This preview shows pages 1–3. Sign up to view the full content.
1
Time series  processes and
realisations
With random sampling the key concepts are population
and sample
The population can be real (Census) or hypothetical
(continuous distribution)
Sample statistics are (imprecise) estimates of
theunderlying population parameters
Random sampling implies:
the probability that a particular observation is
included does not
depend on earlier observations
Prob(x
t
=a)
≠
f(x
s
), for s
≠
t
For time series this is not appropriate:
the probability of observing a particular value
depends on the past history of the series
Prob(x
t
=a) = f(x
s
)
hence if we treated the singleperiod observation as
the primitive event these events would not be
independent
Instead we develop the concepts of the ‘stochastic
process’ (the population) and ‘realisation’ (the
sample)
Stochastic processes
We can regard the sequence
of observations on X
{x
1
, x
2
, .
..x
T
}
as a single (Tdimensional) sample point
The set of all possible sequences of length T is the
(hypothetical) underlying population
The population (the mechanism generating all possible
sequences) is called a stochastic process
A single sample point (an individual sequence) is called
a realisation
Example
: M tosses of a fair coin
A sequence (realisation) is
{H, T, T, T, H.
......
} or {H, H, H, T, T.
......
}
The process (population) generates all such sequences
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document2
The aim of time series analysis
To make valid inferences about the underlying
stochastic process from a single
realisation
If we have N independent realisations (distinct
experiments) we can define ‘ensemble’ estimates of
the moments by averaging over realisations
μ
t
= N
1
Σ
i
x
it
σ
2
t
= (N1)
1
Σ
i
(x
it

μ
t
)
2
for each t
Since these averages are based on random sampling
(repeated identical experiments) they have desirable
properties
Suppose instead we have only one realisation, but that
the moments do not depend on the observation date
E[x
1t
] =
μ
Var[x
1t
] =
σ
2
<
∞
Cov[x
1s
,x
1t
] =
γ
ts
all of these are independent of t
(
γ
ts
is a function of ts, not of t or s separately)
Split this realisation into N subrealisations of length T/N:
This is the end of the preview. Sign up
to
access the rest of the document.
 Spring '10
 Cowell
 Economics

Click to edit the document details