110 The Basics of financial economeTrics
correction.^5 The theory behind them is not trivial. In particular, the error
correction model applies expected values computed conditional on events
(or information).^6
Random Walk
Let us consider some price process given by the series {S}t.^7 The dynamics of
the process are given by
(^) SStt=+− 1 εt (5.5)
or, equivalently, ∆Stt=ε.
In words, tomorrow’s price, St+ 1 , is thought of as today’s price plus some
random shock that is independent of the price. As a consequence, in this
model, known as the random walk, the increments St − St− 1 from t − 1 to
t are thought of as completely undeterministic. Since the εt’s have a mean
of zero, the increments are considered fair.^8 An increase in price is as likely
as a downside movement. At time t, the price is considered to contain all
information available. So at any point in time, next period’s price is exposed
to a random shock.
Consequently, the best estimate for the following period’s price is this
period’s price. Such price processes are called efficient due to their immedi-
ate information processing.
A more general model, for example, AR(p), of the form
SStt=+αα 01 −− 1 ++... αεptS pt+
with several lagged prices could be considered as well. This price process
would permit some slower incorporation of lagged prices into current prices.
Now for the price to be a random walk process, the estimation would have
to produce a 0 = 0, a 1 = 1, a 2 = ... = ap = 0.
(^5) Later in this book we will introduce an additional price process using logarithmic
returns.
(^6) Enders, Applied Econometrics Time Series.
(^7) Here the price of some security at time t, St, ought not be confused with the seasonal
component in equation (5.1).
(^8) Note that since the ε assumes values on the entire real number line, the stock price
could potentially become negative. To avoid this problem, logarithmic returns are
modeled according to equation (5.4) rather than stock prices.