Time series analysis

We are pleased to submit the following report on the “Time Series Analysis”.

By completing the report, we have got acquainted importance and relevance of time series on business application. We also perceived idea on the whole process of Time Series Analysis. We acquired knowledge about the method of measuring trend, growth rate, acceleration rate etc. In spite of limitation of time & opportunity we have tried our level best to complete the report.

We are pleased to provide you with this report with necessary analysis, references and we shall be available for any clarification, if required. Thank you for assigning us in this study.

On behalf of the group
Md. Arif Hasan
ID: 12-150

Table of Contents

Serial No Topic Page No
1 Letter of Transmittal 1
2 Rationale of the study 2
3 Objectives of the report 3
4 Methodology of the report 3
5 What is Time Series 4
6 Uses of Time Series in Business 5
7 Components of a Time Series 5
8 Classical Time Series Model 9
9 Methods of trend measurement 9
10 Least squares method 10
11 The Growth Rate 14
12 The Acceleration Rate 15
13 Rule of 72 16
14 Bibliography 17

Rationale of the study

Having been assigned to prepare a report on Time Series Analysis we are submitting the term paper based on our findings and understandings.

Time series analysis has vast application and is of huge importance in the field of Business and Economics as well as in decision making thereof.

Calculating secular trend we can know whether unit sales or revenues of a particular Business organization are increasing, decreasing or remaining constant over time.

Cyclical variation affords us to track the rise and fall of a time series over a longer period of time. It helps us to realize the different stages the organization is going through over time such as prosperity, depression, recession, recovery etc.

Seasonal variation helps us to clarify in which particular period of time within a year the Business is making good sales or decreasing sales.

Using last square method we can derive a trend equation which dictates us at what particular amount the Business is increasing or decreasing its sales over time.

Calculation of Growth Rate makes us understand at what percentage the Business sales are increasing or decreasing over time and Acceleration Rate lets us know whether the Growth Rate will increase or decrease at an increasing or decreasing rate in the successive years.

In our report, we have identified & described all the components and elements of Time Series Analysis, The Classical Time Series Model, and different methods of measuring trend, application of the Least Square Method and process to calculate growth rate and acceleration rate and their effect on the business thereof.

Necessary graphs and charts have also been attached with an exclusive interpretation of the Rule of 72 in the final part of the report.

Objectives of the Report

? Defining the Time Series Analysis.
? Distinguishing between Time Series Data and Cross-section Data.
? Identifying the elements of Time Series Analysis.
? Developing the Classical Time Series Model.
? Describing the various components of Time Series Analysis.
? Erecting graphical figures for the components of the Time Series Analysis.
? Identifying different methods of measuring Trend.
? Applying the Least Square Method in practical problems.
? Calculating Growth Rate and Acceleration Rate and describing their effect.
? Describing the Rule of 72 and its application in Growth Rate.

Methodology of the Report

Being permitted we are submitting this report on Time Series Analysis. Due to time and fund constraints we could not collect primary data, so we prepared this report based on secondary data. Resultantly our report completely relies in hypothetical data rather than experimental data. We created a hypothetical business situation that exactly coincides with the real business situation.

Time Series: The most popular method of business forecasting is time series analysis. A time series is a collection of observations of well-defined data items obtained through repeated measurements over a period of time- weekly, monthly, quarterly, or yearly.

For example, measuring the value of retail sales each month of the year would comprise a time series. This is because sales revenue is well defined, and consistently measured at equally spaced intervals. Data collected irregularly or only once are not time series.

Distinguish between Trend and Growth rate:

1. Trend indicates the changes of a variable over a period of time in absolute term i.e., in original unit of measurement (Tk. US $, Kg, etc.)

2. Growth rate indicates the changes in a variable over a period of time in relative term i.e., in % (percentage).

3. If two or more variables are given or expressed in different units of measurements, their changes over a period of time are not comparable using the trend but the changes are comparable using growth rate.

4. In calculating the trend all the figures of a time series are taken into account, while in estimating the growth rate, the figures of base and current periods are considered.

Difference between time series data and cross section data:

Time Series Data Cross Section Data
Data collected & arranged on the basis of time are known as time series data. Data on different items given for a specific time period are known as cross-section data.
Year Profit Beximco Textile (crore tk)
2007 12
Company (Textile)
Profit in 2007 (crore tk)
Beximco Textile
Square Textile
Ashraf Textile 22

Uses of Time Series in Business Decision-Making: Various uses of time series in Business Decision-Making are follows:-

? To understand future behaviour: Time series analysis is helpful in predicting the future behaviour. By observing data over a period of time, one can easily understand what changes have taken place in the past & what will be in future.

? Planning of future operations: Time series analysis is helpful in planning future operations. The greatest potential of a time series lies in predicting an unknown value of the series. For capital investment decisions, decisions regarding production and inventory etc are example of planning future operations.

? Evaluations of current activities: The actual performance can be compared with the expected performance and the cause of variation analyzed. For example, if expected sales for 2007 were 15 lacs T-shirts and the actual sales were only 14 lacs; one can investigate the cause in the shortfall in achievement.

? It facilitates comparison: Various time series are often compared and important decisions are taken. One statistician can not forecast 100 percent accuracy of future events. It is needed to compare different time series in the same topic.

Components of a Time Series: The four basic types of variations which account for the changes in the series over a period of time. The four types of patterns, variations or movements are called components or elements of time series.

The four components are
1. Secular Trend
2. Seasonal Variation
3. Cyclical Variation
4. Irregular Variation

Secular Trend: Secular trend means the smooth, regular, long-term movement of the data. Among four components trend is the most important and influential.

The long term trends of sales, employment, stock prices, and other business and economic series follow various patterns. Some move steadily upward, others decline, and still others stay the same over time.

For example, the secular trend of the number of employees in Home Depot (from 1993 to 2002) is shown below.

How to identify trend: Sudden and irregular movements either in upward or in downward direction will not be known as trend.

Classification of trends: Various types of trends are divided under two heads.

? Linear or Straight Line Trends: The linear trend indicates that the increase or decrease of a time series at constant amount. It is the simplest form in describing the trend movement.

? Non Linear Trends: The non-linear trend indicates that a time series may have faster (or slower) increase at early stage and have a slower (or faster increase at more recent times).

Seasonal Variation: Seasonal variations are those periodic movements in business activity which occur regularly every year. These variations tend to repeat themselves each year.

Almost all businesses tend to have frequent seasonal patterns. For example men’s and boy’s clothing has extremely high sales just prior to Christmas and relatively low sales after Christmas and during the summer.

How to identify seasonal variation:

? If the data are given on yearly basis there will be no seasonal basis.

? If the data are given on daily, weekly, monthly, quarterly or half-yearly basis then there might be existence of the influence of the seasonal variation.

Factors affecting seasonal variation: The factors that cause seasonal variations are:

? Climate and weather conditions: The most important factor causing seasonal variation in the climate and weather conditions such as rainfall, humidity, heat etc. act on different product and industries differently.

For example during the winter there is great demand for woolen cloths, whereas in summer cotton clothes have greater sales.

? Customs, traditions & habits: Though climate and weather are primarily responsible seasonal variations in time series, customs, traditions and habits also have their impact.

For example on certain occasions like Eid, Puja the bank withdrawals go up because people want money for shopping and gifts etc.

Cyclical Variation: The rise and fall of a time series over periods longer than one year. Most of the time series relating economics and business show some kind of cyclical variation. A business cycle consists of four well defined periods or phases, namely prosperity, decline, depression and improvement.

For example, the annual unit sale of batteries sold by National Battery from 1984 to 2003 is shown below in the cyclical nature of business.

Irregular Variation: Irregular variations refer to such variations in business activity which does not repeat in a definite pattern. Irregular variations are caused by special occurrences like flood, earthquakes, strikes and wars etc. Irregular variations can not be projected into the future.

Classification of Irregular Variation: Sometimes irregular variations are subdivided into episodic and residual variations.

? Episodic Variation: Episodic fluctuations are unpredictable, but they can be identified. The initial impact on the economy of a major strike or a war can be identified, but a strike or can not predicted.

? Residual Variation: After the episodic fluctuations have been removed, the remaining variation is called the residual variation. The residual fluctuations are often called chance fluctuations. These are unpredictable and can not be identified.
Classical Time Series Model: The model which states that all the components of a time series are in a relationship of multiplicative in nature is known as “Classical Time Series Model”.

Mathematically the model can be written as-

Y = T x S x (C x I)

Y = Value of the variable or result of the four elements
T = Trend
S = Seasonal Variation
C = Cyclical Variation
I = Irregular Variation

Methods of trend measurement: Several methods are used in measuring Trend. Some measures the average of the trend and some measures the absolute value. Here we have listed five main methods for measuring the Trend of the Time Series.

? Linear trend measurement
1. The freehand or Graphic method
2. The Semi-average method
3. The Least Squares method

? Non-linear trend measurement
1. The freehand or Graphic method
2. Moving average method

Figure: Five methods of measuring Trend

Least squares method: The method where time is the independent variable and the value of the time series is the dependent variable is known as the Least Square Method. Here, an equation is developed to estimate future values. It is done by the least square technique. Many researchers suggest that we should not project the future values more than n/2 time periods (where n= number of data points) into the future. Moreover, others suggest the forecast may be for no longer than 2 years, especially in rapidly changing economic times.

I. (Y – YC) = 0
The sum of deviations of the actual values of Y and the computed values of Y is zero.

II. (Y – YC)2 is the least
The sum of the squares of the deviations of the actual and computed values is least from this line. That is why this method is called the method of least squares.

The linear trend is represented by the equation
Yc = a + bx

In spite of these suggestions, the Least Square Method is a very popular and important method for forecasting the future because it finds the best linear relationship between two variables. Besides, it measures the absolute value of the trend.

We have also arranged our report on this Least Square Method. For this purpose we have used both the theory and their applications. We have collected the secondary data from a departmental store named Super Value Shop and calculate the data for application purpose.

Here the collected data has been presented,

Super Value Shop, the agent of Proctor & Gamble Bangladesh Ltd. in Dhanmondi, has experienced the following sales revenues in the past 7 years

Year 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007
Sale(lac Tk.) 6 11 19 22 30 35 39 42 45 50

From the data at first we will develop an equation with the least square techniques.

The Equation:

The equation is-

Ye = a + bt

Where, Y = dependent variable of the time series
a = constant
b = trend (slope)
t = time period

This method is a direct method where-
åY = an + båt
åYt = aåt + båt2

a = and b =

Now, by using the data of the Super Value Shop we will calculate the equation. For to do so we have to find out the a and b.

Year Sales (lac Tk.)y t yt

1998 6 1 6 1
1999 11 2 22 4
2000 19 3 57 9
2001 22 4 88 16
2002 30 5 150 25
2003 35 6 210 36
2004 39 7 273 49
2005 42 8 336 64
2006 45 9 405 81
2007 50 10 500 100
n = 10 299
2047 385

So b = a =


= 3.06

= 4.88

b (Trend) indicates the average amount of change in the variable y due to a unit change in the time period(t).

So the equation is –

Y = 3.06 + 4.88t

This Trend Equation indicates that during the period of 1998 to 2007 sales of the Super Value Shop increased because as an average amount of Tk. 5.679 lac per year.

Now by using this trend equation we can forecast the sale of the Super Value Shop. Suppose, we want to forecast the sale of 2010.

Y2010 = 3.06 + 4.88 *13

= 66.5
So if the above trend equation holds to until 2010 (if the present trend continues) the sales of the Super Value Shop will stand at Tk. 66.5 lac.

The Growth Rate (GR): The rate which indicates the increase of the dependent variable of the time series(Y) within the considering period is known as Growth Rate. For example, the Growth Rate of the Super Value Shop indicates the increase of the rate of sales during the period of 1998 to 2007. Shortly it is noted as GR.

GR =

Here, n = Total number of Years
Yn = Sale of the nth year
Y1 = Sale of the 1st year

Now by using the data we can find out the increase (GR) of sales of the Super Value Shop.

GR =


= 26.56%

This GR = 26.56% indicates that during the period of 1998 to 2007 sales of the Super Value Shop increased at an average rate of 26.56% per year.

The Acceleration Rate (AR): The rate which indicates how much the growth rate will be decreasing in the years to come or the following (future) years is known as the Acceleration Rate. For example, if the GR = 20% then how much (increased or decreased) it will remain in the following years. Shortly it is noted as AR.

AR =

Here, Y1 = Sale of 1st year
Y2 = Sale of 2nd year
Yn = Sale of nth year
Yn-1 = Sale of the previous year of nth year

Now by using the data we can find out the change (increase or decrease) of GR of the Super Value Shop.

AR =


= – 6.07%

This AR = – 6.07% indicates that in the years to come (in future) growth rate of the Super Value Shop will be decreasing at an average rate of 6.07% compared to that of the previous year’s growth rate. So, if the current year GR = 100% (26.56%) then the next year GR = 93.93% (24.95%) of the previous year’s growth rate. This will be continuing until the AR is not changed.

Rule of 72: The rule which estimates how much time the current growth rate will take to make the sales doubled is known as the Rule of 72. It is calculated by 72/GR.
It is an important rule in case of forecasting or estimating the future. Here we have used a few words to make it clear.

If the GR = 1% then it will take 72 years to be doubled.
If the GR = 72% then it will take 1 year to be doubled.
If the GR = 26.56% then it will take 2.71 year to be doubled.


Statistical Techniques in Business & Economics
-by Lind, Marchal, Wathen
Business Statistics
–by Gupta & Gupta
The lectures of our honorable course teacher.