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FORECASTINGANDTIME SERIESThe Method of Least SquaresAmong the methods of fitting a straight line to a series of data, this is the most frequently used method. As we have seen earlier that the equation of a straight line is y = a + bx, where Y is the time period, say, year x is the value of the item measured against time, a is the y-intercept and b is the coefficient of X that shows the slope of the straight line. In order to find out the values of a and b, the following two equations are solved:y = na + b xxy =a x +b x2

E.G. (METHOD OF LEAST SQUARES)A COMPANY THAT MANUFACTURES STEEL OBSERVED THE PRODUCTION OF STEEL IN METRIC TONNES REPRESENTED BY TIME SERIES

FIND THE LINEAR EQUATION THAT DESCRIBES THE TREND IN THE PRODUCTION OF STEEL BY THE COMPANY?ESTIMATE THE PRODUCTION OF STEEL IN YEAR 2003,2004YEARPRODUCTION OF STEEL199660199772199875199965200080200185200295YEARTIME SCALE (X)PRODUCTION (Y)XXY1996160199727219983751990465200058020016852002795Exponential Smoothing MethodIt is a type of moving average forecasting techniqueIt weighs past data from previous time periods with exponentially decreasing importance in the forecast so that the recent data carries more weightExponential Smoothing Method Simple Exponential Smoothing F (t+1) = Xt + (1- ) Ft = Ft + (Xt Ft) WhereF (t+1) = forecast for next time periodFt = forecast for present time period = a weight called exponential smoothing constant (0 1)Xt = actual value for present time periodExample A firm uses simple exponential smoothing with =0.1 to forecast demand. The forecast for the week of Feb 1 was 500 units, whereas actual demand was 450 units.(a) Forecast demand for week of Feb 8 ?

Example A firm uses simple exponential smoothing with =0.1 to forecast demand. The forecast for the week of Feb 1 was 500 units, whereas actual demand was 450 units.(a) Forecast demand for week of Feb 8 Solution F(t+1) = 500 + 0.1(450-500) = 495 units

MEASUREMENT OF SEASONAL EFFECTSSeasonal effects arise as the result of changes in the seasons during the year or may result due to habits , customs , or festival that occur at the same time year after yearWe have three main reasons to study seasonal effects1. The description of seasonal effect provides a better understanding of the impact this component has upon a particular time seriesMEASUREMENT OF SEASONAL EFFECTS2. Once the seasonal pattern that exists is established, seasonal effect can be eliminated from the time series in order to observe the effect of the other components, such as cyclical and irregular components. Elimination of seasonal effect from the series is referred to as deasonalising or seasonal adjusting3. Trend analysis may be adequate for long range forecast , but for short run predictions, knowledge of seasonal effects on time series data is essential for projection of past pattern into the futureMEASUREMENT OF SEASONAL EFFECTSSeasonal IndexAre measured in terms of an index, called seasonal index, attached to each period of time series within a year If monthly data are considered, there are 12 separate indexes of each month Similarly for quarterly data there are 4 separate indexes for each quarter ExampleThe data on prices (Rs in per kg) of a certain commodity during 2000 to 2004 are shown below.Calculate the seasonal indexes by the average percentage method (SIMPLE AVERAGES METHOD) and obtain de-seasonalized valuesExampleYearQuarter 1Quarter 2Quarter 3Quarter 4200045547260200148566356200249637065200352657572200460708466ExampleYearQuarter 1Quarter 2Quarter 3Quarter 4200045547260200148566356200249637065200352657572200460708466QuarterlyTotal254308364319Quarterly AverageSeasonalIndexExampleYearQuarter 1Q 2Q 3Q 4200045547260200148566356200249637065200352657572200460708466QuarterlyTotal254 (45+48+49+52+60)308364319Quarterly Average50.8 (254/5=50.8)61.672.863.8SeasonalIndex15Example Calculation of seasonal indexAverage of quarterly averages = [50.5+61.6+72.8+63.8] / 4 = 62.25Seasonal index for quarter 1 = 50.8/62.25 x 100 = 81.60Seasonal index for quarter 2 = 61.6/62.25 x 100 = 98.95Seasonal index for quarter 3 = 72.8/62.25 x 100= 116.94Seasonal index for quarter 4 = 63.8/62.25 x 100 = 102.48

ExampleYearQuarter 1Quarter 2Quarter 3Quarter 4200045547260200148566356200249637065200352657572200460708466QuarterlyTotal254308364319Quarterly Average50.861.672.863.8SeasonalIndex81.6098.95116.94102.48Example Deasonalized ValuesSeasonalized influences are removed from a time series data by dividing the actual y value for each quarter by its corresponding seasonal indexDeseasonalized value =[ actual quarterly value/seasonal index of corresponding quarter] x 100

ExampleQuarter 1Quarter 2Quarter 3Quarter 4200055.14 (45/81.60) x 10054.5761.5758.54200158.8256.5953.8754.64200260.0963.6659.8563.42200363.7265.6864.1370.25200473.5270.7471.8364.40DESEASONALIZED VALUESTHANK YOU