Partial autocorrelation function: Difference between revisions

Revision as of 17:43, 13 July 2022

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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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[[File:Partial Autocorrelation Function Graph.png|alt=The partial autocorrelation graph has 3 spikes and the rest is close to 0.|thumb|Sample partial autocorrelation function of a simulated AR(3) time series]]

Partial autocorrelation plots areis a commonly used tool for identifying the order of an [[autoregressive model]].<ref name=":0" /> TheAs previously mentioned, the partial autocorrelation of an AR(''p'') process is zero at laglags greater <math>than ''p+1''.<ref name=":1" /math><ref andname=":2" greater./> If the sample autocorrelation plot indicates that an AR model mayis determined to 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.

The estimated partial autocorrelation of lags greater than ''p'' for an AR(''p'') time series is independently and [[Normal distribution|normally distributed]] 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}}</ref> The [[standard error]] is <math display="inline">\frac{1}{\sqrt{n}}</math> and a [[confidence interval]] can be constructed by multiplying the standard error and a selected [[z-score]]. 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.

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 <math>n>30</math>) and that the underlying process has finite second moment.

==References==