Partial autocorrelation function: Difference between revisions

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{{Short description|Partial correlation of a time series with its lagged values}}

[[File:Partial autocorrelation function.png|thumb|Partial autocorrelation function of [[Lake Huron]]'s depth with confidence interval<ref>{{cite book(in |last1=Brockwellblue, |first1=Peterplotted J.around |last2=Davis |first2=Richard A. |title=Introduction to Time Series and Forecasting |date=2016 |publisher=Springer International Publishing |isbn=978-3319298528 |page=132 |edition=Third |url=https://doi.org/10.1007/978-3-319-29854-2 |language=English |chapter=Modeling and Forecasting with ARMA Processes}}</ref>0)]]

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.

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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 [[Stationary process|stationary processesprocess]]es, the coefficients in <math>\hat{z}_{t+k}</math> and <math>\hat{z}_t</math> are the same., but reversed:<ref name=":4">{{Cite book |lastlast1=Shumway |firstfirst1=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=97-9897–99 |language=en |doi=10.1007/978-3-319-52452-8}}</ref> <math display="block">\hat{z}_{t+k}=\beta_1z_{t+k-1}+\cdots+\beta_{k-1}z_{t+1}\qquad\text{and}\qquad\hat{z}_t=\beta_1z_{t+1}+\cdots+\beta_{k-1}z_{t+k-1}.</math><!-- Think of this as reversing the direction of time. Stationary processes are invariant under shifts and reversal of time. -->

== 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 |lastlast1=Shumway |firstfirst1=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 |lastlast1=Box |firstfirst1=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 |lastlast1=Brockwell |firstfirst1=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 partial autocorrelation of [[white noise]] is zero for all lags.

The following table summarizes the partial autocorrelation function of andifferent [[ARMA model|ARMA(''p'', ''q'') model]] also exponentially decays but only after lags greater than ''p''.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>

AR models have nonzero partial autocorrelations for lags less than or equal to its order. In other words, the partial autocorrelation of an AR(''p'') process is zero at lags greater than ''p''.

{| class="wikitable"

!Model

!PACF

|-

|[[White noise]]

|The partial autocorrelation of [[white noise]] is zero0 for all lags.

|-

|[[Autoregressive model]]

|The partial autocorrelation for an AR(''p'') model is nonzero for lags less than or equal to ''p'' and 0 for lags greater than ''p''.

|-

|rowspan=2|[[Moving-average model]]

|If <math>\phi_{1,1} > 0</math>, the partial autocorrelation [[Oscillation (mathematics)|oscillates]] to 0.

|-

|If <math>\phi_{1,1} < 0</math>, the partial autocorrelation [[Exponential decay|geometrically]] decays to 0.

|-

|[[Autoregressive–moving-average model]]

|An ARMA(''p'', ''q'') model's partial autocorrelation geometrically decays to 0 but only after lags greater than ''p''.

|}

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" />

For [[Moving average|moving average (MA)]] models, their partial autocorrelation exponentially decays to 0. For MA models that have <math>\phi_{1,1} > 0</math>, the decay is [[Oscillation (mathematics)|oscillating]] and the other models with <math>\phi_{1,1} < 0</math> have geometric decay.

The partial autocorrelation function of an [[ARMA model|ARMA(''p'', ''q'') model]] also exponentially decays but only after lags greater than ''p''.<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=294-299 |language=en |oclc=1119630068}}</ref>

== 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 estimated partial autocorrelation of lags greater than ''p'' for an AR(''p'') time series isare approximately independentlyindependent and [[Normal distribution|normally distributednormal]] with a [[mean]] of 0 and a [[variance]] of <math display="inline">\frac{1}{n}</math> where <math display="inline">n</math> is the number of observations in the time series.<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|url-access=subscription }}</ref> The [[standard error]] is <math display="inline">\frac{1}{\sqrt{n}}</math> andTherefore, a [[confidence interval]] can be constructed by multiplying the standard error anddividing 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==