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* (i-e)dxt
where 3 represents the difference operator where 3wt = wt-l* ^ random
walk process given as
*t xt-i + 5
is homogenous of order one (first differenced). In fact, Xj. is station
ary and white noise. If a series is white noise, it is also stationary,
but the converse is not necessarily true.
Autoregressive (AR) Process
Many time series can be described as being an autoregressive pro
cess of order p such that xt is expressed as a weighted average of past
observations lagged p periods with a random disturbance on the end
P
*t Ei ^t-i + R + t 0,I,2,...,T
where <(> is the weight on each lagged xt, is the random disturbance, p
is some maximum lag, and R is a constant term associated with the series
mean and drift (R>0 when drift is present). Assuming R=0, this may also
be written in backshift notation as
(1 $jB ... 4>pBP)xt £t
*(B)xt Ct
where 4>( 8) = (l^i & *pBI>) and can be viewed as a polynomial of order
p in lag operator B. The left-hand factor $( 3) acts as a filter on the
time series x resulting in a white noise process Pindyck and Rubin-
feld (1981) state that a necessary condition that x is stationary
requires that the autoregressive process of order p be characterized by