1. Finance 30210: Managerial Economics Demand Forecasting

2. Suppose that you work for a local power company. You have been asked to forecast energy demand for the upcoming year. You have data over the previous 4 years: Time PeriodQuantity (millions of kilowatt hours) 2003:111 2003:215 2003:312 2003:414 2004:112 2004:217 2004:313 2004:416 2005:114 2005:218 2005:315 2005:417 2006:115 2006:220 2006:316 2006:419

3. First, let’s plot the data…what do you see? This data seems to have a linear trend

4. A linear trend takes the following form: Forecasted value at time t (note: time periods are quarters and time zero is 2003:1) Time period: t = 0 is 2003:1 and periods are quarters Estimated value for time zero Estimated quarterly growth (in kilowatt hours)

5. Regression Results VariableCoefficientStandard Errort Stat Intercept11.9.95312.5 Time Trend.394.0994.00 Regression Statistics R Squared.53 Standard Error 1.82 Observations16 Lets forecast electricity usage at the mean time period (t = 8)

6. Here’s a plot of our regression line with our error bands…again, note that the forecast error will be lowest at the mean time period T = 8

7. Sample We can use this linear trend model to predict as far out as we want, but note that the error involved gets worse!

8. Time PeriodActualPredictedError 2003:11112.29-1.29 2003:21512.682.31 2003:31213.08-1.08 2003:41413.47.52 2004:11213.87-1.87 2004:21714.262.73 2004:31314.66-1.65 2004:41615.05.94 2005:11415.44-1.44 2005:21815.842.15 2005:31516.23-1.23 2005:41716.63.37 2006:11517.02-2.02 2006:22017.412.58 2006:31617.81-1.81 2006:41918.20.79 One method of evaluating a forecast is to calculate the root mean squared error Number of Observations Sum of squared forecast errors

9. Lets take another look at the data…it seems that there is a regular pattern… Q2 We are systematically under predicting usage in the second quarter

10. Time PeriodActualPredictedRatioAdjusted 2003:11112.29.8912.29(.87)=10.90 2003:21512.681.1812.68(1.16) = 14.77 2003:31213.08.9113.08(.91) = 11.86 2003:41413.471.0313.47(1.04) = 14.04 2004:11213.87.8713.87(.87) = 12.30 2004:21714.261.1914.26(1.16) = 16.61 2004:31314.66.8814.66(.91) = 13.29 2004:41615.051.0615.05(1.04) = 15.68 2005:11415.44.9115.44(.87) = 13.70 2005:21815.841.1415.84(1.16) = 18.45 2005:31516.23.9216.23(.91) = 14.72 2005:41716.631.0216.63(1.04) = 17.33 2006:11517.02.8817.02(.87) = 15.10 2006:22017.411.1417.41(1.16) = 20.28 2006:31617.81.8917.81(.91) = 16.15 2006:41918.201.0418.20(1.04) = 18.96 Average Ratios Q1 =.87 Q2 = 1.16 Q3 =.91 Q4 = 1.04 We can adjust for this seasonal component…

11. Now, we have a pretty good fit!!

12. Recall our prediction for period 76 ( Year 2022 Q4)

13. Recall, our trend line took the form… This parameter is measuring quarterly change in electricity demand in millions of kilowatt hours. Often times, its more realistic to assume that demand grows by a constant percentage rather that a constant quantity. For example, if we knew that electricity demand grew by g% per quarter, then our forecasting equation would take the form

14. If we wish to estimate this equation, we have a little work to do… Note: this growth rate is in decimal form If we convert our data to natural logs, we get the following linear relationship that can be estimated

15. Regression Results VariableCoefficientStandard Errort Stat Intercept2.49.06339.6 Time Trend.026.0064.06 Regression Statistics R Squared.54 Standard Error.1197 Observations16 Lets forecast electricity usage at the mean time period (t = 8) BE CAREFUL….THESE NUMBERS ARE LOGS !!!

16. The natural log of forecasted demand is 2.698. Therefore, to get the actual demand forecast, use the exponential function Likewise, with the error bands…a 95% confidence interval is +/- 2 SD

17. Again, here is a plot of our forecasts with the error bands T = 8

18. When plotted in logs, our period 76 ( year 2022 Q4) looks similar to the linear trend

19. Again, we need to convert back to levels for the forecast to be relevant!! Errors is growth rates compound quickly!!

20. QuarterMarket Share 120 222 323 424 518 623 719 817 922 1023 1118 1223 Consider a new forecasting problem. You are asked to forecast a company’s market share for the 13 th quarter. There doesn’t seem to be any discernable trend here…

21. Smoothing techniques are often used when data exhibits no trend or seasonal/cyclical component. They are used to filter out short term noise in the data. QuarterMarket Share MA(3)MA(5) 120 222 323 42421.67 51823 6 21.6721.4 71921.6722 8172021.4 92219.6720.2 102319.3319.8 111820.6720.8 12232119.8 A moving average of length N is equal to the average value over the previous N periods

22. The longer the moving average, the smoother the forecasts are…

23. QuarterMarket Share MA(3)MA(5) 120 222 323 42421.67 51823 6 21.6721.4 71921.6722 8172021.4 92219.6720.2 102319.3319.8 111820.6720.8 12232119.8 Calculating forecasts is straightforward… MA(3) MA(5) So, how do we choose N??

24. QuarterMarket Share MA(3)Squared Error MA(5)Squared Error 120 222 323 42421.675.4289 5182325 62321.671.768921.42.56 71921.677.1289229 81720921.419.36 92219.675.428920.23.24 102319.3313.468919.810.24 111820.677.128920.87.84 122321419.810.24 Total = 78.3534Total = 62.48

25. Exponential smoothing involves a forecast equation that takes the following form Forecast for time t+1 Actual value at time t Forecast for time t Smoothing parameter Note: when w = 1, your forecast is equal to the previous value. When w = 0, your forecast is a constant.

26. QuarterMarket Share W=.3W=.5 12021.0 22220.720.5 32321.121.3 42421.722.2 51822.423.1 62321.120.6 71921.721.8 81720.920.4 92219.718.7 102320.4 111821.221.7 122320.219.9 For exponential smoothing, we need to choose a value for the weighting formula as well as an initial forecast Usually, the initial forecast is chosen to equal the sample average

27. As was mentioned earlier, the smaller w will produce a smoother forecast

28. Calculating forecasts is straightforward… W=3 W=5 So, how do we choose W?? QuarterMarket Share W=.3W=.5 12021.0 22220.720.5 32321.121.3 42421.722.2 51822.423.1 62321.120.6 71921.721.8 81720.920.4 92219.718.7 102320.4 111821.221.7 122320.219.9

29. QuarterMarket Share W =.3Squared Error W=.5Squared Error 12021.01 1 22220.71.6920.52.25 32321.13.6121.32.89 42421.75.2922.23.24 51822.419.3623.126.01 62321.13.6120.65.76 71921.77.2921.87.84 81720.915.2120.411.56 92219.75.2918.710.89 102320.46.7620.46.76 111821.210.2421.713.69 122320.27.8419.99.61 Total = 87.19Total = 101.5