Economics 240A
Power Eight
1
Outline
Maximum Likelihood Estimation
 The UC Budget Again
 Regression Models
 The Income Generating Process for an
Asset

2
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
3
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
4
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:

5
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:
6
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
7
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.
8
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
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 a-hat and b-hat.
9

The sum of squared residuals
(estimated)
 eˆ
2
i
10
Regress CA State General Fund Expenditures
on CA Personal Income, Lab Four
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.98873218
R Square
0.97759133
Adjusted R Square 0.97693225
Standard Error
3.38354883
Observations
36
df
Intercept
X Variable 1
2
ˆ
e
 /( n  2)
n
ANOVA
Regression
Residual
Total
Goodness of fit
SS
MS
1 16981.07081 16981.07
34 389.2456906 11.4484
35 17370.3165
F Significance F
1483.27 1.24E-29
2
ˆ
e
i
Coefficients Standard Error t Stat
P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%
-0.2916205 1.014023514 -0.28759 0.775408 -2.35236 1.769122 -2.3523629 1.76912184
0.06329191 0.001643381 38.51324 1.24E-29 0.059952 0.066632 0.0599522 0.06663166
11
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
12
ANOVA
Table of ANOVA
df
Regression
Residual
Total
SS
MS
1 16981.07081 16981.07
34 389.2456906 11.4484
35 17370.3165
F
Significance F
1483.27 1.24373E-29
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
13
Test of the Significance of the
Regression: F-test
F1,n-2 = explained mean square/unexplained
mean square
 example: F1, 34 = 16981.07 /11.8444 = 1483.27

14
The UC Budget
15
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
16
Long Run Political Trends

UC’s Share of CA General Fund
Expenditures
17
UC Budget As Percent of CA Total General fund
8.00%
7.00%
6.00%
4.00%
3.51%
3.00%
y = -0.0009x + 0.0698
R2 = 0.8311
2.00%
1.00%
Fiscal Year
-0
5
04
-0
3
02
-0
1
00
-9
9
98
-9
7
96
-9
5
94
-9
3
92
-9
1
90
-8
9
88
-8
7
86
-8
5
84
-8
3
82
-8
1
80
-7
9
78
-7
7
76
-7
5
74
-7
3
72
-7
1
70
-6
9
0.00%
68
Percent
5.00%
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.
19
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?
20
Ratio of General Fund Expenditures to Personal Income
8.00%
7.00%
6.00%
6.01%
4.00%
3.00%
2.00%
1.00%
Fiscal year
-0
5
04
-0
3
02
-0
1
00
-9
9
98
-9
7
96
-9
5
94
-9
3
92
-9
1
90
-8
9
88
-8
7
86
-8
5
84
-8
3
82
-8
1
80
-7
9
78
-7
7
76
-7
5
74
-7
3
72
-7
1
70
-6
9
0.00%
68
Percent
5.00%
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.
22
Guessing the UC Budget for
2005-06
UC’s Budget Share, 04-05: 0.0351
 Relative Size of CA State Govt.: 0.0601
 Forecast of CA Personal Income for
2005-06

23
04
02
00
98
96
94
92
90
88
86
84
82
80
78
76
74
72
70
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
Billions $
CA Personal Income, $ Nominal Billions, 1968-69 through 2004-05
1400
1200
1000
800
600
400
200
0
Fiscal Year
24
25
26
27
28
Figure ECON-4
Selected California Economic Indicators
2003
Personal income ($ billions)
Nonfarm W&S employment (thousands)
Natural resources and mining
Construction
Manufacturing
High technology
Trade, transportation, & utilities
Information
Financial activities
Professional and business services
Educational and health services
Leisure and hospitality
Other services
Government
Percent
change
2004
Forecast
Percent
change
2005
Percent
change
$1,197.6
3.7%
$1,262.4
5.4%
$1,333.1
5.6%
14,408
22
788
1,543
399
2,715
471
886
2,114
1,538
1,399
505
2,427
-0.3%
-5.2%
1.8%
-5.8%
-9.2%
-0.3%
-5.2%
3.9%
0.0%
2.6%
1.2%
-0.1%
-0.8%
14,525
22
824
1,517
388
2,723
467
904
2,174
1,576
1,424
505
2,391
0.8%
-0.8%
4.5%
-1.7%
-2.9%
0.3%
-0.9%
2.0%
2.8%
2.5%
1.8%
-0.1%
-1.5%
14,832
22
868
1,538
394
2,747
487
926
2,247
1,625
1,453
514
2,408
2.1%
-0.9%
5.3%
1.4%
1.7%
0.9%
4.2%
2.4%
3.4%
3.1%
2.0%
1.8%
0.7%
Guessing the UC Budget for
2005-06
UC’s Budget Share, 04-05: 0.0351
 Relative Size of CA State Govt.: 0.0601
 Forecast of CA Personal Income for 200506: $ 1,333.1 B
 UCBUD(05-06) = 0.035*0.060*$1,333.1B
 UCBUD(05-06) = $ 2.800 B
 compares to UCBUD(04-05) = $ 2.670 B

30
UC Budget, General Fund Component, Millions of Nominal $
4000
y = 81.613x + 19.497
R2 = 0.933
3500
2500
$2670.529
2000
1500
1000
500
Fiscal Year
-0
5
04
-0
3
02
-0
1
00
-9
9
98
-9
7
96
-9
5
94
-9
3
92
-9
1
90
-8
9
88
-8
7
86
-8
5
84
-8
3
82
-8
1
80
-7
9
78
-7
7
76
-7
5
74
-7
3
72
-7
1
70
-6
9
0
68
Millions $
3000
Guessing the UC Budget for
2004-05
UC’s Budget Share 03-04: 0.037
 Relative Size of CA State Govt.: 0.065
 Forecast of CA Personal Income for 200405: $ 1,231.5 B
 UCBUD(04-05) = 0.037*0.065*$1,231.5B
 UCBUD(04-05) = $ 2.962 B
 compares to UCBUD(03-04) = $ 3.039 B

32
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
33
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.

34
CA State Govt Expenditures Vs. Personal Income
90
CA Gen. Fund $ B
80
70
2003=04
60
50
40
30
1993-94
20
10
0
0
500
1000
1500
Personal Income, $ B
Linear Regression
35
CA General Fund Expenditures Vs. CA Personal Income
5
4.331548797
CA Gen. Fund. Ex., ln $B
4.5
4
y = 1.065x - 3.1777
R2 = 0.9891
3.5
3
2.5
2
1.5
1
0.5
0
3
3.5
4
4.5
5
5.5
CA Personal Income, ln $B
Log-Log Regression
6
6.5
7
7.5
37
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  (1.065  1) / 0.0189  3.34

Step # 3: If the null hypothesis were
true, what is the probability of getting a
t-statistic this big?
38
t..050
Appendix B
Table 4
p. B-9
5.0 % in the
upper tail
35
1.69
39
Eviews Output
40
Regression Models

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)]

Engle Curves
– ln y = a + b*lnx + e

Income Generating Process
41
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)

42
 ri(t):
Example
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/ret
irement/performance.html
43
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/
44
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
45
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
46
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 Net47
48
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
49
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
50
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?
51
t..050
Appendix B
Table 4
p. B-9
5.0 % in the
lower tail
95
1.66
52
10
EViews
Chart
Returns Generating Process
UCSTOCKNET
5
0
-5
-10
-15
-20
-10
0
10
SPNET
53
Midterm 2001
54
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?
55
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
R2 = 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
56
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
57