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ApEc 8212
Applied Econometrics

Lecture #28
Time Series Analysis VII: Trends in Time Series Data
(Enders, Chap. 4, Sections 17)
I. Deterministic and Stochastic Trends
In general, stochastic difference equations have 3 parts:
y
t
= trend + stationary component + “noise”
So far we have focused on the stationary component
(Enders, Chapter 2) and on the noise (Enders, Chap. 3),
assuming no trend
. Yet many (most?) times series have
trends and so are not stationary. This lecture shows how
to analyze data that have trends.
We can define a
trend
as
anything that causes a time
series variable not to have a longrun mean
.
Perhaps
the simplest example of a time series with a trend is:
∆
y
t
= a
0
The solution to this linear difference equation is:
y
t
= y
0
+ a
0
t
where y
0
is the initial condition at time period zero.
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Next, add a moving average component to this model:
y
t
= y
0
+ a
0
t + B
(L)ε
t
= y
0
+ a
0
t + ε
t
+
β
1
ε
t1
+ … +
β
p
ε
tp
Question:
Is the moving average part, B
(L)ε
t
, stationary?
In this equation y
t
differs from its trend value (= y
0
+ a
0
t)
by B
(L)ε
t
.
B
(L)ε
t
is stationary, so this deviation from
trend is temporary. This model is called
trend stationary
:
after removing the trend part, what remains is stationary.
Trends can be both “
deterministic
” (nonstochastic) and
stochastic
.
A simple example of the latter is:
∆
y
t
= a
0
+
ε
t
where
ε
t
is white noise (Var
(ε
t
) =
σ
2
). This may seem to be
a small change, but it has a big effect on y
t
. You can verify
that the general solution to this difference equation is:
y
t
= y
0
+
Σ
=
t
1
i
ε
i
+ a
0
t
Think of
Σ
=
t
1
i
ε
t
as a stochastic (random) intercept.
Note:
the effects of past
ε’
s do not “decay” over time; these
3
effects are “permanent”.
Such a process has a
stochastic
trend
(in addition to a
0
t, the deterministic trend):
each ε
t
permanently
changes the (conditional) mean of the series.
Question
: Is the y
0
+
Σ
=
t
1
i
ε
t
part of y
t
stationary?
Four Time Series with Different Trends
8
6
4
2
0
2
4
6
8
0
10
20
30
40
50
60
70
80
90
100
Time
Random walk
0
8
16
24
32
40
48
56
0
10
20
30
40
50
60
70
80
90
100
Time
Random walk plus drift
0
8
16
24
32
40
48
56
0
10
20
30
40
50
60
70
80
90
100
Time
Trend stationary
8
6
4
2
0
2
4
6
8
0
10
20
30
40
50
60
70
80
90
100
Time
Random walk plus noise
1. Random Walk Model
A special case of the stochastic trend model given above
is the case where a
0
= 0.
This gives the
random walk
:
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y
t
= y
t1
+
ε
t
,
which implies
∆
y
t
=
ε
t
This is an AR(1) model, y
t
= a
0
+ a
1
y
t1
+ ε
t
, with a
0
= 0
1
= 1. The general solution, conditional on y
t
, is:
y
t
= y
0
+
Σ
=
t
i
1
ε
i
The unconditional expectation (mean) is E[y
t
] = y
0
.
The
conditional expectations, at time t, of y
t+1
and y
t+s
are:
E
t
[y
t+1
] = E
t
[y
t
+
ε
t+1
] = y
t
E
t
[y
t+s
] = E
t
[y
t
+
Σ
=
s
i
1
ε
t+i
] = y
t
Random walk processes are
not
stationary
because the
(unconditional) variance is not constant over all t:
Var(y
t
) = Var(
ε
t
+
ε
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This note was uploaded on 02/24/2010 for the course ECON 570 taught by Professor Staff during the Fall '08 term at UNC.
 Fall '08
 Staff
 Econometrics

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