Memory characterization of a processWhite noiseLinear Time SeriesAutoregressive modelsCharacterisitics of an AR processMoving Average ModelARMA modelsK. Ensor, STAT 4211Spring 2004Memory 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 4212Spring 2004White 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 4213Spring 2004Linear 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= + jiat-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 4214Spring 2004Autoregressive 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 orr(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 4215Spring 2004Characterisitics 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 4216Spring 2004Moving 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 4217Spring 2004ARMA 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