1 Economics 240A Power Eight. 2 Outline Lab Four Lab Four Maximum Likelihood Estimation Maximum...

Economics 240AEconomics 240A

Power EightPower Eight

OutlineOutline Lab FourLab Four Maximum Likelihood EstimationMaximum Likelihood Estimation The UC Budget AgainThe UC Budget Again Regression ModelsRegression Models The Income Generating Process for an The Income Generating Process for an

Asset Asset

UCBUDGSH(t) = a + b*t + e(t)UCBUDGSH(t) = a + b*t + e(t)UC Budget Share of CA General Fund Expenditure:1968-69 through 2007-08

2007-08

68-69 y = -0.0009x + 0.0704

R2 = 0.8663

Fiscal year

UCBUDSH(t) = a + b*t + e(t)UCBUDSH(t) = a + b*t + e(t)UC Budget Share of CA General Fund Expenditure:68-69 through 2007-08

y = -0.0009x + 0.0694

R2 = 0.8663

0 5 10 15 20 25 30 35 40 45

UC Budget Share of General Fund Expenditure, 1968-69 through 2005-06

1968-69

2005-06

y = -0.0009x + 0.0691

R2 = 0.8449

0 5 10 15 20 25 30 35 40

means: 5.22%, 18.5 yr.

UCBUDSH(t) = a + b*t + e(t)UCBUDSH(t) = a + b*t + e(t)

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

MethodologyMethodology grid searchgrid search differential calculusdifferential calculus likelihood functionlikelihood function

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

Likelihood functionLikelihood function

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

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

)ˆ.....ˆˆˆ( 1210 neeeeg

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

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

This is one of the assumptions of linear This 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

Likelihood functionLikelihood function

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

]ˆ[)2/1(2/2/2

]/)0ˆ[(2/11

*)2/1(*)/1(

)2/1()ˆ...ˆˆ(

eeeegL

The Natural Logarithm Function

0 1 2 3 4 5 6

Log-LikelihoodLog-Likelihood

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

]*ˆˆ[)2/1()2ln(*)2/(]ln[*)2/(ln

ˆ)2/1()2ln(*)2/(]ln[*)2/(ln

xbaynnL

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

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

andand

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

0ˆ)/1(*)2/1()/1(*)2/(/ln1

ii /]ˆ[ˆ

Interpreting Excel OutputInterpreting Excel Output

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

CA Size of Govt. Vs. SIze of Economy

y = 0.0657x - 1.0238

R2 = 0.9902

0 200 400 600 800 1000 1200 1400

CAPY, B Nom.$

CAGFD(t) = a + b*CAPY(t) +e(t): CAGFD(t) = a + b*CAPY(t) +e(t): 1968-69 through 2005-061968-69 through 2005-06

SUMMARY OUTPUT

Regression Statistics

Multiple R 0.99510756

R Square 0.99023905

Adjusted R Square 0.98996792

Standard Error 2.52724563

Observations 38

df SS MS F Significance F

Regression 1 23326.28511 23326.29 3652.16735 8.58311E-38

Residual 36 229.9309379 6.38697

Total 37 23556.21605

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept -1.02377776 0.727626534 -1.40701 0.167999648 -2.499472762 0.451917

X Variable 1 0.06565026 0.001086328 60.43316 8.58311E-38 0.063447085 0.067853

Goodness of fit, R2

Number of Observations, n

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

Estimated CoefficientsEstimated Coefficients

CoefficientsStandard

Error t Stat P-value Lower 95%Upper

Intercept -1.02377776 0.727626534 -1.40701 0.167999648 -2.499472762 0.451917

X Variable 1 0.06565026 0.001086328 60.43316 8.58311E-38 0.063447085 0.067853

41.1727.0/)0204.1(ˆ/)]ˆ(ˆ[ ˆˆ aa aEat

Appendix BTable 4p. B-9

2.5 % in the upper tail

From Power 6:Student’s t-distributionText: pp. 260-2

Table of Analysis of VarianceTable of Analysis of Variance

Mean Mean

SquareSquare

=SS/df=SS/df

Regression 1 23326.28511 23326.29 3652.16735 8.58311E-38

Residual 36 229.9309379 6.38697

Total 37 23556.21605

Degrees ofFreedom

Sum of Squares F1, 37 = EMS/UMS

The Intuition Behind the Table of The Intuition Behind the Table of Analysis of Variance (ANOVA)Analysis of Variance (ANOVA)

y = a + b*x + ey = a + b*x + e the variation in the dependent variable, y, is the variation in the dependent variable, y, is

explained by either the regression, a + b*x, or by explained by either the regression, a + b*x, or by the error, ethe error, e

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

][ yyn

Table of ANOVATable of ANOVA

Source Degrees ofFreedom

Sum ofSquares

MeanSquare

Regression(a + b*x

Error (e) n-2

Total (y) n-1

ANOVAdf SS MS F Significance F

Regression 1 23326.28511 23326.29 3652.167 8.58311E-38Residual 36 229.9309379 6.38697Total 37 23556.21605

][ yyn

By difference

)2/(}ˆ{ 2 nei

SUMMARY OUTPUT

Regression Statistics

Multiple R 0.99510756

R Square 0.99023905

Adjusted R Square 0.98996792

Standard Error 2.52724563

Observations 38

Regression 1 23326.28511 23326.29 3652.16735 8.58311E-38

Residual 36 229.9309379 6.38697

Total 37 23556.21605

Coefficients Standard Error t Stat P-value Lower 95% Upper 95%

Intercept -1.02377776 0.727626534 -1.40701 0.167999648 -2.499472762 0.451917

X Variable 1 0.06565026 0.001086328 60.43316 8.58311E-38 0.063447085 0.067853

Goodness of fit, R2

Number of Observations, n

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

)2/(ˆˆ 2 nee

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

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

squaresquare example: Fexample: F1, 36 1, 36 = = 23326.29 / 6.387= 365223326.29 / 6.387= 3652

Table 6,pp. B-11 throughB-16Text: pp.270-274

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

Fnd. Expenditures/CA Personal IncomeFnd. Expenditures/CA Personal Income

Long Run Political TrendsLong Run Political Trends

UC’s Share of CA General Fund ExpendituresUC’s Share of CA General Fund Expenditures

The Regression Passes Through The Regression Passes Through the Means of y and xthe Means of y and x

UC Budget Share of General Fund Expenditure, 1968-69 through 2005-06

1968-69

2005-06

y = -0.0009x + 0.0691

R2 = 0.8449

0 5 10 15 20 25 30 35 40

means: 5.22%, 18.5 yr.

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

Long Run Political Trends Long Run Political Trends

The Relative size of California GovernmentThe Relative size of California Government The Gann Iniative passed on the ballot in 1979. The Gann Iniative passed on the ballot in 1979.

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

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

The Size of CA State Government Relative to the Economy

Fiscal Year

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

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

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

UC’s Budget Share, 05-06: 0.0327UC’s Budget Share, 05-06: 0.0327 Relative Size of CA State Govt.: 0.0648Relative Size of CA State Govt.: 0.0648 Forecast of CA Personal Income for 2006-07 Forecast of CA Personal Income for 2006-07

California Personal Income, Billions of Nominal $, 1968-69 through 2005-06

2005-06, $1.324B

Fiscal Year

Guessing the UC Budget for 2005-06Guessing the UC Budget for 2005-06 UC’s Budget Share, 05-06: 0.0327UC’s Budget Share, 05-06: 0.0327 Relative Size of CA State Govt.: 0.0648Relative Size of CA State Govt.: 0.0648 Forecast of CA Personal Income for 2006-07: $ Forecast of CA Personal Income for 2006-07: $

1,406.5 B1,406.5 B UCBUD(06-07) = 0.0327*0.0648*$1,406.5BUCBUD(06-07) = 0.0327*0.0648*$1,406.5B UCBUD(06-07) = $ 2.98 BUCBUD(06-07) = $ 2.98 B compares to UCBUD(05-06) = $ 2.81 Bcompares to UCBUD(05-06) = $ 2.81 B An increase of $170 millionAn increase of $170 million

UC Budget in Billions, 1968-69 through 2005-06

y = 0.0805x + 0.1147

R2 = 0.9344

0 5 10 15 20 25 30 35 40

Forecast:$2.98 B

The Relative Size of CA Govt.The Relative Size of CA Govt. Is it determined politically or by economic Is it determined politically or by economic

factors?factors? Economic Perspective: Engle Curve- the Economic Perspective: Engle Curve- the

variation of expenditure on a good or service variation of expenditure on a good or service with incomewith income

lnCAGenFndExp = a + b lnCAPersInc +e lnCAGenFndExp = a + b lnCAPersInc +e b is the elasticity of expenditure with incomeb is the elasticity of expenditure with income

bCAPersIncpCAGenFndEx ln/ln

The elasticity of expenditures The elasticity of expenditures with respect to incomewith respect to income

Note:Note:

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

)/(*)/1(

CAPersIncb

CAPersIncpCAGenFndExpCAGenFndEx

CAPersIncpCAGenFndEx

Logarithms of California General Fund Expenditures and Personal Income, 1968-69 through 2005-06

y = 0.927x - 3.3845

R2 = 0.99

8 8.5 9 9.5 10 10.5 11 11.5 12

LnCAPY(t)

Lncagenfndex(t) = a +b*lncapy(t) + e(t)Lncagenfndex(t) = a +b*lncapy(t) + e(t)

80.30179.0/)1068.1(/)]ˆ(ˆ[

Is the Income Elasticity of CA Is the Income Elasticity of CA State Public Goods >1?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 true, Step # 3: If the null hypothesis were true, what is the probability of getting a t-statistic what is the probability of getting a t-statistic this big?this big?

8.30179.0/)1068.1(/)]ˆ(ˆ[ ˆ b

bEbstatt

5.0 % in the upper tail

t..050

35 1.69

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

Returns Generating ProcessReturns Generating Process

How does the rate of return on an asset vary How does the rate of return on an asset vary with the market rate of return?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 the (t): risk free rate, assumed known for the

period aheadperiod 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)

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

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

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

Example (cont.)Example (cont.)

rrMM(t): rate of return on the market. I used the (t): rate of return on the market. I used the

monthly change in the logarithm of the total monthly change in the logarithm of the total return (dividends reinvested)*100. return (dividends reinvested)*100. http://research.stlouisfed.org/fred2/http://research.stlouisfed.org/fred2/

Returns Generating Process Time Series Data

UC Equity Fund

Standard & Poors 500

UC Money Market Fund

Returns Generating Process, Sept. 95-Sept. 03

-20.00

-15.00

-10.00

-15 -10 -5 0 5 10

Standard & Poors 500, Net

-13.35, 16.09;Ucnet,

S&Pnet

y = 1.0601x - 0.106

R2 = 0.9136

-20.00

-15.00

-10.00

-15 -10 -5 0 5 10

Watch Excel on xy plots!

True x axis: UC Net

Returns Generating Process

y = 1.0601x - 0.106

R2 = 0.9136

-20.00

-15.00

-10.00

-15 -10 -5 0 5 10

Standard & Poors 500, Net

Really the Regression of S&P on UC

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.95580613R Square 0.91356536Adjusted R Square 0.91265552Standard Error 1.31011043Observations 97

ANOVAdf SS MS F Significance F

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

Is the beta for the UC Stock Is the beta for the UC Stock Index Fund <1?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 true, Step # 3: If the null hypothesis were true, what is the probability of getting a t-statistic what is the probability of getting a t-statistic this big?this big?

4.6027.0/)1862.0(/)]ˆ(ˆ[ ˆ b

bEbstatt

5.0 % in the lower tail

t..050

95 1.66

-20 -10 0 10

Returns Generating ProcessEViews Chart

Midterm 2001Midterm 2001

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?

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

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

y = 0.066x - 1.1974

R2 = 0.981

0 200 400 600 800 1000 1200 1400

Personal Income

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

Regression StatisticsMultiple R 0.9904673R Square 0.9810255Adjusted R Square 0.9804325Standard Error 2.9988336Observations 34

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

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