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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 [[Linear combination|linear combinations]] 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 processes]], <math>\hat{z}_{t+k}</math> and <math>\hat{z}_t</math> are the same.<ref>{{Cite book |last=Shumway |first=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=97-98 |language=en |doi=10.1007/978-3-319-52452-8}}</ref>
== Calculation ==
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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 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. 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 be the AR model's order.<ref name=":3" />
==References==
{{Reflist}}
{{Statistics|analysis}}
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