zero. If anything, the variance of the normal distribution two time steps
away increases, and the plausible range of values that the increment can as-
sume actually increases, further increasing our prediction error. Therefore,
knowing the past history of a random walk is not much help in predicting
the forward-looking increments.
The situation is very different for stationary processes. Armed with the
knowledge that stationary processes are mean reverting, one can predict the
increment to be greater than or equal to the difference between the current
value and the mean. The prediction is guaranteed to hold true at some point
in the future realizations of the time series.
However, stock prices are modeled as a log-normal process, and that is
definitely not stationary. So, where does that leave us in terms of making
profits? Definitely not anywhere close to making money. The reader is prob-
ably wondering what the point of this whole chapter is. If the logarithm of
stock prices is assumed to be random walk, there is no need to go at it in a
roundabout way. Just say it is futile trying to predict stock returns and leave
it at that. But all hope is not lost. We shall see in the later chapters that it
may be possible to construct portfolios whose time series are actually sta-
tionary, and the returns for those portfolios are indeed predictable. Let us
stop here with this teaser.
SUMMARY
A time series is constructed by periodically drawing samples from prob-
ability distributions that vary with time.
The white noise process is the most elementary form of time series and
is generated by drawing samples from a fixed distribution at every time
instance.
ARMA time series are generated using fixed linear combinations of
white noise realizations.
Time series forecasting for ARMA processes involves deciphering the
linear combination and the white noise sequence used to generate the
given data and using it to predict the future values.
A random walk process is the time series where the current value is a
simple sum of all the white noise realizations up to the present time.
A random walk is a nonstationary time series.
Nonstationary time series are usually transformed to stationary time se-
ries using differencing.
The logarithm of the stock price series is usually modeled as a random
walk.32 BACKGROUND MATERIAL