Economics 240A

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Economics 240AEconomics 240A

Power EightPower Eight

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OutlineOutline

Maximum Likelihood EstimationMaximum Likelihood Estimation The UC Budget AgainThe UC Budget Again Regression ModelsRegression Models The Income Generating Process for The Income Generating Process for

an Asset an Asset

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How to Find a-hat and b-How to Find a-hat and b-hat?hat?

MethodologyMethodology– grid searchgrid search– differential calculusdifferential calculus– likelihood functionlikelihood function

motivation: the likelihood function connects motivation: the likelihood function connects the topics of the topics of probability probability (especially (especially independence), the practical application of independence), the practical application of random samplingrandom sampling, the , the normal distributionnormal distribution, , and the derivation of estimatorsand the derivation of estimators

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Likelihood functionLikelihood function

The joint density of the estimated The joint density of the estimated residuals can be written as:residuals can be written as:

If the sample of observations on the If the sample of observations on the dependent variable, y, and the dependent variable, y, and the independent variable, x, is random, independent variable, x, is random, then the observations are then the observations are independent of one another. If the independent of one another. If the errors are also identically errors are also identically distributed, f, i.e. i.i.d, thendistributed, f, i.e. i.i.d, then

)ˆ.....ˆˆˆ( 1210 neeeeg

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Likelihood functionLikelihood function Continued: If i.i.d., thenContinued: If i.i.d., then

If the residuals are normally If the residuals are normally distributed:distributed:

Thi is one of the assumptions of linear Thi is one of the assumptions of linear regression: errors are i.i.d normalregression: errors are i.i.d normal

then the joint distribution or likelihood then the joint distribution or likelihood function, L, can be written as:function, L, can be written as:

)ˆ()...ˆ(*)ˆ()ˆ...ˆˆ( 110110 nn efefefeeeg

2]/)0ˆ[(2/12 )2/1(),0(~)ˆ( iei eNef

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Likelihood functionLikelihood function

and taking natural logarithms of both and taking natural logarithms of both sides, where the logarithm is a sides, where the logarithm is a monotonically increasing function so that monotonically increasing function so that if lnL is maximized, so is L:if lnL is maximized, so is L:

1

0

22

2

]ˆ[)2/1(2/2/2

]/)0ˆ[(2/11

0110

*)2/1(*)/1(

)2/1()ˆ...ˆˆ(

n

ii

i

enn

en

in

eL

eeeegL

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The Natural Logarithm Function

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 1 2 3 4 5 6

x

lnx

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Log-LikelihoodLog-Likelihood

Taking the derivative of lnL with respect Taking the derivative of lnL with respect to either a-hat or b-hat yields the same to either a-hat or b-hat yields the same estimators for the parameters a and b estimators for the parameters a and b as with ordinary least squares, except as with ordinary least squares, except now we know the errors are normally now we know the errors are normally distributed.distributed.

21

0

22

1

0

222

]*ˆˆ[)2/1()2ln(*)2/(]ln[*)2/(ln

ˆ)2/1()2ln(*)2/(]ln[*)2/(ln

i

n

ii

n

ii

xbaynnL

ennL

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Log-LikelihoodLog-Likelihood Taking the derivative of lnL with Taking the derivative of lnL with

respect to sigma squared, we obtain an respect to sigma squared, we obtain an estimate for the variance of the errors:estimate for the variance of the errors:

andand

in practice we divide by n-2 since we in practice we divide by n-2 since we used up two degrees of freedom in used up two degrees of freedom in estimating a-hat and b-hat. estimating a-hat and b-hat.

1

0

2422 ˆ)/1(*)2/1()/1(*)2/(/lnn

iienL

nen

ii /]ˆ[ˆ

1

0

22

10

The sum of squared residuals The sum of squared residuals (estimated)(estimated) 2

ie

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SUMMARY OUTPUT

Regression StatisticsMultiple R 0.98873218R Square 0.97759133Adjusted R Square 0.97693225Standard Error 3.38354883Observations 36

ANOVAdf SS MS F Significance F

Regression 1 16981.07081 16981.07 1483.27 1.24E-29Residual 34 389.2456906 11.4484Total 35 17370.3165

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Lower 95.0%Upper 95.0%Intercept -0.2916205 1.014023514 -0.28759 0.775408 -2.35236 1.769122 -2.3523629 1.76912184X Variable 1 0.06329191 0.001643381 38.51324 1.24E-29 0.059952 0.066632 0.0599522 0.06663166

Goodness of fit

Regress CA State General Fund Expenditures on CA Personal Income, Lab Four

2ie

)2/(ˆ2 ne

n

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The Intuition Behind the The Intuition Behind the Table of Analysis of Variance Table of Analysis of Variance

(ANOVA)(ANOVA)

y = a + b*x + ey = a + b*x + e– the variation in the dependent the variation in the dependent

variable, y, is explained by either the variable, y, is explained by either the regression, a + b*x, or by the error, eregression, a + b*x, or by the error, e

The sample sum of deviations in y:The sample sum of deviations in y:

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0

][ yyn

ii

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Table of ANOVATable of ANOVA

Source Degrees ofFreedom

Sum ofSquares

MeanSquare

Regression(a + b*x

1

Error (e) n-2

Total (y) n-1



21

0

][ yyn

ii

2ie

By difference

)2/(}ˆ{ 2 nei

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Test of the Significance of Test of the Significance of the Regression: F-testthe Regression: F-test

FF1,n-2 1,n-2 = explained mean = explained mean square/unexplained mean squaresquare/unexplained mean square

example: Fexample: F1, 34 1, 34 = = 16981.07 /11.8444 = 16981.07 /11.8444 = 1483.271483.27

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The UC BudgetThe UC Budget

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The UC BudgetThe UC Budget The UC Budget can be written as an The UC Budget can be written as an

identity:identity: UCBUD(t)= UC’s Gen. Fnd. Share(t)* UCBUD(t)= UC’s Gen. Fnd. Share(t)*

The Relative Size of CA Govt.(t)*CA The Relative Size of CA Govt.(t)*CA Personal Income(t)Personal Income(t)– where UC’s Gen. Fnd. Share=UCBUD/CA where UC’s Gen. Fnd. Share=UCBUD/CA

Gen. Fnd. ExpendituresGen. Fnd. Expenditures– where the Relative Size of CA Govt.= where the Relative Size of CA Govt.=

CA Gen. Fnd. Expenditures/CA Personal CA Gen. Fnd. Expenditures/CA Personal IncomeIncome

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Long Run Political TrendsLong Run Political Trends

UC’s Share of CA General Fund UC’s Share of CA General Fund ExpendituresExpenditures

UC Budget As Percent of CA Total General fund

3.51%

y = -0.0009x + 0.0698

R2 = 0.8311

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

68-6

9

70-7

1

72-7

3

74-7

5

76-7

7

78-7

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80-8

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82-8

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84-8

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86-8

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88-8

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90-9

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92-9

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94-9

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96-9

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98-9

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00-0

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02-0

3

04-0

5

Fiscal Year

Pe

rce

nt

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UC’s Budget ShareUC’s Budget Share UC’s share of California General Fund UC’s share of California General Fund

expenditure shows a long run expenditure shows a long run downward trend. Like other public downward trend. Like other public universities across the country, UC is universities across the country, UC is becoming less public and more private. becoming less public and more private. Perhaps the most “private” of the public Perhaps the most “private” of the public universities is the University of universities is the University of Michigan. Increasingly, public Michigan. Increasingly, public universities are looking to build up their universities are looking to build up their endowments like private universities.endowments like private universities.

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Long Run Political Trends Long Run Political Trends The Relative size of California The Relative size of California

GovernmentGovernment– The Gann Iniative passed on the ballot The Gann Iniative passed on the ballot

in 1979. The purpose was to limit the in 1979. The purpose was to limit the size of state government so that it size of state government so that it would not grow in real terms per capita.would not grow in real terms per capita.

– Have expenditures on public goods by Have expenditures on public goods by the California state government grown the California state government grown faster than personal income?faster than personal income?

Ratio of General Fund Expenditures to Personal Income

6.01%

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

6.00%

7.00%

8.00%

68-6

9

70-7

1

72-7

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74-7

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76-7

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78-7

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80-8

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82-8

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86-8

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88-8

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90-9

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92-9

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96-9

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98-9

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00-0

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02-0

3

04-0

5

Fiscal year

Pe

rce

nt

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The Relative Size of CA State The Relative Size of CA State Govt.Govt.

California General Fund Expenditure California General Fund Expenditure was growing relative to personal was growing relative to personal income until the Gann initiative income until the Gann initiative passed in 1979. Since then this ratio passed in 1979. Since then this ratio has declined, especially in the has declined, especially in the eighties and early nineties. After eighties and early nineties. After recovery from the last recession, this recovery from the last recession, this ratio recovered, but took a dive in ratio recovered, but took a dive in 2003-04.2003-04.

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Guessing the UC Budget Guessing the UC Budget for 2005-06for 2005-06

UC’s Budget Share, 04-05: 0.0351UC’s Budget Share, 04-05: 0.0351 Relative Size of CA State Govt.: Relative Size of CA State Govt.:

0.06010.0601 Forecast of CA Personal Income for Forecast of CA Personal Income for

2005-06 2005-06

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CA Personal Income, $ Nominal Billions, 1968-69 through 2004-05

0

200

400

600

800

1000

1200

1400

68-6

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70-7

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00-0

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02-0

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04-0

5

Fiscal Year

Bil

lio

ns

$

25

26

27

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ForecastPercent Percent Percent

2003 change 2004 change 2005 change

Personal income ($ billions) $1,197.6 3.7% $1,262.4 5.4% $1,333.1 5.6%

Nonfarm W&S employment (thousands) 14,408 -0.3% 14,525 0.8% 14,832 2.1% Natural resources and mining 22 -5.2% 22 -0.8% 22 -0.9% Construction 788 1.8% 824 4.5% 868 5.3% Manufacturing 1,543 -5.8% 1,517 -1.7% 1,538 1.4% High technology 399 -9.2% 388 -2.9% 394 1.7% Trade, transportation, & utilities 2,715 -0.3% 2,723 0.3% 2,747 0.9% Information 471 -5.2% 467 -0.9% 487 4.2% Financial activities 886 3.9% 904 2.0% 926 2.4% Professional and business services 2,114 0.0% 2,174 2.8% 2,247 3.4% Educational and health services 1,538 2.6% 1,576 2.5% 1,625 3.1% Leisure and hospitality 1,399 1.2% 1,424 1.8% 1,453 2.0% Other services 505 -0.1% 505 -0.1% 514 1.8% Government 2,427 -0.8% 2,391 -1.5% 2,408 0.7%

Figure ECON-4Selected California Economic Indicators

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Guessing the UC Budget for Guessing the UC Budget for 2005-062005-06

UC’s Budget Share, 04-05: 0.0351UC’s Budget Share, 04-05: 0.0351 Relative Size of CA State Govt.: 0.0601Relative Size of CA State Govt.: 0.0601 Forecast of CA Personal Income for 2005-Forecast of CA Personal Income for 2005-

06: $ 1,333.1 B06: $ 1,333.1 B UCBUD(05-06) = 0.035*0.060*$1,333.1BUCBUD(05-06) = 0.035*0.060*$1,333.1B UCBUD(05-06) = $ 2.800 BUCBUD(05-06) = $ 2.800 B compares to UCBUD(04-05) = $ 2.670 Bcompares to UCBUD(04-05) = $ 2.670 B

UC Budget, General Fund Component, Millions of Nominal $

$2670.529

y = 81.613x + 19.497

R2 = 0.933

0

500

1000

1500

2000

2500

3000

3500

4000

68-6

9

70-7

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72-7

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Fiscal Year

Mil

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ns

$

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Guessing the UC Budget for Guessing the UC Budget for 2004-052004-05

UC’s Budget Share 03-04: 0.037UC’s Budget Share 03-04: 0.037 Relative Size of CA State Govt.: 0.065Relative Size of CA State Govt.: 0.065 Forecast of CA Personal Income for 2004-Forecast of CA Personal Income for 2004-

05: $ 1,231.5 B05: $ 1,231.5 B UCBUD(04-05) = 0.037*0.065*$1,231.5BUCBUD(04-05) = 0.037*0.065*$1,231.5B UCBUD(04-05) = $ 2.962 BUCBUD(04-05) = $ 2.962 B compares to UCBUD(03-04) = $ 3.039 Bcompares to UCBUD(03-04) = $ 3.039 B

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The Relative Size of CA The Relative Size of CA Govt.Govt.

Is it determined politically or by Is it determined politically or by economic factors?economic factors?

Economic Perspective: Engle Curve- Economic Perspective: Engle Curve- the variation of expenditure on a the variation of expenditure on a good or service with incomegood or service with income

lnCAGenFndExp = a + b lnCAPersInc lnCAGenFndExp = a + b lnCAPersInc +e +e

b is the elasticity of expenditure with b is the elasticity of expenditure with incomeincome

bCAPersIncpCAGenFndEx ln/ln

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The elasticity of The elasticity of expenditures with respect expenditures with respect

to incometo income

Note:Note:

So, in the log-log regression, So, in the log-log regression, lny = a + b*lnx + e, lny = a + b*lnx + e, the coefficient b is the elasticity of the coefficient b is the elasticity of y with respect to x.y with respect to x.

)/1(*

)/(*)/1(

/ln

CAPersIncb

CAPersIncpCAGenFndExpCAGenFndEx

CAPersIncpCAGenFndEx

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CA State Govt Expenditures Vs. Personal Income

1993-94

2003=04

0

10

20

30

40

50

60

70

80

90

0 500 1000 1500

Personal Income, $ B

CA

Gen

. F

un

d $

B

Linear Regression

CA General Fund Expenditures Vs. CA Personal Income

4.331548797

y = 1.065x - 3.1777

R2 = 0.9891

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

3 3.5 4 4.5 5 5.5 6 6.5 7 7.5

CA Personal Income, ln $B

CA

Ge

n.

Fu

nd

. E

x.,

ln

$B

Log-Log Regression

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Is the Income Elasticity of Is the Income Elasticity of CA State Public Goods >1?CA State Public Goods >1?

Step # 1: Formulate the HypothesesStep # 1: Formulate the Hypotheses– HH0 0 : b = 1: b = 1

– HHa a : b > 1: b > 1

Step # 2: choose the test statisticStep # 2: choose the test statistic

Step # 3: If the null hypothesis were Step # 3: If the null hypothesis were true, what is the probability of true, what is the probability of getting a t-statistic this big?getting a t-statistic this big?

34.30189.0/)1065.1(/)]ˆ(ˆ[ ˆ b

bEbstatt

39

Appendix BTable 4p. B-9

5.0 % in the upper tail

t..050

35 1.69

40

Eviews Output

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Regression ModelsRegression Models Trend AnalysisTrend Analysis

– linear: y(t) = a + b*t + e(t)linear: y(t) = a + b*t + e(t)– exponential: lny(t) = a + b*t + e(t)exponential: lny(t) = a + b*t + e(t)– Y(t) =exp[a + b*t + e(t)]Y(t) =exp[a + b*t + e(t)]

Engle CurvesEngle Curves– ln y = a + b*lnx + eln y = a + b*lnx + e

Income Generating ProcessIncome Generating Process

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Returns Generating Returns Generating ProcessProcess

How does the rate of return on an asset How does the rate of return on an asset vary with the market rate of return?vary with the market rate of return?

rrii(t): rate of return on asset i(t): rate of return on asset i

rrff(t): risk free rate, assumed known for (t): risk free rate, assumed known for the period aheadthe period ahead

rrMM(t): rate of return on the market(t): rate of return on the market

[r[rii(t) - r(t) - rff00(t)] = a +b*[r(t)] = a +b*[rMM(t) - r(t) - rff

00(t)] + e(t) (t)] + e(t)

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ExampleExample rrii(t): monthly rate of return on UC stock (t): monthly rate of return on UC stock

index fund, Sept., 1995 - Sept. 2003index fund, Sept., 1995 - Sept. 2003 rrff(t): risk free rate, assumed known for (t): risk free rate, assumed known for

the period ahead. Usually use Treasury the period ahead. Usually use Treasury Bill Rate. I used monthly rate of return Bill Rate. I used monthly rate of return on UC Money Market Fund on UC Money Market Fund http://atyourservice.ucop.edu/employehttp://atyourservice.ucop.edu/employees/retirement/performance.htmles/retirement/performance.html

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Example (cont.)Example (cont.)

rrMM(t): rate of return on the market. (t): rate of return on the market. I used the monthly change in the I used the monthly change in the logarithm of the total return logarithm of the total return (dividends reinvested)*100. (dividends reinvested)*100. http://research.stlouisfed.org/fred2http://research.stlouisfed.org/fred2//

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Returns Generating Process Time Series Data

-20

-15

-10

-5

0

5

10

15

Se

p-9

5

Se

p-9

6

Se

p-9

7

Se

p-9

8

Se

p-9

9

Se

p-0

0

Se

p-0

1

Se

p-0

2

Se

p-0

3

Date

Mo

thly

Ra

te o

f R

etu

rn

UC Equity Fund

Standard & Poors 500

UC Money Market Fund

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Returns Generating Process, Sept. 95-Sept. 03

-20.00

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

-15 -10 -5 0 5 10

Standard & Poors 500, Net

UC

Sto

ck In

dex

Fu

nd

, Net

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-13.35, 16.09;Ucnet,

S&Pnet

y = 1.0601x - 0.106

R2 = 0.9136

-20.00

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

-15 -10 -5 0 5 10

Watch Excel on xy plots!

True x axis: UC Net

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49

Returns Generating Process

y = 1.0601x - 0.106

R2 = 0.9136

-20.00

-15.00

-10.00

-5.00

0.00

5.00

10.00

15.00

-15 -10 -5 0 5 10

Standard & Poors 500, Net

UC

Sto

ck I

nd

ex F

un

d,

Net

Really the Regression of S&P on UC

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SUMMARY OUTPUT




CoefficientsStandard Error t Stat P-value Lower 95% Upper 95%Lower 95.0%Upper 95.0%Intercept 0.12800497 0.13335 0.959915 0.339535 -0.1367287 0.392739 -0.13673 0.392739X Variable 1 0.86177094 0.027196 31.68748 2.65E-52 0.807780204 0.915762 0.80778 0.915762

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Is the beta for the UC Is the beta for the UC Stock Index Fund <1?Stock Index Fund <1?

Step # 1: Formulate the HypothesesStep # 1: Formulate the Hypotheses– HH0 0 : b = 1: b = 1

– HHa a : b < 1: b < 1

Step # 2: choose the test statisticStep # 2: choose the test statistic

Step # 3: If the null hypothesis were Step # 3: If the null hypothesis were true, what is the probability of true, what is the probability of getting a t-statistic this big?getting a t-statistic this big?

4.6027.0/)1862.0(/)]ˆ(ˆ[ ˆ b

bEbstatt

52

Appendix BTable 4p. B-9

5.0 % in the lower tail

t..050

95 1.66

53

-15

-10

-5

0

5

10

-20 -10 0 10

SPNET

UC

ST

OC

KN

ET

Returns Generating ProcessEViews Chart

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Midterm 2001Midterm 2001

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1. (15 points) The following graph 4-1 shows the results of regressing California

General Fund expenditures, in billions of nominal dollars, against California Personal

Income, in billions of nominal dollars beginning in fiscal year1968-69 and ending in

fiscal year 2001-02.

a. How much of the variance in the dependent variable is explained by personal

income?

b. Interpret the estimated slope.

Table 4-1 follows with the estimated parameters and table of analysis of variance.

c. Is the slope significantly different from zero? What statistic do you use to

answer this question? What distribution do you use to answer this question?

What probability were you willing to accept for a Type I error?

Q. 4

d. What is the ratio of the explained mean square to the unexplained mean square?

56

Calfifornia General Fund Expenditures Vs. California Personal Income, Billions of Nominal $

y = 0.066x - 1.1974

R2 = 0.981

0

10

20

30

40

50

60

70

80

90

0 200 400 600 800 1000 1200 1400

Personal Income

Gen

Fu

nd

Exp

end

itu

res

Q 4

Figure 4-1: California General Fund Expenditures Versus California Personal Income, both in Billions of Nominal Dollars

57


ANOVA

df SS MS F SignificanceF


Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept -1.197411 0.927956018 -1.29037 0.20616709 -3.08759378 0.6927721X Variable 1 0.0659894 0.001622349 40.67523 3.9867E-29 0.062684796 0.069294

Q 4Table 4-1: Summary Output

Economics 240A

Documents

likelihood functionand

derivative of lnl

general fund compnentchart20

yearpercentuc budget

joint distribution

estimated residuals

independent variable

ca personal income