1. Yandell – Econ 216 Chap 16-1 Chapter 16 Time-Series Forecasting

2. Yandell – Econ 216 Chap 16-2 Chapter Goals After completing this chapter, you should be able to: Develop and implement basic forecasting models Identify the components present in a time series Use smoothing-based forecasting models, including single exponential smoothing Apply trend-based forecasting models, including linear trend, nonlinear trend, and seasonally adjusted trend

3. Yandell – Econ 216 Chap 16-3 The Importance of Forecasting Governments forecast unemployment, interest rates, and expected revenues from income taxes for policy purposes Marketing executives forecast demand, sales, and consumer preferences for strategic planning College administrators forecast enrollments to plan for facilities and for faculty recruitment Retail stores forecast demand to control inventory levels, hire employees and provide training

4. Yandell – Econ 216 Chap 16-4 Time-Series Data Numerical data obtained at regular time intervals The time intervals can be annually, quarterly, daily, hourly, etc. Example: Year:1999 2000 2001 2002 2003 Sales: 75.3 74.2 78.5 79.7 80.2

5. Yandell – Econ 216 Chap 16-5 Time Series Plot the vertical axis measures the variable of interest the horizontal axis corresponds to the time periods A time-series plot is a two-dimensional plot of time series data

6. Yandell – Econ 216 Chap 16-6 Time-Series Components Time-Series Cyclical Component Irregular Component Trend Component Seasonal Component

7. Yandell – Econ 216 Chap 16-7 Upward trend Trend Component Long-run increase or decrease over time (overall upward or downward movement) Data taken over a long period of time Sales Time

8. Yandell – Econ 216 Chap 16-8 Downward linear trend Trend Component Trend can be upward or downward Trend can be linear or non-linear Sales Time Upward nonlinear trend Sales Time (continued)

9. Yandell – Econ 216 Chap 16-9 Seasonal Component Short-term regular wave-like patterns Observed within 1 year Often monthly or quarterly Sales Time (Quarterly) Winter Spring Summer Fall

10. Yandell – Econ 216 Chap 16-10 Cyclical Component Long-term wave-like patterns Regularly occur but may vary in length Often measured peak to peak or trough to trough Sales 1 Cycle Year

11. Yandell – Econ 216 Chap 16-11 Irregular Component Unpredictable, random, “residual” fluctuations Due to random variations of Nature Accidents or unusual events “Noise” in the time series

12. Yandell – Econ 216 Chap 16-12 Multiplicative Time-Series Model Used primarily for forecasting Allows consideration of seasonal variation Observed value in time series is the product of components whereT t = Trend value at time t S t = Seasonal value at time t C t = Cyclical value at time t I t = Irregular (random) value at time t

13. Yandell – Econ 216 Chap 16-13 Finding Seasonal Indexes Ratio-to-moving average method: Begin by removing the seasonal and irregular components (S t and I t ), leaving the trend and cyclical components (T t and C t ) To do this, we need moving averages Moving Average: averages of consecutive time series values

14. Yandell – Econ 216 Chap 16-14 Moving Averages Used for smoothing Series of arithmetic means over time Result dependent upon choice of L (length of period for computing means) To smooth out seasonal variation, L should be equal to the number of seasons For quarterly data, L = 4 For monthly data, L = 12

15. Yandell – Econ 216 Chap 16-15 Moving Averages Example: Four-quarter moving average First average: Second average: etc… (continued)

16. Yandell – Econ 216 Chap 16-16 Seasonal Data QuarterSales 1 2 3 4 5 6 7 8 9 10 11 etc… 23 40 25 27 32 48 33 37 50 40 etc… … …

17. Yandell – Econ 216 Chap 16-17 Calculating Moving Averages Each moving average is for a consecutive block of 4 quarters QuarterSales 123 240 325 427 532 648 733 837 9 1050 1140 Average Period 4-Quarter Moving Average 2.528.75 3.531.00 4.533.00 5.535.00 6.537.50 7.538.75 8.539.25 9.541.00 etc…

18. Yandell – Econ 216 Chap 16-18 Centered Moving Averages Average periods of 2.5 or 3.5 don’t match the original quarters, so we average two consecutive moving averages to get centered moving averages Average Period 4-Quarter Moving Average 2.528.75 3.531.00 4.533.00 5.535.00 6.537.50 7.538.75 8.539.25 9.541.00 Centered Period Centered Moving Average 329.88 432.00 534.00 636.25 738.13 839.00 940.13 etc…

19. Yandell – Econ 216 Chap 16-19 Calculating the Ratio-to-Moving Average Now estimate the S t x I t value Divide the actual sales value by the centered moving average for that quarter Ratio-to-Moving Average formula:

20. Yandell – Econ 216 Chap 16-20 Calculating Seasonal Indexes QuarterSales Centered Moving Average Ratio-to- Moving Average 1 2 3 4 5 6 7 8 9 10 11 … 23 40 25 27 32 48 33 37 50 40 … 29.88 32.00 34.00 36.25 38.13 39.00 40.13 etc… … 0.837 0.844 0.941 1.324 0.865 0.949 0.922 etc… …

21. Yandell – Econ 216 Chap 16-21 Calculating Seasonal Indexes QuarterSales Centered Moving Average Ratio-to- Moving Average 1 2 3 4 5 6 7 8 9 10 11 … 23 40 25 27 32 48 33 37 50 40 … 29.88 32.00 34.00 36.25 38.13 39.00 40.13 etc… … 0.837 0.844 0.941 1.324 0.865 0.949 0.922 etc… … Average all of the Fall values to get Fall’s seasonal index Fall Do the same for the other three seasons to get the other seasonal indexes (continued)

22. Yandell – Econ 216 Chap 16-22 Interpreting Seasonal Indexes Suppose we get these seasonal indexes: Season Seasonal Index Spring0.825 Summer1.310 Fall0.920 Winter0.945  = 4.000 -- four seasons, so must sum to 4 Spring sales average 82.5% of the annual average sales Summer sales are 31.0% higher than the annual average sales etc… Interpretation:

23. Yandell – Econ 216 Chap 16-23 Deseasonalizing The data is deseasonalized by dividing the observed value by its seasonal index This smooths the data by removing seasonal variation

24. Yandell – Econ 216 Chap 16-24 Deseasonalizing QuarterSales Seasonal Index Deseasonalized Sales 1 2 3 4 5 6 7 8 9 10 11 … 23 40 25 27 32 48 33 37 50 40 0.825 1.310 0.920 0.945 0.825 1.310 0.920 0.945 0.825 1.310 0.920 … 27.88 30.53 27.17 28.57 38.79 36.64 35.87 39.15 44.85 38.17 43.48 … etc… (continued)

25. Yandell – Econ 216 Chap 16-25 Unseasonalized vs. Seasonalized

26. Yandell – Econ 216 Chap 16-26 Exponential Smoothing A weighted moving average Weights decline exponentially Most recent observation weighted most Used for smoothing and short term forecasting

27. Yandell – Econ 216 Chap 16-27 Exponential Smoothing The weighting factor is W Subjectively chosen Range from 0 to 1 Smaller W gives more smoothing, larger W gives less smoothing The weight is: Close to 0 for smoothing out unwanted cyclical and irregular components Close to 1 for forecasting (continued)

28. Yandell – Econ 216 Chap 16-28 Exponential Smoothing Model Exponential smoothing model where: F t+1 = forecast value for period t + 1 Y t = actual value for period t F t = forecast value for period t W = weight (smoothing constant) or:

29. Yandell – Econ 216 Chap 16-29 Exponential Smoothing Example Suppose we use weight W =.2 Quarter (t) Sales (Y t ) Forecast from prior period Forecast for next period (F t+1 ) 1 2 3 4 5 6 7 8 9 10 etc… 23 40 25 27 32 48 33 37 50 etc… NA 23 26.4 26.12 26.296 27.437 31.549 31.840 32.872 33.697 etc… 23 (.2)(40)+(.8)(23)=26.4 (.2)(25)+(.8)(26.4)=26.12 (.2)(27)+(.8)(26.12)=26.296 (.2)(32)+(.8)(26.296)=27.437 (.2)(48)+(.8)(27.437)=31.549 (.2)(48)+(.8)(31.549)=31.840 (.2)(33)+(.8)(31.840)=32.872 (.2)(37)+(.8)(32.872)=33.697 (.2)(50)+(.8)(33.697)=36.958 etc… F 1 = Y 1 since no prior information exists

30. Yandell – Econ 216 Chap 16-30 Sales vs. Smoothed Sales Seasonal fluctuations have been smoothed NOTE: the smoothed value in this case is generally a little low, since the trend is upward sloping and the weighting factor is only.2

31. Yandell – Econ 216 Chap 16-31 Exponential Smoothing in Excel Use: data analysis / exponential smoothing The “damping factor” is (1 - W)

32. Yandell – Econ 216 Chap 16-32 Trend-Based Forecasting Estimate a trend line using regression analysis Year Time Period (t) Sales (Y) 1999 2000 2001 2002 2003 2004 123456123456 20 40 30 50 70 65 Use time (t) as the independent variable:

33. Yandell – Econ 216 Chap 16-33 Trend-Based Forecasting The linear trend model is: Year Time Period (t) Sales (Y) 1999 2000 2001 2002 2003 2004 123456123456 20 40 30 50 70 65 (continued)

34. Yandell – Econ 216 Chap 16-34 Trend-Based Forecasting Forecast for time period 7: Year Time Period (t) Sales (y) 1999 2000 2001 2002 2003 2004 2005 12345671234567 20 40 30 50 70 65 ?? (continued)

35. Yandell – Econ 216 Chap 16-35 Comparing Forecast Values to Actual Data The forecast error or residual is the difference between the actual value in time t and the forecast value in time t: Error in time t:

36. Yandell – Econ 216 Chap 16-36 Autocorrelation Autocorrelation is correlation of the error terms (residuals) over time (continued) Violates the regression assumption that residuals are random and independent Here, residuals show a cyclic pattern, not random

37. Yandell – Econ 216 Chap 16-37 Testing for Autocorrelation The Durbin-Watson Statistic is used to test for autocorrelation H 0 : ρ = 0 (residuals are not correlated) H 1 : ρ ≠ 0 (autocorrelation is present) Durbin-Watson test statistic:

38. Yandell – Econ 216 Chap 16-38 Testing for Positive Autocorrelation  Calculate the Durbin-Watson test statistic = DW (The Durbin-Watson Statistic can be found using PHStat) Decision rule: reject H 0 if DW < d L H 0 : ρ = 0 (positive autocorrelation does not exist) H 1 : ρ > 0 (positive autocorrelation is present) 0dUdU 2dLdL Reject H 0 Do not reject H 0  Find the values d L and d U from the Durbin-Watson table (for sample size n and number of independent variables p) Inconclusive

39. Yandell – Econ 216 Chap 16-39 Example with n = 25: Durbin-Watson Calculations Sum of Squared Difference of Residuals3296.18 Sum of Squared Residuals3279.98 Durbin-Watson Statistic1.00494 Testing for Positive Autocorrelation (continued) Excel/PHStat output:

40. Yandell – Econ 216 Chap 16-40 Here, n = 25 and there is one independent variable Using the Durbin-Watson table, d L = 1.29 and d U = 1.45 DW = 1.00494 < d L = 1.29, so reject H 0 and conclude that significant positive autocorrelation exists Therefore the linear model is not the appropriate model to forecast sales Testing for Positive Autocorrelation (continued) Decision: reject H 0 since DW = 1.00494 < d L 0 d U =1.45 2 d L =1.29 Reject H 0 Do not reject H 0 Inconclusive

41. Yandell – Econ 216 Chap 16-41 Nonlinear Trend Forecasting A nonlinear regression model can be used when the time series exhibits a nonlinear trend One form of a nonlinear model: Compare r 2 and standard error to that of linear model to see if this is an improvement Can try other functional forms to get best fit

42. Yandell – Econ 216 Chap 16-42 Chapter Summary Discussed the importance of forecasting Addressed component factors present in the time-series model Computed and interpreted index numbers Described least square trend fitting and forecasting linear and nonlinear models Performed smoothing of data series moving averages single and double exponential smoothing