1. 1 Power Nine Econ 240C

2. 2 Outline Lab Three Exercises Lab Three Exercises –Fit a linear trend to retail sales –Add a quadratic term –Use both models to forecast 1 period ahead Lab Five Preview Lab Five Preview –Airline passengers

3. 3 Lab Three Exercises Process Identification Identification –Spreadsheet –Trace –Histogram –Correlogram –Unit root test Estimation Estimation Validation Validation

4. 4 1955.01-1993.12

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10. 10 Lab Three Exercises Process Identification Identification –Spreadsheet: check variable values –Trace: trended series –Histogram: similar to “random walk” –Correlogram: similar to a “random walk” –Unit root test: evolutionary Estimation Estimation Validation Validation

11. 11 Note: low D-W

12. 12 Process Validating the model Validating the model –Actual, fitted, residual –Correlogram of the residuals –Histogram of the residuals

13. 13 Note: autocorrelated residuals

14. 14 Note autocorrelated residuals

15. 15 Surprise: residuals are normal, but not orthogonal

16. 16 Add the quadratic term

17. 17 SER is lower: 3737 Vs. 4860, D-W still low

18. 18 Autocorrelated residual

19. 19 Autocorrelated residual

20. 20 Residual is no longer normal

21. 21 One Period Ahead Forecasts: Linear Retail(1994.01) = retail-fitted(1993.12) + slope Retail(1994.01) = retail-fitted(1993.12) + slope Forecast = 140339 + 215.0 = 140553, with ser = 4860 Forecast = 140339 + 215.0 = 140553, with ser = 4860

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25. 25 One Period Ahead Forecast: Quadratic Retail(t) = a + b*trnd + c*trnd 2 Retail(t) = a + b*trnd + c*trnd 2 d retail(t)/d trend = b + 2*c*trnd = 126.2 + 2*0.190*468 = 304.0 d retail(t)/d trend = b + 2*c*trnd = 126.2 + 2*0.190*468 = 304.0 Retail(1994.01) = retail-fitted(1993.12) + 304 = 147,239 + 304 = 147543 with ser = 3737 Retail(1994.01) = retail-fitted(1993.12) + 304 = 147,239 + 304 = 147543 with ser = 3737

26. 26 SER is lower: 3737 Vs. 4860, D-W still low b = 128.2, c = 0.19

27. 27 Trnd(1994.01) = 468

28. 28 Retail-fitted(1993.12) = 147,239

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31. 31 Now we know another way to forecast First difference retail First difference retail

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33. 33 Looks stationary

34. 34 kurtotic

35. 35 Possibly an ARTWO

36. 36 No unit root

37. 37 Note: the constant 221 is close to the Slope, 215, for the Linear trend model

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39. 39 Q-stats ok until Lag 16

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41. 41 One period ahead forecast Dretailf(1994.01) = -240.66 with ser of 1230 Dretailf(1994.01) = -240.66 with ser of 1230 Retailf(1994.01) = retail(1993.12) + dretailf(1994.01) Retailf(1994.01) = retail(1993.12) + dretailf(1994.01) Retailf(1994.01) = 151,631 – 240.7=151,390.3 Retailf(1994.01) = 151,631 – 240.7=151,390.3 Linear trend forecast: 140,553 with ser = 4860 Linear trend forecast: 140,553 with ser = 4860 Quadratic forecast: 147,543 with ser =3737 Quadratic forecast: 147,543 with ser =3737 Actual observed: retail(1994.01) = 149,918 Actual observed: retail(1994.01) = 149,918 So the ARTWO model is closest So the ARTWO model is closest

42. 42 dretail ARTWO Model

43. 43 dretail ARTWO model forecast

44. 44 One period ahead forecast cont. Add an ar(1) term to the quadratic model: forecast(1994.01) = 151,763 with ser = 1252 Add an ar(1) term to the quadratic model: forecast(1994.01) = 151,763 with ser = 1252 So dretail is still closest So dretail is still closest Add ar(1) ar(2) ar(3) terms to the quadratic: forecast(1994.01) = 151,457 with ser = 1223; Add ar(1) ar(2) ar(3) terms to the quadratic: forecast(1994.01) = 151,457 with ser = 1223; Still closest with ARTWO model for dretail, but not by much Still closest with ARTWO model for dretail, but not by much

45. 45 ARTHREE Model for Quadratic Trend model of Retail

46. 46 ARTHREE Model for Retail

47. 47 Residuals from Quad. trend model plus ARTHREE

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49. 49 Preview of Lab Five A Box-Jenkins famous time series: airline passengers A Box-Jenkins famous time series: airline passengers –Trend in mean –Trend in variance –seasonality Prewhitening Prewhitening –Log transform –First difference –Seasonal difference

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53. 53 Note trend from Spike in pacf at Lag one; seasonal Pattern in ACF

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56. 56 Log transform is fix for trend in Var

57. 57 First difference for trend in mean Looks more stationary but is it?

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59. 59 Note seasonal peaks at, 12 24, etc.

60. 60 No unit root, but Correlogram shows Seasonal Dependence on time

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64. 64 Note: sddlnbjpass is normal

65. 65 Closer to white Noise; proposed Model ma(1), ma(12)

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68. 68 Satisfactory Model from Q-stats

69. 69 And the residuals from the model are normal

70. 70 How to use the model to forecast Forecast sddlnbjpass Forecast sddlnbjpass recolor recolor

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