1. Forecasting & Time Series Minggu 6

2. Learning Objectives Understand the three categories of forecasting techniques available. Become aware of the four components that make up a time series. Understand how to identify which components are present in a specific time series.

3. Learning Objectives, continued Recognize the forecasting methods available for time series with specific components. Learn several ways of identifying the forecasting methods with the least forecasting error. Forecast for time series with specific components using stationary methods, trend methods, and seasonal methods.

4. Introduction to Forecasting Forecasting is the art or science of predicting the future. Forecasting techniques (1) Qualitative techniques: Subjective estimates from informed sources that are used when historical data are scarce or non-existent - Examples: Delphi techniques, scenario writing, and visionary forecast.

5. Introduction to Forecasting, continued (2) Time Series Techniques: Quantitative techniques that use historical data for only the forecast variable to find patterns. - Based on the premise that the factors that influenced patterns of activity in the past will continue to do so in the future. -Examples: moving averages, exponential smoothing, and trend projections

6. Introduction to Forecasting, continued (3) Causal Techniques: Quantitative techniques based on historical data for the variable being forecast, and one or more explanatory variables. - Based on the supposition that a relationship exists between the variable to be forecast and other explanatory time series data. - Examples: regression models, econometric models, and leading indicators

7. Time Series Components Trend: Long-term upward or downward change in a time series Seasonal: Periodic increases or decreases that occur within one year Cyclical: Periodic increases or decreases that occur over more than a single year Irregular: Changes not attributable to the other three components; non-systematic and unpredictable

8. Components of Time Series Data Trend Irregular Seasonal Cyclical

9. Components of Time Series Data 12345678910111213 Year Seasonal Cyclical Trend Irregular fluctuations

10. Composite Time Series Data 12345678910111213 Year

11. Time Series Forecasting Procedure Step 1: Identifying Time Series Form Trend component –time series plot –trend line Seasonal component –folded annual time series plot –autocorrelation

12. Step 2: Select Potential Methods Stationary forecasting methods are effective for a stationary time series, that is one that contains only an irregular component. These methods attempt to eliminate the irregular through averaging. Trend forecasting methods are effective for time series that contain a trend component. These methods asses the trend component and use it to make projections. Seasonal forecasting methods are used for a time series that contains a trend, a seasonal and an irregular component.

13. Step 3: Evaluate Potential Methods Once the appropriate method has been chosen, it is used to forecast the historical data for the time series. The an evaluation is done of how close the estimates approach the actual historical data. Forecasting Error: A single measure of the overall error of a forecast for an entire set of data. Error of an Individual Forecast: The difference between the actual value and the forecast of that value. e t = Y t - F t

14. Reasons for Forecast Failure Failure to examine assumptions Limited expertise Lack of imagination Neglect of constraints Excessive optimism Reliance on mechanical extrapolation Premature closure Over specification

15. Measurement of Forecasting Error  Mean Error (ME): The average of all the errors of forecast for a group of data.  Mean Absolute Deviation (MAD): The mean, or average of the absolute values of the errors.  Mean Square Error (MSE): The average of the squared errors.  Mean Percentage Error (MPE): The average of the percentage errors of a forecast.  Mean Absolute Percentage Error (MAPE): The average of the absolute values of the percentage errors of a forecast.

16. Example: Nonfarm Partnership Tax Returns: Actual and Forecast with  =.7 YearActualForecastError 11402 214581402.056.0 315531441.2111.8 416131519.593.5 516761584.991.1 617551648.7106.3 718071723.183.9 818241781.842.2 918261811.314.7 1017801821.6-41.6 1117591792.5-33.5

17. Mean Error for the Nonfarm Partnership Forecasted Data YearActualForecastError 11402.0 21458.01402.056.0 31553.01441.2111.8 41613.01519.593.5 51676.01584.991.1 61755.01648.7106.3 71807.01723.183.9 81824.01781.842.2 91826.01811.314.7 101780.01821.6-41.6 111759.01792.5-33.5 524.3

18. Mean Absolute Deviation for the Nonfarm Partnership Forecasted Data YearActualForecastError|Error| 11402.0 21458.01402.056.0 31553.01441.2111.8 41613.01519.593.5 51676.01584.991.1 61755.01648.7106.3 71807.01723.183.9 81824.01781.842.2 91826.01811.314.7 101780.01821.6-41.641.6 111759.01792.5-33.533.5 674.5

19. Mean Square Error for the Nonfarm Partnership Forecasted Data YearActualForecastErrorError 2 11402 214581402.056.03136.0 315531441.2111.812499.2 416131519.593.58749.7 516761584.991.18292.3 617551648.7106.311303.6 718071723.183.97038.5 818241781.842.21778.2 918261811.314.7214.6 1017801821.6-41.61731.0 1117591792.5-33.51121.0 55864.2

20. Mean Percentage Error for the Nonfarm Partnership Forecasted Data YearActualForecastErrorError % 11402 214581402.056.03.8% 315531441.2111.87.2% 416131519.593.55.8% 516761584.991.15.4% 617551648.7106.36.1% 718071723.183.94.6% 818241781.842.22.3% 918261811.314.70.8% 1017801821.6-41.6-2.3% 1117591792.5-33.5-1.9% 31.8%

21. Mean Absolute Percentage Error for the Nonfarm Partnership Forecasted Data YearActualForecastError|Error %| 11402 214581402.056.03.8% 315531441.2111.87.2% 416131519.593.55.8% 516761584.991.15.4% 617551648.7106.36.1% 718071723.183.94.6% 818241781.842.22.3% 918261811.314.70.8% 1017801821.6-41.62.3% 1117591792.5-33.51.9% 40.3%

22. Use of Error Measures To identify the best forecasting method Use error measure to identify the best value for the parameters of a specific method. Use error measure to identify the best method. Use MSE and MAD for both of these situations. Note that MSE tends to emphasize large errors.

23. Use of Error Measures, continued Forecast bias is the tendency of a forecasting method to over or under predict. The mean error, ME, measures the forecast bias.

24. Step 4: Make Required Forecasts The best forecasting method is that with the smallest overall error measurement. Using a stationary method will make a forecast for one time into the future, F t+1. This is also the forecast for all future time periods. Forecasts made using a non-stationary method will not be the same for all time periods in the future.

25. Stationary Forecasting Methods Naive Forecasting Method Moving Average Forecasting Method Weighted Moving Average Forecasting Method Exponential Smoothing Forecasting Method

26. Naive Forecasting Simplest of the naive forecasting models Simplest of the naive forecasting models We sold 532 pairs of shoes last week, I predict we’ll sell 532 pairs this week. We sold 532 pairs of shoes last week, I predict we’ll sell 532 pairs this week.

27. Simple Average Forecasting Method The monthly average last 12 months was 56.45, so I predict 56.45 for September. The monthly average last 12 months was 56.45, so I predict 56.45 for September. MonthYear Cents per GallonMonthYear Cents per Gallon January199461.3January199558.2 February63.3February58.3 March62.1March57.7 April59.8April56.7 May58.4May56.8 June57.6June55.5 July55.7July53.8 August55.1August52.8 September55.7September October56.7October November57.2November December58.0December

28. Moving Average Forecasting Method Updated (recomputed) for every new time period May be difficult to choose optimal number of periods May not adjust for trend, cyclical, or seasonal effects Update me each period.

29. Weighted Moving Average Forecasting Method

30. Exponential Smoothing Forecasting Method  is the exponential smoothing constant

31. Trend Forecasting Methods Linear Trend Projection Forecasting Method: Forecasting by fitting a linear equation to a time series Non-linear Trend Projection Forecasting Method: Forecasting by fitting a non-linear equation to a time series

32. Average Hours Worked per Week by Canadian Manufacturing Workers PeriodHoursPeriodHoursPeriodHoursPeriodHours 137.21136.92135.63135.7 237.01236.72235.23235.5 337.41336.72334.83335.6 437.51436.52435.33436.3 537.71536.32535.63536.5 637.71635.92635.6 737.41735.82735.6 837.21835.92835.9 937.31936.02936.0 1037.22035.73035.7

33. Excel Regression Output using Linear Trend Regression Statistics Multiple R0.782 R Square0.611 Adjusted R Square0.5600 Standard Error0.509 Observations35 ANOVA dfSSMSFSignificance F Regression113.4467 51.91.00000003 Residual338.54870.2591 Total3421.9954 CoefficientsStandard Errort StatP-value Intercept37.41610.17582212.81.0000000 Period-0.06140.00852-7.20.00000003

34. Excel Graph of Hours Worked Data with a Linear Trend Line 34.5 35.0 35.5 36.0 36.5 37.0 37.5 38.0 05101520253035 Time Period Work Week

35. Excel Regression Output using Quadratic Trend Regression Statistics Multiple R0.8723 R Square0.761 Adjusted R Square0.747 Standard Error0.405 Observations35 ANOVA dfSSMSFSignificance F Regression216.74838.374151.071.10021E-10 Residual325.24720.1640 Total3421.9954 CoefficientsStandard Errort StatP-value Intercept38.164420.21766175.342.61E-49 Period-0.182720.02788-6.552.21E-07 Period 2 0.00337 0.000754.498.76E-05

36. Excel Graph of Hourly Data with Quadratic Trend Line 34.5 35.0 35.5 36.0 36.5 37.0 37.5 38.0 05101520253035 Period Work Week

37. Exponential Smoothing with Trend Effects: Holt’s Model Holt’s Model adds consideration of a trend component to the basic exponential smoothing relation.

38. Trend Autoregression Method Autoregression Model with two lagged variables Autoregression Model with three lagged variables A multiple regression technique in which the independent variables are time-lagged versions of the dependent variable.

39. Durbin-Watson Test for Autocorrelation

40. Overcoming the Autocorrelation Problem Addition of Independent Variables Transforming Variables –First-differences approach –Percentage change from period to period –Use autoregression

41. Seasonal Forecasting Methods Seasonal Multiple Regression Forecasting Method Seasonal Autoregression Forecasting Method Winter’s Exponential Smoothing Forecasting Model Time Series Decomposition Forecasting Method

42. Exponential Smoothing with Trend and Seasonality: Winter’s Model

43. Time Series Decomposition Forecasting Method Basis for analysis is the multiplicative model Y = T · C · S · I where: T = trend component C = cyclical component S = seasonal component I = irregular component

44. Time Series Decomposition Determine the seasonality of the time series by computing a seasonal index for each season (each quarter, each month, and so on. Divide each time series data value by the appropriate seasonal index to deseasonalize it. Identify a trend model appropriate for the deseasonalized trend model. Forecast deseasonalized values with the trend model Multiply the deseasonalized forecasts times the appropriate seasonal index to compute the final seasonalized forecasts.

45. Demonstration Problem 14.6: Household Appliance Shipment Data YearQuarterShipmentsYearQuarterShipments 114009414595 2432124799 3422434417 4394444258 214123514245 2452224900 3465734585 4403044533 314493 24806 34551 44485 Shipments in $1,000,000.

46. Demonstration Problem 14.6: Graph of Household Appliance Shipment Data 3900 4050 4200 4350 4500 4650 4800 4950 048121620 Quarter Shipments