Introduction to Time Series Analysis 109
as quantities at certain points in time. However, it may sometimes be more
convenient to represent the evolution of {x}t by difference equations of its
components. The four components in difference equation form could be
thought of as
(^) ∆∆xxtt=−xTtt− 1 =+∆∆IStt+ (5.3)
with the change in the linear trend ΔTt = c where c is a constant, and
∆IItt=−φξ()−− 12 Itt+
where ξ are disturbances themselves, and
(^) ∆∆TStt+=ht() (5.4)
where h(t) is some deterministic function of time. The symbol Δ indicates
change in value from one period to the next.
The concept that the disturbance terms are i.i.d. means that the ξ behave
in a manner common to all ξ (i.e., identically distributed) though indepen-
dently of each other. The concept of statistical independence is explained in
Appendix A while for random variables, this is done in Appendix B.
In general, difference equations are some functions of lagged values,
time, and other stochastic variables. In time series analysis, one most often
encounters the task of estimating difference equations such as equation (5.4).
The original intent of time series analysis was to provide some reliable tools
for forecasting.^4
By forecasting, we assume that the change in value of some quantity, say
x, from time t to time t + 1 occurs according to the difference equation (5.3).
However, since we do not know the value of the disturbance in t + 1, ξt+1, at
time t, we incorporate its expected value, that is, zero. All other quantities in
equation (5.3) are deterministic and, thus, known in t. Hence, the forecast
really is the expected value in t + 1 given the information in t.
Application: The Price Process
Time series analysis has grown more and more important for verifying
financial models. Price processes assume a significant role among these mod-
els. In the next two subsections, we discuss two commonly encountered
models for price processes given in a general setting: random walk and error
(^4) See, for example, Walter Enders, Applied Econometrics Time Series (New York:
John Wiley & Sons, 1995).