Time Series Analysis A time series is a collection of data recorded over a period of time – weekly, monthly, quarterly or yearly. Examples: 1. The hourly temperature recorded at a locality for a period of years. 2. The weekly prices of wheat in Dhaka. 3. The monthly consumption of electricity in a certain town. 4. The monthly total passengers carried bay a train. 5. The quarterly sales of a certain fertilizer. 6. The annual rainfall at a place for a number of years. 7. The students enrolment of a college over a number of years.

Application Time Series Analysis is used for many applications such as: Economic ForecastingSales Forecasting Budgetary AnalysisStock Market Analysis Yield ProjectionsProcess and Quality Control Inventory StudiesWorkload Projections Utility StudiesCensus Analysis and many, many more...

Components of a time series There are four components of a time series: 1. The trend 2. The cyclical variation 3. The seasonal variation and 4. The irregular variation. Secular trend The smooth long-term direction of a time series. 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.

Example: Wheat production of Chittagong district from 1991 to Year Production (M. ton)

Example: Number of officers in Sonali bank from 1985 to Year Officer

Cyclical variation A typical business cycle consists of a period of prosperity followed by periods of recession, depression, and then recovery. In a recession, for example, employment, production, the Dow Jones Industrial Average, and many other business and economic series are below the long term trend lines. Conversely, in periods of prosperity they are above their long- term trend lines. Example: Following figure shows the number of batteries sold by National Battery Sales, Inc. from 1988 to The cyclical nature of business is highlighted.

Seasonal variation Patterns of change in a time series within a year. These patterns tend to repeat themselves each year. The unit of time reported is either quarterly or monthly. Example: Men’s and boy’s clothing have extremely high sales just prior to Eid-ul-fitr and relatively low sales after it. Example: Following figure shows the quarterly sales, in millions dollars, of a sporting goods company that specializes in selling baseball and softball equipment for high schools, colleges and youth leagues. There is a distinct seasonal pattern to their business. Most of the sales are in the first and second quarters of the of the year, when schools and organizations are purchasing equipment for the upcoming season.

Irregular variation The irregular variations occur in a completely unpredictable manner as they are caused by some unusual events such as floods, droughts, strikes, fires, earthquakes, wars and political events and so forth. Example: Monthly Value of Building Approvals, Australian Capital Territory (ACT).

Measures of trend To measure a trend which can be represented by a straight line or some type of smooth curve, the following methods are used: 1. Freehand curve 2. Semi average method 3. Moving average method and 4. Method of least squares Freehand curve Plot the given data on a graph paper and join the plotted points by segments of straight line. Draw a straight line freehand passing through the plotted points in a way such that the general direction of change in values is indicated.

Example: Export quantity in ton of a food from 1971 to 1978: Year Export(ton)

Semi average method Divide the values in the series into two equal parts. Find the average values of each part and place the average values against the respective mid points of the two parts. Plot these two average values on the graph of the original values and draw a straight line connecting the two points and extend the line to cover the whole series. Example: Export (Lac ton) information of Bangladesh from 1990 to 1998:

YearExportTotalAverage

Moving average method The moving average method is not only useful in smoothing a time series to see its trend, it is the basic method used in measuring the seasonal fluctuation. The moving average only smooths the fluctuations in the data. This is accomplished by ‘moving’ the average values through the time series. The k period moving averages are defined as the averages calculated using the k consecutive values of the observed series. Each k period moving average is placed against the middle of its time period. These average values are plotted on the graph of the original values and the line connecting these points is the moving average trend.

Example: From the information below draw a 3-year moving average trend. YearExport3-year total 3-year average

Exercise 1: From the information below draw 4-year moving average trend. YearExport4-year mid average4-year average

Method of least squares A trend can be represented by a mathematical equation of the form of a straight line. The straight line equation can be represented by y’ is the projected value of y and y is the time series variable. a is the y intercept. It is the estimated value when x=0. b is the slope of the line, or the average change in y. x is the value of time. If we can convert x is such a way that the sum of converted x is zero, then the value of a and b can be estimated as:

Example: From the information below using least square method draw the trend. Year (X) Produc tion (y) xxyy’ Total

Exercise 2: From the information below draw a trend using least square method. Year (X) Produc tion (y) xxyy’ Total