1
EconometricsQ. He
1
Time Series Data
y
t
=
0
+
1
x
t1
+ . . .+
k
x
tk
+ u
t
2. Further Issues
EconometricsQ. He
2
Stationary Stochastic Process
A stochastic process is stationary if for
every collection of time indices 1
≤
t
1
< …< t
m
the joint distribution of (
x
t1
, … , x
tm
)
is the same as that of (
x
t1+h
, … x
tm+h
) for
h
≥
1
Thus, stationarity implies that the
x
t
’s are
identically distributed and that the nature of
any correlation between adjacent terms is
the same across all periods
EconometricsQ. He
3
Covariance Stationary Process
A stochastic process is covariance
stationary if E(
x
t
) is constant, Var(
x
t
) is
constant and for any
t
,
h
≥
1, Cov(
x
t
,
x
t+h
)
depends only on
h
and not on
t.
Thus, this weaker form of stationarity
requires only that the mean and variance are
constant across time, and the covariance just
depends on the distance across time.
EconometricsQ. He
4
Weakly Dependent Time Series
A stationary time series is weakly
dependent if
x
t
and
x
t+h
are “almost
independent”as
h
increases
If for a covariance stationary process
Corr(
x
t
,
x
t+h
)
→
0 as
h
→ ∞
, we’
ll say this
covariance stationary process is weakly
dependent
Want to still use law of large numbers
EconometricsQ. He
5
An MA(1) Process
A moving average process of order one
[MA(1)] can be characterized as one where
x
t
= e
t
+
1
e
t1
, t
= 1, 2, … with
e
t
being an
iid sequence with mean 0 and variance
2
e
This is a stationary, weakly dependent
sequence as variables 1 period apart are
correlated, but 2 periods apart they are not
EconometricsQ. He
6
An AR(1) Process
An autoregressive process of order one
[AR(1)] can be characterized as one where
y
t
=
y
t1
+ e
t
, t
= 1, 2,… with
e
t
being an
iid sequence with mean 0 and variance
e
2
For this process to be weakly dependent, it
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 Fall '08
 Staff
 Econometrics, Normal Distribution, Regression Analysis, Stochastic process, Autocorrelation, Stationary process

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