NONSTATIONARY PROCESSES Stationary process X t 2 X t 1 t 1 2 1 E(Xt) 2X0
Download ReportTranscript NONSTATIONARY PROCESSES Stationary process X t 2 X t 1 t 1 2 1 E(Xt) 2X0
Slide 1
NONSTATIONARY PROCESSES
Stationary process
X t 2 X t 1 t
1 2 1
E(Xt) 2X0 0
t
1 2
2t
2
Xt
cov X t , X t s
1 2
2
2
2
1
2
2
1 2
s
2
2
1 2
In the last sequence, the process shown at the top was shown to be stationary. The
expected value and variance of Xt were shown to be (asymptotically) independent of time
and the covariance between Xt and Xt+s was also shown to be independent of time.
1
Slide 2
NONSTATIONARY PROCESSES
Random walk
X t X t 1 t
The condition –1 < 2 < 1 was crucial for stationarity. Suppose 2 = 1, as above. Then the
value of X in one time period is equal to its value in the previous time period, plus a random
adjustment. This is known as a random walk.
2
Slide 3
NONSTATIONARY PROCESSES
20
X t X t 1 t
t ~ N 0 ,1
10
0
0
10
20
30
40
50
-1 0
-2 0
The figure shows an example realization of a random walk for the case where t has a
normal distribution with zero mean and unit variance.
3
Slide 4
NONSTATIONARY PROCESSES
20
X t X t 1 t
t ~ N 0 ,1
10
0
-1 0
-2 0
0
10
20
30
40
50
This figure shows the results of a simulation with 50 realizations. It is obvious that the ensemble
distribution is not stationary. The distribution changes as t increases, becoming increasingly
spread out. We will confirm this mathematically.
4
Slide 5
NONSTATIONARY PROCESSES
Random walk
X t X t 1 t
X t 1 X t 2 t 1
If the process is valid for time t, it is valid for time t–1.
5
Slide 6
NONSTATIONARY PROCESSES
Random walk
X t X t 1 t
X t 1 X t 2 t 1
X t X t 2 t 1 t
Hence Xt can be expressed in terms of Xt–2 and the innovations t–1 and t.
6
Slide 7
NONSTATIONARY PROCESSES
Random walk
X t X t 1 t
X t 1 X t 2 t 1
X t X t 2 t 1 t
X t X 0 1 ... t 1 t
Thus, continuing to lag and substitute, Xt is equal to its value at time 0, X0, plus the sum of
the innovations in periods 1 to t.
7
Slide 8
NONSTATIONARY PROCESSES
Random walk
X t X t 1 t
X t X 0 1 ... t 1 t
E ( X t ) X 0 E ( 1 ) ... E ( n ) X 0
If expectations are taken at time 0, the expected value at any future time t is fixed at X0
because the expected values of the future innovations are all 0. Thus E(Xt) is independent
of t and the first condition for stationarity remains satisfied.
8
Slide 9
NONSTATIONARY PROCESSES
20
X t X t 1 t
t ~ N 0 ,1
10
0
-1 0
-2 0
0
10
20
30
40
50
This can be seen from the figure with 50 realizations. The distribution of the values of Xt
spreads out as t increases, but there is no tendency for the mean of the distribution to
change. (In this example X0 = 0, but this is unimportant. It would be true for any value of
X0.)
9
Slide 10
NONSTATIONARY PROCESSES
20
X t X t 1 t
t ~ N 0 ,1
10
0
-1 0
-2 0
0
10
20
30
40
50
However, it is also clear from the figure that the ensemble distribution is not constant over
time, and therefore that the process is nonstationary. The distribution of the values of Xt
spreads out as t increases, so the variance of the distribution is an increasing function of t.
10
Slide 11
NONSTATIONARY PROCESSES
Random walk
X t X t 1 t
X t X 0 1 ... t 1 t
2
Xt
var X 0 1 ... t 1 t
var 1 ... t 1 t
... t
2
2
2
2
We will demonstrate this mathematically. We have seen that Xt is equal to X0 plus the sum
of the innovations 1, ..., t. X0 is an additive constant, so it does not affect the variance.
11
Slide 12
NONSTATIONARY PROCESSES
Random walk
X t X t 1 t
X t X 0 1 ... t 1 t
2
Xt
var X 0 1 ... t 1 t
var 1 ... t 1 t
... t
2
2
2
2
The variance of the sum of the innovations is equal to the sum of their individual variances.
The covariances are all zero because the innovations are assumed to be generated
independently.
12
Slide 13
NONSTATIONARY PROCESSES
Random walk
X t X t 1 t
X t X 0 1 ... t 1 t
2
Xt
var X 0 1 ... t 1 t
var 1 ... t 1 t
... t
2
2
2
2
The variance of each innovation is equal to , by assumption. Hence the population
variance of Xt is directly proportional to t. As we have seen from the figure, its distribution
spreads out as t increases.
13
Slide 14
NONSTATIONARY PROCESSES
Stationary process
X t 1 2 X t 1 t
1 2
1 2 1
t
E(Xt)
1 2
1 2
1 X 0
t
2
2t
2
Xt
cov X t , X t s
1 2
2
2
2
1
1
1 2
2
2
1 2
s
2
2
1 2
A second process considered in the last sequence is shown above. The presence of the
intercept 1 on the right side gave the series a nonzero mean but did not lead to a violation
of the conditions for stationarity.
14
Slide 15
NONSTATIONARY PROCESSES
Random walk with drift
X t 1 X t 1 t
If 2 = 1, however, the series becomes a nonstationary process known as a random walk
with drift.
15
Slide 16
NONSTATIONARY PROCESSES
Random walk with drift
X t 1 X t 1 t
X t 1 1 X t 2 t 1
If the process is valid for time t, it is valid for time t–1.
16
Slide 17
NONSTATIONARY PROCESSES
Random walk with drift
X t 1 X t 1 t
X t 1 1 X t 2 t 1
X t 2 1 X t 2 t 1 t
Hence Xt can be expressed in terms of Xt–2, the innovations t–1 and t, and an intercept. The
intercept is 21. Irrespective of whatever else is happening to the process. a fixed quantity
1 is added in every time period.
17
Slide 18
NONSTATIONARY PROCESSES
Random walk with drift
X t 1 X t 1 t
X t 1 1 X t 2 t 1
X t 2 1 X t 2 t 1 t
X t 1 t X 0 1 ... t 1 t
Thus, lagging and substituting t times, Xt is now equal X0 plus the sum of the innovations,
as before, plus the constant 1 multiplied by t.
18
Slide 19
NONSTATIONARY PROCESSES
Random walk with drift
X t 1 X t 1 t
X t 1 t X 0 1 ... t 1 t
E ( X t ) X 0 1t
As a consequence, the mean of the process becomes a function of time, violating the first
condition for stationarity.
19
Slide 20
NONSTATIONARY PROCESSES
Random walk with drift
X t 1 X t 1 t
X t 1 t X 0 1 ... t 1 t
E ( X t ) X 0 1t
2
Xt
t
2
(The second condition for nonstationarity remains violated since the variance of the
distribution of Xt is proportional to t. It is unaffected by the inclusion of the constant 1.)
20
Slide 21
NONSTATIONARY PROCESSES
Random walk with drift
X t 1 X t 1 t
X t 1 t X 0 1 ... t 1 t
E ( X t ) X 0 1t
2
Xt
t
2
This process is known as a random walk with drift, the drift referring to the systematic
change in the expectation from one time period to the next.
21
Slide 22
NONSTATIONARY PROCESSES
50
X t 0 .5 X t 1 t
40
t ~ N 0 ,1
30
20
10
0
0
10
20
30
40
50
-1 0
-2 0
The figure shows 50 realizations of such a process. The underlying drift line is highlighted
in yellow. It can be seen that the ensemble distribution changes in two ways with time.
22
Slide 23
NONSTATIONARY PROCESSES
50
X t 0 .5 X t 1 t
40
t ~ N 0 ,1
30
20
10
0
0
10
20
30
40
50
-1 0
-2 0
The mean changes. In this case it is drifting upwards because 1 has been taken to be
positive. If 1 were negative, it would be drifting downwards.
23
Slide 24
NONSTATIONARY PROCESSES
50
X t 0 .5 X t 1 t
40
t ~ N 0 ,1
30
20
10
0
0
10
20
30
40
50
-1 0
-2 0
And, as in the case of the random walk with no drift, the distribution spreads out around its
mean.
24
Slide 25
NONSTATIONARY PROCESSES
Deterministic trend
X t 1 2t t
Random walks are not the only type of nonstationary process. Another common example of
a nonstationary time series is one possessing a time trend.
25
Slide 26
NONSTATIONARY PROCESSES
Deterministic trend
X t 1 2t t
This type of trend is described as a deterministic trend, to differentiate it from the trend
found in a model of a random walk with drift.
26
Slide 27
NONSTATIONARY PROCESSES
Deterministic trend
X t 1 2t t
E X t 1 2t
It is nonstationary because the expected value of Xt is not independent of t. Its population
variance is not even defined.
27
Slide 28
NONSTATIONARY PROCESSES
X t 1 2 t ut
20
u t 0 .8 u t 1 t
10
0
0
10
20
30
40
50
-1 0
The figure shows 50 realizations of a variation where the disturbance term is the stationary
process ut = 0.8ut–1 + t. The underlying trend line is shown in white.
28
Slide 29
NONSTATIONARY PROCESSES
Deterministic trend
X t 1 2t t
Random walk with drift
X t 1 t X 0 1 ... t 1 t
Superficially, this model looks similar to the random walk with drift, when the latter is
written in terms of its components from time 0.
29
Slide 30
NONSTATIONARY PROCESSES
20
Deterministic trend
10
50
0
0
10
20
30
40
50
40
X t 1 2t t
Random walk with drift
30
-1 0
20
10
0
0
-1 0
10
20
30
40
50
X t 1 t X 0 1 ... t 1 t
-2 0
The key difference between a deterministic trend and a random walk with drift is that in the
former, the series must keep coming back to a fixed trend line.
30
Slide 31
NONSTATIONARY PROCESSES
20
Deterministic trend
10
50
0
0
10
20
30
40
50
40
X t 1 2t t
Random walk with drift
30
-1 0
20
10
0
0
-1 0
10
20
30
40
50
X t 1 t X 0 1 ... t 1 t
-2 0
In any given observation, Xt will be displaced from the trend line by an amount ut, but,
provided that this is stationary, it must otherwise adhere to the trend line.
31
Slide 32
NONSTATIONARY PROCESSES
20
Deterministic trend
10
50
0
0
10
20
30
40
50
40
X t 1 2t t
Random walk with drift
30
-1 0
20
10
0
0
-1 0
10
20
30
40
50
X t 1 t X 0 1 ... t 1 t
-2 0
By contrast, in a random walk with drift, the displacement from the underlying trend line at
time t is the random walk . Since the displacement is a random walk, there is no reason
why Xt should ever return to its trend line.
32
Slide 33
NONSTATIONARY PROCESSES
Difference-stationarity and trend-stationarity
It is important to make a distinction between the concepts of difference-stationarity and
trend-stationarity.
33
Slide 34
NONSTATIONARY PROCESSES
Difference-stationarity
X t 1 X t 1 t
If a nonstationary process can be transformed into a stationary process by differencing, it is
said to be difference-stationary. A random walk, with or without drift, is an example.
34
Slide 35
NONSTATIONARY PROCESSES
Difference-stationarity
X t 1 X t 1 t
D X t X t X t 1 1 t
The first difference, DXt, is simply equal to the sum of 1 and t.
35
Slide 36
NONSTATIONARY PROCESSES
Difference-stationarity
X t 1 X t 1 t
D X t X t X t 1 1 t
E D X t 1
DX
2
2
t
cov D X t , D X t s 0
This is a stationary process with population mean 1 and variance 2, both independent of
time. It is actually iid and the covariance between DXt and DXt+s is zero.
36
Slide 37
NONSTATIONARY PROCESSES
Difference-stationarity
X t 1 X t 1 t
D X t X t X t 1 1 t
E D X t 1
DX
2
2
t
cov D X t , D X t s 0
If a nonstationary time series can be transformed into a stationary process by differencing
once, as in this case, it is described as integrated of order 1, or I(1).
37
Slide 38
NONSTATIONARY PROCESSES
Difference-stationarity
X t 1 X t 1 t
D X t X t X t 1 1 t
E D X t 1
DX
2
2
t
cov D X t , D X t s 0
The reason that the series is described as 'integrated' is that the shock in each time period
is permanently incorporated in it. There is no tendency for the effects of the shocks to
attenuate with time, as in a stationary process or in a model with a deterministic trend.
38
Slide 39
NONSTATIONARY PROCESSES
Difference-stationarity
X t 1 X t 1 t
D X t X t X t 1 1 t
E D X t 1
DX
2
2
t
cov D X t , D X t s 0
If a series can be made stationary by differencing twice, it is known as I(2), and so on. To
complete the picture, a stationary process, which by definition needs no differencing, is
described as I(0). In practice most series are I(0), I(1), or, occasionally, I(2).
39
Slide 40
NONSTATIONARY PROCESSES
Difference-stationarity
X t 1 X t 1 t
D X t X t X t 1 1 t
E D X t 1
DX
2
2
t
cov D X t , D X t s 0
The stochastic component t is iid. More generally, the stationary process reached after
differencing may be ARMA(p, q): auto-regressive of order p and moving average of order q.
40
Slide 41
NONSTATIONARY PROCESSES
Difference-stationarity
X t 1 X t 1 t
D X t X t X t 1 1 t
E D X t 1
DX
2
2
t
cov D X t , D X t s 0
The original series is then characterized as an ARIMA(p, d, q) time series, where d is the
number of times it has to be differenced to render it stationary.
41
Slide 42
NONSTATIONARY PROCESSES
Trend-stationarity
X t 1 2t t
A nonstationary time series is described as being trend-stationary if it can be transformed
into a stationary process by extracting a time trend.
42
Slide 43
NONSTATIONARY PROCESSES
Trend-stationarity
X t 1 2t t
Xˆ t b1 b 2 t
~
X t X t Xˆ t X t b1 b 2 t
For example, the very simple model given by the first equation can be detrended by fitting it
(second equation) and defining a new variable with the third equation. The new, detrended,
variable is of course just the residuals from the regression of X on t.
43
Slide 44
NONSTATIONARY PROCESSES
Trend-stationarity
X t 1 2t t
Xˆ t b1 b 2 t
~
X t X t Xˆ t X t b1 b 2 t
The distinction between difference-stationarity and trend-stationarity is important for the
analysis of time series.
44
Slide 45
NONSTATIONARY PROCESSES
Trend-stationarity
X t 1 2t t
Xˆ t b1 b 2 t
~
X t X t Xˆ t X t b1 b 2 t
At one time it was conventional to assume that macroeconomic time series could be
decomposed into trend and cyclical components, the former being determined by real
factors, such as the growth of GDP, and the latter being determined by transitory factors,
such as monetary policy.
45
Slide 46
NONSTATIONARY PROCESSES
Trend-stationarity
X t 1 2t t
Xˆ t b1 b 2 t
~
X t X t Xˆ t X t b1 b 2 t
Typically the cyclical component was analyzed using detrended versions of the variables in
the model.
46
Slide 47
NONSTATIONARY PROCESSES
Deterministic trend
X t 1 2t t
Random walk with drift
X t 1 t X 0 1 ... t 1 t
However, this approach is inappropriate if the process is difference-stationary. Although
detrending may remove any drift, it does not affect the increasing variance of the series,
and so the detrended component remains nonstationary.
47
Slide 48
NONSTATIONARY PROCESSES
Deterministic trend
X t 1 2t t
Random walk with drift
X t 1 t X 0 1 ... t 1 t
As will be seen in the next slideshow, this gives rise to problems of estimation and
inference.
48
Slide 49
NONSTATIONARY PROCESSES
Deterministic trend
X t 1 2t t
Random walk with drift
X t 1 t X 0 1 ... t 1 t
Further, because the approach ignores the contribution of real shocks to economic
fluctuations, it causes the role of transitory factors in the cycle to be overestimated.
49
Slide 50
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 13.1 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25
NONSTATIONARY PROCESSES
Stationary process
X t 2 X t 1 t
1 2 1
E(Xt) 2X0 0
t
1 2
2t
2
Xt
cov X t , X t s
1 2
2
2
2
1
2
2
1 2
s
2
2
1 2
In the last sequence, the process shown at the top was shown to be stationary. The
expected value and variance of Xt were shown to be (asymptotically) independent of time
and the covariance between Xt and Xt+s was also shown to be independent of time.
1
Slide 2
NONSTATIONARY PROCESSES
Random walk
X t X t 1 t
The condition –1 < 2 < 1 was crucial for stationarity. Suppose 2 = 1, as above. Then the
value of X in one time period is equal to its value in the previous time period, plus a random
adjustment. This is known as a random walk.
2
Slide 3
NONSTATIONARY PROCESSES
20
X t X t 1 t
t ~ N 0 ,1
10
0
0
10
20
30
40
50
-1 0
-2 0
The figure shows an example realization of a random walk for the case where t has a
normal distribution with zero mean and unit variance.
3
Slide 4
NONSTATIONARY PROCESSES
20
X t X t 1 t
t ~ N 0 ,1
10
0
-1 0
-2 0
0
10
20
30
40
50
This figure shows the results of a simulation with 50 realizations. It is obvious that the ensemble
distribution is not stationary. The distribution changes as t increases, becoming increasingly
spread out. We will confirm this mathematically.
4
Slide 5
NONSTATIONARY PROCESSES
Random walk
X t X t 1 t
X t 1 X t 2 t 1
If the process is valid for time t, it is valid for time t–1.
5
Slide 6
NONSTATIONARY PROCESSES
Random walk
X t X t 1 t
X t 1 X t 2 t 1
X t X t 2 t 1 t
Hence Xt can be expressed in terms of Xt–2 and the innovations t–1 and t.
6
Slide 7
NONSTATIONARY PROCESSES
Random walk
X t X t 1 t
X t 1 X t 2 t 1
X t X t 2 t 1 t
X t X 0 1 ... t 1 t
Thus, continuing to lag and substitute, Xt is equal to its value at time 0, X0, plus the sum of
the innovations in periods 1 to t.
7
Slide 8
NONSTATIONARY PROCESSES
Random walk
X t X t 1 t
X t X 0 1 ... t 1 t
E ( X t ) X 0 E ( 1 ) ... E ( n ) X 0
If expectations are taken at time 0, the expected value at any future time t is fixed at X0
because the expected values of the future innovations are all 0. Thus E(Xt) is independent
of t and the first condition for stationarity remains satisfied.
8
Slide 9
NONSTATIONARY PROCESSES
20
X t X t 1 t
t ~ N 0 ,1
10
0
-1 0
-2 0
0
10
20
30
40
50
This can be seen from the figure with 50 realizations. The distribution of the values of Xt
spreads out as t increases, but there is no tendency for the mean of the distribution to
change. (In this example X0 = 0, but this is unimportant. It would be true for any value of
X0.)
9
Slide 10
NONSTATIONARY PROCESSES
20
X t X t 1 t
t ~ N 0 ,1
10
0
-1 0
-2 0
0
10
20
30
40
50
However, it is also clear from the figure that the ensemble distribution is not constant over
time, and therefore that the process is nonstationary. The distribution of the values of Xt
spreads out as t increases, so the variance of the distribution is an increasing function of t.
10
Slide 11
NONSTATIONARY PROCESSES
Random walk
X t X t 1 t
X t X 0 1 ... t 1 t
2
Xt
var X 0 1 ... t 1 t
var 1 ... t 1 t
... t
2
2
2
2
We will demonstrate this mathematically. We have seen that Xt is equal to X0 plus the sum
of the innovations 1, ..., t. X0 is an additive constant, so it does not affect the variance.
11
Slide 12
NONSTATIONARY PROCESSES
Random walk
X t X t 1 t
X t X 0 1 ... t 1 t
2
Xt
var X 0 1 ... t 1 t
var 1 ... t 1 t
... t
2
2
2
2
The variance of the sum of the innovations is equal to the sum of their individual variances.
The covariances are all zero because the innovations are assumed to be generated
independently.
12
Slide 13
NONSTATIONARY PROCESSES
Random walk
X t X t 1 t
X t X 0 1 ... t 1 t
2
Xt
var X 0 1 ... t 1 t
var 1 ... t 1 t
... t
2
2
2
2
The variance of each innovation is equal to , by assumption. Hence the population
variance of Xt is directly proportional to t. As we have seen from the figure, its distribution
spreads out as t increases.
13
Slide 14
NONSTATIONARY PROCESSES
Stationary process
X t 1 2 X t 1 t
1 2
1 2 1
t
E(Xt)
1 2
1 2
1 X 0
t
2
2t
2
Xt
cov X t , X t s
1 2
2
2
2
1
1
1 2
2
2
1 2
s
2
2
1 2
A second process considered in the last sequence is shown above. The presence of the
intercept 1 on the right side gave the series a nonzero mean but did not lead to a violation
of the conditions for stationarity.
14
Slide 15
NONSTATIONARY PROCESSES
Random walk with drift
X t 1 X t 1 t
If 2 = 1, however, the series becomes a nonstationary process known as a random walk
with drift.
15
Slide 16
NONSTATIONARY PROCESSES
Random walk with drift
X t 1 X t 1 t
X t 1 1 X t 2 t 1
If the process is valid for time t, it is valid for time t–1.
16
Slide 17
NONSTATIONARY PROCESSES
Random walk with drift
X t 1 X t 1 t
X t 1 1 X t 2 t 1
X t 2 1 X t 2 t 1 t
Hence Xt can be expressed in terms of Xt–2, the innovations t–1 and t, and an intercept. The
intercept is 21. Irrespective of whatever else is happening to the process. a fixed quantity
1 is added in every time period.
17
Slide 18
NONSTATIONARY PROCESSES
Random walk with drift
X t 1 X t 1 t
X t 1 1 X t 2 t 1
X t 2 1 X t 2 t 1 t
X t 1 t X 0 1 ... t 1 t
Thus, lagging and substituting t times, Xt is now equal X0 plus the sum of the innovations,
as before, plus the constant 1 multiplied by t.
18
Slide 19
NONSTATIONARY PROCESSES
Random walk with drift
X t 1 X t 1 t
X t 1 t X 0 1 ... t 1 t
E ( X t ) X 0 1t
As a consequence, the mean of the process becomes a function of time, violating the first
condition for stationarity.
19
Slide 20
NONSTATIONARY PROCESSES
Random walk with drift
X t 1 X t 1 t
X t 1 t X 0 1 ... t 1 t
E ( X t ) X 0 1t
2
Xt
t
2
(The second condition for nonstationarity remains violated since the variance of the
distribution of Xt is proportional to t. It is unaffected by the inclusion of the constant 1.)
20
Slide 21
NONSTATIONARY PROCESSES
Random walk with drift
X t 1 X t 1 t
X t 1 t X 0 1 ... t 1 t
E ( X t ) X 0 1t
2
Xt
t
2
This process is known as a random walk with drift, the drift referring to the systematic
change in the expectation from one time period to the next.
21
Slide 22
NONSTATIONARY PROCESSES
50
X t 0 .5 X t 1 t
40
t ~ N 0 ,1
30
20
10
0
0
10
20
30
40
50
-1 0
-2 0
The figure shows 50 realizations of such a process. The underlying drift line is highlighted
in yellow. It can be seen that the ensemble distribution changes in two ways with time.
22
Slide 23
NONSTATIONARY PROCESSES
50
X t 0 .5 X t 1 t
40
t ~ N 0 ,1
30
20
10
0
0
10
20
30
40
50
-1 0
-2 0
The mean changes. In this case it is drifting upwards because 1 has been taken to be
positive. If 1 were negative, it would be drifting downwards.
23
Slide 24
NONSTATIONARY PROCESSES
50
X t 0 .5 X t 1 t
40
t ~ N 0 ,1
30
20
10
0
0
10
20
30
40
50
-1 0
-2 0
And, as in the case of the random walk with no drift, the distribution spreads out around its
mean.
24
Slide 25
NONSTATIONARY PROCESSES
Deterministic trend
X t 1 2t t
Random walks are not the only type of nonstationary process. Another common example of
a nonstationary time series is one possessing a time trend.
25
Slide 26
NONSTATIONARY PROCESSES
Deterministic trend
X t 1 2t t
This type of trend is described as a deterministic trend, to differentiate it from the trend
found in a model of a random walk with drift.
26
Slide 27
NONSTATIONARY PROCESSES
Deterministic trend
X t 1 2t t
E X t 1 2t
It is nonstationary because the expected value of Xt is not independent of t. Its population
variance is not even defined.
27
Slide 28
NONSTATIONARY PROCESSES
X t 1 2 t ut
20
u t 0 .8 u t 1 t
10
0
0
10
20
30
40
50
-1 0
The figure shows 50 realizations of a variation where the disturbance term is the stationary
process ut = 0.8ut–1 + t. The underlying trend line is shown in white.
28
Slide 29
NONSTATIONARY PROCESSES
Deterministic trend
X t 1 2t t
Random walk with drift
X t 1 t X 0 1 ... t 1 t
Superficially, this model looks similar to the random walk with drift, when the latter is
written in terms of its components from time 0.
29
Slide 30
NONSTATIONARY PROCESSES
20
Deterministic trend
10
50
0
0
10
20
30
40
50
40
X t 1 2t t
Random walk with drift
30
-1 0
20
10
0
0
-1 0
10
20
30
40
50
X t 1 t X 0 1 ... t 1 t
-2 0
The key difference between a deterministic trend and a random walk with drift is that in the
former, the series must keep coming back to a fixed trend line.
30
Slide 31
NONSTATIONARY PROCESSES
20
Deterministic trend
10
50
0
0
10
20
30
40
50
40
X t 1 2t t
Random walk with drift
30
-1 0
20
10
0
0
-1 0
10
20
30
40
50
X t 1 t X 0 1 ... t 1 t
-2 0
In any given observation, Xt will be displaced from the trend line by an amount ut, but,
provided that this is stationary, it must otherwise adhere to the trend line.
31
Slide 32
NONSTATIONARY PROCESSES
20
Deterministic trend
10
50
0
0
10
20
30
40
50
40
X t 1 2t t
Random walk with drift
30
-1 0
20
10
0
0
-1 0
10
20
30
40
50
X t 1 t X 0 1 ... t 1 t
-2 0
By contrast, in a random walk with drift, the displacement from the underlying trend line at
time t is the random walk . Since the displacement is a random walk, there is no reason
why Xt should ever return to its trend line.
32
Slide 33
NONSTATIONARY PROCESSES
Difference-stationarity and trend-stationarity
It is important to make a distinction between the concepts of difference-stationarity and
trend-stationarity.
33
Slide 34
NONSTATIONARY PROCESSES
Difference-stationarity
X t 1 X t 1 t
If a nonstationary process can be transformed into a stationary process by differencing, it is
said to be difference-stationary. A random walk, with or without drift, is an example.
34
Slide 35
NONSTATIONARY PROCESSES
Difference-stationarity
X t 1 X t 1 t
D X t X t X t 1 1 t
The first difference, DXt, is simply equal to the sum of 1 and t.
35
Slide 36
NONSTATIONARY PROCESSES
Difference-stationarity
X t 1 X t 1 t
D X t X t X t 1 1 t
E D X t 1
DX
2
2
t
cov D X t , D X t s 0
This is a stationary process with population mean 1 and variance 2, both independent of
time. It is actually iid and the covariance between DXt and DXt+s is zero.
36
Slide 37
NONSTATIONARY PROCESSES
Difference-stationarity
X t 1 X t 1 t
D X t X t X t 1 1 t
E D X t 1
DX
2
2
t
cov D X t , D X t s 0
If a nonstationary time series can be transformed into a stationary process by differencing
once, as in this case, it is described as integrated of order 1, or I(1).
37
Slide 38
NONSTATIONARY PROCESSES
Difference-stationarity
X t 1 X t 1 t
D X t X t X t 1 1 t
E D X t 1
DX
2
2
t
cov D X t , D X t s 0
The reason that the series is described as 'integrated' is that the shock in each time period
is permanently incorporated in it. There is no tendency for the effects of the shocks to
attenuate with time, as in a stationary process or in a model with a deterministic trend.
38
Slide 39
NONSTATIONARY PROCESSES
Difference-stationarity
X t 1 X t 1 t
D X t X t X t 1 1 t
E D X t 1
DX
2
2
t
cov D X t , D X t s 0
If a series can be made stationary by differencing twice, it is known as I(2), and so on. To
complete the picture, a stationary process, which by definition needs no differencing, is
described as I(0). In practice most series are I(0), I(1), or, occasionally, I(2).
39
Slide 40
NONSTATIONARY PROCESSES
Difference-stationarity
X t 1 X t 1 t
D X t X t X t 1 1 t
E D X t 1
DX
2
2
t
cov D X t , D X t s 0
The stochastic component t is iid. More generally, the stationary process reached after
differencing may be ARMA(p, q): auto-regressive of order p and moving average of order q.
40
Slide 41
NONSTATIONARY PROCESSES
Difference-stationarity
X t 1 X t 1 t
D X t X t X t 1 1 t
E D X t 1
DX
2
2
t
cov D X t , D X t s 0
The original series is then characterized as an ARIMA(p, d, q) time series, where d is the
number of times it has to be differenced to render it stationary.
41
Slide 42
NONSTATIONARY PROCESSES
Trend-stationarity
X t 1 2t t
A nonstationary time series is described as being trend-stationary if it can be transformed
into a stationary process by extracting a time trend.
42
Slide 43
NONSTATIONARY PROCESSES
Trend-stationarity
X t 1 2t t
Xˆ t b1 b 2 t
~
X t X t Xˆ t X t b1 b 2 t
For example, the very simple model given by the first equation can be detrended by fitting it
(second equation) and defining a new variable with the third equation. The new, detrended,
variable is of course just the residuals from the regression of X on t.
43
Slide 44
NONSTATIONARY PROCESSES
Trend-stationarity
X t 1 2t t
Xˆ t b1 b 2 t
~
X t X t Xˆ t X t b1 b 2 t
The distinction between difference-stationarity and trend-stationarity is important for the
analysis of time series.
44
Slide 45
NONSTATIONARY PROCESSES
Trend-stationarity
X t 1 2t t
Xˆ t b1 b 2 t
~
X t X t Xˆ t X t b1 b 2 t
At one time it was conventional to assume that macroeconomic time series could be
decomposed into trend and cyclical components, the former being determined by real
factors, such as the growth of GDP, and the latter being determined by transitory factors,
such as monetary policy.
45
Slide 46
NONSTATIONARY PROCESSES
Trend-stationarity
X t 1 2t t
Xˆ t b1 b 2 t
~
X t X t Xˆ t X t b1 b 2 t
Typically the cyclical component was analyzed using detrended versions of the variables in
the model.
46
Slide 47
NONSTATIONARY PROCESSES
Deterministic trend
X t 1 2t t
Random walk with drift
X t 1 t X 0 1 ... t 1 t
However, this approach is inappropriate if the process is difference-stationary. Although
detrending may remove any drift, it does not affect the increasing variance of the series,
and so the detrended component remains nonstationary.
47
Slide 48
NONSTATIONARY PROCESSES
Deterministic trend
X t 1 2t t
Random walk with drift
X t 1 t X 0 1 ... t 1 t
As will be seen in the next slideshow, this gives rise to problems of estimation and
inference.
48
Slide 49
NONSTATIONARY PROCESSES
Deterministic trend
X t 1 2t t
Random walk with drift
X t 1 t X 0 1 ... t 1 t
Further, because the approach ignores the contribution of real shocks to economic
fluctuations, it causes the role of transitory factors in the cycle to be overestimated.
49
Slide 50
Copyright Christopher Dougherty 2011.
These slideshows may be downloaded by anyone, anywhere for personal use.
Subject to respect for copyright and, where appropriate, attribution, they may be
used as a resource for teaching an econometrics course. There is no need to
refer to the author.
The content of this slideshow comes from Section 13.1 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
Additional (free) resources for both students and instructors may be
downloaded from the OUP Online Resource Centre
http://www.oup.com/uk/orc/bin/9780199567089/.
Individuals studying econometrics on their own and who feel that they might
benefit from participation in a formal course should consider the London School
of Economics summer school course
EC212 Introduction to Econometrics
http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx
or the University of London International Programmes distance learning course
20 Elements of Econometrics
www.londoninternational.ac.uk/lse.
11.07.25