1. Econ 240C Lecture 12

2. 2 Part I: Forecasting Time Series Housing Starts Housing Starts Labs 5 and 7 Labs 5 and 7

3. 3 Capacity Utilization, Mfg. First quarter of 1972-First Quarter 2003 First quarter of 1972-First Quarter 2003 Estimate for a sub-sample 1972.1-2000.4 Estimate for a sub-sample 1972.1-2000.4 Test the forecast for sub-sample 2001.1- 2003.1 Test the forecast for sub-sample 2001.1- 2003.1

4. 4 Identification Process Trace Trace Histogram Histogram Correlogram Correlogram Dickey-Fuller Test Dickey-Fuller Test

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9. 9 Estimation and Validation Process First Model: cumfn c ar(1) ar(2) First Model: cumfn c ar(1) ar(2) goodness of fit goodness of fit correlogram of residuals correlogram of residuals histogram of residuals histogram of residuals Second Model: cumfn c ar(1) ar(2) ma(8) Second Model: cumfn c ar(1) ar(2) ma(8) goodness of fit goodness of fit correlogram of residuals correlogram of residuals histogram of residuals histogram of residuals

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18. 18 Additional Validation Forecasting within the sample range for the subsample beyond the data range used for estimation Forecasting within the sample range for the subsample beyond the data range used for estimation

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24. 24 Estimation for the whole Period 1972.1-2003.1 1972.1-2003.1

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26. 26 Correlogram of Residuals from Estimated Model, 1972.1-2003.1

27. 27 Augmenting Data Range for Forecasting Workfile Window in EVIEWS Workfile Window in EVIEWS PROCS menu PROCS menu

28. 28 Equation Window; Forecast Instructions

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30. 30 Generating Forecast and Upper and Lower Bounds for the 95% Confidence Interval, 2003.2-2004.4

31. 31 Compare the Observed to the Forecast Series with Bounds

32. 32 Spreadsheet: Observed, Forecast and Bounds

33. 33 Observed Series, Forecast, and 95% Confidence Interval

34. 34 AdditionalProspective Model Validation How well do the forecasts for 2003-2004 compare to future observations? How well do the forecasts for 2003-2004 compare to future observations? Do future observations lie within the confidence bounds? Do future observations lie within the confidence bounds?

35. 35 Part II. Augmented Dickey-Fuller Tests A second order autoregressive process A second order autoregressive process lecture 7-Powerpoint lecture 7-Powerpoint

36. 36 Lecture 7-Part III. Autoregressive of the Second Order ARTWO(t) = b 1 *ARTWO(t-1) + b 2 *ARTWO(t-2) + WN(t) ARTWO(t) = b 1 *ARTWO(t-1) + b 2 *ARTWO(t-2) + WN(t) ARTWO(t) - b 1 *ARTWO(t-1) - b 2 *ARTWO(t- 2) = WN(t) ARTWO(t) - b 1 *ARTWO(t-1) - b 2 *ARTWO(t- 2) = WN(t) ARTWO(t) - b 1 *Z*ARTWO(t) - b 2 *Z*ARTWO(t) = WN(t) ARTWO(t) - b 1 *Z*ARTWO(t) - b 2 *Z*ARTWO(t) = WN(t) [1 - b 1 *Z - b 2 *Z 2 ] ARTWO(t) = WN(t) [1 - b 1 *Z - b 2 *Z 2 ] ARTWO(t) = WN(t)

37. 37 Triangle of Stable Parameter Space: Heuristic Explanation b 1 = 0 b2b2 1 Draw a line from the vertex, for (b 1 =0, b 2 =1), though the end points for b 1, i.e. through (b 1 =1, b 2 =-1) and (b 1 =-1, b 2 =0), (1, 0)(-1, 0)

38. 38 Triangle of Stable Parameter Space If we are along the right hand diagonal border of the parameter space then we are on the boundary of stability, I.e. there must be a unit root, and from: If we are along the right hand diagonal border of the parameter space then we are on the boundary of stability, I.e. there must be a unit root, and from: [1 - b 1 *Z - b 2 *Z 2 ] ARTWO(t) = WN(t) [1 - b 1 *Z - b 2 *Z 2 ] ARTWO(t) = WN(t) ignoring white noise shocks, ignoring white noise shocks, [1 - b 1 *Z - b 2 *Z 2 ] = [1 -Z][1 + c Z], where multiplying the expressions on the right hand side(RHS), noting that c is a parameter to be solved for and setting the RHS equal to the LHS: [1 - b 1 *Z - b 2 *Z 2 ] = [1 -Z][1 + c Z], where multiplying the expressions on the right hand side(RHS), noting that c is a parameter to be solved for and setting the RHS equal to the LHS:

39. 39 [1 - b 1 *Z - b 2 *Z 2 ] = [1 + (c - 1)Z -c Z 2 ], so - b 1 = c - 1, and - b 2 = -c, or [1 - b 1 *Z - b 2 *Z 2 ] = [1 + (c - 1)Z -c Z 2 ], so - b 1 = c - 1, and - b 2 = -c, or b 1 = 1 - c, (line2) b 1 = 1 - c, (line2) b 2 = c, (line 3) b 2 = c, (line 3) and adding lines 2 and 3: b 1 + b 2 = 1, so and adding lines 2 and 3: b 1 + b 2 = 1, so b 2 = 1 - b 1, the formula for the right hand boundary b 2 = 1 - b 1, the formula for the right hand boundary

40. 40 b 1 + b 2 = 1 is the condition for a unit root for a second order process b 1 + b 2 = 1 is the condition for a unit root for a second order process ARTWO(t) = b 1 *ARTWO(t-1) + b 2 *ARTWO(t- 2) + WN(t) ARTWO(t) = b 1 *ARTWO(t-1) + b 2 *ARTWO(t- 2) + WN(t) add b 2 *ARTWO(t-1) - b 2 *ARTWO(t-1) to RHS add b 2 *ARTWO(t-1) - b 2 *ARTWO(t-1) to RHS ARTWO(t) = (b 1 + b 2 ) *ARTWO(t-1) - b 2 *ARTWO(t-1) + b 2 *ARTWO(t-2) + WN(t) ARTWO(t) = (b 1 + b 2 ) *ARTWO(t-1) - b 2 *ARTWO(t-1) + b 2 *ARTWO(t-2) + WN(t)

41. 41 ARTWO(t) = (b 1 + b 2 ) *ARTWO(t-1) - b 2 *[ARTWO(t-1) - *ARTWO(t-2)] + WN(t) ARTWO(t) = (b 1 + b 2 ) *ARTWO(t-1) - b 2 *[ARTWO(t-1) - *ARTWO(t-2)] + WN(t) ARTWO(t) = (b 1 + b 2 ) *ARTWO(t-1) - b 2  ARTWO(t-1) + WN(t) ARTWO(t) = (b 1 + b 2 ) *ARTWO(t-1) - b 2  ARTWO(t-1) + WN(t) subtract ARTWO(t-1) from both sides subtract ARTWO(t-1) from both sides  ARTWO(t) = (b 1 + b 2 - 1) *ARTWO(t-1) - b 2  ARTWO(t-1) + WN(t)  ARTWO(t) = (b 1 + b 2 - 1) *ARTWO(t-1) - b 2  ARTWO(t-1) + WN(t)

42. 42 Example Capacity utilization manufacturing Capacity utilization manufacturing

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45. 45 ARTWO(t) = (b 1 + b 2 ) *ARTWO(t-1) - b 2 *[ARTWO(t-1) - *ARTWO(t-2)] + WN(t) ARTWO(t) = (b 1 + b 2 ) *ARTWO(t-1) - b 2 *[ARTWO(t-1) - *ARTWO(t-2)] + WN(t) ARTWO(t) = (b 1 + b 2 ) *ARTWO(t-1) - b 2  ARTWO(t-1) + WN(t) ARTWO(t) = (b 1 + b 2 ) *ARTWO(t-1) - b 2  ARTWO(t-1) + WN(t) subtract ARTWO(t-1) from both sides subtract ARTWO(t-1) from both sides  ARTWO(t) = (b 1 + b 2 - 1) *ARTWO(t-1) - b 2  ARTWO(t-1) + WN(t)  ARTWO(t) = (b 1 + b 2 - 1) *ARTWO(t-1) - b 2  ARTWO(t-1) + WN(t)

46. 46 Part III. Lab Seven New privately owned housing units and the 30 year conventional mortgage rate, April 1974-March 2003 New privately owned housing units and the 30 year conventional mortgage rate, April 1974-March 2003 starts(t) = c 0 mort(t) + c 1 mort(t-1) + c 2 mort(t-2) + … + resid(t) starts(t) = c 0 mort(t) + c 1 mort(t-1) + c 2 mort(t-2) + … + resid(t) starts(t) = c 0 mort(t) + c 1 Zmort(t) + c 2 Z 2 mort(t) + … + resid(t) starts(t) = c 0 mort(t) + c 1 Zmort(t) + c 2 Z 2 mort(t) + … + resid(t) starts(t) = [c 0 + c 1 Z + c 2 Z 2 + …] mort(t) + resid(t) starts(t) = [c 0 + c 1 Z + c 2 Z 2 + …] mort(t) + resid(t)

47. 47 Output starts(t) C(Z) Input mort(t) Dynamic relationship + + Resid(t)

48. 48 Identification Process for Mortrate Trace Trace Histogram Histogram Correlogram Correlogram Dickey-Fuller Test Dickey-Fuller Test

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53. 53 Looks like a unit root in the mortgage rate, so prewhiten by differencing Looks like a unit root in the mortgage rate, so prewhiten by differencing  starts(t) = [c 0 + c 1 Z + c 2 Z 2 + …]  mort(t) +  resid(t)  starts(t) = [c 0 + c 1 Z + c 2 Z 2 + …]  mort(t) +  resid(t)

54. 54 Identification Process for dmort Trace Trace Histogram Histogram Correlogram Correlogram Dickey-Fuller Test Dickey-Fuller Test

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59. 59 Estimation of Model for dmort

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64. 64 Dstart(t) = C(Z) dmort(t) + dresid(t) Dstart(t) = C(Z) dmort(t) + dresid(t) dmort(t) = reside(t), where reside(t) = 0.577*reside(t-1) - 0.399*reside(t-2) + N dm dmort(t) = reside(t), where reside(t) = 0.577*reside(t-1) - 0.399*reside(t-2) + N dm dmort(t) = 0.577*dmort(t-1) - 0.399*dmort(t-2) + N dm,, save N dm by GENR resdm=resid dmort(t) = 0.577*dmort(t-1) - 0.399*dmort(t-2) + N dm,, save N dm by GENR resdm=resid dmort(t) - 0.577*Zdmort(t) + 0.399*Z 2 dmort(t) = N dm dmort(t) - 0.577*Zdmort(t) + 0.399*Z 2 dmort(t) = N dm

65. 65 dmort(t) - 0.577*Zdmort(t) + 0.399*Z 2 dmort(t) = N dm dmort(t) - 0.577*Zdmort(t) + 0.399*Z 2 dmort(t) = N dm [1 - 0.577*Z + 0.399*Z 2 ]dmort(t) = N dm [1 - 0.577*Z + 0.399*Z 2 ]dmort(t) = N dm Dstart(t) = C(Z) dmort(t) + dresid(t) Dstart(t) = C(Z) dmort(t) + dresid(t) [1 - 0.577*Z + 0.399*Z 2 ]*Dstart(t) = C(Z)* [1 - 0.577*Z + 0.399*Z 2 ]dmort(t) + [1 - 0.577*Z + 0.399*Z 2 ] dresid(t) [1 - 0.577*Z + 0.399*Z 2 ]*Dstart(t) = C(Z)* [1 - 0.577*Z + 0.399*Z 2 ]dmort(t) + [1 - 0.577*Z + 0.399*Z 2 ] dresid(t) [1 - 0.577*Z + 0.399*Z 2 ]*Dstart(t) = C(Z)* N dm + [1 - 0.577*Z + 0.399*Z 2 ]* dresid(t) [1 - 0.577*Z + 0.399*Z 2 ]*Dstart(t) = C(Z)* N dm + [1 - 0.577*Z + 0.399*Z 2 ]* dresid(t)

66. 66 Transform dstart(t) Transform dstart(t) w(t) = [1 - 0.577*Z + 0.399*Z 2 ]*Dstart(t) w(t) = [1 - 0.577*Z + 0.399*Z 2 ]*Dstart(t) Cross-correlate w(t) and N dm to reveal C(Z) Cross-correlate w(t) and N dm to reveal C(Z) w(t) = c 0 N dm + c 1 N dm (t-1) + c 2 N dm (t-2) +... dresidw(t) w(t) = c 0 N dm + c 1 N dm (t-1) + c 2 N dm (t-2) +... dresidw(t) Then specify and estimate the model: Then specify and estimate the model: w(t) = c 0 N dm + c 1 N dm (t-1) + c 2 N dm (t-2) +... dresidw(t) w(t) = c 0 N dm + c 1 N dm (t-1) + c 2 N dm (t-2) +... dresidw(t) where dresidw(t)=[1 - 0.577*Z + 0.399*Z 2 ]* dresid(t) where dresidw(t)=[1 - 0.577*Z + 0.399*Z 2 ]* dresid(t)

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