where

is assumed to be white noise with

. A simple autoregressive model of order one (an AR(1) model) has the same form as a simple linear regression model, where

is dependent and

is the explanatory variable, but they have different properties. The mean and variance conditionals on past returns are:

and

. Thus given the past return

, the current return is centered around

with variability

. Assuming that the series are weakly stationary,

,

, and

, where

and

are constants. Necessary and sufficient conditions for the AR(1) model to be weakly stationary are

; otherwise the process is explosive. The ACF of

satisfies

for

, but as

,

, and the ACF of a weakly stationary AR(1) series decays exponentially with rate

.

ACF and PACF are powerful tools for time series analysis. Snapshots 1, 2, and 3 show processes that are dependent (the parameter

is large); you can observe slowly decaying strongly significant ACFs, while the PACF shows only one lag strongly significant. On the other hand, snapshots 4, 5, and 6 show a negatively dependent process, where the ACF is decaying exponentially and the PACF has again only one significant lag. Note that the significant lag of the PACF points to the AR(1) dependence in the observed time series, while the sign of the PACF lag corresponds to the sign of

.