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Power NineteenPower Nineteen

Econ 240CEcon 240C

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OutlineOutline

Forecast SourcesForecast Sources Ideas that are transcendingIdeas that are transcending Symbolic SummarySymbolic Summary

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OutlineOutline ForecastingForecasting

Federal: Federal Reserve @ PhilidelphiaFederal: Federal Reserve @ Philidelphia State: CA Department of FinanceState: CA Department of Finance LocalLocal

UCSB: tri-countiesUCSB: tri-counties Chapman College: Orange CountyChapman College: Orange County UCLA: National, CAUCLA: National, CA

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http://www.ucsb-efp.com

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ReviewReview

2. Ideas That Are Transcending2. Ideas That Are Transcending

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Use the Past to Predict the Use the Past to Predict the FutureFuture

A. ApplicationsA. Applications Trend AnalysisTrend Analysis

linear trendlinear trend quadratic trendquadratic trend exponential trendexponential trend

ARIMA ModelsARIMA Models autoregressive modelsautoregressive models moving average modelsmoving average models autoregressive moving average autoregressive moving average

modelsmodels

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Use Assumptions To Cope With Use Assumptions To Cope With ConstraintsConstraints

A. ApplicationsA. Applications 1. Limited number of observations: 1. Limited number of observations:

simple exponential smoothingsimple exponential smoothing assume the model: (p, d, q) = (0, 1, 1)assume the model: (p, d, q) = (0, 1, 1)

2. No or insufficient identifying 2. No or insufficient identifying exogenous variables: interpreting exogenous variables: interpreting VAR impulse response functionsVAR impulse response functions assume the error structure is dominated assume the error structure is dominated

by one pure error or the other, e.g by one pure error or the other, e.g assume assume = 0, then e= 0, then e1 1 = e= edcapudcapu

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Standard VAR (lecture 17)Standard VAR (lecture 17)

dcapu(t) = (dcapu(t) = (/(1- /(1- ) +[ () +[ (+ +

)/(1- )/(1- )] dcapu(t-1) + [ ()] dcapu(t-1) + [ (+ +

)/(1- )/(1- )] dffr(t-1) + [()] dffr(t-1) + [(+ + (1- (1-

)] x(t) + (e)] x(t) + (edcapudcapu(t) + (t) + e edffrdffr(t))/(1- (t))/(1- ))

But if we assume But if we assume thenthendcapu(t) = dcapu(t) = + + dcapu(t-1) + dcapu(t-1) +

dffr(t-1) + dffr(t-1) + x(t) + e x(t) + edcapudcapu(t) + (t) +

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Use Assumptions To Cope With Use Assumptions To Cope With ConstraintsConstraints

A. ApplicationsA. Applications 3. No or insufficient identifying 3. No or insufficient identifying

exogenous variables: simultaneous exogenous variables: simultaneous equationsequations assume the error structure is assume the error structure is

dominated by one error or the other, dominated by one error or the other, tracing out the other curvetracing out the other curve

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SimultaneitySimultaneity

There are two relations that show the There are two relations that show the dependence of price on quantity and dependence of price on quantity and vice versavice versa demand: p = a - b*q +c*y + edemand: p = a - b*q +c*y + epp

supply: q= d + e*p + f*w + esupply: q= d + e*p + f*w + eqq

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demand

price

quantity

Shift in demand with increased income, may trace outi.e. identify or reveal the demand curve

supply

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ReviewReview

2. Ideas That Are Transcending2. Ideas That Are Transcending

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Reduce the unexplained sum of Reduce the unexplained sum of squares to increase the squares to increase the significance of resultssignificance of results A. ApplicationsA. Applications

1. 2-way ANOVA: using randomized 1. 2-way ANOVA: using randomized block designblock design example: minutes of rock music listened example: minutes of rock music listened

to on the radio by teenagers Lecture 1 to on the radio by teenagers Lecture 1 Notes, 240 CNotes, 240 C

we are interested in the variation from day we are interested in the variation from day to dayto day

to get better results, we control for variation to get better results, we control for variation across teenageracross teenager

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Table I. Minutes of Rock Music Listened to On the Radio

Teenager Sunday Monday Tuesday Wednesday Thursday Friday Saturday1 65 40 32 48 60 75 1102 90 85 75 90 78 120 1003 30 30 20 25 30 60 704 72 52 66 100 77 66 945 70 88 47 73 78 67 786 90 51 103 41 57 69 877 43 72 66 39 57 90 738 88 89 82 95 68 105 1259 96 60 80 106 57 81 80

10 60 92 72 45 72 77 90

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Figure 1: Minutes of Rock Music Listened to Per Day

0

10

20

30

40

50

60

70

80

90

Sunday Monday Tuesday Wednesday Thursday Friday Saturday

Day of the Week

Min

ute

s

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Table IV. Two-Way ANOVA, Time Series and Cross-Section

Source of Variation Sum of Squares Degrees of Freedom Mean Square

Explained (Time) 28,673.73 6 4778.96

Explained (Cross-) 209,834.6 199 1054.45

Unexplained 479,125.1 1194 401.28

Total 717,633.5 1399

F6, 1194 = 4778.96/401.28 = 11.91, probability 5.1 x 10-13

Table II. One-Way ANOVA for Rock Minutes by Day of the Week

Source of Variation Sum of Squares Degrees of Freedom Mean Square

Explained (Time) 28,673.73 6 4778.96

Unexplained 688,959.8 1393 494.59

Total 717,633.5 1399

F6, 1393 = 4778.96/494.59 = 9.66, probability 1.93 x 10-10

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Reduce the unexplained sum of Reduce the unexplained sum of squares to increase the squares to increase the significance of resultssignificance of results A. ApplicationsA. Applications

2. Distributed lag models: model 2. Distributed lag models: model dependence of y(t) on a distributed dependence of y(t) on a distributed lag of x(t) andlag of x(t) and model the residual using ARMAmodel the residual using ARMA

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Lab 7 240 C

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Reduce the unexplained sum of Reduce the unexplained sum of squares to increase the squares to increase the significance of resultssignificance of results A. ApplicationsA. Applications

3. Intervention Models: model known 3. Intervention Models: model known changes (policy, legal etc.) by using changes (policy, legal etc.) by using dummy variables, e.g. a step dummy variables, e.g. a step function or pulse functionfunction or pulse function

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Lab 8 240 CLab 8 240 C

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Model with no Intervention Model with no Intervention VariableVariable

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Add seasonal difference of Add seasonal difference of differenced step functiondifferenced step function

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ReviewReview

Symbolic SummarySymbolic Summary

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Autoregressive ModelsAutoregressive Models AR(t) = bAR(t) = b1 1 AR(t-1) + bAR(t-1) + b2 2 AR(t-2) + …. + bAR(t-2) + …. + bp p

AR(t-p) + WN(t)AR(t-p) + WN(t) AR(t) - bAR(t) - b1 1 AR(t-1) - bAR(t-1) - b2 2 AR(t-2) - …. + bAR(t-2) - …. + bp p

AR(t-p) = WN(t)AR(t-p) = WN(t) [1 - b[1 - b1 1 Z + bZ + b2 2 ZZ22 + …. b + …. bp p ZZpp ] AR(t) = ] AR(t) =

WN(t)WN(t) B(Z) AR(t) = WN(t)B(Z) AR(t) = WN(t) AR(t) = [1/B(Z)]*WN(t)AR(t) = [1/B(Z)]*WN(t)

WN(t)1/B(Z)AR(t)

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Moving Average ModelsMoving Average Models

MA(t) = WN(t) + aMA(t) = WN(t) + a1 1 WN(t-1) + aWN(t-1) + a2 2 WN(t-2) + …. WN(t-2) + …. a aq q WN(t-q)WN(t-q)

MA(t) = WN(t) + aMA(t) = WN(t) + a1 1 Z WN(t) + aZ WN(t) + a2 2 ZZ22 WN(t) + WN(t) + …. a…. aq q ZZqq WN(t) WN(t)

MA(t) = [1 + aMA(t) = [1 + a1 1 Z + aZ + a2 2 ZZ22 + …. a + …. aq q ZZqq ] ] WN(t)WN(t)

MA(t) = A(Z)*WN(t) MA(t) = A(Z)*WN(t)

WN(t)A(Z)MA(t)

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ARMA ModelsARMA Models

ARMA(p,q) = [AARMA(p,q) = [Aq q (Z)/B(Z)/Bp p (Z)]*WN(t)(Z)]*WN(t)

WN(t)A(Z)/B(Z)ARMA(t)

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Distributed Lag ModelsDistributed Lag Models

y(t) = hy(t) = h0 0 x(t) + hx(t) + h1 1 x(t-1) + …. hx(t-1) + …. hn n x(t-n) + resid(t)x(t-n) + resid(t) y(t) = hy(t) = h0 0 x(t) + hx(t) + h1 1 Zx(t) + …. hZx(t) + …. hn n ZZnn x(t) + resid(t) x(t) + resid(t) y(t) = [hy(t) = [h0 0 + h + h1 1 Z + …. hZ + …. hn n ZZnn ] x(t) + resid(t) ] x(t) + resid(t) y(t) = h(Z)*x(t) + resid(t)y(t) = h(Z)*x(t) + resid(t) note x(t) = Anote x(t) = Ax x (Z)/B(Z)/Bx x (Z) WN(Z) WNx x (t), or(t), or [B[Bx x (Z) /A(Z) /Ax x (Z)]* x(t) =WN(Z)]* x(t) =WNx x (t), so(t), so [B[Bx x (Z) /A(Z) /Ax x (Z)]* y(t) = h(Z)* [B(Z)]* y(t) = h(Z)* [Bx x (Z) /A(Z) /Ax x (Z)]* x(t) + [B(Z)]* x(t) + [Bx x

(Z) /A(Z) /Ax x (Z)]* resid(t) or (Z)]* resid(t) or W(t) = h(Z)*WNW(t) = h(Z)*WNx x (t) + Resid*(t)(t) + Resid*(t)

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Distributed Lag ModelsDistributed Lag Models

Where w(t) = [BWhere w(t) = [Bx x (Z) /A(Z) /Ax x (Z)]* y(t)(Z)]* y(t) and resid*(t) = [Band resid*(t) = [Bx x (Z) /A(Z) /Ax x (Z)]* resid(t) (Z)]* resid(t) cross-correlation of the orthogonal WNcross-correlation of the orthogonal WNx x (t) (t)

with w(t) will reveal the number of lags n with w(t) will reveal the number of lags n in h(Z), and the signs of the parameters in h(Z), and the signs of the parameters hh0 0 , h, h1 , 1 , etc. for modeling the regression of etc. for modeling the regression of w(t) on a distributed lag of the residual, w(t) on a distributed lag of the residual, WNWNx x (t), from the ARMA model for x(t)(t), from the ARMA model for x(t)

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Distributed Lag ModelDistributed Lag Model

X(t)H(z)Y(t)+

Residual(t)

+

Remember to Model the Residual!

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VAR ModelVAR ModelY1(t) = h1 (t ) Y1 (t) + h2 (t) Y2 (t) +e1 (t)

Y2 (t ) = h3 (t ) Y1 (t) + h4 (t) Y2 (t) +e2 (t)

Y1 (t) h1 (z) Y1 (t)

+

e1 (t)

+ +

h2 (z) Y2 (t)With a similar schematic for Y2 (t)

Note: e1 (t) and e2 (t) are each compound errors, i.e. composed of the pure shock, ey1, to Y1 and the pure shock, ey2, to Y2

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Crime in CaliforniaCrime in California

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Per Capita Crime Rates and Imprisonment Rates, California and US

0.0001

0.001

0.01

0.1

1940 1950 1960 1970 1980 1990 2000 2010

Year

Rat

e

CA Crime Index Per Capita

CA Prisoners Per Capita

FBI Crime Index Per Capita, US

US Prisoners Per Capita

1952-2004

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Use the California ExperienceUse the California Experience

Crime rates Have Fallen. Why Crime rates Have Fallen. Why Haven’t Imprisonment rates?Haven’t Imprisonment rates?

Apply the conceptual tools Apply the conceptual tools Criminal justice system schematicCriminal justice system schematic crime control technologycrime control technology

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Crime Generation

Crime Control

OffenseRate PerCapita

ExpectedCost ofPunishment

Schematic of the Criminal Justice System: Coordinating CJS

Causes ?!!

(detention,deterrence)

Expenditures

Weak Link

“The Driving Force”

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Expenditures Per Capita in 92 $, California .Criminal Justice System, 1967-68 to 1997-98 . .

0

100

200

300

400

500

600

Fiscal Year

$ P

er

Capita

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California Unemployment Rate and Inflation Rate, 1952-2004

-5.00

0.00

5.00

10.00

15.00

20.00

25.00

1940 1950 1960 1970 1980 1990 2000 2010

Year

Pe

r C

en

t

unemployment rate

inflation rate

misery rate

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CA Crime Index Per 1000 and Misery Index (Percent), 1952-2004

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

40.00

45.00

1940 1950 1960 1970 1980 1990 2000 2010

Year

Ra

te

Misery index

CA Crime Index

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California: Crime Index Versus Misery Index .

1970

1992

1975

1998

1980

1952

0

5

10

15

20

25

30

35

40

0.00 5.00 10.00 15.00 20.00 25.00

Misery Index

Cri

me I

nd

ex

Jobs and Crime

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Model SchematicModel SchematicModel SchematicModel Schematic

Crime Generation: California IndexOffenses Per Capita

Causality:California Misery Index

Crime Control: California Prisoners Per Capita

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CA Crime Index Per Capita (t) = 0.039 + 0.00034*Misery Index (t) –3.701*Prisoners Per Capita (t) + e(t)

where e(t) = 0.954*e(t-1)

-0.004

-0.002

0.000

0.002

0.004

0.006

0.00

0.01

0.02

0.03

0.04

55 60 65 70 75 80 85 90 95 00

Residual Actual Fitted

CA Crime Index, CA Prisoners and CA Misery Index

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Ln CA Crime Index Per Capita (t) = -5.25 + 0.17*ln Misery Index (t)-0.22 ln Prisoners Per capita (t) +e(t)

where e(t) = 0.93 e(t-1)

-0.2

-0.1

0.0

0.1

0.2

-5.0

-4.5

-4.0

-3.5

-3.0

55 60 65 70 75 80 85 90 95 00

Residual Actual Fitted

Logarithms of the CA Crime Index Per Capita, CA PrisonersPer Capita, Misery Index