Chapter 5: Demand Forecasting Department of Business Administration FALL 2016-2017 I see that you will get an A this semester. Chapter 5: Demand Forecasting
Outline: What You Will Learn . . . List the elements of a good forecast. Outline the steps in the forecasting process. Describe at least three qualitative forecasting techniques and the advantages and disadvantages of each. Compare and contrast qualitative and quantitative approaches to forecasting. Briefly describe averaging techniques, trend and seasonal techniques, and regression analysis, and solve typical problems. Describe two measures of forecast accuracy. Describe two ways of evaluating and controlling forecasts. Identify the major factors to consider when choosing a forecasting technique
What is meant by Forecasting and Why? Forecasting is the process of estimating a variable, such as the sale of the firm at some future date. Forecasting is important to business firm, government, and non-profit organization as a method of reducing the risk and uncertainty inherent in most managerial decisions. A firm must decide how much of each product to produce, what price to charge, and how much to spend on advertising, and planning for the growth of the firm.
The aim of forecasting The aim of forecasting is to reduce the risk or uncertainty that the firm faces in its short-term operational decision making and in planning for its long term growth. Forecasting the demand and sales of the firm’s product usually begins with macroeconomic forecast of general level of economic activity for the economy as a whole or GNP.
The aim of forecasting The firm uses the macro-forecasts of general economic activity as inputs for their micro-forecasts of the industry’s and firm’s demand and sales. The firm’s demand and sales are usually forecasted on the basis of its historical market share and its planned marketing strategy (i.e., forecasting by product line and region). The firm uses long-term forecasts for the economy and the industry to forecast expenditure on plant and equipment to meet its long-term growth plan and strategy.
Forecasting Process Map Demand History Causal Factors Statistical Model Marketing Sales Product Management & Finance Executive Production & Inventory Control Consensus Process Consensus Forecast
Uses of Forecasts Accounting Cost/profit estimates Finance Cash flow and funding Human Resources Hiring/recruiting/training Marketing Pricing, promotion, strategy MIS IT/IS systems, services Operations Schedules, MRP, workloads Product/service design New products and services
Features of Forecasts Assumes causal system past ==> future Forecasts rarely perfect because of randomness Forecasts more accurate for groups vs. individuals Forecast accuracy decreases as time horizon increases or may otherway around I see that you will get an A this semester.
Elements of a Good Forecast Timely Accurate Reliable Meaningful Written Easy to use
Steps in the Forecasting Process Step 1 Determine purpose of forecast Step 2 Establish a time horizon Step 3 Select a forecasting technique Step 4 Obtain, clean and analyze data Step 5 Make the forecast Step 6 Monitor the forecast “The forecast”
Forecasting Techniques A wide variety of forecasting methods are available to management. These range from the most naïve methods that require little effort to highly complex approaches that are very costly in terms of time and effort such as econometric systems of simultaneous equations. Mainly these techniques can break down into three parts: Qualitative approaches (Judgmental forecasts) and Quantitative approaches (Time-series forecasts) and Associative model forecasts).
Forecasting Techniques Judgmental - uses subjective inputs such as opinion from consumer surveys, sales staff etc.. Time series - uses historical data assuming the future will be like the past Associative models - uses explanatory variables to predict the future
Qualitative Forecasts or Judgmental Forecasts Survey Techniques Some of the best-know surveys Planned Plant and Equipment Spending Expected Sales and Inventory Changes Consumers’ Expenditure Plans Opinion Polls Business Executives Sales Force Consumer Intentions
What are qualitative forecast ? Qualitative forecast estimate variables at some future date using the results of surveys and opinion polls of business and consumer spending intentions. The rational is that many economic decisions are made well in advance of actual expenditures. For example, businesses usually plan to add to plant and equipment long before expenditures are actually incurred.
Qualitative Forecasts or Judgmental Forecasts Surveys and opinion pools are often used to make short-term forecasts when quantitative data are not available. Usually based on judgments about causal factors that underlie the demand of particular products or services. Do not require a demand history for the product or service, therefore are useful for new products/services. Approaches vary in sophistication from scientifically conducted surveys to intuitive hunches about future events. The approach/method that is appropriate depends on a product’s life cycle stage.
Qualitative Forecasts or Judgmental Forecasts Polls can also be very useful in supplementing quantitative forecasts, anticipating changes in consumer tastes or business expectations about future economic conditions, and forecasting the demand for a new product. Firms conduct opinion polls for economic activities based on the results of published surveys of expenditure plans of businesses, consumers and governments.
Qualitative Forecasts or Judgmental Forecasts Survey Techniques– The rationale for forecasting based on surveys of economic intentions is that many economic decisions are made in advance of actual expenditures (Ex: Consumer’s decisions to purchase houses, automobiles, TV sets, furniture, vocation, education etc. are made months or years in advance of actual purchases) Opinion Polls– The firm’s sales are strongly dependent on the level of economic activity and sales for the industry as a whole, but also on the policies adopted by the firm. The firm can forecast its sales by pooling experts within and outside the firm.
Qualitative Forecasts or Judgmental Forecasts Executive Polling- Firm can poll its top management from its sales, production, finance for the firm during the next quarter or year. Bandwagon effect (opinions of some experts might be overshadowed by some dominant personality in their midst). Delphi Method – experts are polled separately, and then feedback is provided without identifying the expert responsible for a particular opinion.
Consumers intentions polling- Qualitative Forecasts or Judgmental Forecasts Consumers intentions polling- Firms selling automobiles, furniture, etc. can pool a sample of potential buyers on their purchasing intentions. By using results of the poll a firm can forecast its sales for different levels of consumer’s future income. Sales force polling – Forecast of the firm’s sales in each region and for each product line, it is based on the opinion of the firm’s sales force in the field (people working closer to the market and their opinion about future sales can provide essential information to top management).
Quantitative Forecasting Approaches Based on the assumption, the “forces” that generated the past demand will generate the future demand, i.e., history will tend to repeat itself. Analysis of the past demand pattern provides a good basis for forecasting future demand. Majority of quantitative approaches fall in the category of time series analysis.
Time Series Analysis A time series (naive forecasting) is a set of numbers where the order or sequence of the numbers is important, i.e., historical demand Attempts to forecasts future values of the time series by examining past observations of the data only. The assumption is that the time series will continue to move as in the past Analysis of the time series identifies patterns Once the patterns are identified, they can be used to develop a forecast
Forecast Horizon Short term Medium term Long term Up to a year One to five years Long term More than five years
Reasons for Fluctuations in Time Series Data Secular Trend are noted by an upward or downward sloping line- long-term movement in data (e.g. Population shift, changing income and cultural changes). Cycle fluctuations is a data pattern that may cover several years before it repeats itself- wavelike variations of more than one year’s duration (e.g. Economic, political and agricultural conditions). Seasonality is a data pattern that repeats itself over the period of one year or less- short-term regular variations in data (e.g. Weekly or daily restaurant and supermarket experiences). Irregular variations caused by unusual circumstances (e.g. Severe weather conditions, strikes or major changes in a product or service). Random influences (noise) or variations results from random variation or unexplained causes. (e.g. residuals)
Forecast Variations Trend Cycles Irregular variation 90 89 88 Seasonal variations
Uses for Naïve Forecasts Stable time series data F(t) = A(t-1) Seasonal variations F(t) = A(t-n) Data with trends F(t) = A(t-1) + (A(t-1) – A(t-2))
Techniques for Averaging Moving average Weighted moving average Exponential smoothing
At-n + … At-2 + At-1 Ft = MAn= n n Moving Averages Moving average – A technique that averages a number of recent actual values, updated as new values become available. At-n + … At-2 + At-1 Ft = MAn= n Weighted moving average – More recent values in a series are given more weight in computing the forecast. wnAt-n + … wn-1At-2 + w1At-1 Ft = WMAn= n n=total amount of number of weights
Simple Moving Average Actual MA5 MA3 At-n + … At-2 + At-1 Ft = MAn= n
Simple Moving Average An averaging period (AP) is given or selected The forecast for the next period is the arithmetic average of the AP most recent actual demands It is called a “simple” average because each period used to compute the average is equally weighted It is called “moving” because as new demand data becomes available, the oldest data is not used By increasing the AP, the forecast is less responsive to fluctuations in demand (low impulse response and high noise dampening) By decreasing the AP, the forecast is more responsive to fluctuations in demand (high impulse response and low noise dampening)
Exponential Smoothing Ft = Ft-1 + (At-1 - Ft-1) Ft = forecast for period t Ft-1 = forecast for the previous period = smoothing constant At-1 = actual data for the previous period Premise--The most recent observations might have the highest predictive value. Therefore, we should give more weight to the more recent time periods when forecasting. Weighted averaging method based on previous forecast plus a percentage of the forecast error A-F is the error term, is the % feedback
Exponential Smoothing Forecasts The weights used to compute the forecast (moving average) are exponentially distributed. The forecast is the sum of the old forecast and a portion (a) of the forecast error (A t-1 - Ft-1). The smoothing constant, , must be between 0.0 and 1.0. A large provides a high impulse response forecast. A small provides a low impulse response forecast.
Example-Moving Average Days Call Volume 1 159 2 217 3 186 4 161 5 173 6 157 7 203 8 195 9 188 10 168 11 198 12 Central Call Center (CCC) wishes to forecast the number of incoming calls it receives in a day from the customers of one of its clients, BMI. CCC schedules the appropriate number of telephone operators based on projected call volumes. CCC believes that the most recent 12 days of call volumes (shown on the next slide) are representative of the near future call volumes.
Example-Moving Average Use the moving average method with an AP = 3 days to develop a forecast of the call volume in Day 13 (The 3 most recent demands) compute a three-period average forecast given scenario above: F13 = (168 + 198 + 159)/3 = 175.0 calls
Example-Weighted Moving Average Weighted Moving Average (Central Call Center ) Use the weighted moving average method with an AP = 3 days and weights of .1 (for oldest datum), .3, and .6 to develop a forecast of the call volume in Day 13. compute a weighted average forecast given scenario above: F13 = .1(168) + .3(198) + .6(159) = 171.6 calls Note: The WMA forecast is lower than the MA forecast because Day 13’s relatively low call volume carries almost twice as much weight in the WMA (.60) as it does in the MA (.33). 1
Example-Exponential Smoothing Exponential Smoothing (Central Call Center) Suppose a smoothing constant value of .25 is used and the exponential smoothing forecast for Day 11 was 180.76 calls. what is the exponential smoothing forecast for Day 13? F12 = 180.76 + .25(198 – 180.76) = 185.07 F13 = 185.07 + .25(159 – 185.07) = 178.55 Ft = Ft-1 + (At-1 - Ft-1)
Example 2-Exponential Smoothing Suppose a smoothing constant value of .10 is used and the exponential smoothing forecast for the previous period was 42 units (actual demand was 40 units). what is the exponential smoothing forecast for the next periods? Illustrate its graphical presentation on a diagram.
Example 2-Exponential Smoothing what is the exponential smoothing forecast for the next periods? F3 = 42 + .10(40 – 42) = 41.8 F4 = 41.8 + .10(43 – 41.8) = 41.92
Example 2-Exponential Smoothing Graphical presentation .1 .4 Actual
Trend Projection The simplest form of time series is projecting the past trend by fitting a straight line to the data either visually or more precisely by regression analysis. Linear regression analysis establishes a relationship between a dependent variable and one or more independent variables. In simple linear regression analysis there is only one independent variable. If the data is a time series, the independent variable is the time period. The dependent variable is whatever we wish to forecast.
Linear Trend Equation Ft = a + bt Ft = Forecast for period t t = Specified number of time periods a = Value of Ft at t = 0 b = Slope of the line
Trend Projection Linear Trend: St = S0 + b t b = Growth per time period Constant Growth Rate(Non-linear) St = S0 (1 + g)t g = Growth rate Estimation of Growth Rate ln St = ln S0 + t ln (1 + g)
Trend Projection- Simple Linear Regression Regression Equation This model is of the form: Y = a + bX Y = dependent variable (the value of time series to be forecasted for period t) X = independent variable ( time period in which the time series is to be forecasted) a = y-axis intercept (estimated value of the time series, the constant of the regression) b = slope of regression line (absolute amount of growth per period)
The correlation coefficient, determination of coefficient and standard deviation Standard deviation Correlation Coefficient Determination of coefficient
Trend Projection- Calculating a and b Constants a and b The constants a and b are computed using the equations given: Once the a and b values are computed, a future value of X can be entered into the regression equation and a corresponding value of Y (the forecast) can be calculated.
Trend Projection- Calculating a and b Or If formula b is used first, it may be used formula a in the following format:
Example 1 for Trend Projection- Electricity sales Year ELECSALE (Y) 1997Q1 11 1997Q2 15 1997Q3 12 1997Q4 14 1998Q1 1998Q2 17 1998Q3 13 1998Q4 16 1999Q1 1999Q2 18 1999Q3 1999Q4 2000Q1 2000Q2 20 2000Q3 2000Q4 19 Suppose we have the electricity sales data in a city between 1997.1 and 2000.4. The data are shown in the following table. Construct the forecast equation. Briefly explain the relationship in the forecast equation. Calculate the next four quarters. Compute the correlation coefficient, determination of coefficient and standard deviation.
Example1 for Trend Projection Year Trent (t) ELECSALE (Y) (t )SQ Y*t (y) SQ (Σt) SQ (ΣY) SQ 1997Q1 1 11 121 1997Q2 2 15 4 30 225 1997Q3 3 12 9 36 144 1997Q4 14 16 56 196 1998Q1 5 25 60 1998Q2 6 17 102 289 1998Q3 7 13 49 91 169 1998Q4 8 64 128 256 1999Q1 81 126 1999Q2 10 18 100 180 324 1999Q3 165 1999Q4 204 2000Q1 195 2000Q2 20 280 400 2000Q3 240 2000Q4 19 304 361 (SUM) Σ 136 244 1496 2208 3820 18496 59536 a 11.9 b 0.394 Av 15.25
Example1 for Trend Projection Y = 11.90 + 0.394X Y17 = 11.90 + 0.394(17) = 18.60 in the first quarter of 2001 Y18 = 11.90 + 0.394(18) = 18.99 in the second quarter of 2001 Y19 = 11.90 + 0.394(19) = 19.39 in the third quarter of 2001 Y20 = 11.90 + 0.394(20) = 19.78 in the fourth quarter of 2001 Note: Electricity sales are expected to increase by 0.394 mn kilowatt-hours per quarter.
The correlation coefficient, determination of coefficient and standard deviation Std.dev Sxy = SQRT( [3820-(11.9) (244)- (0.394) (2208)]/(16-2))=1.82 Sxy is a measure of how historical data points have been dispersed about the trend line. If it is large (reference point in mean of the data) , the historical data points have been spread widely about the trend line and if otherway around, the data points have been grouped tightly about the trend.
The correlation coefficient, determination of coefficient and standard deviation Corr. Coeff. r= ((16) (2208)- (136) (244))/SQRT( [(16) (1496)-(18496)*((16)(3820-59536)]=0.73 r lies between -1 and 1, -1 is strong negative whereas 1 is strong positive. 0 means that there is no relationship between the two variables (x and y). In this case, there is a strong positive relationship between the two variables and if an increase in independent variable, it will be a rise in dependent variable.
The correlation coefficient, determination of coefficient and standard deviation R2=0.533. It varies between 0 and 1. 0 means that there is no relationship between the two variables whereas 1 indicates that there is a perfect relationship. 53.3% variation in dependent variable can be explained by the variation happened in the independent variable. It is worth to emphasize that 46.7% shows unexplained part of the relationship.
Example for Trend Projection- Petrol sales Year PETROLSALE (Y) 2004 1 2005 3 2006 4 2007 2 2008 2009 2010 5 2011 Suppose we have the petrol sales data in a city between 2004 and 2011. The data are shown in the following table. Construct the forecast equation. Briefly explain Calculate the next four years. Briefly explain. Compute the correlation coefficient, determination of coefficient and standard deviation. Briefly explain.
Example for Trend Projection Year Trent (t) PETROLSALE (Y) (t )SQ Y*t (y) SQ (Σt) SQ (ΣY) SQ 2004 1 2005 2 3 4 6 9 2006 12 16 2007 8 2008 5 25 2009 36 18 2010 7 49 35 2011 64 24 (SUM) Σ 22 204 109 74 1296 484 a 1.678 b 0.238
Example1 for Trend Projection Y = 1.678 + 0.238X Y9 = 1.678 + 0.238(9) = 3.83 in 2012 Y10 = 1.678 + 0.238(10) = 4.06 in 2013 Y11 = 1.678 + 0.238(11) = 4.30 in 2014 Y12 = 1.678 + 0.238(12) = 4.54 in 2015 Note: Petrol sales are expected to increase by 0.238 mn gallons per year.
The correlation coefficient, determination of coefficient and standard deviation Std.dev Sxy = Sxy is a measure of how historical data points have been dispersed about the trend line. If it is large (reference point in mean of the data) , the historical data points have been spread widely about the trend line and if otherway around, the data points have been grouped tightly about the trend. 1.36
The correlation coefficient, determination of coefficient and standard deviation Corr. coef r=0.42 r lies between -1 and 1, -1 is strong negative whereas 1 is strong positive. 0 means that there is no relationship between the two variables (x and y). In this case, there is a strong positive relationship between the two variables and if an increase in independent variable, it will be a rise in dependent variable.
The correlation coefficient, determination of coefficient and standard deviation R2=0.18. It varies between 0 and 1. 0 means that there is no relationship between the two variables whereas 1 indicates that there is a perfect relationship. 18.0% variation in dependent variable can be explained by the variation happened in the independent variable. It is worth to emphasize that 82% shows unexplained part of the relationship.
Example for Trend Projection:Sales Year Time Period (t) Sales (F) 2003 2004 2005 2006 2007 2008 1 2 3 4 5 6 20 40 30 50 70 65 Estimate the forecast equation Predict the next period Compute the correlation coefficient, determination of coefficient and standard deviation.
Example 2 for Trend Projection (continued) The linear trend model is: Year Time Period (t) Sales (y) 2003 2004 2005 2006 2007 2008 1 2 3 4 5 6 20 40 30 50 70 65
Example 2 for Trend Projection Year Time Period (t) Sales (F) 2003 2004 2005 2006 2007 2008 2009 1 2 3 4 5 6 7 20 40 30 50 70 65 ?? (continued) Forecast for time period 7: Std dev=8.91 R2=0.83 r=0.91
Evaluating Forecast-Model Performance Accuracy Accuracy is the typical criterion for judging the performance of a forecasting approach Accuracy is how well the forecasted values match the actual values Accuracy of a forecasting approach needs to be monitored to assess the confidence you can have in its forecasts and changes in the market may require reevaluation of the approach Accuracy can be measured in several ways Standard error of the forecast (SEF) Mean absolute deviation (MAD) Mean squared error (MSE) Mean absolute percent error (MAPE) Root mean squared error (RMSE)
Forecast Accuracy Error - difference between actual value and predicted value Mean Absolute Deviation (MAD) Average absolute error Mean Squared Error (MSE) Average of squared error Mean Absolute Percent Error (MAPE) Average absolute percent error Root Mean Squared Error (RMSE) Root Average of squared error
MAD, MSE, and MAPE Actual forecast MAD = n ( Actual forecast) 2 ( Actual forecast) MSE = n - 1 MAPE = Actual forecast n / Actual)*100) (
MAD, MSE and MAPE MAD Easy to compute Weights errors linearly MSE Squares error More weight to large errors MAPE Puts errors in perspective RMSE Root of Squares error
Example-MAD, MSE, and MAPE Compute MAD, MSE and MAP for the following data showing actual and the predicted numbers of account serviced.
Example-MAD, MSE, and MAPE Compute MAD, MSE and MAP for the following data showing actual and the predicted numbers of account serviced. 22/8=2.75 76/8-1=10.86 10.26/8=1.28 %
Example: MA and ES Techniques Electricity sales data from 2000 Example: MA and ES Techniques Electricity sales data from 2000.1 to 2002.4 (t=12)-Forecast Accuracy - RMSE Using MA3 and MA5 to forecast next period. b) Conduct RMSE technique to check which model measures the forecasting results more sensitive. c) Using ESW3 and ESW5 to forecast next period. d) Conduct RMSE technique to check which model measures the forecasting results more sensitive. e) Briefly explain the case.
Example-For MA Techniques Electricity sales data from 2000. 1 to 2002 Example-For MA Techniques Electricity sales data from 2000.1 to 2002.4 (t=12)-Forecast Accuracy - RMSE AP = 3 moving average AP = 5 moving average
Example-For MA Techniques Electricity sales data from 2000. 1 to 2002 Example-For MA Techniques Electricity sales data from 2000.1 to 2002.4 (t=12)-Forecast Accuracy - RMSE RMSE for 3-qma=2.95 Sqroot of 78.33/9=2.95 RMSE for 5-qma=2.99 Sqroot of 62.48/7=2.98 Thus three-quarter moving average forecast is marginally better than the corresponding five- moving average forecast.
Example-Exponential Smoothing Forecast Accuracy - RMSE Ft = Ft-1 + (At-1 - Ft-1) Example-Exponential Smoothing Forecast Accuracy - RMSE F2= 21+(0.3) (20-21)=20.7 with w=α=0.3 F2= 21+(0.5) (20-21)=20.5 with w=α=0.5
Example-Exponential Smoothing Forecast Accuracy - RMSE RMSE= SQRT(87.19/12)= 2.6955 RMSE= SQRT(101.5/12)=2.908 RMSE with α=0.3 is 2.6955 RMSE with α=0.5 is 2.908 Both exponential forecasts are better than the previous techniques in terms of average values.
Example-Exponential Smoothing Forecast Accuracy - RMSE
Example for MA, WMA and ES (a) Use a simple three-month moving average to find the next period (b) Use a weight average method conducting 0.50 (for most recent datum), 0.30 , and 0.20 to find the next period. (c) Use single exponential smoothing technique to find the next period employing smoothing constant and 5. period forecast value are 0.4 and 58.60 respectively. (d) Use RMSE error model and decide which technique is better explain the data (MA and ES). (e) Plot the monthly data, three-month moving average estimates as as well as exponential smoothing estimates. Briefly explain the patterns. period No of law case 1 60 2 64 3 55 4 58 5
Example for MA, WMA and ES a-b-c period No of law case MA3 WMA3 ES 0.4 1 60 60.2 2 64 60.12 3 55 61.672 4 58 59.66667 59.0032 5 59 58.60192 next period 60.4 60.761152
Example for MA, WMA and ES -d- ErrorMA3 sq(ErrorMA3) ErrorES0.4 sq(ErrorES0.4) -0.2 0.04 3.88 15.0544 -6.672 44.515584 -1.666666667 2.777777778 -1.0032 1.00641024 5 25 5.39808 29.13926769 sum 27.77777778 89.76 RMSE 3.726779962 4.24 Simple moving average technique is better than exponential smoothing technique because the former one gives less error than the latter one.
Example for MA, WMA and ES -e-
Example-Forecast Sales A company records indicate that monthly sales for a seven-month period are as follows:.
Example-Forecast Sales a) Use a simple two-month moving average and single exponential smoothing technique to find the next period employing smoothing constant and 7. period forecast value are 21.3 and 0.4 respectively. Use a simple two-month moving average and single exponential smoothing technique to find all periods. First month forecast value is 15. c) Use RMSE error model and decide which technique is better explain the data (MA and ES). d) Plot the monthly data, two-month moving average estimates as well as exponential smoothing estimates. Briefly explain the patterns.
Example-Forecast Sales
Example-Forecast Sales MA02: F3= 15+23/2=19 F8=21+24/2=22.5 ***ES(α=0.4) Ft = F (t-1)+α (A(t-1)-F(t-1)) F1= 15 (it is the average of series if it is not given) F2= 15+0.4(15-15)=15 F8=21.3+0.4(21-21.3)=21.2
Example-Forecast Sales
Example-Forecast Sales (A - F )2 RMSE n = Σ RMSE(MA02) = 3,11 RMSE(ES04) = 2,37 ES 04 is better explain the pattern of the data than MA 02 because ES 04 gives less error compared to MA 02.
Example-Forecast Sales
Seasonal Variation
Seasonal Variation Select a representative historical data set. Develop a seasonal index for each season. Use the seasonal indexes to deseasonalize the data. Perform linear regression analysis on the deseasonalized data. Use the regression equation to compute the forecasts. Use the seasonal indexes to reapply the seasonal patterns to the forecasts.
Unseasonalized vs. Seasonalized Quarter Seasonalized Sales Seasonal Index Deseasonalized Sales 1 2 3 4 5 6 7 8 9 10 11 … 23 40 25 27 32 48 33 37 50 0.825 1.310 0.920 0.945 27.88 30.53 27.17 28.57 38.79 36.64 35.87 39.15 44.85 38.17 43.48
Deflating a Time Series Observed values can be adjusted to base year equivalent Allows uniform comparison over time Deflation formula: where = adjusted time series value at time t yt = value of the time series at time t It = index value at time t
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