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TM 745 Forecasting for Business & Technology

TM 745 Forecasting for Business & Technology. ARIMA (Box-Jenkins). =. Y. observatio. n. (. realizatio. n. ). at. t. t. =. +. e. Pattern. t. ARIMA Models. =. Y. function. of. past. values. t. +. random. shocks. =. e. +. e. f. (. Y. ,. ). -. -. t. k. t. k.

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TM 745 Forecasting for Business & Technology

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  1. TM 745Forecasting for Business & Technology ARIMA(Box-Jenkins)

  2. = Y observatio n ( realizatio n ) at t t = + e Pattern t ARIMA Models

  3. = Y function of past values t + random shocks = e + e f ( Y , ) - - t k t k t Assumptions

  4. Notation AR I MA p d q p = order of autoregressive d = order of integration (differencing) q = order of moving average

  5. = q + f + e Y Y - 1 1 t o t t $ = q + f Y Y - 1 1 t o t Processes • ARIMA (1, 0, 0)

  6. = q + f + e Y Y - 1 1 t o t t f £ | | 1 f Þ 1 drift / trend f Processes • ARIMA (1, 0, 0) • Autoregressive • Stationarity

  7. AutoRegressive Process

  8. = m - q e + e Y - 1 1 t t t $ = m - q e Y - 1 1 t t Moving Average Process • ARIMA (0, 0, 1)

  9. Moving Average Process

  10. = + e Y Y - 1 t t t $ = Y Y - 1 t t Integrated Processes • ARIMA (0, 1, 0)

  11. Integrated Process

  12. = + q + e Y Y - 1 t t o t $ = + q Y Y - 1 t t o Deterministic Trend • ARIMA (0, 1, ,0) 1

  13. Deterministic Trend

  14. Model Identification • (Figure 7-1, p. 277) • AutoRegressive ARIMA(1, 0, 0), f=0.8 ACF(1) = f = 0.8 ACF(2) = f2 = 0.64 ACF(3) = f3 = 0.512 ACF(4) = f4 = 0.410

  15. Model Identification • (Figure 7-1, p. 277) • AutoRegressive ARIMA(1, 0, 0), f=0.4 ACF(1) = f = 0.4 ACF(2) = f2 = 0.16 ACF(3) = f3 = 0.064 ACF(4) = f4 = 0.0256

  16. Model Identification • (Figure 7-1, p. 277) • AutoRegressive ARIMA(1, 0, 0), f=-0.8 ACF(1) = f = -0.8 ACF(2) = f2 = 0.64 ACF(3) = f3 = -0.512 ACF(4) = f4 = 0.410

  17. Example; Stock Prices

  18. Example; ACF

  19. Example; PACF

  20. = + e Y Y - 1 t t t $ = Y Y - 1 t t e = - Y Y - 1 t t t Example; Stock Prices • Based on ACF and PACF, appears to be a random walk

  21. Example; Stock Prices

  22. Example; Stock Prices

  23. Non-Stationarity • Two Primary Types • Non-Stationary in Level (Mean) • Non-Stationary Variance

  24. Example; FAD Apparel

  25. Example; FAD Apparel ACF and PACF indicate random walk or more likely, a non- stationary process

  26. = - z Y Y - 1 t t t FAD Apparel (First Differences)

  27. Fad Apparel (1st Diff) ACF and PACF suggest MA model on zt.

  28. Fad Ex; Fitted Model

  29. Fad Ex; ARIMA(0, 1, 1)

  30. FAD Residuals

  31. Non-Stationary Variance

  32. Power Consumption (1st diff)

  33. 2 s µ Y Y t = Y KY - 1 t t Power Consumption • Variance of series is proportional to level • Common in product demand, economy, stock market

  34. = + ln( Y ) ln( Y ) ln( K ) - 1 t t - = ln( Y ) ln( Y ) ln( K ) - = 1 t t Y KY - 1 t t = - ¢ z ln( Y ) ln( Y ) - 1 t t t Transformation

  35. Power Transformed

  36. Power Transformed

  37. Power Transformed

  38. Power Transformed (diff)

  39. AutoCorrelation (Zt) Suggests MA(1) with q = -0.15 ARIMA(0,1,1)

  40. = e - q e ¢ z - 1 1 t t t - = e - q e ln( Y ) ln( Y ) - - 1 1 1 t t t t = + ln( Y ) ln( Y ) 0 . 15 e - - 1 1 t t t Power Model

  41. Fitted Model

  42. Residuals

  43. Fitted Model

  44. = f + f + e Y Y Y - - t 1 t 1 2 t 2 t = f + f + e 2 Y Y Y Y Y Y - - - - - t 1 t 1 t 1 2 t 1 t 2 t t 1 = f + f + e 2 E [ Y Y ] E [ Y ] E [ Y Y ] E [ Y ] - - - - - t 1 t 1 t 1 2 t 1 t 2 t t 1 PACF’s (Partial Derivation) Consider an AR(2) process, Multiply both sides by Yt-1 Take Expectations

  45. = f + f Cov ( Y Y ) Var ( Y ) Cov( Y Y ) - - - - 1 1 1 2 1 2 t t t t t = f + f + e 2 E [ Y Y ] E [ Y ] E [ Y Y ] E [ Y ] - - - - - t 1 t 1 t 1 2 t 1 t 2 t t 1 = f + f Cov ( Y Y ) Var ( Y ) Cov ( Y Y ) - - - 1 1 1 2 1 t t t t t PACF’s (Partial Derivation) by stationarity, independence by stationarity, Cov(Yt-1Yt-2) = Cov(YtYt-1)

  46. f Cov ( Y Y ) cov( Y Y ) = f + - - 1 2 1 t t t t 1 Var ( Y ) Var ( Y ) t t = f + f ACF ( 1 ) ACF ( 1 ) 1 2 = f + f Cov ( Y Y ) Var ( Y ) Cov ( Y Y ) - - - 1 1 1 2 1 t t t t t PACF’s (Partial Derivation) Dividing both sides by Var(Yt-1)

  47. = f + f ACF ( 1 ) ACF ( 1 ) 1 2 PACF’s (Partial Derivation) Total AC = Direct AC + Indirect AC ACF = PACF + Indirect

  48. = f + f + e Y Y Y - - t 1 t 1 2 t 2 t = f f + e 2 Y Y Y Y Y Y + - - - - - t 2 t 1 t 2 t 1 t 2 t t 2 2 PACF’s (Partial Derivation) Consider an AR(2) process, Multiply both sides by Yt-2

  49. = f f + e 2 Y Y Y Y Y Y + - - - - - t 2 t 1 t 2 t 1 t 2 t t 2 2 = f ACF ( 2 ) + ACF f ( 2 ) 1 2 PACF’s (Partial Derivation) Taking expectations, miracle 37b

  50. = f + f ACF ( 1 ) ACF ( 1 ) 1 2 = f ACF ( 2 ) ACF + f ( 2 ) 1 2 PACF’s (Partial Derivation) Two eqs, two unknowns Solve for f1, f2 from ACF(1), ACF(2) Estimate f1 from PACF(1)

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