104 The Basics of financial economeTrics
variables as well as lagged values of the series itself or other quantities. These
latter quantities may be given exogenously or also depend on time.
To visualize this, we plot the graph of 20 daily closing values of the
German stock market index, the DAX (Deutscher Aktien IndeX), in
Figure 5.1. The values are listed in Table 5.1. The time points of observation
t with equidistant increments are represented by the horizontal axis while
the DAX index values are represented by the vertical axis.
Decomposition of Time Series
Each point in Figure 5.1 is a pair of the components, time and value. In this
section, the focus is on the dynamics of the observations; that is, one wants
to know what the values are decomposable into at each point in time. A time
series with observations xt, t = 1, 2,... , n is usually denoted by {x}t.^1 In the
context of time series analysis, for any value xt, the series is thought of as a
composition of several quantities. The most traditional decomposition is of
the form
FIguRE 5.1 DAX Index Values: May 3 to May 31, 2007
0 2 4 6 8 10 12 14 16 18 2074007450750075507600765077007750780078507900tDAX