Wold's decomposition theorem
The most fundamental justification for time series analysis (as described in this text) is due to Wold's decomposition theorem, where it is explicitly proved that any (stationary) time series can be decomposed into two different parts. The first (deterministic) part can be exactly described by a linear combination of its own past, the second part is a MA component of a finite order.
A slightly adapted version of Wold's decomposition theorem states that any real-valued stationary process Yt can be written as
are not correlated.
has an uncorrelated error with zero mean).
of its importance for time series analysis in general, and in
practice, we will discuss the proof of Wold's decomp. theorem
there seems to be much confusion in literature about this theorem,
we will only discuss the proof of another version (than that
described above) which can be proved quite easily. In order to make
a distinction with the previous description, we will explicitly use
a stationary time series Wt with zero mean and finite
order to forecast the time series by means of a linear combination
of its own past
criterion is used to optimize the parameter values. This criterion
is called the sum of squared residuals (SSR).
normal equations (c.q. eq.
(V.I.1-176) differentiated w.r.t. the parameters) is easily found to
matrix notation eq. (V.I.1-177) becomes
is symmetric about both diagonals due to the stationary of Wt
adding an error component et,n to eq.
(V.I.1-175) it can be shown that
first part of (V.I.1-181) is almost trivial
second part of (V.I.1-181) is
a RHS equal to zero since the parameters satisfy (V.I.1-177)
repeating the previous procedure we obtain
that the error components (V.I.1-181) and (V.I.1-184) are
uncorrelated due to (V.I.1-181), from which we obviously find
substituting (V.I.1-184) into (V.I.1-175) it is obvious that
depends on the past of Wt-1
it is obvious from (V.I.1-183) that
from (II.II.1-27) and the fact that et,n
is independent from the other regressors of (V.I.1-186), that
repeating the step described above it is easy to obtain
applying (V.I.1-189) and (V.I.1-190) on Wt-i we obtain
always depends on it's own past only, and where evidently