Partial autocorrelation function: Difference between revisions

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where <math>P_{t,k}(x)</math> is the surjective operator of orthogonal projection of <math>x</math> onto the linear subspace of Hilbert space spanned by <math> z_{t+1}, \dots, z_{t+k-1}</math>.

There are algorithms for estimating the partial autocorrelation based on the sample autocorrelations.<ref (name=":0">{{Cite book |last=Box, Jenkins,|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 and|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, 2009)|first2=Richard A. |publisher=Springer |year=1991 |isbn=9781441903198 |edition=2nd |___location=New York, NY |language=en}}</ref> These algorithms derive from the exact theoretical relation between the partial autocorrelation function and the autocorrelation function.

Partial autocorrelation plots (Box and Jenkins, Chapter 3.2, 2008) are a commonly used tool for identifying the order of an [[autoregressive model]].<ref name=":0" /> The partial autocorrelation of an AR(''p'') process is zero at lag ''p''&nbsp;+&nbsp;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 <math>\pm 1.96/\sqrt{n}</math>, 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.

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==References==

{{Reflist}}

*{{cite book |last=Enders |first=Walter |title=Applied Econometric Time Series |___location=New York |publisher=John Wiley |year=2004 |edition=Second |pages=[https://archive.org/details/appliedeconometr00ende_0/page/65 65–67] |isbn=0-471-23065-0 |url-access=registration |url=https://archive.org/details/appliedeconometr00ende_0/page/65 }}

*{{cite book |last=Box |first=G. E. P. |last2=Jenkins |first2=G. M. |last3=Reinsel |first3=G. C. |year=2008 |title=Time Series Analysis, Forecasting and Control |edition=4th |publisher=Wiley |___location=Hoboken, NJ |isbn=9780470272848 }}

*{{cite book |last=Brockwell |first=Peter |last2=Davis |first2=Richard |year=2009 |title=Time Series: Theory and Methods |edition=2nd |publisher=Springer |___location=New York |isbn=9781441903198 }}

*{{cite book |last=Enders |first=Walter |title=Applied Econometric Time Series |___location=New York |publisher=John Wiley |year=2004 |edition=Second |pages=[https://archive.org/details/appliedeconometr00ende_0/page/65 65–67] |isbn=0-471-23065-0 |url-access=registration |url=https://archive.org/details/appliedeconometr00ende_0/page/65 }}

{{NIST-PD|http://www.itl.nist.gov/div898/handbook/pmc/section4/pmc4463.htm}}