1. 1 Pertemuan 23 Deret Berkala, Peramalan, dan Angka Indeks-1 Matakuliah: A0064 / Statistik Ekonomi Tahun: 2005 Versi: 1/1

2. 2 Learning Outcomes Pada akhir pertemuan ini, diharapkan mahasiswa akan mampu : Menghubungkan beberapa deret berkala bagi penyusunan peramalan dengan menggunakan analisis trend, variasi musim, dan perilaku siklus

3. 3 Outline Materi Anailis Trend Variasi Musim dan Perilaku Siklus

4. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 12-4 Using Statistics Trend Analysis Seasonality and Cyclical Behavior The Ratio-to-Moving-Average Method Exponential Smoothing Methods Index Numbers Summary and Review of Terms Time Series, Forecasting, and Index Numbers 12

5. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 12-5 A time series is a set of measurements of a variable that are ordered through time. Time series analysis attempts to detect and understand regularity in the fluctuation of data over time. Regular movement of time series data may result from a tendency to increase or decrease through time - trend- or from a tendency to follow some cyclical pattern through time - seasonality or cyclical variation. Forecasting is the extrapolation of series values beyond the region of the estimation data. Regular variation of a time series can be forecast, while random variation cannot. A time series is a set of measurements of a variable that are ordered through time. Time series analysis attempts to detect and understand regularity in the fluctuation of data over time. Regular movement of time series data may result from a tendency to increase or decrease through time - trend- or from a tendency to follow some cyclical pattern through time - seasonality or cyclical variation. Forecasting is the extrapolation of series values beyond the region of the estimation data. Regular variation of a time series can be forecast, while random variation cannot. 12-1 Using Statistics

6. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 12-6 A random walk: Z t - Z t-1 =a t or equivalently: Z t = Z t-1 +a t The difference between Z in time t and time t-1 is a random error. A random walk: Z t - Z t-1 =a t or equivalently: Z t = Z t-1 +a t The difference between Z in time t and time t-1 is a random error. 50403020100 15 10 5 0 t Z Random Walk a There is no evident regularity in a random walk, since the difference in the series from period to period is a random error. A random walk is not forecastable. The Random Walk

7. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 12-7 A Time Series as a Superposition of Two Wave Functions and a Random Error (not shown) In contrast with a random walk, this series exhibits obvious cyclical fluctuation. The underlying cyclical series - the regular elements of the overall fluctuation - may be analyzable and forecastable. A Time Series as a Superposition of Cyclical Functions

8. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 12-8 12-2 Trend Analysis: Example 12-1 The following output was obtained using the template. Z t = 696.89 + 109.19t Note:The template contains forecasts for t = 9 to t = 20 which corresponds to years 2002 to 2013. Note: The template contains forecasts for t = 9 to t = 20 which corresponds to years 2002 to 2013.

9. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 12-9 12-2 Trend Analysis: Example 12-1 Straight line trend.

10. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 12-10 Example 12-2 Observe that the forecast model is Z t = 82.96 + 26.23t The forecast for t = 10 (year 2002) is 345.27

11. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 12-11 888786 11000 10000 9000 8000 7000 6000 t P a s s e n g e r s Monthly Numbers of Airline Passengers 89909192939495 95949392919089888786858483828180 20 15 10 5 Year E a r n i n g s Gross Earnings: Annual AJJMAMFJDNOSAJJMAMFJDNOSAJJMAMFJ 200 100 0 Time S a l e s Monthly Sales of Suntan Oil When a cyclical pattern has a period of one year, it is usually called seasonal variation. A pattern with a period of other than one year is called cyclical variation. 12-3 Seasonality and Cyclical Behavior

12. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 12-12 Types of Variation Trend (T) Seasonal (S) Cyclical (C) Random or Irregular (I) Additive Model  Z t = T t + S t + C t + I t Multiplicative Model  Z t = (T t )(S t )(C t )(I t ) Time Series Decomposition

13. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 12-13 An additive regression model with seasonality: Z t =  0 +  1 t+  2 Q 1 +  3 Q 2 +  4 Q 3 + a t where Q 1 =1 if the observation is in the first quarter, and 0 otherwise Q 2 =1 if the observation is in the second quarter, and 0 otherwise Q 3 =1 if the observation is in the third quarter, and 0 otherwise An additive regression model with seasonality: Z t =  0 +  1 t+  2 Q 1 +  3 Q 2 +  4 Q 3 + a t where Q 1 =1 if the observation is in the first quarter, and 0 otherwise Q 2 =1 if the observation is in the second quarter, and 0 otherwise Q 3 =1 if the observation is in the third quarter, and 0 otherwise Estimating an Additive Model with Seasonality

14. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 12-14 A moving average of a time series is an average of a fixed number of observations that moves as we progress down the series. Time, t: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Series, Z t : 15 12 11 18 21 16 14 17 20 18 21 16 14 19 Five-period moving average:15.415.616.017.217.617.018.018.417.817.6 Time, t: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Series, Z t : 15 12 11 18 21 16 14 17 20 18 21 16 14 19 (15 + 12 + 11 + 18 + 21)/5=15.4 (12 + 11 + 18 + 21 + 16)/5=15.6 (11 + 18 + 21 + 16 + 14)/5=16.0..... (18 + 21 + 16 + 14 + 19)/5=17.6 12-4 The Ratio-to-Moving- Average Method

15. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 12-15 Moving Average: – “Smoother” – Shorter – Deseasonalized – Removes seasonal and irregular components – Leaves trend and cyclical components Moving Average: – “Smoother” – Shorter – Deseasonalized – Removes seasonal and irregular components – Leaves trend and cyclical components Comparing Original Data and Smoothed Moving Average

16. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 12-16 Ratio-to-Moving Average for Quarterly Data  Compute a four-quarter moving-average series.  Center the moving averages by averaging every consecutive pair and placing the average between quarters.  Divide the original series by the corresponding moving average. Then multiply by 100.  Derive quarterly indexes by averaging all data points corresponding to each quarter. Multiply each by 400 and divide by sum. Ratio-to-Moving Average for Quarterly Data  Compute a four-quarter moving-average series.  Center the moving averages by averaging every consecutive pair and placing the average between quarters.  Divide the original series by the corresponding moving average. Then multiply by 100.  Derive quarterly indexes by averaging all data points corresponding to each quarter. Multiply each by 400 and divide by sum. Ratio-to-Moving Average

17. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 12-17 151050 170 160 150 140 130 S a l e s Time Actual Smoothed Actual Smoothed MSD: MAD: MAPE: Length: Moving Average 152.574 10.955 7.816 4 Four-Quarter Moving Averages Simple Centered Ratio Moving Moving to Moving QuarterSales Average Average Average 1998W170*** 1998S148*** 1998S141*151.12593.3 1998F150152.25148.625100.9 1999W161150.00146.125110.2 1999S137147.25146.00093.8 1999S132145.00146.50090.1 1999F158147.00147.000107.5 2000W157146.00147.500106.4 2000S145148.00144.000100.7 2000S128147.00141.37590.5 2000F134141.00141.00095.0 2002W160141.75140.500113.9 2002S139140.25142.00097.9 2002S130140.75** 2002F144143.25** Simple Centered Ratio Moving Moving to Moving QuarterSales Average Average Average 1998W170*** 1998S148*** 1998S141*151.12593.3 1998F150152.25148.625100.9 1999W161150.00146.125110.2 1999S137147.25146.00093.8 1999S132145.00146.50090.1 1999F158147.00147.000107.5 2000W157146.00147.500106.4 2000S145148.00144.000100.7 2000S128147.00141.37590.5 2000F134141.00141.00095.0 2002W160141.75140.500113.9 2002S139140.25142.00097.9 2002S130140.75** 2002F144143.25** Ratio-to-Moving Average: Example 12-3

18. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 12-18 Quarter Year Winter Spring Summer Fall 199893.3100.9 1999110.293.890.1107.5 2000106.4100.790.595.0 2002113.997.9 Sum330.5292.4273.9303.4 Average110.1797.4791.3101.13 Sum of Averages = 400.07 Seasonal Index = (Average)(400)/400.07 Seasonal Index110.1597.4591.28101.11 Quarter Year Winter Spring Summer Fall 199893.3100.9 1999110.293.890.1107.5 2000106.4100.790.595.0 2002113.997.9 Sum330.5292.4273.9303.4 Average110.1797.4791.3101.13 Sum of Averages = 400.07 Seasonal Index = (Average)(400)/400.07 Seasonal Index110.1597.4591.28101.11 Seasonal Indexes: Example 12-3

19. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 12-19 Seasonal Deseasonalized Quarter Sales Index (S) Series(Z/S)*100 1998W170110.15154.34 1998S14897.45151.87 1998S14191.28154.47 1998F150101.11148.35 1999W161110.15146.16 1999S13797.45140.58 1999S13291.28144.51 1999F158101.11156.27 2000W157110.15142.53 2000S14597.45148.79 2000S12891.28140.23 2000F134101.11132.53 2002W160110.15145.26 2002S13997.45142.64 2002S13091.28142.42 2002F144101.11142.42 Seasonal Deseasonalized Quarter Sales Index (S) Series(Z/S)*100 1998W170110.15154.34 1998S14897.45151.87 1998S14191.28154.47 1998F150101.11148.35 1999W161110.15146.16 1999S13797.45140.58 1999S13291.28144.51 1999F158101.11156.27 2000W157110.15142.53 2000S14597.45148.79 2000S12891.28140.23 2000F134101.11132.53 2002W160110.15145.26 2002S13997.45142.64 2002S13091.28142.42 2002F144101.11142.42 Deseasonalized Series: Example 12-3

20. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 12-20 180 170 160 150 140 130 120 1992F1992S1992S1992W t S a l e s Trend Line and Moving Averages The cyclical component is the remainder after the moving averages have been detrended. In this example, a comparison of the moving averages and the estimated regression line: illustrates that the cyclical component in this series is negligible. The cyclical component is the remainder after the moving averages have been detrended. In this example, a comparison of the moving averages and the estimated regression line: illustrates that the cyclical component in this series is negligible. The Cyclical Component: Example 12-3

21. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 12-21 The centered moving average, ratio to moving average, seasonal index, and deseasonalized values were determined using the Ratio-to-Moving-Average method. Example 12-3 using the Template This displays just a partial output for the quarterly forecasts.

22. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 12-22 Example 12-3 using the Template Graph of the quarterly Seasonal Index

23. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 12-23 Example 12-3 using the Template Graph of the Data and the quarterly Forecasted values

24. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 12-24 The centered moving average, ratio to moving average, seasonal index, and deseasonalized values were determined using the Ratio-to-Moving-Average method. Example 12-3 using the Template This displays just a partial output for the monthly forecasts.

25. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 12-25 Example 12-3 using the Template Graph of the monthly Seasonal Index

26. COMPLETE 5 t h e d i t i o n BUSINESS STATISTICS Aczel/Sounderpandian McGraw-Hill/Irwin © The McGraw-Hill Companies, Inc., 2002 12-26 Example 12-3 using the Template Graph of the Data and the monthly Forecasted values

27. 27 Penutup Pembahasan materi dilanjutkan dengan Materi Pokok 24 (Deret Berkala, Peramalan, dan Angka Indeks-2)