=Time Series Analysis and Its Applications: With R Examples |last2=Stoffer |first2=David S. |date=2017 |publisher=Springer International Publishing |isbn=978-3-319-52451-1 |series=Springer Texts in Statistics |___location=Cham |pages=97-9997–99 |language=en |doi=10.1007/978-3-319-52452-8}}</ref>

== Calculation ==

The theoretical partial autocorrelation function of a stationary time series can be calculated by using the Durbin–Levinson Algorithm:<math display="block">\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) }</math>where <math>\phi_{n,k} = \phi_{n-1, k} - \phi_{n,n} \phi_{n-1,n-k}</math> for <math>1 \leq k \leq n - 1</math> and <math>\rho(n)</math> is the autocorrelation function.<ref>{{Cite journal |last=Durbin |first=J. |date=1960 |title=The Fitting of Time-Series Models |url=https://www.jstor.org/stable/1401322 |journal=Revue de l'Institut International de Statistique / Review of the International Statistical Institute |volume=28 |issue=3 |pages=233–244 |doi=10.2307/1401322 |issn=0373-1138}}</ref><ref>{{Cite book |last=Shumway |first=Robert H. |url=http://link.springer.com/10.1007/978-3-319-52452-8 |title=Time Series Analysis and Its Applications: With R Examples |last2=Stoffer |first2=David S. |date=2017 |publisher=Springer International Publishing |isbn=978-3-319-52451-1 |series=Springer Texts in Statistics |___location=Cham |pages=103-104103–104 |language=en |doi=10.1007/978-3-319-52452-8}}</ref><ref name=":1">{{Cite book |last=Enders |first=Walter |url=https://www.worldcat.org/oclc/52387978 |title=Applied econometric time series |date=2004 |publisher=J. Wiley |isbn=0-471-23065-0 |edition=2nd |___location=Hoboken, NJ |pages=65-6765–67 |language=en |oclc=52387978}}</ref>

The formula above can be used with sample autocorrelations to find the sample partial autocorrelation function of any given time series.<ref name=":0">{{Cite book |last=Box |first=George E. P. |title=Time Series Analysis: Forecasting and Control |last2=Reinsel |first2=Gregory C. |last3=Jenkins |first3=Gwilym M. |publisher=John Wiley |year=2008 |isbn=9780470272848 |edition=4th |___location=Hoboken, New Jersey |language=en}}</ref><ref>{{Cite book |last=Brockwell |first=Peter J. |title=Time Series: Theory and Methods |last2=Davis |first2=Richard A. |publisher=Springer |year=1991 |isbn=9781441903198 |edition=2nd |___location=New York, NY |pages=102, 243-245243–245 |language=en}}</ref>

== Examples ==

The following table summarizes the partial autocorrelation function of different models:<ref name=":1" /><ref name=":2">{{Cite book |last=Das |first=Panchanan |url=https://www.worldcat.org/oclc/1119630068 |title=Econometrics in Theory and Practice : Analysis of Cross Section, Time Series and Panel Data with Stata 15. 1 |date=2019 |publisher=Springer |year=2019 |isbn=978-981-329-019-8 |edition= |___location=Singapore |pages=294-299294–299 |language=en |oclc=1119630068}}</ref>

{| class="wikitable"

!Model

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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.<ref name=":4" />

== Autoregressive model identification ==

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Partial autocorrelation is a commonly used tool for identifying the order of an autoregressive model.<ref name=":0" /> As previously mentioned, the partial autocorrelation of an AR(''p'') process is zero at lags greater than ''p''.<ref name=":1" /><ref name=":2" /> 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 than ''p'' for an AR(''p'') time series are approximately independent and [[Normal distribution|normal]] with a [[mean]] of 0.<ref>{{Cite journal |last=Quenouille |first=M. H. |date=1949 |title=Approximate Tests of Correlation in Time-Series |url=https://onlinelibrary.wiley.com/doi/10.1111/j.2517-6161.1949.tb00023.x |journal=Journal of the Royal Statistical Society:, Series B (Methodological) |language=en |volume=11 |issue=1 |pages=68–84 |doi=10.1111/j.2517-6161.1949.tb00023.x}}</ref> Therefore, a [[confidence interval]] can be constructed by dividing a selected [[Standard score|z-score]] by <math>\sqrt{n}</math>. 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.<ref name=":3" />

==References==