View source for Auto-regressive process

A stochastic process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a0139801.png" /> whose values satisfy an [[Auto-regression|auto-regression]] equation with certain constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a0139802.png" />:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a0139803.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a0139804.png" /> is some positive number and where the variables <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a0139805.png" /> are usually assumed to be uncorrelated and identically distributed around their average value 0 with a variance <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a0139806.png" />. If all the zeros of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a0139807.png" /> of a complex variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a0139808.png" /> lie inside the unit circle, then equation (*) has the solution

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a0139809.png" /></td> </tr></table>

where the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a01398010.png" /> are connected with the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a01398011.png" /> by the relation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a01398012.png" /></td> </tr></table>

For example, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a01398013.png" /> be a [[White noise|white noise]] process with spectral density <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a01398014.png" />; in such a case the only kind of auto-regressive process satisfying equation (*) will be a process <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a01398015.png" /> with spectral density

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a01398016.png" /></td> </tr></table>

which is stationary in the wide sense if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a01398017.png" /> has no real zeros. The autocovariances (cf. [[Autocovariance|Autocovariance]]) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a01398018.png" /> of the process satisfy the recurrence relation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a01398019.png" /></td> </tr></table>

and, in terms of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a01398020.png" />, have the form

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a01398021.png" /></td> </tr></table>

The parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a01398022.png" /> of the auto-regression are connected with the [[Auto-correlation|auto-correlation]] coefficients <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a01398023.png" /> of the process by the matrix relation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a01398024.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a01398025.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a01398026.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a013/a013980/a01398027.png" /> is the matrix of auto-correlation coefficients (the Yule–Walker equation).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  U. Grenander,   M. Rosenblatt,   "Statistical analysis of stationary time series" , Wiley  (1957)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.J. Hannan,   "Time series analysis" , Methuen , London  (1960)</TD></TR></table>