Partial autocorrelation function: Difference between revisions

Revision as of 17:24, 21 July 2022

Revision as of 01:12, 24 July 2022

Line 16:

== Examples ==

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

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

{| class="wikitable"

!Model

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''.

!PACF

|-

For [[Moving-average model|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.

|[[White noise]]

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

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]]

|AR(''p'') 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''.

|-

|[[Moving-average model]]

For|The [[Moving-averagepartial model|movingautocorrelation averagefor MA(MA''q'')]] models, their partial autocorrelation exponentially decaysdecay to 0. Foras MAlags models thatincrease. haveIf <math>\phi_{1,1} > 0</math>, the decay is [[Oscillation (mathematics)|oscillating]] andor the other models withif <math>\phi_{1,1} < 0</math>, have geometricthe decay is geometric.

|-

|[[Autoregressive–moving-average model]]

|ARMA(''p'', ''q'') models have exponentially decaying partial autocorrelation to 0 but only after lags greater than ''p''.

|}

== Autoregressive model identification ==