FIN 271
FINANCIAL MODELING AND ECONOMETRICS
TIME SERIES MODELING
LECTURE SET 2
REFIK SOYER
THE GEORGE WASHINGTON UNIVERSITY

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2
STOCHASTIC TIME
SERIES MODELS
In the general model
] oe 0Ð Ñ
t
t
t
%
when the effect of time is embodied in
we have a "stochastic process" (SP) with
%
t
Ê
a probabilistic structure.
What is a SP ? It is a sequence of random variables (RV's) indexed by time.
In analyzing stochastic TS, we have a long sequence of random variables:
á
]
]
]
]
]
. ,
,
,
,
,
, . . . .
2
1
0
1
2
Ê
an infinite series going indefinitely into the future and past.
The sequence of n observations
,
,
,
is a sample from this long series.
]
]
á ]
1
2
n
Goal
longer
:
to make inference about the probability structure of the
series (on the
basis of sample).
1

3
LINEAR STOCHASTIC PROCESS
Consider an
, that is,
autoregressive process of order one
] œ
]
t
t
1
t
9
%
Ö] ×
t
- observed values of a random variable.
Ö
×
%
t
- white-noise error terms; unobserved random variables.
9
- unknown constant.
We can write from the above that
]
œ
]
] œ
]
œ
]
t-1
t-2
t-1
t
t-1
t
t-2
t-1
t
9
%
9
%
9 9
%
%
,
(
)
and if we continue this way
we obtain
] œ
t
t
t-1
t-2
t-3
2
3
%
9%
9
%
9 %
. . . . .
Ê
]
the effect of time is embodied in
, that is,
's are correlated as a result of sharing
%
t
t
common error terms.
A stochastic process generated by taking a linear combination of (moving average of)
the white noise terms is known as a
.
linear process
2

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STATIONARY PROCESSES
Wold's theorem: any stationary process can be represented as a decomposition of a
linear process and a deterministic part. We will consider some special cases of linear
processes.
We need some simplifying assumptions about the probability structure of the longer
series (about the underlying process):
Stationarity Assumption.
Ê
• What is a Stationary TS ?
The behavior of a set of random variables at one point in time
is the
probabilistically
same as the behavior of a set at another point in time
the probabilistic structure of the TS is not time-dependent.
Ê
• Why do we need this ?
You can take your
from any portion of the long series and the probability
finite
sample
structure will be the same. Note that what we have (the sample) is one realization from
the long-series.
3