K. Ensor, STAT 4211

Spring 2004

Memory characterization of a process

• How would the ACF behave for a process with no memory?

• What is a short memory series?– Autocorrelation function decays exponentially as a

function of lag e.g. if X(t) is given by

X(t)- = (X(t-1)- ) + (t) then Corr(X(t),X(t+h))= |h| for all h

• In contrast, the autocorrelation function for a long memory process decays at a polynomial rate.

• A nonstationary process – the autocorrelation function does not decay to zero.

K. Ensor, STAT 4212

Spring 2004

White noise

• Uncorrelated OR independent random variables.

• Identically distributed – Usually with mean 0, but must be finite – And variance finite variance 2

• Notation rt~ WN(0, 2)

• What if we computed the ACF or PACF?

K. Ensor, STAT 4213

Spring 2004

Linear Time Series

• A time series rt if it can be written as a linear function of present and past values of a white noise series.

rt= + jiat-j where j=0 to infinity

and at is a white noise series.

• The coefficients define the behavior of the series.

• Let’s take a look at the mean and covariance for a covariance stationary (or weakly stationary) linear time series.

K. Ensor, STAT 4214

Spring 2004

Autoregressive models

• Just as the name implies, an autoregressive model is derived by regressing our process of interest on its on past.

• Consider an autoregressive model of order 1, or AR(1) model or

r(t)=0 + 1 r(t-1) + a(t) with a(t) representing a white noise

process• Or more generally the AR(p) model where

r(t)=0 + 1 r(t-1) + p r(t-p) a(t)

K. Ensor, STAT 4215

Spring 2004

Characterisitics of an AR process

• The behavior of the difference equation associated with the process determines the behavior of the process. Solutions to this equation are referred to as the characteristic roots.

• Same comment about the behavior of the equation characterizing the autocorrelations.

• The ACF decays exponentially to zero.– Recall ACF for AR(1)

• The PACF is zero after the lag of the AR process (see section 2.4.2.)

K. Ensor, STAT 4216

Spring 2004

Moving Average Model

• Weighted average of present and past shocks to the system.

r(t)=0 + 1 a(t-1) + a(t) with a(t) representing a white noise process

• Or more generally the MA(q) model where

r(t)= 0 + 1 a(t-1) + q a(t-q) + a(t) • Can also be viewed as a representation of an

infinite or AR model.• Basic properties

– Autocorrelation is zero after the largest lag of the process.

– Partial autocorrelation decays to zero.

K. Ensor, STAT 4217

Spring 2004

ARMA models

• The series r(t) is a function of past values of itself plus current and past values of the noise or shocks to the system.

• See page 50.• More next class period.