108 The Basics of financial economeTrics
The coefficient of the irregular term is φ = 0.2850 indicating that the
previous period’s value is weighted by about one third in the computation of
this period’s value. The overall model of the returns, then, looks like
yTtt=+SItt+=TStt+ ()weekdayI−+0 2850. tt− 1 UThe technique used to estimate the times series model is the moving
average method. Since it is beyond the scope of this chapter, we will not
discuss it here.
As can be seen by Figure 5.4, it might appear difficult to detect a linear
trend, at least when one does not exclude the first 15 observations. If there really
is no trend, most of the price is contained in the other components rather than
any deterministic term. Efficient market theory that is central in financial theory
does not permit any price trend since this would indicate that today’s price does
not contain all information available. By knowing that the price grows deter-
ministically, this would have to be already embodied into today’s price.
Representation of Time Series with Difference Equations
Rather than representing {x}t by equation (5.1) or (5.2), often the dynamics of
the components of the series are given. So far, the components are considered
FIguRE 5.4 Daily S&P 500 Stock Index Prices with Daily Changes Extending Vertically
0 5 10 15 20 25 30 35 40 45
1030104010501060107010801090110011101120tS&P 500 Value