Partial autocorrelation function: Difference between revisions

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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]]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]]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 |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=97–99 |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>

== Calculation ==