1 Economics 240A Power Eight

2 Outline Lab Four Lab Four Maximum Likelihood Estimation Maximum Likelihood Estimation The UC Budget Again The UC Budget Again Regression Models Regression Models The Income Generating Process for an Asset The Income Generating Process for an Asset

3 UCBUDGSH(t) = a + b*t + e(t)

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

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

6 How to Find a-hat and b-hat? Methodology Methodology grid search grid search differential calculus differential calculus likelihood function likelihood function motivation: the likelihood function connects the topics of probability (especially independence), the practical application of random sampling, the normal distribution, and the derivation of estimators motivation: the likelihood function connects the topics of probability (especially independence), the practical application of random sampling, the normal distribution, and the derivation of estimators

7 Likelihood function The joint density of the estimated residuals can be written as: The joint density of the estimated residuals can be written as: If the sample of observations on the dependent variable, y, and the independent variable, x, is random, then the observations are independent of one another. If the errors are also identically distributed, f, i.e. i.i.d, then If the sample of observations on the dependent variable, y, and the independent variable, x, is random, then the observations are independent of one another. If the errors are also identically distributed, f, i.e. i.i.d, then

8 Likelihood function Continued: If i.i.d., then Continued: 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 regression: errors are i.i.d normal This is one of the assumptions of linear regression: errors are i.i.d normal then the joint distribution or likelihood function, L, can be written as: then the joint distribution or likelihood function, L, can be written as:

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

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

12 Log-Likelihood Taking the derivative of lnL with respect to sigma squared, we obtain an estimate for the variance of the errors: Taking the derivative of lnL with respect to sigma squared, we obtain an estimate for the variance of the errors: and and in practice we divide by n-2 since we used up two degrees of freedom in estimating a- hat and b-hat. in practice we divide by n-2 since we used up two degrees of freedom in estimating a- hat and b-hat.

13 Interpreting Excel Output

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

15 CAGFD(t) = a + b*CAPY(t) +e(t): through

16 SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations38 ANOVA dfSSMSFSignificance F Regression E-38 Residual Total CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept X Variable E Goodness of fit, R 2 Number of Observations, n Regress CA State General Fund Expenditures on CA Personal Income, Lab Four

17 Estimated Coefficients Coefficients Standard Errort StatP-valueLower 95% Upper 95% Intercept X Variable E

18 Appendix B Table 4 p. B % in the upper tail From Power 6: Student’s t-distribution Text: pp

19 Table of Analysis of Variance ANOVAMeanSquare=SS/df dfSSMSFSignificance F Regression E-38 Residual Total Degrees of Freedom Sum of Squares F 1, 37 = EMS/UMS

20 The Intuition Behind the Table of Analysis of Variance (ANOVA) y = a + b*x + e y = a + b*x + e the variation in the dependent variable, y, is explained by either the regression, a + b*x, or by the error, e the variation in the dependent variable, y, is explained by either the regression, a + b*x, or by the error, e The sample sum of deviations in y: The sample sum of deviations in y:

21 Table of ANOVA By difference

22 SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations38 ANOVA dfSSMSFSignificance F Regression E-38 Residual Total CoefficientsStandard Errort StatP-valueLower 95%Upper 95% Intercept X Variable E Goodness of fit, R 2 Number of Observations, n Regress CA State General Fund Expenditures on CA Personal Income, Lab Four

23 Test of the Significance of the Regression: F-test F 1,n-2 = explained mean square/unexplained mean square F 1,n-2 = explained mean square/unexplained mean square example: F 1, 36 = / 6.387= 3652 example: F 1, 36 = / 6.387= 3652

24 Table 6, pp. B-11 through B-16 Text: pp

25 The UC Budget

26 The UC Budget The UC Budget can be written as an identity: The UC Budget can be written as an identity: UCBUD(t)= UC’s Gen. Fnd. Share(t)* The Relative Size of CA Govt.(t)*CA Personal Income(t) UCBUD(t)= UC’s Gen. Fnd. Share(t)* The Relative Size of CA Govt.(t)*CA Personal Income(t) where UC’s Gen. Fnd. Share=UCBUD/CA Gen. Fnd. Expenditures where UC’s Gen. Fnd. Share=UCBUD/CA Gen. Fnd. Expenditures where the Relative Size of CA Govt.= CA Gen. Fnd. Expenditures/CA Personal Income where the Relative Size of CA Govt.= CA Gen. Fnd. Expenditures/CA Personal Income

27 Long Run Political Trends UC’s Share of CA General Fund Expenditures UC’s Share of CA General Fund Expenditures

28 The Regression Passes Through the Means of y and x

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

30 Long Run Political Trends The Relative size of California Government The Relative size of California Government The Gann Iniative passed on the ballot in The purpose was to limit the size of state government so that it would not grow in real terms per capita. The Gann Iniative passed on the ballot in The purpose was to limit the size of state government so that it would not grow in real terms per capita. Have expenditures on public goods by the California state government grown faster than personal income? Have expenditures on public goods by the California state government grown faster than personal income?

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32 The Relative Size of CA State Govt. California General Fund Expenditure was growing relative to personal income until the Gann initiative passed in Since then this ratio has declined, especially in the eighties and early nineties. After recovery from the last recession, this ratio recovered, but took a dive in California General Fund Expenditure was growing relative to personal income until the Gann initiative passed in Since then this ratio has declined, especially in the eighties and early nineties. After recovery from the last recession, this ratio recovered, but took a dive in

33 Guessing the UC Budget for UC’s Budget Share, 05-06: UC’s Budget Share, 05-06: Relative Size of CA State Govt.: Relative Size of CA State Govt.: Forecast of CA Personal Income for Forecast of CA Personal Income for

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40 Guessing the UC Budget for UC’s Budget Share, 05-06: UC’s Budget Share, 05-06: Relative Size of CA State Govt.: Relative Size of CA State Govt.: Forecast of CA Personal Income for : $ 1,406.5 B Forecast of CA Personal Income for : $ 1,406.5 B UCBUD(06-07) = *0.0648*$1,406.5B UCBUD(06-07) = *0.0648*$1,406.5B UCBUD(06-07) = $ 2.98 B UCBUD(06-07) = $ 2.98 B compares to UCBUD(05-06) = $ 2.81 B compares to UCBUD(05-06) = $ 2.81 B An increase of $170 million An increase of $170 million

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42 The Relative Size of CA Govt. Is it determined politically or by economic factors? Is it determined politically or by economic factors? Economic Perspective: Engle Curve- the variation of expenditure on a good or service with income Economic Perspective: Engle Curve- the variation of expenditure on a good or service with income lnCAGenFndExp = a + b lnCAPersInc +e lnCAGenFndExp = a + b lnCAPersInc +e b is the elasticity of expenditure with income b is the elasticity of expenditure with income

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

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

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46 Is the Income Elasticity of CA State Public Goods >1? Step # 1: Formulate the Hypotheses Step # 1: Formulate the Hypotheses H 0 : b = 1 H 0 : b = 1 H a : b > 1 H a : b > 1 Step # 2: choose the test statistic Step # 2: choose the test statistic Step # 3: If the null hypothesis were true, what is the probability of getting a t-statistic this big? Step # 3: If the null hypothesis were true, what is the probability of getting a t-statistic this big?

47 Appendix B Table 4 p. B % in the upper tail t

48 Regression Models Trend Analysis Trend 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 Curves Engle Curves ln y = a + b*lnx + e ln y = a + b*lnx + e Income Generating Process Income Generating Process

49 Returns Generating Process How does the rate of return on an asset vary with the market rate of return? How does the rate of return on an asset vary with the market rate of return? r i (t): rate of return on asset i r i (t): rate of return on asset i r f (t): risk free rate, assumed known for the period ahead r f (t): risk free rate, assumed known for the period ahead r M (t): rate of return on the market r M (t): rate of return on the market [r i (t) - r f 0 (t)] = a +b*[r M (t) - r f 0 (t)] + e(t) [r i (t) - r f 0 (t)] = a +b*[r M (t) - r f 0 (t)] + e(t)

50 Example r i (t): monthly rate of return on UC stock index fund, Sept., Sept r i (t): monthly rate of return on UC stock index fund, Sept., Sept r f (t): risk free rate, assumed known for the period ahead. Usually use Treasury Bill Rate. I used monthly rate of return on UC Money Market Fund nt/performance.html r f (t): risk free rate, assumed known for the period ahead. Usually use Treasury Bill Rate. I used monthly rate of return on UC Money Market Fund nt/performance.html

51 Example (cont.) r M (t): rate of return on the market. I used the monthly change in the logarithm of the total return (dividends reinvested)* r M (t): rate of return on the market. I used the monthly change in the logarithm of the total return (dividends reinvested)*100.

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54 Watch Excel on xy plots! True x axis: UC Net

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56 Really the Regression of S&P on UC

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58 Is the beta for the UC Stock Index Fund <1? Step # 1: Formulate the Hypotheses Step # 1: Formulate the Hypotheses H 0 : b = 1 H 0 : b = 1 H a : b < 1 H a : b < 1 Step # 2: choose the test statistic Step # 2: choose the test statistic Step # 3: If the null hypothesis were true, what is the probability of getting a t-statistic this big? Step # 3: If the null hypothesis were true, what is the probability of getting a t-statistic this big?

59 Appendix B Table 4 p. B % in the lower tail t

60 EViews Chart

61 Midterm 2001

62 Q. 4

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

64 Q 4 Table 4-1: Summary Output