Time Series Analysis PDF

# Chapter 7: Time Series Analysis ## Course Information: - Course: Managerial Statistics - Course Code: BSMS2209 - Specialization: College Requirement - College: College of Economics and Business Administration ## Course Objective This course emphasizes the development of practical computing skill...

# Chapter 7: Time Series Analysis ## Course Information: - Course: Managerial Statistics - Course Code: BSMS2209 - Specialization: College Requirement - College: College of Economics and Business Administration ## Course Objective This course emphasizes the development of practical computing skills and problem-solving skills in the field of business, using appropriate statistical tools and graphs, as well as the use of spreadsheets for data management and analysis. ## Learning Outcomes - **Learning Outcome 7**: Demonstrate the knowledge of time series concepts by applying them to real-world data and interpreting the results of the analysis. - **Learning Outcome 8**: Practice the fundamentals of descriptive and inferential data analysis using Excel spreadsheet and SPSS. ## Chapter Outline: - Time Series - Introduction - Importance of Time Series Analysis in Business - Components of Time Series - Measurement of Trend - Time Series Analysis Using - Excel Spread Sheet - Time Series Analysis Using - SPSS ## Time Series - Introduction - A time series is a collection of values of a variable taken at different time periods. - The values of the concerned variable is not expected to be same for every time period. - For example, if we consider the price of 1 kg tea of a particular brand, for over twenty years, we will note that the price is not the same for every year. - What has caused the price to vary? ## Meaning Time series data is a sequence of observations - collected from a process - with equally spaced periods of time ## Definition - According to Merriam Webster Dictionary, time series is a set of data collected sequentially and usually at fixed intervals of time. - **Example**: The number of packets of milk sold in a small shop. | Day | No. of packets of milk sold | | ----------- | ----------- | | Monday | 90 | | Tuesday | 88 | | Wednesday | 85 | | Thursday | 75 | | Friday | 72 | | Saturday | 90 | | Sunday | 102 | ## Importance of Time Series Analysis in Business - Profit Planning - Sales Forecasting - Stock Market Analysis - Process and Quality Control - Economic Forecasting - Risk Analysis & Evaluation of changes ## Components of Time Series Any time series can contain some or all of the following components: - Secular trend or Long-term Variation - Seasonal variation - Cyclical variation - Irregular variation ## Secular Trend - The increase or decrease in the movements of a time series is called Secular trend. - Secular trend is a long term movement in a time series. - A time series data may show upward trend or downward trend for period of years and this may be due to factors like: - Increase in population - Change in technological progress, - Large scale shift in consumers' demands ## Seasonal Variation - Seasonal variation are short-term fluctuation in a time series which occur periodically in a year. - In a time series, seasonal variations occur quite regularly. - The major factors are weather conditions and customs of people. - **Examples**: - More woolen clothes are sold in winter than the season of summer. - 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. - Even Banks have not escaped from seasonal variations. Banks observe heavy withdrawals in the first week of every month ## Cyclical Variation - Cyclical variations are recurrent upward or downward movements in a time series but period of cycle is greater than a year. - Cyclical variations are seldom periodic and they may or may not follow same pattern after equal interval of time. - Also these variations are not regular as Seasonal variation. - In business and economic time series, business cycles are example of cyclical variations. - There are four phases of a business cycle: - Depression - Boom - Recovery - Decline ## Irregular Variation - Irregular variation are fluctuations in time series that are short in duration, erratic in nature and follow no regularity in the occurrence pattern. - Irregular fluctuations results due to the occurrence of unforeseen factors. - This component is unpredictable. - Floods - Earthquakes - Wars - Famines ## Measurement of Trend - Free hand Graphic method - Semi average method - Moving average method - Least Square method ## Free Hand Graphic Method - In this method the data is denoted on graph paper. - Show "Time" on "X" axis and "Data" on the "Y" axis. - On graph there will be a point for every point of time. - Draw a smooth hand curve with the help of this plotted points. **Exercise**: Draw a free hand curve on the basis of the following data: | Year | Profit (in '000) | | ----------- | ----------- | | 1989 | 148 | | 1990 | 149 | | 1991 | 149.5 | | 1992 | 150.5 | | 1993 | 152.2 | | 1994 | 153.7 | | 1995 | 153.7 | | 1996 | 153 | ## Semi Average Method - In this method the given data are divided into two parts, preferably with the equal number of years. - If the number of items is odd, then we make two equal parts by leaving the middle most value. - An average is obtained for each part. - Each such average is shown against the mid-point of the half period. - Obtain two points on a graph paper based on the averages of each part. - By joining these points, a straight line trend is obtained. **Exercise**: Find the trend line from the following data by Semi-Average method: | Year | Production | | ----------- | ----------- | | 1989 | 150 | | 1990 | 152 | | 1991 | 153 | | 1992 | 151 | | 1993 | 154 | | 1994 | 153 | | 1995 | 156 | | 1996 | 158 | ## Moving Average Method - This method is based on a series of arithmetic means as shown in the below example. **Exercise 1**: Calculate 3-yearly moving average for the following data: | Year | Production | | ----------- | ----------- | | 2000-2001 | 40 | | 2001-2002 | 45 | | 2002-2003 | 40 | | 2003-2004 | 42 | | 2004-2005 | 46 | | 2005-2006 | 52 | | 2006-2007 | 56 | | 2007-2008 | 61 | **Exercise 2**: Following figures relate to output of cloth in a factory (output in lakhs of metres): | Year | Output | | ----------- | ----------- | | 1967 | 72 | | 1968 | 68 | | 1969 | 64 | | 1970 | 60 | | 1971 | 68 | | 1972 | 72 | | 1973 | 72 | | 1974 | 76 | | 1975 | 72 | | 1976 | 68 | **Exercise 3**: Calculate 5-yearly, 7- yearly, 9-yearly moving average for the following data: | Year | Value of the Variable | | ----------- | ----------- | | 1955 | 8 | | 1956 | 10 | | 1957 | 11 | | 1958 | 10 | | 1959 | 10 | | 1960 | 9 | | 1961 | 9 | | 1962 | 11 | | 1963 | 7 | | 1964 | 9 | | Year | Value of the Variable | | ----------- | ----------- | | 1965 | 9 | | 1966 | 11 | | 1967 | 13 | | 1968 | 9 | | 1969 | 10 | | 1970 | 8 | | 1971 | 11 | | 1972 | 9 | | 1973 | 12 | | 1974 | 11 | ## Least Square Method (Linear Trend) - This is a mathematical method. - By using this method, we can find linear trend as well as non-linear trend of the corresponding data. - The equation of the required trend line can be expressed as: $ Y = a + bX$ where Y is the actual value, X is time, a, b are constants - The constants 'a' and 'b' are estimated by solving the following two normal equations: $ΣY = n a + b ΣX$ $ΣXY = a ∑X + b ΣΧ2$ where 'n' = number of years given in the data. - By taking the mid-point of the time as the origin, we get $ΣX = 0$ - When $ΣX = 0$, the two normal equations reduces to: $ΣY = n a + b (0)$ ; $a = \frac{ΣY}{n}$ $ΣXY = a(0) + b ΣΧ2$; $b = \frac{ΣXY}{ΣΧ^2}$ - The constant 'a' gives the mean of Y and 'b' gives the rate of change (slope). - By substituting the values of 'a' and 'b' in the trend equation, we get the Line of Best Fit. **Exercise 1**: Given below are the data relating to the production of sugarcane in a district. Fit a straight line trend by the method of least squares and tabulate the trend values. | Year | Prod. of Sugarcane | | ----------- | ----------- | | 2000 | 40 | | 2001 | 45 | | 2002 | 46 | | 2003 | 42 | | 2004 | 47 | | 2005 | 50 | | 2006 | 46 | **Exercise 2**: Given below are the data relating to the production of sugarcane in a district. Fit a straight line trend by the method of least squares and tabulate the trend values. | Year | Sales | | ----------- | ----------- | | 1995 | 6.7 | | 1996 | 5.3 | | 1997 | 4.3 | | 1998 | 6.1 | | 1999 | 5.6 | | 2000 | 7.9 | | 2001 | 5.8 | | 2002 | 6.1 | ## References - Divyansh Choudhary. (2021). Business Statistics: Vol. First edition. Laxmi Publications Pvt Ltd. PP. 82-101. Retrieved from: https://search-ebscohost-com.masader.idm.oclc.org/login.aspx?direct=true&db=nlebk&AN=3103348&site=ehost-live. ## Contact Information: - Dr. S. Porkodi - Office: BS032, Business Department - Email: [email protected] ## Version History | Version No | Date Approved | Changes incorporated | | ----------- | ----------- | ----------- | | New Outcome Version: 01 | Sem. (1) 2022/2023 | |