1.6
Seasonal variation
The time plot should be examined to see which model, whether additive or
multiplicative is likely to give a better description. The seasonal indices
cfw_st
are
usually assumed to change slowly through time so that st st s , where '

3.2
Autoregressive Processes
Autoregressive processes as their name suggests regressions on themselves.
A pth order autoregressive process cfw_Yt is given by
Yt = 1Yt 1 + 2Yt 2 + . pYt p + Zt
(3.2.1)
The current value of the series Yt is a linear combina

2.
STATIONARY PROCESSES
We shall describe the fundamental concepts in theory of time series models.
Introduce the concepts of stochastic processes, mean and covariance functions,
stationary processes and autocorrelation functions.
2.1
Stochastic Processes

3.3
ARMA
An ARMA model, of order ( p, q ) is defined by
Yt = 1Yt 1 + . + pYt p + Z t + 1Z t 1 + . + q Z t q
(3.3.1)
ie
p
q
i =1
i =0
Yt = iYt i + i Z t i
which is an autoregressive moving average process of order
( p, q )
or an
ARMA ( p, q ) .
In other wo

3.
MODELS FOR STATIONARY TIME SERIES
The process cfw_Yt is called a linear process if it has a representation of the form
Yt = +
i Z t i
i =
where:
is a common mean
cfw_ i
cfw_Z t
is a sequence of fixed constants
are independent random variables with

1.
INTRODUCTION
Data obtained from observations collected sequentially over time are extremely
common. For example:
In business, we observe weekly interest rates, daily closing stock prices,
monthly price indices, yearly sales figures.
In meteorology, w

3.4
Partial Autocorrelation Function (PACF)
Consider the general statement:- the partial correlation coefficient between X and Y
with dependence on a third variable Z removed is given by
Corr ( X , Y | Z ) = XY .Z =
XY XZ YZ
(1 )(1 )
2
XZ
2
YZ
In our cas

1.5
Serial Dependence
Recall that the y ' s are not independent but are serially dependent. We can describe
the nature of the dependence using a set of autocorrelations.
1.5.1 Autocorrelation
( y1,., yn ) on a time series, we can form n 1 pairs
namely, (

1.
INTRODUCTION
Data obtained from observations collected sequentially over time are extremely
common. For example:
In business, we observe weekly interest rates, daily closing stock prices,
monthly price indices, yearly sales figures.
In meteorology, w