We have discussed in Example 10.1 the AR(1) process is
 |
(11.43) |
Given
, the process is stationary when
or after the ``decaying process''. The case
where
corresponds to the random walk which is
non-stationary. The relationship between a stationary AR(1)
process and
close to one is so similar to a random walk
that it is often tested whether we have the case
or
. To do this the so called unit root tests have
been developed.
The unit root test developed by Dickey and Fuller tests the null
hypothesis of a unit root, that is, there
is a root for the characteristic equation (11.6) of the
AR(1) process with
, against the alternative hypothesis that
the process has no unit roots. As a basis for the test the
following regression used is
 |
(11.44) |
which is obtained by rearranging (10.43) with
. If
is a random walk, then the coefficient of
is equal to
zero. If, on the other hand,
is a stationary AR(1) process,
then the coefficient is negative. The standard
-statistic is
formed
 |
(11.45) |
where
and
are the least squares
estimators for
and the variance
of
. For increasing
the statistic
(10.45) converges not to a standard normal
distribution but instead to the distribution of a functional of
Wiener process,
where
is a standard Wiener process. The critical value of the
distribution are, for example, at the 1%, 5% and 10%
significance levels, -2.58, -1.95, and -1.62 respectively.
A problem with this test is that the normal test significance
level (for example 5%) is not reliable when the error terms
in (10.44) are autocorrelated. The larger
the autocorrelation of
, the larger the distortion
in general will be of the test significance. Ignoring then
autocorrelations could lead to rejecting the null hypothesis of a
unit root at low significance levels of 5%, when in reality the
significance level lies at, for example, 30%. In order to
prohibit these negative effects, Dickey and Fuller suggest another
regression which contains lagged differences. The regression of
this augmented Dickey Fuller Test (ADF) is thus
 |
(11.46) |
where as with the simple Dickey-Fuller Test the null hypothesis of
a unit root is rejected when the test statistic
(10.45) is smaller than the critical value (which have
been summarized in a table). Problematic is naturally the choice
of
. In general it holds that the size of the test is better
when
gets larger, but which causes the test to lose power. This is illustrated in a simulated process. The errors
are correlated through the relationship
where
are i.i.d.
. In the next chapter these
processes will be referred to as moving average processes of
order 1, MA(1). It holds that
, and
for
. For the ACF of
we then get
 |
(11.47) |
For the process
 |
(11.48) |
simulations of the ADF Tests were done and are summarized in an
abbreviated form in Table 10.3.
Table 10.3:
ADF-Test: Simulated rejection probabilities for the
process (10.48) at a nominal significance level of
5% (according to Friedmann (1992)).
|
|
 |
 |
 |
-0.99 |
-0.9 |
0 |
0.9 |
1 |
3 |
0.995 |
0.722 |
0.045 |
0.034 |
|
11 |
0.365 |
0.095 |
0.041 |
0.039 |
0.9 |
3 |
1.000 |
0.996 |
0.227 |
0.121 |
|
11 |
0.667 |
0.377 |
0.105 |
0.086 |
|
As one can see, the nominal significance level of 5% under the
null hypothesis (
) is held better, if
is larger.
However the power of the test decreases, i.e., the test is no
longer capable of distinguishing between a process with unit roots
and a stationary process with
. Thus in choosing
there is also the conflict between validity and power of the test.
If
is a trend-stationary process as in (10.41),
the ADF test likewise does not reject often enough the (false)
null hypothesis of a unit root. Asymptotically the probability of
rejecting goes to zero. The ADF regression (10.46) can be
extended by a linear time trend, i.e., run the regression
 |
(11.49) |
and test the significance of
. The critical values are
contained in tables. The ADF test with a time trend
(10.49) has power against a trend-stationary process. On
the other hand, it loses power as compared to the simple ADF test
(10.46), when the true process, for example, is a
stationary AR(1) process.
As an empirical example, consider the daily stock prices of the 20
largest German stock companies from Jan. 2, 1974 to Dec. 30, 1996.
Table 10.4 displays the ADF test statistics for the logged
stock prices for
and
. The tests were run with and
without a linear time trend. In every regression a constant was
included in estimation.
Table 10.4:
Unit root tests: ADF Test (Null hypothesis: unit root)
and KPSS Test (Null hypothesis: stationary). The augmented portion
of the ADF regression as order
and
. The KPSS statistic
was calculated with the reference point
and
. The
asterisks indicate significance at the 10% (*) and 1% (**)
levels.
|
ADF |
KPSS |
|
|
|
|
|
without time trend |
. | with time trend |
without time trend |
with time trend |
|
|
|
and |
0 |
4 |
. |
0 |
. | 4 |
. | 8 |
12 |
8 |
12 |
|
ALLIANZ |
-0 |
. | 68 |
-0 |
. | 62 |
2 |
. | 44 |
2 |
. | 59 |
24.52** |
16.62** |
2.36** |
1.61** |
BASF |
0 |
. | 14 |
0 |
. | 34 |
2 |
. | 94 |
3 |
. | 13 |
23.71** |
16.09** |
1.39** |
0.95** |
BAYER |
-0 |
. | 11 |
0 |
. | 08 |
2 |
. | 96 |
3 |
. | 26 |
24.04** |
16.30** |
1.46** |
1.00** |
BMW |
-0 |
. | 71 |
-0 |
. | 66 |
2 |
. | 74 |
2 |
. | 72 |
23.92** |
16.22** |
2.01** |
1.37** |
COMMERZ- |
|
. | |
|
. | |
|
. | |
|
. | |
|
|
|
|
BANK |
-0 |
. | 80 |
-0 |
. | 67 |
1 |
. | 76 |
1 |
. | 76 |
22.04** |
14.96** |
1.43** |
0.98** |
DAIMLER |
-1 |
. | 37 |
-1 |
. | 29 |
2 |
. | 12 |
2 |
. | 13 |
22.03** |
14.94** |
3.34** |
2.27** |
DEUTSCHE |
|
. | |
|
. | |
|
. | |
|
. | |
|
|
|
|
BANK |
-1 |
. | 39 |
-1 |
. | 27 |
2 |
. | 05 |
1 |
. | 91 |
23.62** |
16.01** |
1.70** |
1.16** |
DEGUSSA |
-0 |
. | 45 |
-0 |
. | 36 |
1 |
. | 94 |
1 |
. | 88 |
23.11** |
15.68** |
1.79** |
1.22** |
DRESDNER |
-0 |
. | 98 |
-0 |
. | 94 |
1 |
. | 90 |
1 |
. | 77 |
22.40** |
15.20** |
1.79** |
1.22** |
HOECHST |
0 |
. | 36 |
0 |
. | 50 |
3 |
. | 24 |
3 |
. | 37 |
23.80** |
16.15** |
1.42** |
0.97** |
KARSTADT |
-1 |
. | 18 |
-1 |
. | 17 |
1 |
. | 15 |
1 |
. | 15 |
20.40** |
13.84** |
3.33** |
2.26** |
LINDE |
-1 |
. | 69 |
-1 |
. | 44 |
2 |
. | 74 |
2 |
. | 70 |
24.40** |
16.54** |
3.14** |
2.15** |
MAN |
-1 |
. | 78 |
-1 |
. | 58 |
1 |
. | 66 |
1 |
. | 61 |
21.97** |
14.91** |
1.59** |
1.08** |
MANNES- |
|
. | |
|
. | |
|
. | |
|
. | |
|
|
|
|
MANN |
-0 |
. | 91 |
-0 |
. | 80 |
2 |
. | 73 |
2 |
. | 55 |
21.97** |
14.93** |
1.89** |
1.29** |
PREUSSAG |
-1 |
. | 40 |
-1 |
. | 38 |
2 |
. | 21 |
2 |
. | 03 |
23.18** |
15.72** |
1.53** |
1.04** |
RWE |
-0 |
. | 09 |
-0 |
. | 04 |
2 |
. | 95 |
2 |
. | 84 |
24.37** |
16.52** |
1.66** |
1.14** |
SCHERING |
0 |
. | 11 |
0 |
. | 04 |
2 |
. | 37 |
2 |
. | 12 |
24.20** |
16.40** |
2.35** |
1.60** |
SIEMENS |
-1 |
. | 35 |
-1 |
. | 20 |
2 |
. | 13 |
1 |
. | 84 |
23.24** |
15.76** |
1.69** |
1.15** |
THYSSEN |
-1 |
. | 45 |
-1 |
. | 34 |
1 |
. | 92 |
1 |
. | 90 |
21.97** |
14.90** |
1.98** |
1.35** |
VOLKS- |
|
. | |
|
. | |
|
. | |
|
. | |
|
|
|
|
WAGEN |
-0 |
. | 94 |
-0 |
. | 81 |
1 |
. | 89 |
1 |
. | 73 |
21.95** |
14.89** |
1.11** |
0.76** |
SFEAdfKpss.xpl
|
Only for RWE with a linear time trend does the ADF test reject the
null hypothesis of a unit root by a significance level of 10%.
Since in all other cases no unit root is rejected, it appears that
taking differences of stock prices is a necessary operation in
order to obtain a stationary process, i.e., to get log returns
that can be investigated further. These results will be put into
question in the next section using another test.
The KPSS Test from Kwiatkowski et al. (1992) tests
for stationarity, i.e., for a unit root. The hypotheses are thus
exchanged from those of the ADF test. As with the ADF test, there
are two cases to distinguish between, whether to estimate with or
without a linear time trend. The regression model with a time
trend has the form
 |
(11.50) |
with stationary
and i.i.d.
with an expected value
0 and variance 1. Obviously for
the process is
integrated and for
trend-stationary. The null hypothesis is
, and the alternative hypothesis is
.
Under
the regression (10.50) is run with the
method of the least squares obtaining the residuals
. Using these residuals the partial sum
is built which under
is integrated of order 1, i.e., the
variance
increases linearly with
. The KPSS test
statistic is then
 |
(11.51) |
where
is an estimator of the spectral density at a frequency of zero
where
is the variance estimator of
and
is the covariance estimator. The problem
again is to determine the reference point
: for
that are
too small the test is biased when there is autocorrelation, for
that is too large it loses power.
The results of the KPSS tests in Table 10.4 clearly
indicate that the investigated stock prices are not stationary or
trend-stationary, since in every case the null hypothesis at a
significance level of 1% was rejected. Even RWE, which was
significant under the ADF test at a significance level of 10 %,
implies a preference of the hypothesis of unit roots here at a
lower significance level.
If one wants to test whether a time series follows a random walk,
one can take advantage of the fact that the variance of a random
walk increases linearly with time, see (10.4).
Considering the log prices of a financial time series,
,
the null hypothesis would be
with log returns
, constant
and
white noise. An alternative hypothesis is, for
example, that
is stationary and autocorrelated. The sum over
the returns is formed
and the variance of
is determined. For
it holds
that, for example,
where taking advantage of the stationarity of
, generally
 |
(11.52) |
Under
it holds that
for all
, so
that under
A test statistic can now be constructed where the consistent
estimator
for
,
for
and
for
are substituted into
(10.52). The test statistic is then
It can be shown that the asymptotic distribution is
The asymptotic variance can be established through the following
approximation: Assume that
and
. Then we
have that
and
Since under
the estimated autocorrelation
scaled with
is asymptotically standard normal and is
independent, see Section 11.5, the asymptotic variance
is thus:
With the summation formulas
and
we finally obtain