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Econ 240CEcon 240C
Lecture 12Lecture 12
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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
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27Augmenting Data Range for Augmenting Data Range for ForecastingForecasting
Workfile Window in EVIEWSWorkfile Window in EVIEWSPROCS menuPROCS menu
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28Equation Window; Forecast Instructions
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30Generating Forecast and Upper and Lower Boundsfor the 95% Confidence Interval, 2003.2-2004.4
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31Compare the Observed to the Forecast Series with Bounds
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32Spreadsheet: Observed, Forecast and Bounds
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33Observed Series, Forecast, and 95% Confidence Interval
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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?
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35Part II. Augmented Dickey-Part II. Augmented Dickey-Fuller TestsFuller Tests
A second order autoregressive A second order autoregressive processprocesslecture 7-Powerpointlecture 7-Powerpoint
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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)
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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)
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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)
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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)
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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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