BUSINESS STATISTICS NOTES
B.COM 2ND AND 3RD SEM NEW SYLLABUS (CBCS PATTERN)
TIME SERIES ANALYSIS
Meaning of Time Series:
One of the most important tasks of any
businessman is to make estimates of future demand of his product so that he can
adjust his production according to the future demand. For this purpose, it is
necessary to gather information from the past. In this connection one usually
deals with statistical data which are collected, observed or recorded at
successive intervals of time. Such data are generally referred to as Time
series.
In the words of Morris Hamburg,”A time series
is a set of statistical observations arranged is chronological order.”
In the words of Wessel & Wellet,” When
quantitative data are arranged in the order of their occurrence, the resulting
statistical series is called a time series.”
From the above explanation we can say that
the time series consists of data arranged chronologically.
Table of
Contents |
1. Meaning of Time Series 2. Significance of Time Series
Analysis 3. Components of Time Series
Analysis 4. Different Models of Time Series
Analysis 5. Different Methods of Times Series
Analysis i)
Graphic
method ii) Semi-average method iii) Moving average method iv) Method of least squares. 6. Seasonal Index method i)
Method
of simple averages ii) Ratio to trend method iii) Ratio to moving average iv) Link relative method. 6. Shifting of base and Deseasonalised
Value ALSO READ: TIME SERIES ANALYSIS
COMPLETE FORMULA (COMING SOON) ALSO READ: TIME SERIES ANALYSIS MCQs |
Utility of Time Series Analysis
The analysis of Time Series is of great significance not only
to the economist and businessman but also to the scientist, geologist,
biologist, research worker, etc., for the reasons given below:
a)
It helps in understanding past behaviors: By observing data over a period of time one can easily
understanding what changes have taken place in the past, Such analysis will be
extremely helpful in producing future behavior.
b)
It helps in planning future operations: Plans for the future cannot be made without
forecasting events and relationship they will have. Statistical techniques have
been evolved which enable time series to be analyzed in such a way that the
influences which have determined the form of that series to be analyzed in such
a way that the influences which have determined the form of that series may be
ascertained.
c)
It helps in evaluating current
accomplishments: The performance can be compared
with the expected performance and the cause of variation analyzed. For example,
if expected sale for 1995 was 10,000 refrigerators and the actual sale was only
9,000, one can investigate the cause for the shortfall in achievement. Time
series analysis will enable us to apply the scientific procedure for such
analysis.
d)
It facilitates comparison: Different time series are often compared and important
conclusions drawn there from. However,
one should not be led to believe that by time series analysis one can foretell
with 100percnet accuracy the course of future events.
Components of Time Series Analysis
The four components of time series
are: (FACTORS RESPONSIBLE FOR TREND IN TIMES SERIES)
1. Secular trend
2. Seasonal variation
3. Cyclical variation
4. Irregular variation
Secular
trend: A time series data may show upward trend or downward trend for a
period of years and this may be due to factors like increase in population,
change in technological progress, large scale shift in consumer’s demands etc.
For example, population increases over a period of time, price increases over a
period of years, production of goods on the capital market of the country
increases over a period of years. These are the examples of upward trend. The
sales of a commodity may decrease over a period of time because of better
products coming to the market. This is an example of declining trend or
downward trend. The increase or decrease in the movements of a time series is
called Secular trend. Examples of Trend
or secular trend: Increase in demand of two wheeler, decrease on death rate due
to advancement of medical science, increase in food production due to increase
in population.
Seasonal
variation: Seasonal variations are short-term fluctuation in a time series
which occur periodically in a year. This continues to repeat year after year.
The major factors that are responsible for the repetitive pattern of seasonal
variations are weather conditions and customs of people. More woolen clothes
are sold in winter than in the season of summer .Regardless of the trend we can
observe that in each year more ice creams are sold in summer and very little in
winter season. The sales in the departmental stores are more during festive
seasons that in the normal days. Examples
of seasonal variation: sale of woolen clothes during winter, decline in
ice-cream sales during winter, demand of TV during international games.
Cyclical
variations: Cyclical variations are recurrent upward or downward movements in
a time series but the period of cycle is greater than a year. Also these
variations are not regular as seasonal variation. There are different types of
cycles of varying in length and size. The ups and downs in business activities
are the effects of cyclical variation. A business cycle showing these
oscillatory movements has to pass through four phases-prosperity, recession,
depression and recovery. In a business, these four phases are completed by
passing one to another in this order. It has four important characteristics: i) Prosperity ii) Decline iii)
Depression iv) Improvement. Examples of cyclical variation: Recession,
Boom, Depression, Recovery, balancing of demand and supply.
Irregular
variation: Irregular variations are fluctuations in time series that are
short in duration, erratic in nature and follow no regularity in the occurrence
pattern. These variations are also referred to as residual variations since by
definition they represent what is left out in a time series after trend,
cyclical and seasonal variations. Irregular fluctuations results due to the
occurrence of unforeseen events like floods, earthquakes, wars, famines, etc. Examples of irregular variation: Flood, fire,
strike, lockout, earthquake, hot wave in winter, rain in desert.
Different models of time series
Times series model are of two types.
One is multiplicative model and other one is additive model.
Multiplicative Model: In Traditional
time series analysis, it is ordinarily assumed that there is a multiplicative
relationship between the components of time series. Symbolically, Y=T X S X C X I
Where
T= Trend
S=
Seasonal component
C=
Cyclical component
I= Irregular
component
Y=
Result of four components.
Additive Model: Another approach is to
treat each observation of a time series as the sum of these four components
Symbolically, Y=T + S+ C + I
Different methods of measuring trend and seasonal variation
The following four methods are commonly
used for measuring trends:
i)
Graphic method
ii)
Semi-average method
iii)
Moving average method
iv)
Method of least squares.
Again, the following methods are
commonly used for measuring seasonal variation:
i)
Method of simple averages
ii)
Ratio to trend method
iii)
Ratio to moving average
iv)
Link relative method.
i)
Graphic method: This is the simplest method of studying trend.
The procedure of obtaining a straight line trend is:
a) Plot the time series on a Graph.
b) Examine the direction of the trend based
on the plotted information.
c) Draw a straight line which shows
the direction of the trend.
The trend line thus obtained can be
extended to predict future values.
Merits:
i) This method is simplest method of
measuring trend.
ii) This method is very flexible. I
can be used regardless of whether the trend is a straight line or curve.
Demerits:
i) This
method is highly subjective because it depends on the personal judgement of the
investigator.
ii) Since this method is subjective in nature it
cannot be used for predictions.
ii) Semi-average method: - Under
this method, the given data is divided into two parts. After that an average of
each part is obtained which gives two points. Each point is plotted at the
mid-point of the class interval covered by the respective part and then the two
points are joined by a straight line which gives the required trend line.
Merits:
i) This method is simple to understand
as compared to the moving average method and the method of least square.
ii) This is an objective method of measuring
trend as everyone who applies this method gets the same result.
Demerits:
i) It is affected by extreme values.
ii) This method assumes straight
relationship between the plotted points whether this exist or not.
iii)
Method of moving average: - Under this method the average value for a
certain time span is secured and this average is taken as the trend value for
the unit of time falling at the middle of the period covered in the calculation
of the average. While using this method it is necessary to select a period for
moving average.
Merits:
i) This
method is simple to understand and apply.
ii) It is
particularly effective if the trend of a series is very irregular.
iii) It is
a flexible method of measuring trend because all figures are not changed if a
few figures are added to the data.
Demerits:
i) Trend
values cannot be computed for all years.
ii) No
there is no hard and fast rule for selecting the period of moving average.
iii) This
method is not appropriate if the trend situation is not linear.
iv) Method of Least Square: This
method is most commonly used method of measuring trend. It is a mathematical
method and a trend line is fitted to the data in such a manner that the
following two conditions are satisfied:-
i) the sum
of deviation of the actual values from their respective mean is zero.
ii) the
sum of square of the deviations of the actual and compute values is least from
this line. That is why this method is called method of least square.
The
straight line trend is represented by the equation: Y = a + bx
Where, y =
denotes the trend values
a
= represents the intercept on y axis.
b=
represents slope of the trend line.
Merits:
i) This is
a mathematical method of measuring trend.
ii) Trend
values can be obtained for all the given time periods in the series.
Demerits:
i) This
method is more tedious and time consuming.
ii) This
method cannot be used to fit the growth curves.
Measurement of Seasonal Variations
Almost
every business show seasonal patterns. If data are expression annually there is
no seasonal variation but if data are expressed monthly or quarterly then data
show strong seasonal movements. In order to analyse seasonal variation, it is
necessary to assume that seasonal pattern is superimposed on a series of
values. Before attempting to measure seasonal variation certain preliminary
decision like whether weekly, quarterly or monthly indexes are required. This
will be decided in the light of the nature of the problem and the type of data
available. To obtain a static description of a pattern of seasonal variation it
will be desirable to eliminate the effect of trend, cycles and irregular
variation from the seasonal data then calculate the seasonal variation in the
form of index which is known as seasonal indexes (percent).
The
following methods are commonly used for measuring seasonal variation:
1) Method of simple averages: This is
the simplest method of obtaining a seasonal index. The following steps are
followed while calculating seasonal index:
a)
Arrange the unadjusted data by years and months or quarters.
b)
Find the monthly or quarterly total.
c)
Divided each total by the number of years for which data are given.
d)
Obtain an average of monthly averages by dividing the total of monthly averages
by 12.
e)
Taking the average of monthly averages as 100, compute the percentage of
various monthly averages as follows:
Seasonal
Index = (Monthly average /Average of monthly average)*100
Main
advantage of this method is that is the simplest of all methods of measuring
seasonal variation. But it is useful only when there are no definite trend
exists. This method assumes that there is no trend component in the series
i.e., 0 = CSI.
2) Ratio-to-trend Method: This
method of calculating a seasonal index is relatively simple and an improvement
over the method of simple average. This method assumes that seasonal variation
for a given month is constant fraction of trend. The ratio-to-trend method
presumably eliminates the seasonal factor in the following manner:
(TXSXCXI)/T
= SXCXI
Merits:
a)
This method is simple to compute and easy to understand.
b)
This method is useful when time series data are for a short period.
c)
There is no loss of data in this method.
Demerits:
a)
This method cannot be used when there is a pronounced cyclical swing in the
series, the trend – whether a straight line or a curve – can never follow the
actual data as closely as a 12-month moving average does.
3) Ratio to moving average: The ratio
to moving average method is the most widely used method of measuring seasonal
variations. This method assumes that all the four components of a time series
i.e., TSCI are present and, therefore, widely used for measuring seasonal
variations. However, the seasonal variations are not completely eliminated if
the cycles of these variations are not of regular nature. Further, some
information is always lost at ends of the time series.
Merits:
a)
The index obtained by the ratio-to-moving average method ordinarily does not
fluctuate so such as the index based on the straight-line trends.
b)
This method is more flexible as compared to other method of seasonal index.
Demerits:
a)
Seasonal index cannot be calculated for each month for which data are available.
4) Link relative method: Amongst
all methods of measuring seasonal variation, link relative method is the most
difficult one. This method is based on the assumption that the trend is linear
and cyclical variations are of uniform pattern. The link relatives are
percentages of the current period (quarter or month) as compared with the
previous period. With the computations of the link relatives and their average,
the effect of cyclical and the random components are minimized. Further, the
trend gets eliminated in the process of adjustment of chain relatives.
Uses of Seasonal Index
a)
A seasonal index is employed to adjust original data in order to yield
deseasonalised data that permit the study of short-run fluctuations of a series
not associated with seasonal variations.
b)
A seasonal index may be used for economic forecasting and managerial control.
Management usually benefits from examining the seasonal pattern of its own
business, patterns that directly influence its employment, production, purchase,
sales and inventory policies.
c)
Seasonal index helps in diversification. It benefits not only the firm but also
society at large.
Limitations
of Seasonal Index
a)
No technique can measure seasonal variations precisely because various methods
are based on unrealistic assumptions.
b)
Seasonal Index only shows an average pattern during a number of years which may
have no significance for a particular period.
Shifting of trend origin and deseasonalised data
Shifting: Shifting
of trend origin means replacing the origin with new base. Shifting can be done
by using the following formula: Y = a + b (X + k)
Where k is the number of time units
shifted. If the origin is shifted forward in time, k is positive and if shifted
backward in time, k is negative.
Deseasonalised
Value: The value which shows how things would have been or would be if
there were no seasonal fluctuations is called Deseasonalised data. In order to
obtain Deseasonalised data, the effect of seasonal variations have to be
removed. For this purpose, the actual data is divided by the appropriate
seasonal indices.
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