1. TESTS OF NONSTATIONARITY: TRENDED DATA
1
In this slideshow we consider testing for nonstationarity when an inspection of the graph of
a process reveals evidence of a trend.
t
t
t t
Y
Y
1
2
1
1
1 2
1
2
0
0
or
or
1
2
0
1
2
0
1
0
1
2
0
1
0
1
2
0
*
1
*
1
1
2
0
*
1
General model
Alternatives
Case (a)
Case (b)
Case (c)
Case (d)
Case (e)
2. 2
Cases (a) and (b), considered in the previous slideshow, can be eliminated because they do
not give rise to trends. Case (e) has been eliminated because it implies a quadratic trend.
TESTS OF NONSTATIONARITY: TRENDED DATA
t
t
t t
Y
Y
1
2
1
1
1 2
1
2
0
0
or
or
1
2
0
1
2
0
1
0
1
2
0
1
0
1
2
0
*
1
*
1
1
2
0
*
1
General model
Alternatives
Case (a)
Case (b)
Case (c)
Case (d)
Case (e)
3. 3
So we are left with Cases (c) and (d).
TESTS OF NONSTATIONARITY: TRENDED DATA
t
t
t t
Y
Y
1
2
1
1
1 2
1
2
0
0
or
or
1
2
0
1
2
0
1
0
1
2
0
1
0
1
2
0
*
1
*
1
1
2
0
*
1
General model
Alternatives
Case (a)
Case (b)
Case (c)
Case (d)
Case (e)
4. 4
We need to consider whether the process is better characterized as a random walk with
drift, as in Case (c), or a deterministic trend, as in Case (d).
TESTS OF NONSTATIONARITY: TRENDED DATA
t
t
t Y
Y
1
1
t
t
t t
Y
Y
1
2
1
1
1 2
1
2
0
0
or
or
General model
Alternatives
0
0
1
Case (c)
Case (d) 0
1
2
1
2
*
1
t
t
t t
Y
Y
1
2
1
5. 5
To do this, we fit the general model, as in Case (d), with no assumption about the
parameters. We can then test H0: b2 = 1 using as our test statistic either T(b2 – 1) or the t
statistic for b2, as before.
TESTS OF NONSTATIONARITY: TRENDED DATA
t
t
t Y
Y
1
1
t
t
t t
Y
Y
1
2
1
1
1 2
1
2
0
0
or
or
General model
Alternatives
0
0
1
Case (c)
Case (d) 0
1
2
1
2
*
1
t
t
t t
Y
Y
1
2
1
6. 6
The inclusion of the time trend in the specification causes the critical values under the null
hypothesis to be different from those in the untrended case. They are determined by
simulation methods, as before.
TESTS OF NONSTATIONARITY: TRENDED DATA
t
t
t Y
Y
1
1
t
t
t t
Y
Y
1
2
1
1
1 2
1
2
0
0
or
or
General model
Alternatives
0
0
1
Case (c)
Case (d) 0
1
2
1
2
*
1
t
t
t t
Y
Y
1
2
1
7. 7
We can also perform an F test. We have argued that a process cannot combine a random
walk with drift and a time trend, so we can test the composite hypothesis H0: b2 = 1, d = 0.
Critical values for the three tests are given in Table A.7 at the end of the text.
TESTS OF NONSTATIONARITY: TRENDED DATA
t
t
t Y
Y
1
1
t
t
t t
Y
Y
1
2
1
1
1 2
1
2
0
0
or
or
General model
Alternatives
0
0
1
Case (c)
Case (d) 0
1
2
1
2
*
1
t
t
t t
Y
Y
1
2
1
8. 8
If the null hypothesis is false, and Yt is therefore a stationary autoregressive process about
a deterministic trend, the OLS estimators of the parameters are √T consistent, and the
conventional test statistics are asymptotically valid.
TESTS OF NONSTATIONARITY: TRENDED DATA
t
t
t Y
Y
1
1
t
t
t t
Y
Y
1
2
1
1
1 2
1
2
0
0
or
or
General model
Alternatives
0
0
1
Case (c)
Case (d) 0
1
2
1
2
*
1
t
t
t t
Y
Y
1
2
1
9. 9
Two special cases should be mentioned, if only as econometric curiosities.
TESTS OF NONSTATIONARITY: TRENDED DATA
t
t
t Y
Y
1
1
t
t
t t
Y
Y
1
2
1
1
1 2
1
2
0
0
or
or
General model
Alternatives
0
0
1
Case (c)
Case (d) 0
1
2
1
2
*
1
t
t
t t
Y
Y
1
2
1
10. 10
In general, if a plot of the process exhibits a trend, we will not know whether it is caused by
a deterministic trend or a random walk with drift, and we have to allow for both by fitting the
general case, as in Case (d), with no restriction on the parameters.
TESTS OF NONSTATIONARITY: TRENDED DATA
t
t
t Y
Y
1
1
t
t
t t
Y
Y
1
2
1
1
1 2
1
2
0
0
or
or
General model
Alternatives
0
0
1
Case (c)
Case (d) 0
1
2
1
2
*
1
t
t
t t
Y
Y
1
2
1
11. 11
But if, for some reason, we know that the process is a deterministic trend or, alternatively,
we know that it is a random walk with drift, and if we fit the model appropriately, there is a
spectacular improvement in the properties of the OLS estimator of the slope coefficient.
TESTS OF NONSTATIONARITY: TRENDED DATA
t
t
t Y
Y
1
1
t
t
t t
Y
Y
1
2
1
1
1 2
1
2
0
0
or
or
General model
Alternatives
0
0
1
Case (c)
Case (d) 0
1
2
1
2
*
1
t
t
t t
Y
Y
1
2
1
12. 12
In the special case where b2 = 0 and the process is just a simple deterministic trend, we
encounter a surprising result.
0
50
100
0 0.1 0.2 0.3 0.4
T = 25
T = 50
T = 100
0
50
100
0 0.1 0.2 0.3 0.4
T = 25
T = 50
T = 100
TESTS OF NONSTATIONARITY: TRENDED DATA
Distribution of d Distribution of d
dt
b
Yt
1
ˆ dt
Y
b
b
Y t
t
1
2
1
ˆ
Fitted model:
Special case: process is a deterministic trend Yt = b1 +dt + et
13. 13
If it is known that there is no autoregressive component, and the regression model is
correctly specified with t as the only explanatory variable, the OLS estimator of d is
hyperconsistent, its variance being inversely proportional to T3
.
TESTS OF NONSTATIONARITY: TRENDED DATA
0
50
100
0 0.1 0.2 0.3 0.4
T = 25
T = 50
T = 100
0
50
100
0 0.1 0.2 0.3 0.4
T = 25
T = 50
T = 100
Distribution of d Distribution of d
dt
b
Yt
1
ˆ dt
Y
b
b
Y t
t
1
2
1
ˆ
Fitted model:
Special case: process is a deterministic trend Yt = b1 +dt + et
14. 14
This is illustrated for the case d = 0.2 in the left chart in the figure. Since the standard
deviation of the distribution is inversely proportional to T3/2
, the height is proportional to T3/2
,
and so it more than doubles when the sample size is doubled.
TESTS OF NONSTATIONARITY: TRENDED DATA
0
50
100
0 0.1 0.2 0.3 0.4
T = 25
T = 50
T = 100
0
50
100
0 0.1 0.2 0.3 0.4
T = 25
T = 50
T = 100
Distribution of d Distribution of d
dt
b
Yt
1
ˆ dt
Y
b
b
Y t
t
1
2
1
ˆ
Fitted model:
Special case: process is a deterministic trend Yt = b1 +dt + et
15. 15
If Yt–1 is mistakenly included in the regression model, the loss of efficiency is dramatic. The
estimator of d reverts to being only √T consistent. Further, it is subject to finite-sample
bias. This is illustrated in the right chart in the figure.
TESTS OF NONSTATIONARITY: TRENDED DATA
0
50
100
0 0.1 0.2 0.3 0.4
T = 25
T = 50
T = 100
0
50
100
0 0.1 0.2 0.3 0.4
T = 25
T = 50
T = 100
Distribution of d Distribution of d
dt
b
Yt
1
ˆ dt
Y
b
b
Y t
t
1
2
1
ˆ
Fitted model:
Special case: process is a deterministic trend Yt = b1 +dt + et
16. 16
In this special case, if the regression model is correctly specified, and the disturbance term
is normally distributed, OLS t and F tests are valid for finite samples, despite the
hyperconsistency of the estimator of d.
TESTS OF NONSTATIONARITY: TRENDED DATA
0
50
100
0 0.1 0.2 0.3 0.4
T = 25
T = 50
T = 100
0
50
100
0 0.1 0.2 0.3 0.4
T = 25
T = 50
T = 100
Distribution of d Distribution of d
dt
b
Yt
1
ˆ dt
Y
b
b
Y t
t
1
2
1
ˆ
Fitted model:
Special case: process is a deterministic trend Yt = b1 +dt + et
17. 17
If the disturbance term is not normal, but has constant variance and finite fourth moment,
the t and F tests are asymptotically valid.
TESTS OF NONSTATIONARITY: TRENDED DATA
0
50
100
0 0.1 0.2 0.3 0.4
T = 25
T = 50
T = 100
0
50
100
0 0.1 0.2 0.3 0.4
T = 25
T = 50
T = 100
Special case: process is a deterministic trend Yt = b1 +dt + et
Distribution of d Distribution of d
dt
b
Yt
1
ˆ dt
Y
b
b
Y t
t
1
2
1
ˆ
Fitted model:
18. 18
Similarly, in the special case where the process is a random walk with drift, so that b2 = 1
and d = 0, and the model is correctly specified with Yt–1 as the only explanatory variable, the
OLS estimator of b2 is hyperconsistent.
0
20
40
60
80
100
120
0.6 0.7 0.8 0.9 1 1.1
T = 200
T = 200
T = 50
T = 50
T = 100
T = 100
T = 25 T = 25
Red: time trend added
TESTS OF NONSTATIONARITY: TRENDED DATA
Special case: process is a random walk with drift Yt = b1 +Yt–1 + et
Distribution of b2
1
2
1
ˆ
t
t Y
b
b
Y
t
t
t Y
Y
1
1
19. 19
If a time trend is added to the specification by mistake, there is a loss of efficiency, but it is
not as dramatic as in the other special case.
TESTS OF NONSTATIONARITY: TRENDED DATA
0
20
40
60
80
100
120
0.6 0.7 0.8 0.9 1 1.1
T = 200
T = 200
T = 50
T = 50
T = 100
T = 100
T = 25 T = 25
Red: time trend added
Special case: process is a random walk with drift Yt = b1 +Yt–1 + et
Distribution of b2
1
2
1
ˆ
t
t Y
b
b
Y
t
t
t Y
Y
1
1
20. 20
The estimator is still superconsistent (variance inversely proportional to T2
). The
distributions for the various sample sizes for this case are shown as the red lines in the
figure.
TESTS OF NONSTATIONARITY: TRENDED DATA
0
20
40
60
80
100
120
0.6 0.7 0.8 0.9 1 1.1
T = 200
T = 200
T = 50
T = 50
T = 100
T = 100
T = 25 T = 25
Red: time trend added
Special case: process is a random walk with drift Yt = b1 +Yt–1 + et
Distribution of b2
1
2
1
ˆ
t
t Y
b
b
Y
t
t
t Y
Y
1
1
21. 21
The conventional t and F tests are asymptotically valid, but not valid for finite samples
because the process is autoregressive.
TESTS OF NONSTATIONARITY: TRENDED DATA
0
20
40
60
80
100
120
0.6 0.7 0.8 0.9 1 1.1
T = 200
T = 200
T = 50
T = 50
T = 100
T = 100
T = 25 T = 25
Red: time trend added
Special case: process is a random walk with drift Yt = b1 +Yt–1 + et
Distribution of b2
1
2
1
ˆ
t
t Y
b
b
Y
t
t
t Y
Y
1
1
22. 22
We need to generalize the discussion to higher order processes. We will start with the
second-order process shown.
TESTS OF NONSTATIONARITY: TRENDED DATA
t
t
t
t Y
Y
Y
2
3
1
2
1
Augmented Dickey–Fuller tests: second-order autoregressive process
Main condition for stationarity:
1
3
2
23. 23
To be stationary, the parameters now need to satisfy several conditions. The most
important in practice is |b2 + b3| < 1. To test this, it is convenient to reparameterize the
model.
TESTS OF NONSTATIONARITY: TRENDED DATA
t
t
t
t Y
Y
Y
2
3
1
2
1
Augmented Dickey–Fuller tests: second-order autoregressive process
Main condition for stationarity:
1
3
2
24. 24
Subtract Yt–1 from both sides, add and subtract b3Yt–1 on the right side, and group terms
together.
TESTS OF NONSTATIONARITY: TRENDED DATA
t
t
t
t Y
Y
Y
2
3
1
2
1
t
t
t
t
t
t
t
t
t
t
t
t
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
2
1
3
1
3
2
1
2
3
1
3
1
3
1
1
2
1
1
1
Augmented Dickey–Fuller tests: second-order autoregressive process
Main condition for stationarity:
1
3
2
25. 25
Thus we obtain a model where DYt = Yt – Yt–1 is related to Yt–1 and DYt–1, with b2
*
= b2 + b3 and
b3
*
= b3.
TESTS OF NONSTATIONARITY: TRENDED DATA
t
t
t
t Y
Y
Y
2
3
1
2
1
t
t
t
t
t
t
t
t
t
t
t
t
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
2
1
3
1
3
2
1
2
3
1
3
1
3
1
1
2
1
1
1
t
t
t
t
t
t
t
Y
Y
Y
Y
Y
1
*
3
1
*
2
1
1
3
1
3
2
1
1
1
3
2
*
2
3
*
3
2
1
1
t
t
t Y
Y
Y
Augmented Dickey–Fuller tests: second-order autoregressive process
Main condition for stationarity:
1
3
2
26. 26
Under the null hypothesis H0: b2
*
= 1, the process is nonstationary. Given the
reparameterization, H0 may be tested by testing whether the coefficient of Yt–1 is
significantly different from zero.
TESTS OF NONSTATIONARITY: TRENDED DATA
t
t
t
t Y
Y
Y
2
3
1
2
1
t
t
t
t
t
t
t
t
t
t
t
t
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
2
1
3
1
3
2
1
2
3
1
3
1
3
1
1
2
1
1
1
t
t
t
t
t
t
t
Y
Y
Y
Y
Y
1
*
3
1
*
2
1
1
3
1
3
2
1
1
1
3
2
*
2
3
*
3
2
1
1
t
t
t Y
Y
Y
Augmented Dickey–Fuller tests: second-order autoregressive process
Main condition for stationarity:
1
3
2
27. 27
One may usually perform a one-sided test with alternative hypothesis H1: b2
*
< 1 since b2
*
> 1
implies an explosive process.
TESTS OF NONSTATIONARITY: TRENDED DATA
t
t
t
t Y
Y
Y
2
3
1
2
1
t
t
t
t
t
t
t
t
t
t
t
t
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
2
1
3
1
3
2
1
2
3
1
3
1
3
1
1
2
1
1
1
t
t
t
t
t
t
t
Y
Y
Y
Y
Y
1
*
3
1
*
2
1
1
3
1
3
2
1
1
1
3
2
*
2
3
*
3
2
1
1
t
t
t Y
Y
Y
Augmented Dickey–Fuller tests: second-order autoregressive process
Main condition for stationarity:
1
3
2
28. 28
Under the null hypothesis, the estimator of b2
*
is superconsistent and the test statistics
T(b2
*
– 1), t, and F have the same distributions, and therefore critical values, as before.
TESTS OF NONSTATIONARITY: TRENDED DATA
t
t
t
t Y
Y
Y
2
3
1
2
1
t
t
t
t
t
t
t
t
t
t
t
t
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
2
1
3
1
3
2
1
2
3
1
3
1
3
1
1
2
1
1
1
t
t
t
t
t
t
t
Y
Y
Y
Y
Y
1
*
3
1
*
2
1
1
3
1
3
2
1
1
1
3
2
*
2
3
*
3
2
1
1
t
t
t Y
Y
Y
Augmented Dickey–Fuller tests: second-order autoregressive process
Main condition for stationarity:
1
3
2
29. 29
t
t
t
t Y
Y
Y
2
3
1
2
1
t
t
t
t
t
t
t
t
t
t
t
t
Y
Y
Y
Y
Y
Y
Y
Y
Y
Y
2
1
3
1
3
2
1
2
3
1
3
1
3
1
1
2
1
1
1
t
t
t
t
t
t
t
Y
Y
Y
Y
Y
1
*
3
1
*
2
1
1
3
1
3
2
1
1
1
3
2
*
2
3
*
3
2
1
1
t
t
t Y
Y
Y
If a deterministic time trend is suspected, it may be included and the critical values are
those for the first-order specification with a time trend.
TESTS OF NONSTATIONARITY: TRENDED DATA
Augmented Dickey–Fuller tests: second-order autoregressive process
Main condition for stationarity:
1
3
2
30. 30
Generalizing to the case where Yt depends on Yt–1, ..., Yt–p, a condition for stationarity is that
|b2 + ...+ bp+1| < 1.
TESTS OF NONSTATIONARITY: TRENDED DATA
1
2
*
2 ...
p
t
p
t
p
t
t
t Y
Y
Y
Y
*
1
1
*
3
1
*
2
1 ...
1
Augmented Dickey–Fuller tests: general autoregressive process
t
p
t
p
t
t Y
Y
Y
1
1
2
1 ...
Main condition for stationarity:
1
... 1
2
p
31. 31
It is convenient to reparameterize the model as shown, where b2
*
= b2 + ...+ bp+1 and the other
b*
coefficients are appropriate linear combinations of the original b coefficients.
TESTS OF NONSTATIONARITY: TRENDED DATA
1
2
*
2 ...
p
t
p
t
p
t
t
t Y
Y
Y
Y
*
1
1
*
3
1
*
2
1 ...
1
Augmented Dickey–Fuller tests: general autoregressive process
t
p
t
p
t
t Y
Y
Y
1
1
2
1 ...
Main condition for stationarity:
1
... 1
2
p
32. 32
Under the null hypothesis of non-explosive nonstationarity, the test statistics T(b2
*
– 1), t,
and F asymptotically have the same distributions and critical values as before.
TESTS OF NONSTATIONARITY: TRENDED DATA
1
2
*
2 ...
p
t
p
t
p
t
t
t Y
Y
Y
Y
*
1
1
*
3
1
*
2
1 ...
1
Augmented Dickey–Fuller tests: general autoregressive process
t
p
t
p
t
t Y
Y
Y
1
1
2
1 ...
Main condition for stationarity:
1
... 1
2
p
33. 33
In practice, the t test is particularly popular and is generally known as the augmented
Dickey–Fuller (ADF) test.
TESTS OF NONSTATIONARITY: TRENDED DATA
1
2
*
2 ...
p
t
p
t
p
t
t
t Y
Y
Y
Y
*
1
1
*
3
1
*
2
1 ...
1
Augmented Dickey–Fuller tests: general autoregressive process
t
p
t
p
t
t Y
Y
Y
1
1
2
1 ...
Main condition for stationarity:
1
... 1
2
p
34. 34
There remains the issue of the determination of p. Two main approaches have been
proposed and both start by assuming that one can hypothesize some maximum value pmax.
TESTS OF NONSTATIONARITY: TRENDED DATA
1
2
*
2 ...
p
t
p
t
p
t
t
t Y
Y
Y
Y
*
1
1
*
3
1
*
2
1 ...
1
Augmented Dickey–Fuller tests: general autoregressive process
t
p
t
p
t
t Y
Y
Y
1
1
2
1 ...
Main condition for stationarity:
1
... 1
2
p
35. 35
In the F test approach, the reparameterized model is fitted with p = pmax and a t test is
performed on the coefficient of DYt–pmax. If this is not significant, this term may be dropped.
TESTS OF NONSTATIONARITY: TRENDED DATA
1
2
*
2 ...
p
t
p
t
p
t
t
t Y
Y
Y
Y
*
1
1
*
3
1
*
2
1 ...
1
Augmented Dickey–Fuller tests: general autoregressive process
t
p
t
p
t
t Y
Y
Y
1
1
2
1 ...
Main condition for stationarity:
1
... 1
2
p
36. 36
Next, an F test is performed on the joint explanatory power of DYt–pmax and DYt–pmax–1. If this is
not significant, both terms may be dropped.
TESTS OF NONSTATIONARITY: TRENDED DATA
1
2
*
2 ...
p
t
p
t
p
t
t
t Y
Y
Y
Y
*
1
1
*
3
1
*
2
1 ...
1
Augmented Dickey–Fuller tests: general autoregressive process
t
p
t
p
t
t Y
Y
Y
1
1
2
1 ...
Main condition for stationarity:
1
... 1
2
p
37. 37
The process continues, including further lagged differences in the F test until the null
hypothesis of no joint explanatory power is rejected.
TESTS OF NONSTATIONARITY: TRENDED DATA
1
2
*
2 ...
p
t
p
t
p
t
t
t Y
Y
Y
Y
*
1
1
*
3
1
*
2
1 ...
1
Augmented Dickey–Fuller tests: general autoregressive process
t
p
t
p
t
t Y
Y
Y
1
1
2
1 ...
Main condition for stationarity:
1
... 1
2
p
38. 38
The last lagged difference included in the test becomes the term with the maximum lag.
Higher order lags may be dropped because the previous F test was not significant.
TESTS OF NONSTATIONARITY: TRENDED DATA
1
2
*
2 ...
p
t
p
t
p
t
t
t Y
Y
Y
Y
*
1
1
*
3
1
*
2
1 ...
1
Augmented Dickey–Fuller tests: general autoregressive process
t
p
t
p
t
t Y
Y
Y
1
1
2
1 ...
Main condition for stationarity:
1
... 1
2
p
39. 39
Provided that the disturbance term is iid, the normalized coefficient of Yt–1 and its t statistic
will have the same (non-standard) distributions as for the Dickey–Fuller test.
TESTS OF NONSTATIONARITY: TRENDED DATA
1
2
*
2 ...
p
t
p
t
p
t
t
t Y
Y
Y
Y
*
1
1
*
3
1
*
2
1 ...
1
Augmented Dickey–Fuller tests: general autoregressive process
t
p
t
p
t
t Y
Y
Y
1
1
2
1 ...
Main condition for stationarity:
1
... 1
2
p
40. 40
The other method is to use an information criterion such as the Bayes Information Criterion
(BIC), also known as the Schwarz Information Criterion (SIC). This requires the
computation of the BIC statistic shown and choosing p so as to minimize the expression.
TESTS OF NONSTATIONARITY: TRENDED DATA
1
2
*
2 ...
p
t
p
t
p
t
t
t Y
Y
Y
Y
*
1
1
*
3
1
*
2
1 ...
1
T
T
p
T
RSS
T
T
k
T
RSS
BIC
log
2
log
log
log
Augmented Dickey–Fuller tests: general autoregressive process
t
p
t
p
t
t Y
Y
Y
1
1
2
1 ...
Main condition for stationarity:
1
... 1
2
p
41. 41
The first term falls as p increases, but the second term increases, and the trade-off is such
that asymptotically the criterion will select the true value of p.
TESTS OF NONSTATIONARITY: TRENDED DATA
1
2
*
2 ...
p
t
p
t
p
t
t
t Y
Y
Y
Y
*
1
1
*
3
1
*
2
1 ...
1
T
T
p
T
RSS
T
T
k
T
RSS
BIC
log
2
log
log
log
Augmented Dickey–Fuller tests: general autoregressive process
t
p
t
p
t
t Y
Y
Y
1
1
2
1 ...
Main condition for stationarity:
1
... 1
2
p
42. 42
A common alternative is the Akaike Information Criterion (AIC) shown. This imposes a
smaller penalty on overparameterization and will therefore tend to select a larger value of p,
but simulation studies suggest that it may produce better results in practice.
TESTS OF NONSTATIONARITY: TRENDED DATA
1
2
*
2 ...
p
t
p
t
p
t
t
t Y
Y
Y
Y
*
1
1
*
3
1
*
2
1 ...
1
T
T
p
T
RSS
T
T
k
T
RSS
BIC
log
2
log
log
log
T
k
T
RSS
AIC
2
log
Augmented Dickey–Fuller tests: general autoregressive process
t
p
t
p
t
t Y
Y
Y
1
1
2
1 ...
Main condition for stationarity:
1
... 1
2
p
43. 43
Whether one uses the F test approach or information criteria, it is necessary to check that
the residuals are not subject to autocorrelation, for example, using a Breusch–Godfrey
lagrange multiplier test.
TESTS OF NONSTATIONARITY: TRENDED DATA
1
2
*
2 ...
p
t
p
t
p
t
t
t Y
Y
Y
Y
*
1
1
*
3
1
*
2
1 ...
1
T
T
p
T
RSS
T
T
k
T
RSS
BIC
log
2
log
log
log
T
k
T
RSS
AIC
2
log
Augmented Dickey–Fuller tests: general autoregressive process
t
p
t
p
t
t Y
Y
Y
1
1
2
1 ...
Main condition for stationarity:
1
... 1
2
p
44. 44
Autocorrelation would provide evidence that there remain dynamics in the model not
accounted for by the specification and that the model does not include enough lags.
TESTS OF NONSTATIONARITY: TRENDED DATA
1
2
*
2 ...
p
t
p
t
p
t
t
t Y
Y
Y
Y
*
1
1
*
3
1
*
2
1 ...
1
T
T
p
T
RSS
T
T
k
T
RSS
BIC
log
2
log
log
log
T
k
T
RSS
AIC
2
log
Augmented Dickey–Fuller tests: general autoregressive process
t
p
t
p
t
t Y
Y
Y
1
1
2
1 ...
Main condition for stationarity:
1
... 1
2
p
45. 45
The 1979 and 1981 Dickey–Fuller papers were truly seminal in that they have given rise to a
very extensive research literature devoted to the improvement of testing for nonstationarity
and of the representation of nonstationary processes.
TESTS OF NONSTATIONARITY: TRENDED DATA
1
2
*
2 ...
p
t
p
t
p
t
t
t Y
Y
Y
Y
*
1
1
*
3
1
*
2
1 ...
1
T
T
p
T
RSS
T
T
k
T
RSS
BIC
log
2
log
log
log
T
k
T
RSS
AIC
2
log
Augmented Dickey–Fuller tests: general autoregressive process
t
p
t
p
t
t Y
Y
Y
1
1
2
1 ...
Main condition for stationarity:
1
... 1
2
p
46. 46
The low power of the Dickey–Fuller tests was acknowledged in the original papers and
much effort has been directed to the problem of distinguishing between nonstationary
processes and highly autoregressive stationary processes.
TESTS OF NONSTATIONARITY: TRENDED DATA
1
2
*
2 ...
p
t
p
t
p
t
t
t Y
Y
Y
Y
*
1
1
*
3
1
*
2
1 ...
1
T
T
p
T
RSS
T
T
k
T
RSS
BIC
log
2
log
log
log
T
k
T
RSS
AIC
2
log
Augmented Dickey–Fuller tests: general autoregressive process
t
p
t
p
t
t Y
Y
Y
1
1
2
1 ...
Main condition for stationarity:
1
... 1
2
p
47. 47
Remarkably, the original Dickey–Fuller tests, particularly the t test in augmented form, are
still widely used, perhaps even dominant.
TESTS OF NONSTATIONARITY: TRENDED DATA
1
2
*
2 ...
p
t
p
t
p
t
t
t Y
Y
Y
Y
*
1
1
*
3
1
*
2
1 ...
1
T
T
p
T
RSS
T
T
k
T
RSS
BIC
log
2
log
log
log
T
k
T
RSS
AIC
2
log
Augmented Dickey–Fuller tests: general autoregressive process
t
p
t
p
t
t Y
Y
Y
1
1
2
1 ...
Main condition for stationarity:
1
... 1
2
p
48. 48
Other tests with superior asymptotic properties have been proposed, but some
underperform in finite samples, as far as this can be established by simulation.
TESTS OF NONSTATIONARITY: TRENDED DATA
1
2
*
2 ...
p
t
p
t
p
t
t
t Y
Y
Y
Y
*
1
1
*
3
1
*
2
1 ...
1
T
T
p
T
RSS
T
T
k
T
RSS
BIC
log
2
log
log
log
T
k
T
RSS
AIC
2
log
Augmented Dickey–Fuller tests: general autoregressive process
t
p
t
p
t
t Y
Y
Y
1
1
2
1 ...
Main condition for stationarity:
1
... 1
2
p
49. 49
The augmented Dickey–Fuller t test has retained its popularity on account of robustness
and, perhaps, theoretical simplicity.
TESTS OF NONSTATIONARITY: TRENDED DATA
1
2
*
2 ...
p
t
p
t
p
t
t
t Y
Y
Y
Y
*
1
1
*
3
1
*
2
1 ...
1
T
T
p
T
RSS
T
T
k
T
RSS
BIC
log
2
log
log
log
T
k
T
RSS
AIC
2
log
Augmented Dickey–Fuller tests: general autoregressive process
t
p
t
p
t
t Y
Y
Y
1
1
2
1 ...
Main condition for stationarity:
1
... 1
2
p
50. 50
However, a refinement, the ADF–GLS (generalized least squares) test due to Elliott,
Rothenberg, and Stock (1996) appears to be gaining in popularity and is implemented in
major regression applications.
TESTS OF NONSTATIONARITY: TRENDED DATA
1
2
*
2 ...
p
t
p
t
p
t
t
t Y
Y
Y
Y
*
1
1
*
3
1
*
2
1 ...
1
T
T
p
T
RSS
T
T
k
T
RSS
BIC
log
2
log
log
log
T
k
T
RSS
AIC
2
log
Augmented Dickey–Fuller tests: general autoregressive process
t
p
t
p
t
t Y
Y
Y
1
1
2
1 ...
Main condition for stationarity:
1
... 1
2
p
51. 51
1
2
*
2 ...
p
Simulations indicate that its power to discriminate between a nonstationary process and a
stationary autoregressive process is uniformly closer to the theoretical limit than the
standard tests, irrespective of the degree of autocorrelation.
t
p
t
p
t
t
t Y
Y
Y
Y
*
1
1
*
3
1
*
2
1 ...
1
T
T
p
T
RSS
T
T
k
T
RSS
BIC
log
2
log
log
log
T
k
T
RSS
AIC
2
log
TESTS OF NONSTATIONARITY: TRENDED DATA
Augmented Dickey–Fuller tests: general autoregressive process
t
p
t
p
t
t Y
Y
Y
1
1
2
1 ...
Main condition for stationarity:
1
... 1
2
p
52. Copyright Christopher Dougherty 2013.
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The content of this slideshow comes from Section 13.4 of C. Dougherty,
Introduction to Econometrics, fourth edition 2011, Oxford University Press.
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2013.08.29
