Economics 20 - Prof,Anderson 1
Stationary Stochastic Process
A stochastic process is stationary if for
every collection of time indices 1 ≤ t1 < …<
tm the joint distribution of (xt1,…,xtm) is the
same as that of (xt1+h,… xtm+h) for h ≥ 1
Thus,stationarity implies that the xt’s are
identically distributed and that the nature of
any correlation between adjacent terms is
the same across all periods
Economics 20 - Prof,Anderson 2
Covariance Stationary Process
A stochastic process is covariance
stationary if E(xt) is constant,Var(xt) is
constant and for any t,h ≥ 1,Cov(xt,xt+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
Economics 20 - Prof,Anderson 3
Weakly Dependent Time Series
A stationary time series is weakly
dependent if xt and xt+h are,almost
independent” as h increases
If for a covariance stationary process
Corr(xt,xt+h) → 0 as h → ∞,we’ll say this
covariance stationary process is weakly
dependent
Want to still use law of large numbers
Economics 20 - Prof,Anderson 4
An MA(1) Process
A moving average process of order one
[MA(1)] can be characterized as one where
xt = et + a1et-1,t = 1,2,… with et being an
iid sequence with mean 0 and variance s2e
This is a stationary,weakly dependent
sequence as variables 1 period apart are
correlated,but 2 periods apart they are not
Economics 20 - Prof,Anderson 5
An AR(1) Process
An autoregressive process of order one
[AR(1)] can be characterized as one where
yt = ryt-1 + et,t = 1,2,… with et being an
iid sequence with mean 0 and variance se2
For this process to be weakly dependent,it
must be the case that |r| < 1
Corr(yt,yt+h) = Cov(yt,yt+h)/(sysy) = r1h
which becomes small as h increases
Economics 20 - Prof,Anderson 6
Trends Revisited
A trending series cannot be stationary,
since the mean is changing over time
A trending series can be weakly dependent
If a series is weakly dependent and is
stationary about its trend,we will call it a
trend-stationary process
As long as a trend is included,all is well
Economics 20 - Prof,Anderson 7
Assumptions for Consistency
Linearity and Weak Dependence
A weaker zero conditional mean
assumption,E(ut|xt) = 0,for each t
No Perfect Collinearity
Thus,for asymptotic unbiasedness
(consistency),we can weaken the
exogeneity assumptions somewhat relative
to those for unbiasedness
Economics 20 - Prof,Anderson 8
Large-Sample Inference
Weaker assumption of homoskedasticity,
Var (ut|xt) = s2,for each t
Weaker assumption of no serial correlation,
E(utus| xt,xs) = 0 for t ? s
With these assumptions,we have
asymptotic normality and the usual standard
errors,t statistics,F statistics and LM
statistics are valid
Economics 20 - Prof,Anderson 9
Random Walks
A random walk is an AR(1) model where
r1 = 1,meaning the series is not weakly
dependent
With a random walk,the expected value of
yt is always y0 – it doesn’t depend on t
Var(yt) = se2t,so it increases with t
We say a random walk is highly persistent
since E(yt+h|yt) = yt for all h ≥ 1
Economics 20 - Prof,Anderson 10
Random Walks (continued)
A random walk is a special case of what’s
known as a unit root process
Note that trending and persistence are
different things – a series can be trending
but weakly dependent,or a series can be
highly persistent without any trend
A random walk with drift is an example of
a highly persistent series that is trending
Economics 20 - Prof,Anderson 11
Transforming Persistent Series
In order to use a highly persistent series
and get meaningful estimates and make
correct inferences,we want to transform it
into a weakly dependent process
We refer to a weakly dependent process as
being integrated of order zero,[I(0)]
A random walk is integrated of order one,
[I(1)],meaning a first difference will be I(0)
Stationary Stochastic Process
A stochastic process is stationary if for
every collection of time indices 1 ≤ t1 < …<
tm the joint distribution of (xt1,…,xtm) is the
same as that of (xt1+h,… xtm+h) for h ≥ 1
Thus,stationarity implies that the xt’s are
identically distributed and that the nature of
any correlation between adjacent terms is
the same across all periods
Economics 20 - Prof,Anderson 2
Covariance Stationary Process
A stochastic process is covariance
stationary if E(xt) is constant,Var(xt) is
constant and for any t,h ≥ 1,Cov(xt,xt+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
Economics 20 - Prof,Anderson 3
Weakly Dependent Time Series
A stationary time series is weakly
dependent if xt and xt+h are,almost
independent” as h increases
If for a covariance stationary process
Corr(xt,xt+h) → 0 as h → ∞,we’ll say this
covariance stationary process is weakly
dependent
Want to still use law of large numbers
Economics 20 - Prof,Anderson 4
An MA(1) Process
A moving average process of order one
[MA(1)] can be characterized as one where
xt = et + a1et-1,t = 1,2,… with et being an
iid sequence with mean 0 and variance s2e
This is a stationary,weakly dependent
sequence as variables 1 period apart are
correlated,but 2 periods apart they are not
Economics 20 - Prof,Anderson 5
An AR(1) Process
An autoregressive process of order one
[AR(1)] can be characterized as one where
yt = ryt-1 + et,t = 1,2,… with et being an
iid sequence with mean 0 and variance se2
For this process to be weakly dependent,it
must be the case that |r| < 1
Corr(yt,yt+h) = Cov(yt,yt+h)/(sysy) = r1h
which becomes small as h increases
Economics 20 - Prof,Anderson 6
Trends Revisited
A trending series cannot be stationary,
since the mean is changing over time
A trending series can be weakly dependent
If a series is weakly dependent and is
stationary about its trend,we will call it a
trend-stationary process
As long as a trend is included,all is well
Economics 20 - Prof,Anderson 7
Assumptions for Consistency
Linearity and Weak Dependence
A weaker zero conditional mean
assumption,E(ut|xt) = 0,for each t
No Perfect Collinearity
Thus,for asymptotic unbiasedness
(consistency),we can weaken the
exogeneity assumptions somewhat relative
to those for unbiasedness
Economics 20 - Prof,Anderson 8
Large-Sample Inference
Weaker assumption of homoskedasticity,
Var (ut|xt) = s2,for each t
Weaker assumption of no serial correlation,
E(utus| xt,xs) = 0 for t ? s
With these assumptions,we have
asymptotic normality and the usual standard
errors,t statistics,F statistics and LM
statistics are valid
Economics 20 - Prof,Anderson 9
Random Walks
A random walk is an AR(1) model where
r1 = 1,meaning the series is not weakly
dependent
With a random walk,the expected value of
yt is always y0 – it doesn’t depend on t
Var(yt) = se2t,so it increases with t
We say a random walk is highly persistent
since E(yt+h|yt) = yt for all h ≥ 1
Economics 20 - Prof,Anderson 10
Random Walks (continued)
A random walk is a special case of what’s
known as a unit root process
Note that trending and persistence are
different things – a series can be trending
but weakly dependent,or a series can be
highly persistent without any trend
A random walk with drift is an example of
a highly persistent series that is trending
Economics 20 - Prof,Anderson 11
Transforming Persistent Series
In order to use a highly persistent series
and get meaningful estimates and make
correct inferences,we want to transform it
into a weakly dependent process
We refer to a weakly dependent process as
being integrated of order zero,[I(0)]
A random walk is integrated of order one,
[I(1)],meaning a first difference will be I(0)
