# Autocorrelation and Partial Autocorrelation Functions of AR(1) Process

Autocorrelation and Partial Autocorrelation Functions of AR(1) Process

This Demonstration shows realizations of an autoregressive process of order one (AR(1)), its autocorrelation function (ACF), and its partial autocorrelation (PACF) function. The first tab, "sample AR(1)" shows realization of :

r

t

r=ϕ+ϕr+ϵ

t

0

1

t-1

t

where is assumed to be white noise with . Necessary and sufficient conditions for the AR(1) model to be weakly stationary are ; otherwise, it is said to be explosive. In this Demonstration you can vary the parameters to observe the behavior of the process.

{ϵ}

t

ϵ∼N(0,σ)

t

2

|ϕ|<1

1

Choosing "ACF function" shows the autocorrelation function of the simulated AR(1) process; choosing "PACF function" shows the partial autocorrelation function of the simulated AR(1) process. These two choices help us to study the dependence between the lags of the process. The ACF and PACF functions are plotted together with standard deviation bands that determine significance.