Edited economic statistics note

1
UNIT ONE
INTRODUCTION
1.1. MEANING OF ECONOMIC STATISTICS
Statistics: is a science that deals with the methods of collecting, organizing, and analyzing of
data and interpretation of results. It is a science of decision making under uncertainty.
There are 4 stages of statistics;
 Collection of data,
 Presentation of data,
 Analysis and
 Interpretation.
Economic Statistics is an area which uses statistical methods in presenting, analyzing and
interpretation of economic data.
It includes those statistical methods which are frequently used in economics.
1.2. FUNCTIONS OF ECONOMIC STATISTICS
 It presents facts in a definite form
 It simplifies a mass of figures
 It facilitates comparison
 It helps in formulating and testing hypotheses
 It helps in prediction
 It helps in formulation of suitable policies
1.3. TYPES OF STATISTICAL DATA
Data are row facts about a phenomenon. Data are records of the actual state of some measurable
aspect of the universe at a particular point in time. Data are not abstract; they are concrete, they
are measurements or the tangible and countable features of the world. When data are processed,
they generate information. There are different types of statistical data based on the reference in
which they are measured.
A) Based on Scale of measurement
Based on the scale of measurement, there are different types of data. There are four basic
measurement scales: Nominal, ordinal, interval and ratio.
The most accepted basis for scaling has three characteristics:
1. Numbers are ordered. One number is less than, greater than, or equal to another number.
2. Differences between numbers are ordered. The difference between any pair of numbers is
greater than, less than, or equal to the difference between any other pair of numbers.
3. The number series has a unique origin indicated by the number zero.

2
Combination of these characteristics of order, distance, and origin provide the following widely
used classification of measurement scales.
Nominal Scales: When we use nominal scale, we partition a set into categories that are mutually
exclusive and collectively exhaustive. The counting of members is the only possible arithmetic
operation and as a result the researcher is restricted to the use of the mode as the measure of
central tendency. If we use numbers to identify categories, they are recognized as labels only and
have no quantitative value. Nominal scales are the least powerful of the four types. They
suggest no order or distance relationship and have no arithmetic origin. Examples can be
respondents’ marital status, gender, students’ Id number, etc.
Ordinal Scales: Ordinal scales include the characteristics of the nominal scale plus an indicator
of order. The use of an ordinal scale implies a statement of ‘greater than’ or ‘less than’ (an
equality statement is also acceptable) without stating how much greater or less. Thus the real
difference between ranks 1 and 2 may be more or less than the difference between ranks 2 and 3.
The appropriate measure of central tendency for ordinal scales is the median. Examples of
ordinal scales include opinion or preference scales.
Interval Scales: The interval scale has the powers of nominal and ordinal scales plus one
additional strength: It incorporates the concept of equality of interval (the distance between 1
and 2 equals the distance between 2 and 3). When a scale is interval, you use the arithmetic
mean as the measure of central tendency. Calendar time is such a scale. For example, the elapsed
time between 4 and 6 A.M. equals the time between 5 and 7 A.M. One cannot say, however, 6
A.M is twice as late as 3 A.M. because zero time is an arbitrary origin. Centigrade and
Fahrenheit temperature scales are other examples of classical interval scales.
Ratio Scales: Ratio scales incorporate all of the powers of the previous ones plus the provision
for absolute zero or origin. The ratio scale represents the actual amounts of a variable.
Multiplication and division can be used with this scale but not with the other mentioned.
Money values, population counts, distances, return rates, weight, height, and area can be
examples for ratio scales.
Summary of measurement scales
Type of scale Characteristics Basic empirical operation
Nominal No order, distance, or origin Determination of equality
Ordinal Order but no distance or
unique origin
Determination of greater or lesser values
Interval Both order and distance but no
unique origin
Determination of equality of intervals or
differences
Ratio Order, distance, and unique
origin
Determination of equality of ratios

3
B) Based on Time Reference
On the basis of time reference, statistical data are of four types;
Time Series Data: These are data collected over periods of time. Data which can take different
values in different periods of time are normally referred as time series data.
Cross-Sectional Data: Data collected at a point of time from different places. Data collected at a
single time are known as cross-sectional data.
Pooled Data: Data collected over periods of time from different places. It is the combination of
both time series and cross-sectional data.
Panel Data: It is also known as longitudinal data. It is a time series data collected from the same
sample over periods of time.
C) Based on the Sources
Depending on the source, the type of data collected could be primary or secondary in nature.
Primary data are those which are collected afresh and for the first time, and thus happen to be
original in character. Its advantage is its relevance to the user, but it is also likely to be expensive
in time and money terms to collect.
Secondary data are those which have already been collected by someone else and which have
already been passed through the statistical process. It is information extracted from an existing
source, probably published or held on a computer database. From Practical point of view this
type of information is collected for any purpose other than the current research objectives and is
not always up-to-date. For this reason it may not precisely meet the needs of the secondary user.
However, it is less expensive and time-consuming to obtain. Therefore, it provides a good
starting point and very often can help the investigator to formulate and generate ideas which can
later be refined further by collecting primary data.

4
UNIT TWO
OVERVIEW OF DATA PRESENTATION AND ANALYSIS
TECHNIQUES
2.1DATA PRESENTATION TECHNIQUES
Data Presentation: is the process of summarizing the collected data in a meaningful and
suitable form. Presentation can be done in two basic forms: Statistical tables and Statistical
charts.
Statistical table is the presentation of numbers in a logical arrangement with some brief
explanation to show what the data representing.
Statistical Charts or graphs on the other hand are pictorial devices of presenting data.
2.1.1 Presentation of Quantitative Data
The important tools for presenting quantitative data are:
1. Frequency Distribution
2. Histograms
3. Frequency Polygons
4. Cumulative Frequency Curve or Ogives.
Frequency Distribution
It is the method of arranging data in some order and counting the number of times each
observation appears in the data set. Frequency is the number of times that each observation
appears in the data set. Important elements of frequency distribution are:
1. Class interval: it is the difference between the upper limit and the lower limit
2. Class limit: It is the lowest and the highest values that can be included in the class.
3. Class frequency: Is the number of observations corresponding to the particular class.
4. Class mark: Is the midpoint of the class interval.
2
limlim itclassloweritclassupper
markclass


5. Class width: Is the size of the class.
Histogram
Is a graph consisting of rectangles having:
1. Bases on horizontal axis with centers at class marks and lengths
2. Areas proportional to class frequencies.
Frequency Polygon
It is a graph of frequency distribution. It facilitates comparison of two or more frequency
distributions on the same graph. It can be drawn with or without histogram.
Ogive Curve

5
It is a graph that shows the cumulative frequency less than any upper class boundary or more
than any lower class boundary. There are two types of ogive curves; the less than ogive and more
than ogive curves. In less than ogive cure we should use the upper class boundaries of class in
the horizontal axis and cumulative class frequencies on the vertical axis but in the case of more
than ogive cure we use lower class boundaries on the horizontal axis and cumulative class
frequencies on the vertical axis.
2.1.2 PRESENTATION OF QUALITATIVE DATA
Important methods of presenting qualitative data include:
1. Bar charts
2. Categorical distribution
3. Pie-charts
Bar charts: are one sided rectangular shaped diagrams used to present qualitative data. Bar
charts are drawn in such a way that the height of the par is proportional to the amount of a given
category and the width of each bar must be equal for all bars and the space between any two bars
must be the same with the space between any other two bars. There are different types of bar
graphs such as simple, sub divided and special bar graphs.
Categorical Distribution: is a distribution used to present categorical data. It is the frequency
distribution counter part of a categorical data. In this method of presenting data, categories
should be defined in such a manner that they should be mutually exclusive and collectively
exhaustive.
Ex. Employees of Organization X by level of education.
No Name of Employee Level of Education
1 Alemu Diploma
2. Yalew Cirtificate
3. Chala Bachelor
4. Toga MSc.
. .
. . .
60 Gebremedhin PhD
Present the data using categorical distribution.
Solution: first, we have to establish the level of education into mutually exclusive and
collectively exhaustive categories in such a way that we should ensure that each employee
should have only one category and does not have more than one category and there must be a
category for any employee to be belonged for.

6
Second, we have to count the number of employees that are belonging to each category.
Categorical distribution of Employees of Organization X by LOE
Education Number
Certificate 5
Diploma 15
Bachelor 25
Master 12
PhD 3
Total 60
Pie-chart: is a type of circle used to display the percentage of total number of measurements
falling into each category. It is a method of presenting data in a manner of dividing 360 degrees
of a circle into a degree that is allocated to each category proportional to the share of each
category in the total data.
Ex. Present the data of employees of organization X by LOE using pie-chart.
Solution: first, calculate the share of each category from the total data and secondly divide the
circle into a degree that each category should have from the circle.
Employees of Organization X by the LOE
Education Number Percentage Degree of a category
Certificate 5 8.33 30
Diploma 15 25 90
Bachelor 25 41.67 150
Master 12 20 72
PhD 3 5 18
Total 60 100 360
2.2 DATA ANALYTICAL TECHNIQUES
REGRESSION AND CORRELATION ANALYSIS
Very often data are given in pairs of measurements where one variable is dependent on the other
variable.
Ex. Income and years of service of workers
Saving and family size
Food consumption and weight of people
University to high school level performance Etc.
Regression and correlation analysis will show us how to determine both the nature and strength
of relationship between a series of paired observations of two or more variables. Regression
deals with the mathematical method that depicts the relationship while correlation concerned
with measuring and expressing the closeness of the relationship between variables.

7
2.2.1 Correlation Analysis
Correlation is the degree of relationship that exists between two or more variables. It is the
measure of degree of co-variation or association between two or more variables. Two variables
are said to be correlated if an increase or a decrease on average in one variable is accompanied
by an average increase or decrease of the other otherwise they are not.
Types of Correlation: There are different types of correlation investigated from showing the
nature of relationship that variables has. Correlation may be:
 Positive or negative
 Simple, partial or multiple
 Linear or non-linear
Positive or Negative Correlation
If an increase or a decrease in one variable is accompanied by the increase or decrease in the
other changing with the same direction of both variables, then we will have a positive
correlation. There are many economic variables which are positively correlated. Some examples
of these include; quantity supplied and price of a commodity, income of a consumer and demand
for a normal good, consumption and family size, saving and disposable income, etc.
If an increase or a decrease in one variable is accompanied by the decrease or increase in the
other changing with opposite directions of both variables, we will have a negative correlation.
There are also several economic variables which are negatively correlated. Some of these
include; quantity demanded and price of a commodity, interest rate and investment, saving and
family size, supply and price of an input, etc.
Simple, Partial or Multiple Correlations
A correlation is said to be simple if it studies or if it exists between two variables only. A
correlation is said to partial if it exists between two variables when all other variables connected
to those two are kept constant and a correlation is said to multiple if it exists between more than
two variables. Simple and partial correlations can take any value positive, zero or negative but
multiple correlations cannot be negative.
Linear or Non-linear Correlation: A correlation is said to be linear if a change in one variable
brings on average a constant change on the other variable. A correlation is said to be non-linear if
a change in one variable brings on average a different change on the other variable.
METHODS OF STUDYING CORRELATION
Correlation is studied in one of the following three methods
1. The scatter diagram(Graphic method)
2. Simple linear correlation coefficient
3. The coefficient of rank correlation
The Scatter Diagram: It is the rectangular diagram which helps us to visualize the relationship
between two phenomena. We can plot the data by an X-Y plane starting from the minimum
values of X and Y variables. When there is a strong correlation (Either positive or negative) the
dots are condensed each other and as the degree correlation decreases they become more and

8
more scatter. If the two variables are positively correlated the scatter diagram shows that the
points will be moving from left bottom to right top. If the two variables are negatively correlated,
the scatter diagram shows that the points will be moving from right bottom to left top. If the two
variables are uncorrelated, the points do not show any pattern but they show a non-patterned
plot.
Simple Linear Correlation Coefficient: The sample correlation coefficient is denoted by r. It is
the measure of degree of relationship that exists between two variables. It is only applied in
linear relationship as well as in simple correlation. The values of correlation coefficient cannot
be less than -1 and cannot be greater than +1. In other words, the values of correlation coefficient
always range between -1 and +1. A good measure of correlation coefficient is a one which
supplies the answer in pure number, independent of the unit of measurement and indicates the
direction and extent of correlation.
2222
2
__
2
__
____
22
____
)(())((
)()(
))((
.
YYnXXn
YXXYn
r
YYXX
YYXX
r
yx
xy
r
YYyandXXxwhere
n
xy
r
yx













If the correlation coefficient is ranging between -1 and zero, there is a negative correlation.
Movement from zero to -1 increases the degree of negative correlation and vice versa. If the
correlation coefficient takes the value of -1, there is a perfect or exact negative correlation
between the two variables.
If the correlation coefficient is ranging between zero and +1, there is a positive correlation.
Movement from zero to +1 increases the degree of positive correlation and vice versa. If the
correlation coefficient takes the value of +1, there is a perfect or exact positive correlation
between the two variables.
If 0r , there is no any linear correlation between the two variables
Properties of Correlation Coefficient
1. The values of a correlation coefficient range between -1 and +1.
2. Correlation coefficient is symmetric. XYYX rr 
3. Correlation coefficient is the geometric mean of two regression coefficients. XYYX bbr .
4. Correlation coefficient has the same sign with regression coefficients. If the two regression
coefficients are positive, correlation coefficient will be positive and vice verssa.
5. Correlation coefficient is independent of change of origin and change of scale. By change of
origin we mean that adding or subtracting any constant from the values of the variables.

9
Independent of change of origin indicates that adding or subtracting any constant value from the
values of the two variables does not change the correlation coefficient. By change of scale we
mean that multiplying or dividing values of the two variables by any constant. Independent of
change of scale indicates that multiplying or dividing values of the two variables by any constant
does not change the correlation coefficient.
Example: Calculate Karl Pearson’s correlation coefficient from the following paired data
X: 28 41 40 38 35 33 40 32 36 33
Y: 23 34 33 34 30 26 28 31 36 38
Solution: We can apply the formula for computing correlation coefficient after we compute the
arithmetic means of the two variables. We can also compute the correlation coefficient by using
property five. Let we subtract the value 35 from all values of X and the value 31 from all values
of Y and compute the coefficient.
X Y )35( X 2
)35( X )31( Y 2
)31( Y (X-35)(Y-31)
28 23 -7 49 -8 64 56
41 34 6 36 3 9 18
40 33 5 25 2 4 10
38 34 3 9 3 9 9
35 30 0 0 -1 1 0
33 26 -2 4 -5 25 10
40 28 5 25 -3 9 -15
32 31 -3 9 0 0 0
36 36 1 1 5 25 5
33 38 -2 4 7 49 -14
162)35( 2
 X 195)31( 2
 Y 79
ncorrelatiopositiveaveragelyiswhichr
YX
YX
r
.44.0
195.162
79
)31(.)35(
)31)(35(
22





Rank correlation Coefficient
The Karl Pearson’s coefficient of correlation can’t be used in cases where the direct quantitative
measurement of phenomenon under study is not possible. In such cases one may rank the
different items and apply the Spearman’s method of rank differences for finding out the degree
of correlation. The rank correlation coefficient is denoted by R. Its value also ranges from -1 to
+1.
)1(
6
1 2
2



nn
D
R i

10
Ex. A group of workers of a factory are ranked according to their efficiency by judges as ff;
Name of Worker Judgment of Judge A Judgment of Judge B
A 4 3
B 8 9
C 6 6
D 7 5
E 1 1
F 3 2
G 2 4
H 5 7
I 10 8
J 9 10
Compute the rank correlation coefficient and interpret your result. Solution:
Name of Worker 1R 2R iD 2
iD
A 4 3 1 1
B 8 9 -1 1
C 6 6 0 0
D 7 5 2 4
E 1 1 0 0
F 3 2 1 1
G 2 4 -2 4
H 5 7 -2 4
I 10 8 -2 4
J 9 10 1 1
202
 iD
88.0
990
120
1
)1100(10
20.6
1
)1(
6
1 2
2







R
R
R
nn
D
R i
Therefore, we can interpret the result as the opinion of two judges with regard to the efficiency
of workers shows greater similarity.
2.2.2 Regression Analysis

11
Regression describes the average relationship between variables in a sense that the change in one
or more variables brings a certain change on the other variable. A variable or group of variables
that makes a certain cause for the change of the other variable is called independent or
explanatory variable. A variable which is affected by the change of other variables is called
dependent or explained variable. Regression describes the cause and effect relationship among
variables. A regression can be simple or multiple or it can be also linear or non-linear. A
regression is said to be linear if it studies the relationship between one independent and one
dependent variable while a regression is said to multiple if it studies the relationship between one
dependent and more than one independent variable. A correlation is said to be linear if the
change in an independent variable brings a constant change on the dependent variable while a
correlation is said to non-linear if the change on an independent variable brings a non-constant
change on the dependent variable.
Estimating the Parameters of A function
Ordinary least Square Estimating Method (OLS): There are different methods of estimating
the unknown parameters of a regression function in which the ordinary least squares is the most
prominent method which is frequently used by statisticians because of its simplicity and having
the desirable statistical properties that a good estimator should have to be a reliable estimator.
For the matter of simplicity and scope here we will discuss only simple linear regression in
which there are two variables in the model (one dependent and one independent variable) and the
function is linear.
iii UXY  10  . There are five elements in this function; iUandXY 10 ,,,  . Y, X and U
are known as variables and 10  and are known as parameters. Y and X are known variables
whose values are collected from the field or from secondary sources. Since iUand10 , are not
observed, the above function cannot be estimated as it is. Thus, we have to get the estimators of
the unobserved elements and try to estimate the parameters.
iii eXbbY  10
The estimated regression line is given by the equation as;
ii XbbY 10 

.The difference between the actual value of Y and its estimated value (

iY ) is the
error term, which can be given as;

 iii YYe . But we can replace

iY by iXbb 10  and find the
equation for the error term.
iii XbbYe 10  . The ordinary least squares method is designed to compute the estimated
values of the parameters in such a way that the error term is the minimum possible. To minimize
the error term function, first we have to find the aggregate of second degree function of the error
term and apply the classical optimization criteria for minimization. This can be done as follows;
2
10
2
)( iii XbbYe  and partially differentiate this error term function with respect to
10 bandb and set it with equal to zero to find the critical points.

12
0))((2
0)1)((2
10
1
2
10
0
2






iii
i
ii
i
XXbbY
b
e
XbbY
b
e
After manipulating certain algebraic operations, we can get values of parameter estimators that
can minimize the error term as follows;
221
2_
2
____
121
____
0
)(
.
XXn
XYXYn
b
XnX
XYnXY
bor
x
xy
b
XbYb










It should be also noted that YXbb 1 that is the regression coefficient of Y dependent on X. This
regression can also be given in terms of the correlation coefficient.
Xofdeviationdardstheisand
Yofdeviationdardstheisandtcoefficienncorrelatiotheisrwhererb
X
Y
X
Y
YX
tan
tan.





The inverse of the function )(XfY  which can be given as )(1
YfX 
 and its regression is
known as inverse regression. The inverse regression is given as
ii dYcX  . Where c and d are parameter estimates. By the same token c and are computed as
follows;
22__
2
____
2
____
)(
.
2
XYn
YXXYn
YnY
YXnXY
dor
x
xy
d
YdXc










The regression coefficient d is also known as the regression coefficient of X dependent on Y.
Y
X
XY rbd


.
Properties of Regression Coefficients
1. The regression coefficients are not symmetric. That is the value of the regression coefficient
of Y dependent on X is not equal with the value of the regression coefficient of X on Y.
XYYX bb 
2. The regression coefficients must be having the same sign. If the regression of Y on X is
negative, then should be the regression coefficient of X on Y and vice-versa.

13
3. If one of the regression coefficient is greater than one the other should be less than one and
vice-versa.
4. Regression coefficients are independent of change of origin but not independent of change of
scale.
The Coefficient of Determination
After we estimate the unknown parameters, we have check to what extent the estimators are the
reliable representatives of the parameters. We can test them using either t-test or z-test. The
second test that can help us in testing the goodness of fit is the coefficient of determination.
The coefficient of determination is the measure of the explanatory power of the model . It
measures the proportion or percentage of the total variation of the dependent variable explained
or determined by the model. The coefficient of determination is denoted by the symbol 2
R .

14
UNIT FIVE
TIME SERIES ANALYSIS
1.1 Introduction
Time series Analysis
Definitely four types of data may be available for empirical analysis: time series, cross-section
panel and pooled (combination of time series and cross section) data. A time series is a set of
observations on the value that a variable takes at different times. Cross section data are data on
one or more variables collected at the same point of time.
There are two major methods of analyzing time series data: Conventional and econometric
methods. Econometric method of analysis can also be divided into two; frequency domain
approach or spectral analysis and time domain approach. For ease of understanding, we are going
to discuss the conventional method of time series analysis.
A time series data is a set of observations taken at specified times, usually at “equal intervals”,
Mathematically, a time series is defined by the values Y1,Y2, . . . Yt , thus Y is a function of time,
symbolically Y=f(t). Thus, when we observe numerical data at different points of time and the
set of observations is known as time series. A good example is the production of teff in each
production year.
Role of Time series Analysis
Time series analysis is great significance in business decision making for the following reasons:
1. It helps in the understanding of post behavior by observing data over the period of time; one
can easily understand what changes have taken place in the past. Such exercise will be
important in understanding and predicting the future.
2. It helps in planning future operations if the regularity of occurrence of any feature over a
sufficient long period could be clearly established, then, prediction of probable future
variations would become possible.
3. It helps in evaluating current accomplishments. Times series analysis helps comparing the
actual performance with that of the expected performance and the cause of variation is
analyzed.
4. It facilitates comparison – Different time series are often compared and important
conclusions drawn from them.
5.2 Components of Time Series

15
Time series elements are classified in to four basic types of variations which account for the
changes in the series over a period of time. These four types of patterns, variations, movements
are often called components or elements of time series. These are:
1) Secular trend
2) Seasonal variations
3) Cyclical variations
4) Irregular variations
In traditional or classical time series analysis, it is ordinarily assumed that there is a
multiplicative relationship between these four components. That is, it is assumed that any
particular value in series is the product of factors that can be attributed to the various
components. Symbolically, it is given as; Y= T*S*C*I
Where; T= Trend, S= Seasonal, C= Cyclical and I= Irregular
If the above model is employed, the seasonal, cyclical and irregular items are not viewed as
absolute amounts, but rather as relative magnitude.
1. Secular Trend
Trend is the variation of value of a variable that can be observed in a long period of time. It is
the general tendency of the data to grow or to decline over a long period of time. Trend is
broadly divided under two heads: linear (what we going to see) and non – linear trends.
Methods of measuring Trend
The following methods are used for measuring trend:
1) Graphic method
2) The semi – average method
3) The method of least squares
Graphic method: - This is the simplest method of studying trend. Under this method the given
data are plotted on graph paper and a trend line is fitted to the data just by inspecting the graph
of the series. There is no formal statistical criterion where by the adequacy of such a line can be
judged and the judgment depends on the discretion of the individual researcher. However, as a
rough guide, the line should be drawn in such a way that it passes between the plotted points in
such a manner that the fluctuations in one direction are approximately equal to those in the other
direction and that it shows a general movement. This method is not frequently used since its
approach is subjective and no statistical method is used.
Methods of semi – Averages: This method is used in such a way that the given data are divided
in to two parts, preferably, with equal number of years. For example, if we are given data from
1982 to 1999, that is, over a period of 18 years, the two equal parts will be first nine years, i.e.,
from 1982 to 1990 and from 1991 to 1999. In the case of odd number of years like 9, 13, 17,
etc, two equal parts can be made simply by ignoring the middle year. For example, if the data
are given for 19 years from 1981 to 1999, the two equal parts would be from 1981 to 1989 and
from 1991 to 1999, the middle year 1990 would be ignored.
Example: fit a trend line to the following data by the method of semi-averages:

16
Year sales
1994 102
1995 105
1996 114
1997 110
1998 108
1999 116
2000 112
Solution: since seven years are given, the middle year should be omitted and an average of the
first three years and the last three years shall be obtained. The averages of the first three years is
112
3
336
3
112116108
107
3
321
3
114105102




isyearsthreelasttheofaveragetheand
Thus, we get two points, 107 and 112, which shall be plotted corresponding to their respective
middle years, i.e. 1995 and 1999. By joining these two points; we obtain the required trend line.
Y
Trend Line
112
107
Time
1994 1995 1996 1997 1998 1999 2000
Method of least squares
This method is most widely used in practice. When this method is applied, a trend line is fitted to
the data in such a manner that the following two conditions are satisfied:
1) 0)(  CYY . The sum of deviations of the actual values of Y and the computed values
of Y is zero.
2) 2
)( cyy  Is the least, that is, the sum of the squares of the deviations of the actual and
computed values is the least one. The method of least squares can be used either to fit a

17
straight line trend or a parabolic trend. The straight line trend is represented by the
equation
bXaYC 
In order to determine the value of the constants a and b, the following two normal equations are
to be solved.
XbanY  .
2
XbXaYX  Where n represents number of years and X is the time period.
We can measure the variable x from any point of time in origin such as the first year. However,
this calculations are very much simplified when the midpoint in time is taken as the origin
because in that case, the negative values in the first half of the series balances the positive
values in the second half so that x=0, the above two normal equations would take the form:
2
2
.
x
xy
b
n
Y
a
xbxy
naY







The constant ‘a’ gives the arithmetic mean of Y and constant ‘b’ indicates the rate of change.
Example: - based on the following figures of production of a sugar factory (in thousand
quintals), fit a straight line trend and estimate the likely sales of the company in 1990.
Year 1983 1984 1985 1986 1987 1988 1989
Production 80 90 92 83 94 99 92
Solution
Year Production (Y) Time (X) XY X2
1983 80 -3 -240 9
1984 90 -2 -180 4
1985 92 -1 - 92 1
1986 83 0 0 0
1987 84 1 94 1
1988 99 2 198 4
1989 92 3 276 9
Y=630 X=0 XY=56 X2
=28
bXaYC 

18
2
28
56
90
7
630
2







x
xy
b
n
Y
a
XYC 290 
2. Forecasting for the year 1990. Since 1990 is four years later than the base year, x=4.
Therefore, we have to find the value of the YC when x=4
YC = 90 + 2 (4) = 98 units i.e. the likely production of sugar factory in 1990 is 98,000 quintals.
Example 2:- calculate the trend values by the method of least squares from the data given below
and estimate the sales for the year 2003.
Year 1996 1997 1998 1999 2000
Sales 12 18 20 23 27
Solution
Year Sales Time XY X2
YC
1996 12 2 24 4 13
1997 18 -1 -18 1 16.5
1998 20 0 0 0 20
1999 23 1 23 1 23.5
2000 27 2 54 4 27
y=100 x=0 xy=35 x2
=10
bXaYC 
5.3
10
35
20
5
100
2







x
xy
b
n
Y
a
YC = 20 + 3.5 X
Forecasting for the sales of 2003 is carried out by substituting x = 5 since 2003 is found five
years later than the base year 1998.
Y2003 = 20 + 3.5 (5) = 37.5
Example 3:- Fit a straight line trend to the following data
Year 1995 1996 1997 1998 1999 2000
Production 64 70 75 82 88 95
Year Sales Time XY X2
1995 64 - 3 - 192 9
1996 70 -2 -140 4
1997 75 -1 -75 1
1998 82 0 0 0

19
1999 88 1 88 1
2000 95 2 190 4
y=474 x=-3 yx=-129 x2
=19
Y = n. a + bx
Xy = ax + bx2
474 = 6a – 3b
-129 = -3a + 196
474 = 6a – 3b
-258 = -6a + 38b
216 = 35 b
b = 216 = 6.17
35
474 = 6a – 3 (6.17)
6a = 474+18.51
6a = 492.51
085.82
6
51.492
a
YC = 82.085 + 6.17X
Seasonal variations
Seasonal variations are periodic movements in business activity which occurs, regularly every
year and have their origin in the nature of the year itself. It exists only when data are given in a
period which is less than a year (monthly, semi-annually, quarterly, weekly, daily, etc).
However, it does not exist in data which are given in annual basis or more than a year period
internal. Nearly every type of business activity is liable to seasonal influence to a greater or
lesser degree and, as such, these variations are regarded as normal phenomenon recurring every
year. Although the word ‘seasonal’ seems to imply a connection with the season of the year, the
term is meant to include any kind of variation which is of periodic nature and whose repeating
cycles are of relatively short duration. The factors that cause seasonal variations are:
1) Climate and weather conditions. The most important factor causing seasonal variation is the
climate changes in the climate and weather conditions such as rainfall, humidity, heat, etc,
act on different product and industry differently.
2) Customs, traditions and habits – Though nature is mainly responsible for seasonal variations
in time series, customs and traditions also have their impact.
Measurement of seasonal variations
When data are expressed annually there is no seasonal variation. However, monthly or quarterly
data frequently exhibit strong seasonal movements and considerable interest attaches to devise a

20
pattern of average seasonal variation. There are several methods of measuring seasonal
variation. However, the following methods are popularly used in practice:
1. Method of simple averages
2. Ratio to trend method
3. Ratio to moving average method
4. Link relatives method
Method of simple averages
This is the simplest method of obtaining a seasonal index. The following steps are necessary for
computing the index:
1) Average the unadjusted data by years and months or quarters if the data are given quarterly.
2) Find the totals of the data in each month, quarter or a period in which the data are given.
3) Divide each total by the number of years for which data are given.
4) Obtain an average of monthly averages by dividing the total of monthly averages by 12.
5) Taking the average of monthly averages as 100, compute the percentage.
100)(
averagesmonthlyofAverage
JanuaruforaverageMonthly
JanuaryforIndexSeasonal 
Example: consumption of monthly electric power in KW hours of for street lighting in
Haramaya University from 1995 – 1999.
Year Jan Feb Mar Apri may Jun Jul Aug Sep Oct
1995 318 281 278 250 231 216 223 245 269 302
1996 342 309 299 268 249 236 242 262 288 321
1997 367 328 320 287 269 251 259 284 309 345
1998 392 349 342 311 290 273 282 305 328 364
1999 420 378 370 334 314 296 305 330 356 396
Year Nov Dec
1995 325 347
1996 342 364
1997 367 394
1998 389 417
1999 422 452

21
Find out seasonal variation by the method of monthly averages?
Solution:
Month 1995 1996 1997 1998 1999 Total Average %
Jan 318 342 367 392 420 1839 367.8 116.1
Feb 281 309 328 349 378 1645 329 103.9
Mar 278 299 320 342 370 1609 321.8 101.6
April 250 268 287 311 334 1450 290 91.6
May 231 249 269 290 314 1353 270.6 85.4
Jun 216 236 251 273 296 1272 254.4 80.3
July 223 242 259 282 305 1311 262.2 82.8
Aug 245 262 284 305 330 1426 285.2 90.1
Sep 269 288 309 328 356 1550 310 97.9
Oct 302 321 345 364 396 1728 345.6 109.
Nov 325 342 367 389 422 1845 369 116.
Dec 347 364 394 417 452 1974 394.8 124.7
Total 19002 3800.4 1200
Average 1583.5 316.7 100
8.82100
7.316
2.262
9.103100
7.316
329
1.116100
7.316
8.367



JulayforindexSeasonal
FebruraryforindexSeasonal
JanuaryforindexSeasonal
Ratio – to- Trend method
This method of calculating a seasonal index in relatively simple and yet an improvement over
the method of simple average explained in the preceding section. The method assumes that the
seasonal variation for a given month is a constant fraction of the trend. It first eliminates the
trend component by dividing the original data with the trend value.
ICS
T
ICST


Random elements are supposed to disappear when the ratios are averaged. A careful selection of
the period of years used in the computation is expected to cause the influences of prosperity or
depression to offset each other and thus removes the cycle.

22
This method requires the following steps:
1. Compute the trend values by applying the method of least squares;
2. Divide the original data month by month by the corresponding trend values and multiply the
ratio by 100. The values obtained are now free from trend;
3. In order to free form irregular and cyclical movements, the irregular given for various years
for January, February, etc should be averaged and
4. The seasonal index for each month is expressed as a percentage of the average month. The
sum of 12 values must equal 1,200 or 100%. If it does not, an adjustment is made by
multiplying each index by a suitable factor (1200). This gives the final seasonal index.
Example: - find the seasonal variations by ratio to trend method from the data given below
Year 1st
q 2nd
q 3rd
q 4th
quarter
1996 30 40 36 34
1997 34 52 50 44
1998 40 58 54 48
1999 74 76 68 42
2000 80 92 86 82
Solution: - To determine seasonal variation by ratio to trend method, first we will determine
the trend of yearly data and then convert it to quarterly data. First calculate the trend values;
Year Yearly total Yearly average (Y) Time (X) XY X2
trend values
1996 140 35 -2 -70 4 32
1997 180 45 -1 - 45 1 44
1998 200 50 0 0 0 56
1999 260 65 1 65 1 68
2000 340 85 2 170 4 80
Y=280 X= 0 xY=120 X2
=10
bXaYC 
3
4
12
12
10
120
56
5
280
2








incrementQuarterly
x
xy
b
n
Y
a
Calculation of Quarterly trend values
Consider 1997. The trend value of 1997 indicates the trend value of the middle quarter of the
year. The middle quarter is found half of 2nd
and half of 3rd
quarter. Therefore, trend value of the
2nd quarter is given as 44 – 3/2 = 42.5 and the trend value of the 3rd
quarter is 44+3/2 = 45.5.

23
After this, subtract 3 from the 2nd
quarter trend value to get the trend value of the first quarter
and add 3 to get the trend value of the 4th
quarter to the trend value of the 3rd quarter.
Quarterly Trend Values
Year 1st
quarter 2nd
quarter 3rd
quarter 4th
quarter
1996 27.5 30.5 33.5 36.5
1997 39.5 42.5 45.5 48.5
1998 51.5 54.5 57.5 60.5
1999 63.5 66.5 69.5 72.5
2000 75.5 78.5 81.5 84.5
The ratio to trend values can be found by dividing the original data by the trend values
expressed in percentage.
Quarterly values as percentage of trend values
Year 1st
quarter 2nd
quarter 3rd
quarter 4th
quarter
1996 109.1 131.1 107.5 93.1
1997 86.1 122.4 109.9 90.7
1998 77.7 106.4 93.9 79.3
1999 85.0 114.3 97.8 85.5
2000 106.0 117.1 105.5 84.5
Total 463.9 591.3 514.6 445.6
Average 92.78 118.26 102.92 89.12
Since 92.78+118.26+102.92+89.12=403.08 is greater than 400, we have to find the correction
factor and multiply each seasonal index by the correction factor.
08.403
400
4
400

quartersofvaluesofsum
CF ,
Then the adjusted seasonal index will be given as follows;
1st
quarter = 92.0
2nd
quarter = 117.4
3rd
quarter = 102.2
4th
quarter = 88.4
Ratio-to- moving average method
The ratio to moving average is the most widely used method of measuring seasonal variations.
The following steps are important in measuring seasonal variations using the ratio to moving
average method.
1) Compute the centered 12 – month moving average from the original data. This contains
trend and cyclical variations.
2) Express the original data for each month as percentage of the centered 12 – month moving
average

24
3) Divide each month data by the corresponding 12- centered moving average and list the
quotient
IS
CT
ICST



4) Compute the average of each month for the quotient that we obtained in step 3. By doing so,
the irregular component will be removed.
S
I
IS


The sum of seasonal index should be 1200. If the sum is different from 1200, compute the
correction factor and multiply each month’s seasonal index by the correction factor. The
correction factor is obtained as,
CF = 1200 ______________
The total mean for 12 months
Link Relatives Method
This is also one of the methods of measuring seasonal variations. When this method is adopted,
the following steps need to be considered;
1. Calculate the link relatives of seasonal figures.
100
'Pr
'

figuresseasonevious
figuresseasonCurrent
LR
2. Calculate the average of the link relatives for each season
3. Convert the averages in to chain relatives on the base of the last season
4. Calculate the chain relatives of the first season on the base of the last season
5. For correction, the chain relative of the first season calculated by the first method is deducted
from the chain relative of the first season calculated by the second method
6. Express corrected chain relatives as percentage of their averages. These provide the required
seasonal indices by the method of link relatives.
Example: Apply the method of link relatives to the following data and calculate seasonal indices
Quarter 1998 1999 2000 2001 2002
I 6.0 5.4 6.8 7.2 6.6
II 6.5 7.9 6.5 5.8 7.3
III 7.8 8.4 9.3 7.5 8.0
IV 8.7 7.3 6.4 8.5 7.1
Solution: Calculation of Seasonal Indices by Link Relatives
Year I II III IV
1998 - 108.3 120.0 111.5
1999 62.1 146.3 106.3 86.9
2000 93.2 95.6 143.1 68.8

25
2001 112.5 80.6 129.3 113.3
2002 77.6 110.6 109.6 88.8
Mean 86.35 108.28 121.66 93.86
Chin relative 100
28.108
100
28.108100


73.131
100
28.10866.121


64.123
73.131
100
86.93


Corrected chain
relative
100
605.106
675.128.108


38.128
35.373.131


615.118
025.564.123


Seasonal Indices 100
00.94
100
4.113
605.106


21.113
100
4.113
38.128


60.104
4.113
100615.118


Correction factor= firstfromrelativeLinklastfromrelativeLink 
LR in the first season= 100
675.1
4
76.6
76.610076.106
76.10664.123
100
35.86



quarterperDifference
relativechainthebetweenDifference
seasonlasttheinLR
Adjusted chain relatives are obtained by subtracting 675.11 from the second quarter, 675.12
from third quarter and 675.13 from the fourth quarter. Seasonal indices can be calculated as;
100
4.113
4.113
4
6.453
4
615.11838.128605.106100



relativechainCorrected
indexSeasonal
Cyclical Variations
The term cycle refers to recurrent variations in time series that usually last longer than a year and
regular, neither in amplitude nor in length. Cyclical fluctuations are long term movements that
represent consistently recurring rises and declines in activity. They are resulted mainly from
business cycles. A business cycle consists of the recurrence of the up down movements of business
activity from some sort of statistical trend. There are four well defined periods or phases in the
business cycle. These are prosperity, decline, depression and improvement. The study of cyclical
variations is extremely useful in framing suitable policies for stabilizing the level of business
activity, i.e. for avoiding periods of booms and depressions as both are bad for the economy.
Measurement of Cyclical Variations
Business cycles are important types of fluctuations in economic data. Definitely, they are receiving
a lot of attention in economic literature. Despite the importance of business cycles, they are most

26
difficult types of fluctuations to measure. This is because successive cycles vary widely in timing,
amplitude and pattern. Because of such reason, it is impossible to construct meaningful typical
cycle indices of curves similar to those that have been developed for trends and seasonality. The
important methods used for measuring cyclical variations are:
1. Residual Method
2. Reference Cycle Analysis Method
3. Direct Method
4. Harmonic Analysis Method
Because of the frequent usage and convenience of time, only the first method is discussed.
Residual Method: Among all the methods of arriving at estimates of the cyclical movements of
time series, the residual method is most commonly used. This method consists of eliminating
seasonal and then trend variations to obtain the cyclical and irregular movements.
IC
T
ICT
ICT
S
ICST




The data are usually smoothed in order to obtain cyclical movements, which are sometimes termed
as the cyclical relatives since they are always expressed in percentages. This is because cyclical,
irregular or the cyclical movements remain residuals. As a result, this procedure is referred to as the
residual method.
Irregular Variations
Irregular variations refer to such variations in business activities which do not repeat in a definite
pattern. It includes all types of variations other than those accounting for the trend, seasonal and
cyclical movements. Irregular movements are considered to be largely random, being the result of
chance factors, which like the fall of a coin, that are wholly unpredictable. Irregular variations are
caused by such special occurrences as flood, earthquakes, strikes and wars. Sudden changes in
demand or rapid technological progress may also be included in this category. By their nature, these
movements are very irregular and unpredictable. Quantitatively it is almost impossible to separate
out the irregular movements and the cyclical movements. Therefore, while analyzing time series, the
trend and seasonal variations are measured separately and the cyclical and irregular variations are
left altogether.
Measurement of Irregular Variations
The irregular component in a time series represents the residue of fluctuations after trend, seasonal
and cyclical movements have been accounted for. Thus, if original data is divided by T, S and C, we
get I. .I
TSC
TSCI
 In practice, the cycle itself is so erratic and interwoven with irregular movements
that it is impossible to separate them. In the analysis of time series into its components, trend and
seasonal movements are usually measured directly, while cyclical and irregular fluctuations are left
altogether after the other elements have been removed.

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