Partial autocorrelation function: Difference between revisions

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Latest revision as of 12:09, 18 July 2025

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== 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 |jstor=1401322 |issn=0373-1138|url-access=subscription }}</ref><ref>{{Cite book |last1=Shumway |first1=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=103–104 |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=65–67 |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 |last1=Box |first1=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 |last1=Brockwell |first1=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, 243–245 |language=en}}</ref>

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== 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 |publisher=Springer |year=2019 |isbn=978-981-329-019-8 |edition= |___location=Singapore |pages=294–299 |language=en |oclc=1119630068}}</ref>

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!Model