We have seen that econometric models, at least with stock prices
and exchange rates, motivate using a random walk as a statistical
model. With exchange rates we saw that as a consequence of the
uncovered interest rate parity and the assumption of risk
neutrality of forward and future speculators the model in
(10.14) follows a random walk. Assuming a geometric Brownian
motion for stock price as in (10.18), then it follows from
Itô's lemma that the log of stock price follows a Wiener
process with a constant drift rate:
 |
(11.37) |
where
. If one observes (10.37)
in time intervals of length
, i.e., at discrete points
in time
, then one obtains
 |
(11.38) |
with independent, standard normally distributed
. The process (10.38) is a random walk with a drift
for the logged stock prices. The log returns (see Definition
10.15) over the time interval of length
are
also independently normally distributed with expected value
and variance
.
With long interest rate time series the random walk appears to be
less plausible, since it is assumed that in the long-run there is
a stationary level around which interest rates fluctuate in the
short run. Let's consider once again the process for the short
rate in (10.16), the Cox-Ingersoll-Ross (CIR) model. A
discrete approximation is
or
 |
(11.39) |
If
in (10.39) is negative (and larger than
-2), then the process is a stationary AR(1) process with
heteroscedastic error terms. In Example 10.1 we encountered
such a process with heteroscedastic error terms.
There is also the interpretation that interest rates are, at least
in the short-run, explained well by a random walk. It is therefore
of general interest to test whether a random walk exists. In the
following we show the distinguishes through the three versions of
the random walk hypothesis. In general we consider a random walk
with a drift
 |
(11.40) |
- The stochastic errors in (10.40) are independent
and identically distributed (i.i.d.) with expectation zero and
variance
. This hypothesis has already been tested on
multiple data sets in the sixties and was empirically determined
to be unsupported. For example, distinct volatility clusters were
discovered which under the i.i.d. hypothesis are statistically not
expected.
- The stochastic errors in (10.40) are independent
but not necessarily identically distributed with an expectation of
zero. This hypothesis is weaker than the first one since, for
example, it allows for heteroscedasticity. Nonetheless, even here
the empirical discoveries were that a dependence between the error
terms must be assumed.
- The stochastic errors in (10.40) are uncorrelated,
i.e.,
for every
.
This is the weakest and most often discussed random walk
hypothesis. Empirically it is most often tested through the
statistical (in)significance of the estimated autocorrelations of
.
The discussion of the random walk hypotheses deals with, above
all, the predicability of financial time series. Another
discussion deals with the question of whether the model
(10.40) with independent, or as the case may be with
uncorrelated, error terms is even a reasonable model for financial
time series or whether it would be better to use just a model with
a deterministic trend. Such a trend-stationary model has the
form
 |
(11.41) |
with constant parameters
and
. The process
(10.41) is non-stationary since, for example,
, the expected value is time dependent. If the
linear time trend is filtered from
, then the stationary
process
is obtained.
To compare the difference stationary random walk with a drift to
the trend stationary process (10.41) we write the random
walk from (10.40) through recursive substitution as
 |
(11.42) |
with a given initial value
. One sees that the random walk
with a drift also implies a linear time trend, but the cumulative
stochastic increments (
) in
(10.42) are not stationary, unlike the stationary
increments (
) in (10.41). Due to the
representation (10.42) the random walk with or without a
drift will be described as integrated, since the deviation
from a deterministic trend is the sum of error terms. Moreover,
every error term
has a permanent influence on all
future values of the process. For the best forecast in the sense
of the mean squared error it holds for every
,
In contrast, the impact of a shock
on the forecast
of the trend-stationary process (10.41) could be zero,
i.e.,
It is thus of most importance to distinguish between a difference
stationary and a trend-stationary process. Emphasized here is that
the random walk is only a special case of a difference stationary
process. If, for example, the increasing variables in
(10.40) are stationary but autocorrelated, then we have a
general difference stationary process. There are many statistical
tests which test whether a process is difference stationary or
not. Two such tests are discussed in the next section.