1 Economics 240A Power Eight

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

3 How to Find a-hat and b-hat? n Methodology –grid search –differential calculus –likelihood function n 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

4 Likelihood function n The joint density of the estimated residuals can be written as: n 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

5 Likelihood function n Continued: If i.i.d., then n If the residuals are normally distributed: n Thi is one of the assumptions of linear regression: errors are i.i.d normal n then the joint distribution or likelihood function, L, can be written as:

6 Likelihood function n 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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8 Log-Likelihood n 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.

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

10 n The sum of squared residuals (estimated)

11 Goodness of fit Regress CA State General Fund Expenditures on CA Personal Income, Lab Four n

12 The Intuition Behind the Table of Analysis of Variance (ANOVA) n 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 n The sample sum of deviations in y:

13 Table of ANOVA By difference

14 Test of the Significance of the Regression: F-test n F 1,n-2 = explained mean square/unexplained mean square n example: F 1, 34 = / =

15 The UC Budget

16 The UC Budget n The UC Budget can be written as an identity: n 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 the Relative Size of CA Govt.= CA Gen. Fnd. Expenditures/CA Personal Income

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

19 UC’s Budget Share n 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.

20 Long Run Political Trends n 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. –Have expenditures on public goods by the California state government grown faster than personal income?

22 The Relative Size of CA State Govt. n 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

23 Guessing the UC Budget for n UC’s Budget Share, 04-05: n Relative Size of CA State Govt.: n Forecast of CA Personal Income for

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30 Guessing the UC Budget for n UC’s Budget Share, 04-05: n Relative Size of CA State Govt.: n Forecast of CA Personal Income for : $ 1,333.1 B n UCBUD(05-06) = 0.035*0.060*$1,333.1B n UCBUD(05-06) = $ B n compares to UCBUD(04-05) = $ B

32 Guessing the UC Budget for n UC’s Budget Share 03-04: n Relative Size of CA State Govt.: n Forecast of CA Personal Income for : $ 1,231.5 B n UCBUD(04-05) = 0.037*0.065*$1,231.5B n UCBUD(04-05) = $ B n compares to UCBUD(03-04) = $ B

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

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

35 Linear Regression

Log-Log Regression

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

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

40 Eviews Output

41 Regression Models n Trend Analysis –linear: y(t) = a + b*t + e(t) –exponential: lny(t) = a + b*t + e(t) –Y(t) =exp[a + b*t + e(t)] n Engle Curves –ln y = a + b*lnx + e n Income Generating Process

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

43 Example n r i (t): monthly rate of return on UC stock index fund, Sept., Sept n 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 irement/performance.html

44 Example (cont.) n 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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47 Watch Excel on xy plots! True x axis: UC Net

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49 Really the Regression of S&P on UC

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

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

53 EViews Chart

54 Midterm 2001

55 Q. 4

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

57 Q 4 Table 4-1: Summary Output