Introduction to Time Series Analysis 105
(^) xTtt=+ZStt++Ut (5.1)
where Tt = trend
Zt = cyclical term
St = seasonal term
Ut = disturbance (or error)
While the trend and seasonal terms are assumed to be deterministic
functions of time (i.e., their respective values at some future time t are known
at any lagged time t – d, which is d units of time prior to t), the cyclical and
disturbance terms are random. One also says that the last two terms are
(^1) The number of dates n may theoretically be infinite. We will restrict ourselves to
finite lengths.
TAblE 5.1 DAX Values of the Period May 3 to
May 31, 2007
Date t Level
5/3/2007 1 7883.04
5/4/2007 2 7764.97
5/7/2007 3 7781.04
5/8/2007 4 7739.20
5/9/2007 5 7697.38
5/10/2007 6 7735.88
5/11/2007 7 7659.39
5/14/2007 8 7619.31
5/15/2007 9 7607.54
5/16/2007 10 7499.50
5/17/2007 11 7481.25
5/18/2007 12 7505.35
5/21/2007 13 7459.61
5/22/2007 14 7479.34
5/23/2007 15 7415.33
5/24/2007 16 7475.99
5/25/2007 17 7442.20
5/29/2007 18 7525.69
5/30/2007 19 7516.76
5/31/2007 20 7476.69
Source: Deutsche Börse, http://deutsche-boerse.com/.