Final_1.pdf

chastic time series. Fittingly, their methodology for time series forecasting is
referred to as the Box-Jenkins approach. In this chapter, we will describe the
Box-Jenkins approach. Instead of doing this by definition, we will attempt
to do this by way of construction and examples.
We begin by introducing some basic notation. Throughout the chapter
the value of a time series at time tis denoted as yt. It then follows that the gen-
eral time series is the set of values yt,t= 0, 1, 2, 3...T. We denote this as yt.

Autocorrelation

Let us begin the discussion by introducing the notion of the autocorrelation.
Given a stochastic time series, the first question one tends to ask in the
process of analysis is, “Is there a relationship between the value now and the
value observed one time step in the past?” We can choose to answer the
question by measuring the correlation between the time series values one
time interval apart. The strength of the (linear) relationship is reflected in the
correlation number. And what about the relationship of the current value to
the value two time steps in the past? What about three time steps in the past?
The question seems to repeat itself naturally for the whole range of time
steps. The answer to these questions, spanning the entire range of time steps,
could very well be the autocorrelation function.
The autocorrelation function is the plot of the correlation between val-
ues in the time series based on the time interval between them. The x-axis de-
notes the length of the time lag between the current value and the value in
the past. The y-axis value for a time lag t, (x=t) is the correlation between
the values in the time series ttime units apart. This correlation is estimated
using the formula

(2.1)

wherey–is the calculated average of variable y.

The plot of the estimated correlation against time intervals forms an es-
timation of the autocorrelation function, called the correlogram. It serves as
a proxy for the autocorrelation function of the time series.
We shall see in the ensuing discussions that the autocorrelation function
serves as a signature or fingerprint for a time series and plays a key role in
characterizing various cases of the time series that we describe in the fol-
lowing sections.

ˆ()

[][ ]

()

ρτ

τ

= τ

−−

−

−

=+

=

∑

∑

1

1

1 2

1

T tt

t

T

T t

t

T

yyy y

yy

Time Series 15