A
set of ordered observations of a quantitative variable taken at successive
points in time is known as “Time Series”. In other words, arrangement of
statistical data in chronological order, i.e. in accordance with occurrence of
time, is known as ‘Time series’. Time could be in terms of years, months, days or hours, is simply a device
that enables one to relate all phenomenon to a set of common, stable reference
points.

“A
time series may be defined as a collection of readings belonging to different
time periods, of some economic variable or composite of variables.”

Objective of the Analysis of Time Series

An
analysis of time series may enable them to set up a model showing how the
economy works and indicating the main forces which determine whether we have
boom or depression. Thus one of the objects of the analysis of time series is
to forecast how these series will behave in future on the basis of how they
behaved in the past.

**Mathematical Relationship**

It is defined by the functional
relationship

**Components of Time Series**

The various forces at work,
affecting the values of a phenomenon in a time series, can be broadly
classified into the following four categories, commonly known as the components
of a time series:

2.1 Seasonal
variations

2.2 Cyclic variations

3. Random or irregular movements

**Main Problems in the Analysis of Time Series**

**The
main problems in the analysis of time series are:**

- To identify the forces or
components at work, the net effect of whose interaction is exhibited by
the movement of a time series, and
- To isolate, study, analyse and
measure them independently, i.e., by holding other things constant.

**Basic Assumptions in the Analysis of Time
Series**

The immediate objective of the
analysis of time series is to break down the series into the main components
which reflect the secular trend, the periodic movements and the erratic
movements. The method of analysis depends very largely on the hypothesis as to
how components of the series are combined and interact. The simplest hypothesis
is to assume that the separate influences have values which are additive and
independent of each other. The latter assumption means that the seasonal
influence will be the same irrespective of which phase of the cycle obtains.
Thus we have

**Measurement of Trend**

**Trend can be measured by following
methods:**

- Graphic method
- Method of semi-averages
- Method of curve fitting by Principle of least squares
- Method of moving averages

**Graphical Method**

A
free hand smooth curve obtained on plotting
the values zt against t enables us to form an idea about the general ‘trend’ of
the series. Smoothing of the curve eliminates other components, regular and
irregular fluctuations. This method does not involve any complex mathematical
techniques and can be used to describe all type of trend, linear and
non-linear.

**Method of Semi-Averages**

The
whole data is divided into two parts with respect to time, e.g. if we are given
zt for t from 1881-1992, i.e. over a period of 12 years, the two equal parts
will be the data from 1881 to 1886 and 1887 to 1992. In case of odd number of
years the two parts are obtained by omitting
the value corresponding to the middle year, e.g. for the data from 1881 to
1992, the value corresponding to middle year 1886 being omitted. Next we
compute the arithmetic mean for each part and plot these two averages against
the mid values of the respective time periods covered by each part. The line
obtained on joining these two points is the required trend line and may be
extended both ways to estimate intermediate or future values.

**Principle of Least Squares**

It is the most popular and widely
used method of fitting mathematical functions to a given set of data. The
method yields very correct results if sufficiently good appraisal of the form
of the function to the fitted is obtained either by a scrutiny of the graphical
plot of the values over time or by a theoretical understanding of the mechanism
of the variable change. The various types of curves that may be used in
describing data are

- A straight line:
**z**_{t}= a + bt - Second degree parabola:
**z**_{t}= a + bt + ct^{2} - Kth-degree polynomial:
**z**_{t}= a_{0}+ a_{1}t + a_{2}t^{2}+ ….. + a_{k}t^{k} - Exponential curves:
**z**_{t}= ab^{t } - Second degree curve fitted to
logarithms:
**z**_{t}= ab^{t}c^{t^2} - Growth curves:
**z**_{t}= a + b^{c^t}

**Time Series in Correlation and Regression**

Very
often the only data available are in the form of time series, and special care
is needed in correlating data which are in this form. It may happen that two
variables exhibit a high degree of correlation. Over time not because they are
related in any way but because other factors have produced persistent trends
causing both series to rise together or the one to rise and the other to fall
steadily.

**The removal of trend from a series is quite straightforward. It we regard the variable as being compounded as follows**

Similarly
if we are interested in the regression of a dependent variable Z on an independent variable Y,
where the data are in the form of time series, the regression should be
performed in terms of trend free data, so that it will be

**Measurement of Seasonal Variations**

When
data are expressed annually there is no seasonal variation. but
monthly or quarterly data frequently exhibit strong seasonal movements and
considerable interest attaches to devising a pattern of average seasonal
variation. It may be desired to compare the seasonal patterns of different
series, but more often we may want to know the extent to which we should
discount the most recently available statistics for seasonal factors.

**Example of Measurement of Seasonal Variations**

Average
earnings in Australia were $143.60 per week in
the March quarter 1974-75 and then rose to $156.30 in the June quarter. Was
this due to an underlying upward tendency or simply because the June quarter is
usually seasonally higher than the March quarter? If we knew how much the June
quarter is usually above or below the March quarter for seasonal reasons.

**Uses of Time Series**

The
time series analysis is of greater importance not only to businessman or an
economist but also to people working in various disciplines in natural, social
and physical sciences. Some of its uses are enumerated below:

- It enables us to study the past
behavior of the phenomenon under consideration, i.e., to determine the
type and nature of the variations in the data.

The
segregation and study of the various components is of paramount importance to a
businessman in the planning of future operations and in the formulation of
executive and policy decision.