Non stationary processes If X is not stationary: transform X to find related stationary series. In this course: two sorts of non-stationarity - non constant mean and integration. Non constant mean: If E(Xt ) is not constant we will hope to model t = E(Xt
STAT 804: 2004-1
1. Suppose X and Y are stationary independent processes with respective
spectra fX and fY . Compute the spectrum of Z = aX + Y .
Solution We have
Cov(Zt , Zt+h )
= a2 Cov(Xt , Xt+h + aCov(Xt , Yt+h ) + aCov(Yt , Xt+h ) + Cov(
be a Gaussian white noise process. Define
Compute and plot the autocovariance function of X.
2. Suppose that
sequence is iid.
are uncorrelated and have mean 0 with finite variance. Verify
is stationary and that it is wid
is an ARMA(1,1) process
Suppose we mistakenly fit an AR(1) model (mean 0) to X using the YuleWalker estimate
In terms of , and
what is close to?
Solution: This is a ratio of two sample covariances. Each converges,
to the co
Assignment 2 Solutions
Consider the ARIMA(1,0,1) process
Show that the autocorrelation function is
Plot the autocorrelation functions for the ARMA(1,1) process above, the AR(1)
and the MA(1) process
on the same plot when
is a Uniform
random variable. Define
Show that X is weakly stationary. (In fact it is strongly stationary so show that
if you can.) Compute the autocorrelation function of X.
and you get
Basic jargon Defn: Stochastic process family cfw_Xi; i I of random variables indexed by a set I. In practice the jargon is used only when the Xi are not independent. If I Real Line, then we often call cfw_Xi; i I a time series. Of course the usual situati
Model Order Selection: formal methods Outline of topics: 1) Likelihood Ratio Tests 2) Form of Likelihood Ratio Tests for ARMA(p, q) models. 3) Final Prediction Error 4) Akaike's Information Criterion 5) Example of use in model selection General set up: da
Fitting ARIM A(p, d, q) models to data Fitting I part easy: difference d times. Same for seasonal multiplicative model. Thus to fit an ARIMA(p, d, q) model to X compute Y = (I - B)dX. Note: shortens data set by d observations. Then fit an ARM A(p, q) mode
Defn: If cfw_t is a white noise series and and b0, . . . , bp are constants then Xt = + b0t + b1t1 + + bptp is a moving average of order p; write M A(p). Q: From observations on X can we estimate the bs and 2 = Var(t) accurate
Stationarity Goal: nd assumptions on a discrete time process which will permit us to make reasonable estimates of the parameters. Intuition: need some notion of replication. Defn: Stochastic process Xt; t = 0, 1, . . . is stationary if joint distribution