Econ 240C Lecture 12. 2 Part I: Forecasting Time Series Housing Starts Housing Starts Labs 5 and 7...

Econ 240CEcon 240C

Lecture 12Lecture 12

2Part I: Forecasting Time Part I: Forecasting Time SeriesSeries

Housing StartsHousing StartsLabs 5 and 7Labs 5 and 7

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Capacity Utilization, Mfg.Capacity Utilization, Mfg.

First quarter of 1972-First Quarter First quarter of 1972-First Quarter 20032003

Estimate for a sub-sample 1972.1-Estimate for a sub-sample 1972.1-2000.42000.4

Test the forecast for sub-sample Test the forecast for sub-sample 2001.1-2003.12001.1-2003.1

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Identification ProcessIdentification Process

TraceTraceHistogramHistogramCorrelogramCorrelogramDickey-Fuller TestDickey-Fuller Test

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9Estimation and Validation Estimation and Validation ProcessProcess

First Model: cumfn c ar(1) ar(2)First Model: cumfn c ar(1) ar(2)goodness of fitgoodness of fitcorrelogram of residualscorrelogram of residualshistogram of residualshistogram of residuals

Second Model: cumfn c ar(1) ar(2) Second Model: cumfn c ar(1) ar(2) ma(8)ma(8)goodness of fitgoodness of fitcorrelogram of residualscorrelogram of residualshistogram of residuals histogram of residuals

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Additional ValidationAdditional Validation

Forecasting within the sample range Forecasting within the sample range for the subsample beyond the data for the subsample beyond the data range used for estimationrange used for estimation

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24Estimation for the whole Estimation for the whole PeriodPeriod

1972.1-2003.11972.1-2003.1

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26Correlogram of Residuals from Estimated Model, 1972.1-2003.1

27Augmenting Data Range for Augmenting Data Range for ForecastingForecasting

Workfile Window in EVIEWSWorkfile Window in EVIEWSPROCS menuPROCS menu

28Equation Window; Forecast Instructions

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30Generating Forecast and Upper and Lower Boundsfor the 95% Confidence Interval, 2003.2-2004.4

31Compare the Observed to the Forecast Series with Bounds

32Spreadsheet: Observed, Forecast and Bounds

33Observed Series, Forecast, and 95% Confidence Interval

34AdditionalProspective Model AdditionalProspective Model ValidationValidation

How well do the forecasts for 2003-How well do the forecasts for 2003-2004 compare to future 2004 compare to future observations?observations?

Do future observations lie within the Do future observations lie within the confidence bounds?confidence bounds?

35Part II. Augmented Dickey-Part II. Augmented Dickey-Fuller TestsFuller Tests

A second order autoregressive A second order autoregressive processprocesslecture 7-Powerpointlecture 7-Powerpoint

36Lecture 7-Part III. Lecture 7-Part III. Autoregressive of the Second Autoregressive of the Second

Order Order ARTWO(t) = bARTWO(t) = b1 1 *ARTWO(t-1) + b*ARTWO(t-1) + b2 2

*ARTWO(t-2) + WN(t)*ARTWO(t-2) + WN(t)ARTWO(t) - bARTWO(t) - b1 1 *ARTWO(t-1) - b*ARTWO(t-1) - b2 2

*ARTWO(t-2) = WN(t)*ARTWO(t-2) = WN(t)ARTWO(t) - bARTWO(t) - b1 1 *Z*ARTWO(t) - b*Z*ARTWO(t) - b2 2

*Z*ARTWO(t) = WN(t)*Z*ARTWO(t) = WN(t)[1 - b[1 - b1 1 *Z - b*Z - b2 2 *Z*Z22] ARTWO(t) = WN(t)] ARTWO(t) = WN(t)

37Triangle of Stable Parameter Triangle of Stable Parameter Space: Heuristic ExplanationSpace: Heuristic Explanation

b1 = 0

b2

1

-1

Draw a line from the vertex, for (b1=0, b2=1), though theend points for b1, i.e. through (b1=1, b2=-1) and (b1=-1, b2=0),

(1, 0)(-1, 0)

38Triangle of Stable Parameter Triangle of Stable Parameter SpaceSpace

If we are along the right hand diagonal If we are along the right hand diagonal border of the parameter space then we border of the parameter space then we are on the boundary of stability, I.e. are on the boundary of stability, I.e. there must be a unit root, and from:there must be a unit root, and from:

[1 - b[1 - b1 1 *Z - b*Z - b2 2 *Z*Z22] ARTWO(t) = WN(t)] ARTWO(t) = WN(t) ignoring white noise shocks,ignoring white noise shocks,[1 - b[1 - b1 1 *Z - b*Z - b2 2 *Z*Z22] = [1 -Z][1 + c Z], ] = [1 -Z][1 + c Z],

where multiplying the expressions on where multiplying the expressions on the right hand side(RHS), noting that c is the right hand side(RHS), noting that c is a parameter to be solved for and setting a parameter to be solved for and setting the RHS equal to the LHS: the RHS equal to the LHS:

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[1 - b[1 - b1 1 *Z - b*Z - b2 2 *Z*Z22] = [1 + (c - 1)Z -c Z] = [1 + (c - 1)Z -c Z22], ], so - b so - b1 1 = c - 1, and - b= c - 1, and - b2 2 = -c, or= -c, or

bb1 1 = 1 - c , (line2)= 1 - c , (line2)

bb2 2 = c , (line 3)= c , (line 3)

and adding lines 2 and 3: band adding lines 2 and 3: b1 1 + b+ b2 2 = 1, so= 1, so

bb2 2 = 1 - b= 1 - b1 1 , the formula for the right , the formula for the right hand boundary hand boundary

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bb1 1 + b+ b2 2 = 1 is the condition for a unit root = 1 is the condition for a unit root for a second order processfor a second order process

ARTWO(t) = bARTWO(t) = b1 1 *ARTWO(t-1) + b*ARTWO(t-1) + b2 2

*ARTWO(t-2) + WN(t)*ARTWO(t-2) + WN(t)add badd b2 2 *ARTWO(t-1) - b*ARTWO(t-1) - b2 2 *ARTWO(t-1) to *ARTWO(t-1) to

RHSRHSARTWO(t) = (bARTWO(t) = (b1 1 + b+ b22)) *ARTWO(t-1) - b*ARTWO(t-1) - b2 2

*ARTWO(t-1) + b*ARTWO(t-1) + b2 2 *ARTWO(t-2) + WN(t)*ARTWO(t-2) + WN(t)

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ARTWO(t) = (bARTWO(t) = (b1 1 + b+ b22)) *ARTWO(t-1) - *ARTWO(t-1) - b b2 2 *[ARTWO(t-1) -*[ARTWO(t-1) - *ARTWO(t-2)] + *ARTWO(t-2)] + WN(t)WN(t)

ARTWO(t) = (bARTWO(t) = (b1 1 + b+ b22)) *ARTWO(t-1) - *ARTWO(t-1) - b b2 2 ARTWO(t-1) + WN(t)ARTWO(t-1) + WN(t)

subtract ARTWO(t-1) from both sidessubtract ARTWO(t-1) from both sidesARTWO(t) = (bARTWO(t) = (b1 1 + b+ b22 - 1) - 1) *ARTWO(t-*ARTWO(t-

1) - b1) - b2 2 ARTWO(t-1) + WN(t)ARTWO(t-1) + WN(t)

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ExampleExample

Capacity utilization manufacturingCapacity utilization manufacturing

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ARTWO(t) = (bARTWO(t) = (b1 1 + b+ b22)) *ARTWO(t-1) - *ARTWO(t-1) - b b2 2 *[ARTWO(t-1) -*[ARTWO(t-1) - *ARTWO(t-2)] + *ARTWO(t-2)] + WN(t)WN(t)

ARTWO(t) = (bARTWO(t) = (b1 1 + b+ b22)) *ARTWO(t-1) - *ARTWO(t-1) - b b2 2 ARTWO(t-1) + WN(t)ARTWO(t-1) + WN(t)

subtract ARTWO(t-1) from both sidessubtract ARTWO(t-1) from both sidesARTWO(t) = (bARTWO(t) = (b1 1 + b+ b22 - 1) - 1) *ARTWO(t-*ARTWO(t-

1) - b1) - b2 2 ARTWO(t-1) + WN(t)ARTWO(t-1) + WN(t)

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Part III. Lab SevenPart III. Lab SevenNew privately owned housing units and New privately owned housing units and

the 30 year conventional mortgage rate, the 30 year conventional mortgage rate, April 1974-March 2003April 1974-March 2003

starts(t) = cstarts(t) = c00mort(t) + cmort(t) + c11mort(t-1) + mort(t-1) + cc22mort(t-2) + … + resid(t)mort(t-2) + … + resid(t)

starts(t) = cstarts(t) = c00mort(t) + cmort(t) + c11 Zmort(t) + c Zmort(t) + c2 2 ZZ22 mort(t) + … + resid(t)mort(t) + … + resid(t)

starts(t) = [cstarts(t) = [c00 + c + c11 Z + c Z + c22 Z Z22 + …] mort(t) + …] mort(t) + resid(t)+ resid(t)

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Outputstarts(t)

C(Z) Inputmort(t)

Dynamic relationship+

+

Resid(t)

48Identification Process for Identification Process for MortrateMortrate

TraceTraceHistogramHistogramCorrelogramCorrelogramDickey-Fuller TestDickey-Fuller Test

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Looks like a unit root in the mortgage Looks like a unit root in the mortgage rate, so prewhiten by differencingrate, so prewhiten by differencing

starts(t) = [cstarts(t) = [c00 + c + c11 Z + c Z + c22 Z Z22 + …] + …] mort(t) + mort(t) + resid(t)resid(t)

54Identification Process for Identification Process for dmortdmort

TraceTraceHistogramHistogramCorrelogramCorrelogramDickey-Fuller TestDickey-Fuller Test

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Estimation of Model for dmortEstimation of Model for dmort

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Dstart(t) = C(Z) dmort(t) + dresid(t)Dstart(t) = C(Z) dmort(t) + dresid(t)dmort(t) = reside(t), where reside(t) dmort(t) = reside(t), where reside(t)

= 0.577*reside(t-1) - 0.399*reside(t-= 0.577*reside(t-1) - 0.399*reside(t-2) + N2) + Ndmdm

dmort(t) = 0.577*dmort(t-1) - dmort(t) = 0.577*dmort(t-1) - 0.399*dmort(t-2) + N0.399*dmort(t-2) + Ndm, dm, , save N, save Ndmdm by by GENR resdm=residGENR resdm=resid

dmort(t) - 0.577*Zdmort(t) + dmort(t) - 0.577*Zdmort(t) + 0.399*Z0.399*Z22 dmort(t) = N dmort(t) = Ndmdm

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dmort(t) - 0.577*Zdmort(t) + 0.399*Zdmort(t) - 0.577*Zdmort(t) + 0.399*Z22 dmort(t) = Ndmort(t) = Ndmdm

[1 - 0.577*Z + 0.399*Z[1 - 0.577*Z + 0.399*Z22 ]dmort(t) = N ]dmort(t) = Ndmdm

Dstart(t) = C(Z) dmort(t) + dresid(t)Dstart(t) = C(Z) dmort(t) + dresid(t)[1 - 0.577*Z + 0.399*Z[1 - 0.577*Z + 0.399*Z22 ]*Dstart(t) = ]*Dstart(t) =

C(Z)* [1 - 0.577*Z + 0.399*ZC(Z)* [1 - 0.577*Z + 0.399*Z22 ]dmort(t) ]dmort(t) + [1 - 0.577*Z + 0.399*Z+ [1 - 0.577*Z + 0.399*Z22 ] dresid(t) ] dresid(t)

[1 - 0.577*Z + 0.399*Z[1 - 0.577*Z + 0.399*Z22 ]*Dstart(t) = ]*Dstart(t) = C(Z)* NC(Z)* Ndmdm + [1 - 0.577*Z + 0.399*Z + [1 - 0.577*Z + 0.399*Z22 ]* ]* dresid(t)dresid(t)

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Transform dstart(t)Transform dstart(t)w(t) = [1 - 0.577*Z + 0.399*Zw(t) = [1 - 0.577*Z + 0.399*Z22 ]*Dstart(t) ]*Dstart(t)

Cross-correlate w(t) and NCross-correlate w(t) and Ndm dm to reveal to reveal C(Z)C(Z)w(t) = cw(t) = c00 N Ndmdm + c + c11 N Ndmdm(t-1) + c(t-1) + c22 N Ndmdm(t-2) + ... (t-2) + ...

dresidw(t)dresidw(t)Then specify and estimate the model:Then specify and estimate the model:

w(t) = cw(t) = c00 N Ndmdm + c + c11 N Ndmdm(t-1) + c(t-1) + c22 N Ndmdm(t-2) + ... (t-2) + ... dresidw(t)dresidw(t)

where dresidw(t)=[1 - 0.577*Z + where dresidw(t)=[1 - 0.577*Z + 0.399*Z0.399*Z22 ]* dresid(t) ]* dresid(t)

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