TIME SERIES
by
H.V.S. DE SILVA
DEPARTMENT OF MATHEMATICS
Contents:

Introduction to time series

Fundamentals of time series analysis

Basic theory of stationary processes
Time series Models:

---

MA model
AR model
ARIMA model
Seasonal Adjustment
Some applications of social and Physical Sciences
References:
• The analysis of time series: Theroy & Practice
- Chatfield , C
• The statistical analysis of time series
- Anderson,T.W.
• Time series
- Kondall, Maurice
• The analysis of time series an introduction
- Chatfield, C.
1. Introduction to time series
A time series is a collection of observations
made sequentially in time.
Examples:



In Economics, daily closing stock prices, weekly interest rates,
monthly price indices and yearly earnings.
In Meteorology, hourly wind speed, daily temperature and
annual rainfall
In Social Sciences annual birth rates, mortality rates, accident
rates and various crime rates.
Ctd…




In Engineering electric signals, voltage and
sound.
In Geophysics ocean waves, earth noise in an
area.
EEG and EKG tracings in Medicine.
In Agriculture annual crop production prices.
1.1 Discrete & Continuous time series


A time series is said to be discrete when
observations are taken only at specific time points.
A time series is said to be continuous when
observations are made continuously.
During this lecture we consider only discrete time series that is
when observations are taken at equal intervals.
1.2 Objectives of Time series Analysis

To understand the variability of the time series.

To identify the regular and irregular oscillations of the
time series.

To describe the characteristics of the oscillations.

To understand physical processes that give rise to each
of these oscillations.
1.3
Components of a time series
A time series is a collection of data obtained by observing a
response variable at periodic points in time. Xt is used to
denote the value of the variable x at time t.
A time series is made up of one or more components of the following :
 1.
-
Trend
Measures the average change in the
variable per unit time. In other words
it measures the change in the mean
level of the time series.
Ctd…
 2. Seasonal variations
- Periodic variations that occur with some degree of
regulations within a year or shorter.

3. Cyclical variations
- Recurring up and down movement which
are extended over a long period.( usually 2
years or more)

4. Irregular variations
- Random Fluctuations which happen due to errors.
Time series models can be classified into two types:
Additive model & multiplicative model
The four components are combined in the additive time series
model & multiplicative model as follows;
Yt = T+S+C+R(additive)
Yt = T.S.C.R(multiplicative
Where Yt is the actual value
T – trend or long-term movement
S – seasonal movement
C – cyclical movement
R - residual or random movement
(irregular variation)
The components of a time series are easily
Identified and explained pictorially in a time
series plot.
A time series plot is a sequence plot with the
time series variable Yt the vertical axis and time t
on the horizontal axis.
The figure 1.1 shows a trend in the time series values.
The trend component describes the tendency of the
value of the variable to increase or decrease over a
long period of time.
Figure 1.1 : Trend in a time series
A cyclical effect in a time series as shown in the
figure 1.2 describes the fluctuation about the
trend line.
In business the fluctuations are called business
cycles.
figure 1.2 : cyclic variations in a time series
A seasonal variation in a series describes the
fluctuations that recur during specific portions of each
year ( monthly or seasonally), You can see in the figure
1.3 that the pattern of change in the time series within a
year tends to be repeated from year to year producing a
wavelike or oscillating curve.
Figure 1.4 : seasonal variations in a time series
The final component, the residual effect is
what remains after the trend, cyclical and
seasonal components have been removed.
This component is not systematic and may be
attributed to unpredictable influences.
Thus the residual effect represents the
random error component of a time series.
• One of the objectives of time series is to
forecast some future values of the series.
• To obtain forecast some type of model that can
be projected into the future must be used to
describe the time series.
• One of the most widely used models is the
additive model.
Filtering( Smoothing techniques)
• A procedure to remove random variation revealing more
clearly the underlying trend and cyclic components in a set of
time series data, converting one time series to another by
performing a linear operation for the use of forecasting.

Moving Average
- Smooth out (reduce) fluctuations and gives
trend values to a fair degree of accuracy.
- The average value of a number of adjacent time
series values is taken as the trend value for the
middle point.
Using linear regression technique the equation of
the straight line trend can be estimated.
 Exponential Smoothing
- This is a very popular scheme to produce a
smooth time series.
- weigh past observations with exponentially
decreasing weights to forecast future values.

Differencing
- This is the common method of filtering.
- A popular and effective method of removing trend from a
time series.
- Differencing of a time series { x } in discrete time t is the
transformation of the series {x } to a new series { x }
where the values x
are the differences between
values of consecutive x .
Time series operators

•

Difference operator
- First difference operator
-Second difference operator
- Higher order difference operator
Lag operator (Backward shift operator)
-Transforms an observation of a time series to the
previous one.
Autocorrelation


This is the correlation(relationship) between members of a time
series.
Autocorrelation coefficient at lag k is given by

Suppose we have a set of n values, {xt}, which
represent measurements taken at different time
periods, t=1,2,3,4,…n, of the closing daily price of a
stock or commodity. The following Figure shows a
typical stock price time series: the blue line is the
closing stock price on each trading day; the red and
black looped lines highlight the time series for 7 and
14 day intervals or ‘lags’, i.e. the sets
{xt,xt+7,xt+14,xt+21,...} and
{xt,xt+14,xt+28,xt+42,...}.

Time series of stock price and volume data

The pattern of values recorded and graphed might show that
commodity prices, exhibits some regularity over time. For
example, it might show that days of high commodity prices are
commonly followed by another day of high commodity prices ,
and days of low commodity prices are also often followed by
days of low commodity prices . In this case there would be a
strong positive correlation between commodity prices on
successive days, i.e. on days that are one step or lag apart. We
could regard the set of “day 1” values as one series, {xt,1}
t=1,2,3…n-1, and set of “day 2” values as a second series {xt,2}
t=2,3…n, and compute the correlation coefficient for these two
series in the same manner as for the r expression above.