1. Economics 240A
Power Eight

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

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

4. UCBUDSH(t) = a + b*t + e(t)
5.105
19.5

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

6. How to Find a-hat and b-hat?
Methodology
grid search
differential calculus
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

7. Likelihood function
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

8. Likelihood function Continued: If i.i.d., then
If the residuals are normally distributed:
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:

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:

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.

12. Log-Likelihood
Taking the derivative of lnL with respect to sigma squared, we obtain an estimate for the variance of the errors:
and
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)

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

16. Regression Statistics
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations 38
ANOVA df SS MS F
Significance F
Regression 1
E-38
Residual 36
Total 37
Coefficients
t Stat
P-value
Lower 95%
Upper 95%
Intercept
X Variable 1
Regress CA State General Fund Expenditures
on CA Personal Income, Lab Four
Goodness of fit, R2
Number of Observations, n

17. Estimated Coefficients
Coefficients
Standard Error
t Stat
P-value
Lower 95%
Upper 95%
Intercept
X Variable 1
E-38

18. Appendix B Table 4 p. B-9 2.5 % in the upper tail From Power 6:
Student’s
t-distribution
Text: pp
Appendix B
Table 4
p. B-9
2.5 % in the
upper tail

19. Table of Analysis of Variance
ANOVA
Mean
Square
=SS/df df SS MS F
Significance F
Regression 1
E-38
Residual 36
Total 37
Degrees of
Freedom
F1, 37 =
EMS/UMS
Sum of Squares

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

21. Table of ANOVA By
difference

22. Regression Statistics
SUMMARY OUTPUT
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations 38
ANOVA df SS MS F
Significance F
Regression 1
E-38
Residual 36
Total 37
Coefficients
t Stat
P-value
Lower 95%
Upper 95%
Intercept
X Variable 1
Regress CA State General Fund Expenditures
on CA Personal Income, Lab Four
Goodness of fit, R2
Number of Observations, n

23. Test of the Significance of the Regression: F-test
F1,n-2 = explained mean square/unexplained mean square
example: F1, 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:
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

27. Long Run Political Trends
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.

30. Long Run Political Trends
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?

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

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

40. Guessing the UC Budget for 2005-06
UC’s Budget Share, 05-06:
Relative Size of CA State Govt.:
Forecast of CA Personal Income for : $ 1,406.5 B
UCBUD(06-07) = *0.0648*$1,406.5B
UCBUD(06-07) = $ 2.98 B
compares to UCBUD(05-06) = $ 2.81 B
An increase of $170 million

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

43. The elasticity of expenditures with respect to income
Note:
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)

46. Is the Income Elasticity of CA State Public Goods >1?
Step # 1: Formulate the Hypotheses
H0 : b = 1
Ha : b > 1
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?

47. t..050
Appendix B
Table 4
p. B-9
5.0 % in the
upper tail
1.69 35

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

49. Returns Generating Process
How does the rate of return on an asset vary with the market rate of return?
ri(t): rate of return on asset i
rf(t): risk free rate, assumed known for the period ahead
rM(t): rate of return on the market
[ri(t) - rf0(t)] = a +b*[rM(t) - rf0(t)] + e(t)

50. Example
ri(t): monthly rate of return on UC stock index fund, Sept., Sept. 2003
rf(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

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

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

56. Really the Regression of S&P on UC

58. Is the beta for the UC Stock Index Fund <1?
Step # 1: Formulate the Hypotheses
H0 : b = 1
Ha : b < 1
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?

59. t..050
Appendix B
Table 4
p. B-9
5.0 % in the
lower tail
1.66 95

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