Economics 240A
Power Eight
1
Outline





Lab Four
Maximum Likelihood Estimation
The UC Budget Again
Regression Models
The Income Generating Process for an
Asset
2
UCBUDSH(t) = a + b*t + e(t)
UC Budget Share of General Fund Expenditure, 1968-69 through 2005-06
8.00%
1968-69
7.00%
6.00%
Percent
5.00%
means:
5.22%, 18.5
yr.
4.00%
3.00%
2005-06
y = -0.0009x + 0.0691
R2 = 0.8449
2.00%
1.00%
0.00%
0
5
10
15
20
25
30
35
40
Year
3
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
4
Likelihood function

The joint density of the estimated residuals
can be written as:
g (eˆ0  eˆ1  eˆ2 .....  eˆn1 )

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

Continued: If i.i.d., then
g (eˆ0  eˆ1...  eˆn1 )  f (eˆ0 ) * f (eˆ1 )... f (eˆn1 )

If the residuals are normally distributed:
f (eˆi ) ~ N (0,  )  (1 /  2 )e
2


1/ 2[( eˆi 0 ) /  ]2
Thi 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:
6
Likelihood function
n 1
L  g (eˆ0  eˆ1...  eˆn 1 )   (1 /  2 )e
1/ 2[( eˆi  0 ) /  ]2
i 0
 (1 / 2 )
2
L  (1 /  )
2 n/2

* (1 / 2 )
n/2
*e
n1
[ eˆi
]2
i 0
and taking natural logarithms of both sides,
where the logarithm is a monotonically
increasing function so that if lnL is
maximized, so is L:
7
The Natural Logarithm Function
2
1.5
1
0.5
lnx
0
-0.5
0
1
2
3
4
5
6
-1
-1.5
-2
-2.5
-3
x
8
Log-Likelihood
n 1
ln L  (n / 2) * ln[  ]  (n / 2) * ln( 2 )  (1 / 2 ) eˆi
2
2
2
i 0
n 1
ln L  (n / 2) * ln[  ]  (n / 2) * ln( 2 )  (1 / 2 ) [ yi  aˆ  bˆ * xi ]2
2
2
i 0

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

Taking the derivative of lnL with respect to
sigma squared, we obtain an estimate for
the variance of the errors:
n 1
 ln L /   (n / 2) * (1 /  )  (1 / 2) * (1 /  ) eˆi2  0
2
2
4
i 0

and
n 1
2
2
ˆ
ˆ
  [ ei ] / n
i 0

in practice we divide by n-2 since we used
up two degrees of freedom in estimating ahat and b-hat.
10
Interpreting Excel Output
11

The sum of squared residuals (estimated)
2
ˆ
e
i
12
CAGFD(t) = a + b*CAPY(t) +e(t)
CA Size of Govt. Vs. SIze of Economy
100
90
y = 0.0657x - 1.0238
Gen. Fund Ex. B Nom. $
80
2
R = 0.9902
70
60
50
40
30
20
10
0
0
200
400
600
800
1000
1200
1400
CAPY, B Nom.$
13
SUMMARY OUTPUT
Regress CA State General Fund Expenditures
on CA Personal Income, Lab Four
Regression Statistics
Multiple R
0.99510756
R Square
0.99023905
Goodness of fit, R2
Adjusted R Square
0.98996792
Standard Error
2.52724563
Observations
Number of Observations, n
38
ANOVA
df
SS
Regression
MS
1
23326.28511
23326.29
Residual
36
229.9309379
6.38697
Total
37
23556.21605
Coefficients
Intercept
X Variable 1
Standard Error
t Stat
F
3652.16735
P-value
Significance F
8.58311E-38
Lower 95%
2
ˆ
e
i
Upper 95%
-1.02377776
0.727626534
-1.40701
0.167999648
-2.499472762
0.451917
0.06565026
0.001086328
60.43316
8.58311E-38
0.063447085
0.067853
14
Estimated Coefficients
taˆ  [aˆ  E (aˆ )] / ˆ aˆ  (1.204  0) / 0.727  1.41
Coefficients
Standard
Error
t Stat
P-value
Lower 95%
Upper
95%
-1.02377776
0.727626534
-1.40701
0.167999648
-2.499472762
0.451917
0.06565026
0.001086328
60.43316
8.58311E-38
0.063447085
0.067853
â
Intercept
b̂
X Variable 1
15
From Power 6:
Student’s
t-distribution
Text: pp. 260-2
Appendix B
Table 4
p. B-9
2.5 % in the
upper tail
16
Table of Analysis of Variance
ANOVA
Degrees of
Sum of Squares
Freedom
Mean
Square
=SS/df
df
F1, 37 =
EMS/UMS
SS
MS
F
1
23326.28511
23326.29
3652.16735
Residual
36
229.9309379
6.38697
Total
37
23556.21605
Regression
Significance F
8.58311E-38
17
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:
n 1
[ y
i 0
i
 y ]2
18
Table of ANOVA
ANOVA
df
Regression
Residual
Total
SS
MS
F
Significance F
1 23326.28511 23326.29 3652.167 8.58311E-38
36 229.9309379 6.38697
37 23556.21605
Source
Degrees of Sum of Mean
Freedom
Squares Square
By
Regression 1
difference
(a + b*x
Error (e)
n-2
{ eˆ } /( n  2)
 eˆ
2
i
Total (y)
n-1
n 1
[ y
i 0
i
2
i
 y ]2
19
SUMMARY OUTPUT
Regress CA State General Fund Expenditures
on CA Personal Income, Lab Four
Regression Statistics
Multiple R
0.99510756
R Square
0.99023905
Goodness of fit, R2
Adjusted R Square
0.98996792
Standard Error
2.52724563
Observations
ˆ e 
38
Number of Observations, n
ANOVA
df
SS
Regression
MS
1
23326.28511
23326.29
Residual
36
229.9309379
6.38697
Total
37
23556.21605
Coefficients
Intercept
X Variable 1
2
ˆ
e
 /( n  2)
Standard Error
t Stat
F
3652.16735
P-value
Significance F
8.58311E-38
Lower 95%
2
ˆ
e
i
Upper 95%
-1.02377776
0.727626534
-1.40701
0.167999648
-2.499472762
0.451917
0.06565026
0.001086328
60.43316
8.58311E-38
0.063447085
0.067853
20
Test of the Significance of the
Regression: F-test


F1,n-2 = explained mean square/unexplained mean
square
example: F1, 36 = 23326.29 / 6.387= 3652
21
Table 6,
pp. B-11 through
B-16
Text: pp.270-274
22
The UC Budget
23
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
24
Long Run Political Trends

UC’s Share of CA General Fund Expenditures
25
The Regression Passes Through
the Means of y and x
UC Budget Share of General Fund Expenditure, 1968-69 through 2005-06
8.00%
1968-69
7.00%
6.00%
Percent
5.00%
means:
5.22%, 18.5 yr.
4.00%
3.00%
2005-06
y = -0.0009x + 0.0691
R2 = 0.8449
2.00%
3.27%
1.00%
0.00%
0
5
10
15
20
25
30
35
40
Year
26
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.
27
Long Run Political Trends

The Relative size of California Government


The Gann Iniative passed on the ballot in 1979.
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?
28
20
04
20
02
20
00
19
98
19
96
19
94
19
92
19
90
19
88
19
86
19
84
19
82
19
80
19
78
19
76
19
74
19
72
19
70
19
68
-0
5
-0
3
-0
1
-9
9
-9
7
-9
5
-9
3
-9
1
-8
9
-8
7
-8
5
-8
3
-8
1
-7
9
-7
7
-7
5
-7
3
-7
1
-6
9
Percent
The Size of CA State Government Relative to the Economy
8.00%
7.00%
6.48%
6.00%
5.00%
4.00%
3.00%
2.00%
1.00%
0.00%
Fiscal Year
29
The Relative Size of CA State Govt.

California General Fund Expenditure was
growing relative to personal income until
the Gann initiative passed in 1979. 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 2003-04.
30
Guessing the UC Budget for
2005-06



UC’s Budget Share, 05-06: 0.0327
Relative Size of CA State Govt.: 0.0648
Forecast of CA Personal Income for 2006-07
31
California Personal Income, Billions of Nominal $, 1968-69 through 2005-06
2005-06, $1.324B
1400
1200
800
600
400
200
5
04
-0
3
02
-0
1
00
-0
9
98
-9
7
96
-9
5
94
-9
3
92
-9
1
90
-9
9
88
-8
7
86
-8
5
84
-8
3
82
-8
1
80
-8
9
78
-7
7
76
-7
5
74
-7
3
72
-7
1
70
-7
9
0
68
-6
Billions of $
1000
Fiscal Year
32
33
34
35
36
37
Guessing the UC Budget for 2005-06







UC’s Budget Share, 05-06: 0.0327
Relative Size of CA State Govt.: 0.0648
Forecast of CA Personal Income for 2006-07: $
1,406.5 B
UCBUD(06-07) = 0.0327*0.0648*$1,406.5B
UCBUD(06-07) = $ 2.98 B
compares to UCBUD(05-06) = $ 2.81 B
An increase of $170 million
38
UC Budget in Billions, 1968-69 through 2005-06
4
3.5
Forecast:
$2.98 B
y = 0.0805x + 0.1147
R2 = 0.9344
3
$
2.5
2
1.5
1
0.5
0
0
5
10
15
20
25
30
35
40
Year
39
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
 ln CAGenFndExp /  ln CAPersInc  b
40
The elasticity of expenditures
with respect to income
Note:
 ln CAGenFndExp / CAPersInc 
(1 / CAGenFndExp ) * (CAGenFndExp / CAPersInc) 
b * (1 / CAPersInc)
 So, in the log-log regression,
lny = a + b*lnx + e,
the
coefficient b is the elasticity of y with respect
to x.

41
Lncagenfndex(t) = a +b*lncapy(t) + e(t)
Logarithm s of California General Fund Expenditures and Personal Incom e, 1968-69 through 2005-06
7.5
LnCAGenFndEx(t)
7
y = 0.927x - 3.3845
R2 = 0.99
6.5
6
5.5
5
4.5
4
8
8.5
9
9.5
10
10.5
11
11.5
12
LnCAPY(t)
42
H 0 : b  1, H a : b  1
t  [bˆ  E (bˆ)] /   (1.068  1) / 0.0179  3.80
bˆ
bˆ
43
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
t  stat  [bˆ  E (bˆ)] /  bˆ  (1.068  1) / 0.0179  3.8

Step # 3: If the null hypothesis were true,
what is the probability of getting a t-statistic
this big?
44
t..050
Appendix B
Table 4
p. B-9
5.0 % in the
upper tail
35
1.69
45
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)]
ln y = a + b*lnx + e
Income Generating Process
46
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)
47
Example


ri(t): monthly rate of return on UC stock index
fund, Sept., 1995 - 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
http://atyourservice.ucop.edu/employees/retireme
nt/performance.html
48
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.
http://research.stlouisfed.org/fred2/
49
Returns Generating Process Time Series Data
15
5
Sep-03
Sep-02
Sep-01
Sep-00
Sep-99
Sep-98
Sep-97
-5
Sep-96
0
Sep-95
Mothly Rate of Return
10
-10
-15
UC Equity Fund
Standard & Poors 500
UC Money Market Fund
-20
Date
50
Returns Generating Process, Sept. 95-Sept. 03
15.00
UC Stock Index Fund, Net
10.00
5.00
0.00
-15
-10
-5
0
5
10
-5.00
-10.00
-15.00
-20.00
Standard & Poors 500, Net
51
Watch Excel on xy plots!
15.00
10.00
y = 1.0601x - 0.106
2
R = 0.9136
5.00
0.00
-15
-10
-5
0
5
10
-5.00
-10.00
-13.35,
16.09;Ucnet,
S&Pnet
-15.00
-20.00
True x axis: UC Net 52
53
Returns Generating Process
15.00
UC Stock Index Fund, Net
y = 1.0601x - 0.106
10.00
2
R = 0.9136
5.00
0.00
-15
-10
-5
0
5
10
-5.00
-10.00
-15.00
-20.00
Standard & Poors 500, Net
Really the Regression of S&P on UC
54
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.95580613
R Square
0.91356536
Adjusted R Square 0.91265552
Standard Error
1.31011043
Observations
97
ANOVA
df
Regression
Residual
Total
Intercept
X Variable 1
SS
MS
F
Significance F
1 1723.42 1723.42 1004.096 2.65348E-52
95 163.057 1.716389
96 1886.477
CoefficientsStandard Error t Stat
P-value
Lower 95% Upper 95%Lower 95.0%
Upper 95.0%
0.12800497 0.13335 0.959915 0.339535
-0.1367287 0.392739 -0.13673 0.392739
0.86177094 0.027196 31.68748 2.65E-52 0.807780204 0.915762 0.80778 0.915762
55
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
t  stat  [bˆ  E (bˆ)] /  bˆ  (0.862  1) / 0.027  6.4

Step # 3: If the null hypothesis were true,
what is the probability of getting a t-statistic
this big?
56
t..050
Appendix B
Table 4
p. B-9
5.0 % in the
lower tail
95
1.66
57
10
EViews
Chart
Returns Generating Process
UCSTOCKNET
5
0
-5
-10
-15
-20
-10
0
10
SPNET
58
Midterm 2001
59
Q. 4
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?
60
Q4
Calfifornia General Fund Expenditures Vs. California Personal Income, Billions of Nominal $
90
80
Gen Fund Expenditures
70
60
50
y = 0.066x - 1.1974
2
R = 0.981
40
30
20
10
0
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
61
Q4
Table 4-1: Summary Output
Regression Statistics
Multiple R
0.9904673
R Square
0.9810255
Adjusted R Square
0.9804325
Standard Error
2.9988336
Observations
34
ANOVA
df
Regression
Residual
Total
Intercept
X Variable 1
SS
1
32
33
MS
F
Significance
F
14878.68965 14878.69 1654.47398 3.98668E-29
287.7761003 8.993003
15166.46575
Coefficients Standard Error
t Stat
P-value
Lower 95% Upper 95%
-1.197411
0.927956018 -1.29037 0.20616709 -3.08759378 0.6927721
0.0659894
0.001622349 40.67523 3.9867E-29 0.062684796
0.069294
62