1 economics 240a power eight. 2 outline n maximum likelihood estimation n the uc budget again n...
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Economics 240AEconomics 240A
Power EightPower Eight
2
OutlineOutline
Maximum Likelihood EstimationMaximum Likelihood Estimation The UC Budget AgainThe UC Budget Again Regression ModelsRegression Models The Income Generating Process for The Income Generating Process for
an Asset an Asset
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How to Find a-hat and b-How to Find a-hat and b-hat?hat?
MethodologyMethodology– grid searchgrid search– differential calculusdifferential calculus– likelihood functionlikelihood function
motivation: the likelihood function connects motivation: the likelihood function connects the topics of the topics of probability probability (especially (especially independence), the practical application of independence), the practical application of random samplingrandom sampling, the , the normal distributionnormal distribution, , and the derivation of estimatorsand the derivation of estimators
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Likelihood functionLikelihood function
The joint density of the estimated The joint density of the estimated residuals can be written as:residuals can be written as:
If the sample of observations on the If the sample of observations on the dependent variable, y, and the dependent variable, y, and the independent variable, x, is random, independent variable, x, is random, then the observations are then the observations are independent of one another. If the independent of one another. If the errors are also identically errors are also identically distributed, f, i.e. i.i.d, thendistributed, f, i.e. i.i.d, then
)ˆ.....ˆˆˆ( 1210 neeeeg
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Likelihood functionLikelihood function Continued: If i.i.d., thenContinued: If i.i.d., then
If the residuals are normally If the residuals are normally distributed:distributed:
Thi is one of the assumptions of linear Thi is one of the assumptions of linear regression: errors are i.i.d normalregression: errors are i.i.d normal
then the joint distribution or likelihood then the joint distribution or likelihood function, L, can be written as:function, L, can be written as:
)ˆ()...ˆ(*)ˆ()ˆ...ˆˆ( 110110 nn efefefeeeg
2]/)0ˆ[(2/12 )2/1(),0(~)ˆ( iei eNef
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Likelihood functionLikelihood function
and taking natural logarithms of both and taking natural logarithms of both sides, where the logarithm is a sides, where the logarithm is a monotonically increasing function so that monotonically increasing function so that if lnL is maximized, so is L:if lnL is maximized, so is L:
1
0
22
2
]ˆ[)2/1(2/2/2
]/)0ˆ[(2/11
0110
*)2/1(*)/1(
)2/1()ˆ...ˆˆ(
n
ii
i
enn
en
in
eL
eeeegL
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The Natural Logarithm Function
-3
-2.5
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
0 1 2 3 4 5 6
x
lnx
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Log-LikelihoodLog-Likelihood
Taking the derivative of lnL with respect Taking the derivative of lnL with respect to either a-hat or b-hat yields the same to either a-hat or b-hat yields the same estimators for the parameters a and b estimators for the parameters a and b as with ordinary least squares, except as with ordinary least squares, except now we know the errors are normally now we know the errors are normally distributed.distributed.
21
0
22
1
0
222
]*ˆˆ[)2/1()2ln(*)2/(]ln[*)2/(ln
ˆ)2/1()2ln(*)2/(]ln[*)2/(ln
i
n
ii
n
ii
xbaynnL
ennL
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Log-LikelihoodLog-Likelihood Taking the derivative of lnL with Taking the derivative of lnL with
respect to sigma squared, we obtain an respect to sigma squared, we obtain an estimate for the variance of the errors:estimate for the variance of the errors:
andand
in practice we divide by n-2 since we in practice we divide by n-2 since we used up two degrees of freedom in used up two degrees of freedom in estimating a-hat and b-hat. estimating a-hat and b-hat.
1
0
2422 ˆ)/1(*)2/1()/1(*)2/(/lnn
iienL
nen
ii /]ˆ[ˆ
1
0
22
10
The sum of squared residuals The sum of squared residuals (estimated)(estimated) 2
ie
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SUMMARY OUTPUT
Regression StatisticsMultiple R 0.98873218R Square 0.97759133Adjusted R Square 0.97693225Standard Error 3.38354883Observations 36
ANOVAdf SS MS F Significance F
Regression 1 16981.07081 16981.07 1483.27 1.24E-29Residual 34 389.2456906 11.4484Total 35 17370.3165
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Lower 95.0%Upper 95.0%Intercept -0.2916205 1.014023514 -0.28759 0.775408 -2.35236 1.769122 -2.3523629 1.76912184X Variable 1 0.06329191 0.001643381 38.51324 1.24E-29 0.059952 0.066632 0.0599522 0.06663166
Goodness of fit
Regress CA State General Fund Expenditures on CA Personal Income, Lab Four
2ie
)2/(ˆ2 ne
n
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The Intuition Behind the The Intuition Behind the Table of Analysis of Variance Table of Analysis of Variance
(ANOVA)(ANOVA)
y = a + b*x + ey = a + b*x + e– the variation in the dependent the variation in the dependent
variable, y, is explained by either the variable, y, is explained by either the regression, a + b*x, or by the error, eregression, a + b*x, or by the error, e
The sample sum of deviations in y:The sample sum of deviations in y:
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0
][ yyn
ii
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Table of ANOVATable of ANOVA
Source Degrees ofFreedom
Sum ofSquares
MeanSquare
Regression(a + b*x
1
Error (e) n-2
Total (y) n-1
ANOVAdf SS MS F Significance F
Regression 1 16981.07081 16981.07 1483.27 1.24373E-29Residual 34 389.2456906 11.4484Total 35 17370.3165
21
0
][ yyn
ii
2ie
By difference
)2/(}ˆ{ 2 nei
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Test of the Significance of Test of the Significance of the Regression: F-testthe Regression: F-test
FF1,n-2 1,n-2 = explained mean = explained mean square/unexplained mean squaresquare/unexplained mean square
example: Fexample: F1, 34 1, 34 = = 16981.07 /11.8444 = 16981.07 /11.8444 = 1483.271483.27
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The UC BudgetThe UC Budget
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The UC BudgetThe UC Budget The UC Budget can be written as an The UC Budget can be written as an
identity:identity: UCBUD(t)= UC’s Gen. Fnd. Share(t)* UCBUD(t)= UC’s Gen. Fnd. Share(t)*
The Relative Size of CA Govt.(t)*CA The Relative Size of CA Govt.(t)*CA Personal Income(t)Personal Income(t)– where UC’s Gen. Fnd. Share=UCBUD/CA where UC’s Gen. Fnd. Share=UCBUD/CA
Gen. Fnd. ExpendituresGen. Fnd. Expenditures– where the Relative Size of CA Govt.= where the Relative Size of CA Govt.=
CA Gen. Fnd. Expenditures/CA Personal CA Gen. Fnd. Expenditures/CA Personal IncomeIncome
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Long Run Political TrendsLong Run Political Trends
UC’s Share of CA General Fund UC’s Share of CA General Fund ExpendituresExpenditures
UC Budget As Percent of CA Total General fund
3.51%
y = -0.0009x + 0.0698
R2 = 0.8311
0.00%
1.00%
2.00%
3.00%
4.00%
5.00%
6.00%
7.00%
8.00%
68-6
9
70-7
1
72-7
3
74-7
5
76-7
7
78-7
9
80-8
1
82-8
3
84-8
5
86-8
7
88-8
9
90-9
1
92-9
3
94-9
5
96-9
7
98-9
9
00-0
1
02-0
3
04-0
5
Fiscal Year
Pe
rce
nt
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UC’s Budget ShareUC’s Budget Share UC’s share of California General Fund UC’s share of California General Fund
expenditure shows a long run expenditure shows a long run downward trend. Like other public downward trend. Like other public universities across the country, UC is universities across the country, UC is becoming less public and more private. becoming less public and more private. Perhaps the most “private” of the public Perhaps the most “private” of the public universities is the University of universities is the University of Michigan. Increasingly, public Michigan. Increasingly, public universities are looking to build up their universities are looking to build up their endowments like private universities.endowments like private universities.
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Long Run Political Trends Long Run Political Trends The Relative size of California The Relative size of California
GovernmentGovernment– The Gann Iniative passed on the ballot The Gann Iniative passed on the ballot
in 1979. The purpose was to limit the in 1979. The purpose was to limit the size of state government so that it size of state government so that it would not grow in real terms per capita.would not grow in real terms per capita.
– Have expenditures on public goods by Have expenditures on public goods by the California state government grown the California state government grown faster than personal income?faster than personal income?
Ratio of General Fund Expenditures to Personal Income
6.01%
0.00%
1.00%
2.00%
3.00%
4.00%
5.00%
6.00%
7.00%
8.00%
68-6
9
70-7
1
72-7
3
74-7
5
76-7
7
78-7
9
80-8
1
82-8
3
84-8
5
86-8
7
88-8
9
90-9
1
92-9
3
94-9
5
96-9
7
98-9
9
00-0
1
02-0
3
04-0
5
Fiscal year
Pe
rce
nt
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The Relative Size of CA State The Relative Size of CA State Govt.Govt.
California General Fund Expenditure California General Fund Expenditure was growing relative to personal was growing relative to personal income until the Gann initiative income until the Gann initiative passed in 1979. Since then this ratio passed in 1979. Since then this ratio has declined, especially in the has declined, especially in the eighties and early nineties. After eighties and early nineties. After recovery from the last recession, this recovery from the last recession, this ratio recovered, but took a dive in ratio recovered, but took a dive in 2003-04.2003-04.
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Guessing the UC Budget Guessing the UC Budget for 2005-06for 2005-06
UC’s Budget Share, 04-05: 0.0351UC’s Budget Share, 04-05: 0.0351 Relative Size of CA State Govt.: Relative Size of CA State Govt.:
0.06010.0601 Forecast of CA Personal Income for Forecast of CA Personal Income for
2005-06 2005-06
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CA Personal Income, $ Nominal Billions, 1968-69 through 2004-05
0
200
400
600
800
1000
1200
1400
68-6
9
70-7
1
72-7
3
74-7
5
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7
78-7
9
80-8
1
82-8
3
84-8
5
86-8
7
88-8
9
90-9
1
92-9
3
94-9
5
96-9
7
98-9
9
00-0
1
02-0
3
04-0
5
Fiscal Year
Bil
lio
ns
$
25
26
27
28
ForecastPercent Percent Percent
2003 change 2004 change 2005 change
Personal income ($ billions) $1,197.6 3.7% $1,262.4 5.4% $1,333.1 5.6%
Nonfarm W&S employment (thousands) 14,408 -0.3% 14,525 0.8% 14,832 2.1% Natural resources and mining 22 -5.2% 22 -0.8% 22 -0.9% Construction 788 1.8% 824 4.5% 868 5.3% Manufacturing 1,543 -5.8% 1,517 -1.7% 1,538 1.4% High technology 399 -9.2% 388 -2.9% 394 1.7% Trade, transportation, & utilities 2,715 -0.3% 2,723 0.3% 2,747 0.9% Information 471 -5.2% 467 -0.9% 487 4.2% Financial activities 886 3.9% 904 2.0% 926 2.4% Professional and business services 2,114 0.0% 2,174 2.8% 2,247 3.4% Educational and health services 1,538 2.6% 1,576 2.5% 1,625 3.1% Leisure and hospitality 1,399 1.2% 1,424 1.8% 1,453 2.0% Other services 505 -0.1% 505 -0.1% 514 1.8% Government 2,427 -0.8% 2,391 -1.5% 2,408 0.7%
Figure ECON-4Selected California Economic Indicators
30
Guessing the UC Budget for Guessing the UC Budget for 2005-062005-06
UC’s Budget Share, 04-05: 0.0351UC’s Budget Share, 04-05: 0.0351 Relative Size of CA State Govt.: 0.0601Relative Size of CA State Govt.: 0.0601 Forecast of CA Personal Income for 2005-Forecast of CA Personal Income for 2005-
06: $ 1,333.1 B06: $ 1,333.1 B UCBUD(05-06) = 0.035*0.060*$1,333.1BUCBUD(05-06) = 0.035*0.060*$1,333.1B UCBUD(05-06) = $ 2.800 BUCBUD(05-06) = $ 2.800 B compares to UCBUD(04-05) = $ 2.670 Bcompares to UCBUD(04-05) = $ 2.670 B
UC Budget, General Fund Component, Millions of Nominal $
$2670.529
y = 81.613x + 19.497
R2 = 0.933
0
500
1000
1500
2000
2500
3000
3500
4000
68-6
9
70-7
1
72-7
3
74-7
5
76-7
7
78-7
9
80-8
1
82-8
3
84-8
5
86-8
7
88-8
9
90-9
1
92-9
3
94-9
5
96-9
7
98-9
9
00-0
1
02-0
3
04-0
5
Fiscal Year
Mil
lio
ns
$
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Guessing the UC Budget for Guessing the UC Budget for 2004-052004-05
UC’s Budget Share 03-04: 0.037UC’s Budget Share 03-04: 0.037 Relative Size of CA State Govt.: 0.065Relative Size of CA State Govt.: 0.065 Forecast of CA Personal Income for 2004-Forecast of CA Personal Income for 2004-
05: $ 1,231.5 B05: $ 1,231.5 B UCBUD(04-05) = 0.037*0.065*$1,231.5BUCBUD(04-05) = 0.037*0.065*$1,231.5B UCBUD(04-05) = $ 2.962 BUCBUD(04-05) = $ 2.962 B compares to UCBUD(03-04) = $ 3.039 Bcompares to UCBUD(03-04) = $ 3.039 B
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The Relative Size of CA The Relative Size of CA Govt.Govt.
Is it determined politically or by Is it determined politically or by economic factors?economic factors?
Economic Perspective: Engle Curve- Economic Perspective: Engle Curve- the variation of expenditure on a the variation of expenditure on a good or service with incomegood or service with income
lnCAGenFndExp = a + b lnCAPersInc lnCAGenFndExp = a + b lnCAPersInc +e +e
b is the elasticity of expenditure with b is the elasticity of expenditure with incomeincome
bCAPersIncpCAGenFndEx ln/ln
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The elasticity of The elasticity of expenditures with respect expenditures with respect
to incometo income
Note:Note:
So, in the log-log regression, So, in the log-log regression, lny = a + b*lnx + e, lny = a + b*lnx + e, the coefficient b is the elasticity of the coefficient b is the elasticity of y with respect to x.y with respect to x.
)/1(*
)/(*)/1(
/ln
CAPersIncb
CAPersIncpCAGenFndExpCAGenFndEx
CAPersIncpCAGenFndEx
35
CA State Govt Expenditures Vs. Personal Income
1993-94
2003=04
0
10
20
30
40
50
60
70
80
90
0 500 1000 1500
Personal Income, $ B
CA
Gen
. F
un
d $
B
Linear Regression
CA General Fund Expenditures Vs. CA Personal Income
4.331548797
y = 1.065x - 3.1777
R2 = 0.9891
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
3 3.5 4 4.5 5 5.5 6 6.5 7 7.5
CA Personal Income, ln $B
CA
Ge
n.
Fu
nd
. E
x.,
ln
$B
Log-Log Regression
37
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Is the Income Elasticity of Is the Income Elasticity of CA State Public Goods >1?CA State Public Goods >1?
Step # 1: Formulate the HypothesesStep # 1: Formulate the Hypotheses– HH0 0 : b = 1: b = 1
– HHa a : b > 1: b > 1
Step # 2: choose the test statisticStep # 2: choose the test statistic
Step # 3: If the null hypothesis were Step # 3: If the null hypothesis were true, what is the probability of true, what is the probability of getting a t-statistic this big?getting a t-statistic this big?
34.30189.0/)1065.1(/)]ˆ(ˆ[ ˆ b
bEbstatt
39
Appendix BTable 4p. B-9
5.0 % in the upper tail
t..050
35 1.69
40
Eviews Output
41
Regression ModelsRegression Models Trend AnalysisTrend 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 CurvesEngle Curves– ln y = a + b*lnx + eln y = a + b*lnx + e
Income Generating ProcessIncome Generating Process
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Returns Generating Returns Generating ProcessProcess
How does the rate of return on an asset How does the rate of return on an asset vary with the market rate of return?vary with the market rate of return?
rrii(t): rate of return on asset i(t): rate of return on asset i
rrff(t): risk free rate, assumed known for (t): risk free rate, assumed known for the period aheadthe period ahead
rrMM(t): rate of return on the market(t): rate of return on the market
[r[rii(t) - r(t) - rff00(t)] = a +b*[r(t)] = a +b*[rMM(t) - r(t) - rff
00(t)] + e(t) (t)] + e(t)
43
ExampleExample rrii(t): monthly rate of return on UC stock (t): monthly rate of return on UC stock
index fund, Sept., 1995 - Sept. 2003index fund, Sept., 1995 - Sept. 2003 rrff(t): risk free rate, assumed known for (t): risk free rate, assumed known for
the period ahead. Usually use Treasury the period ahead. Usually use Treasury Bill Rate. I used monthly rate of return Bill Rate. I used monthly rate of return on UC Money Market Fund on UC Money Market Fund http://atyourservice.ucop.edu/employehttp://atyourservice.ucop.edu/employees/retirement/performance.htmles/retirement/performance.html
44
Example (cont.)Example (cont.)
rrMM(t): rate of return on the market. (t): rate of return on the market. I used the monthly change in the I used the monthly change in the logarithm of the total return logarithm of the total return (dividends reinvested)*100. (dividends reinvested)*100. http://research.stlouisfed.org/fred2http://research.stlouisfed.org/fred2//
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Returns Generating Process Time Series Data
-20
-15
-10
-5
0
5
10
15
Se
p-9
5
Se
p-9
6
Se
p-9
7
Se
p-9
8
Se
p-9
9
Se
p-0
0
Se
p-0
1
Se
p-0
2
Se
p-0
3
Date
Mo
thly
Ra
te o
f R
etu
rn
UC Equity Fund
Standard & Poors 500
UC Money Market Fund
46
Returns Generating Process, Sept. 95-Sept. 03
-20.00
-15.00
-10.00
-5.00
0.00
5.00
10.00
15.00
-15 -10 -5 0 5 10
Standard & Poors 500, Net
UC
Sto
ck In
dex
Fu
nd
, Net
47
-13.35, 16.09;Ucnet,
S&Pnet
y = 1.0601x - 0.106
R2 = 0.9136
-20.00
-15.00
-10.00
-5.00
0.00
5.00
10.00
15.00
-15 -10 -5 0 5 10
Watch Excel on xy plots!
True x axis: UC Net
48
49
Returns Generating Process
y = 1.0601x - 0.106
R2 = 0.9136
-20.00
-15.00
-10.00
-5.00
0.00
5.00
10.00
15.00
-15 -10 -5 0 5 10
Standard & Poors 500, Net
UC
Sto
ck I
nd
ex F
un
d,
Net
Really the Regression of S&P on UC
50
SUMMARY OUTPUT
Regression StatisticsMultiple R 0.95580613R Square 0.91356536Adjusted R Square 0.91265552Standard Error 1.31011043Observations 97
ANOVAdf SS MS F Significance F
Regression 1 1723.42 1723.42 1004.096 2.65348E-52Residual 95 163.057 1.716389Total 96 1886.477
CoefficientsStandard Error t Stat P-value Lower 95% Upper 95%Lower 95.0%Upper 95.0%Intercept 0.12800497 0.13335 0.959915 0.339535 -0.1367287 0.392739 -0.13673 0.392739X Variable 1 0.86177094 0.027196 31.68748 2.65E-52 0.807780204 0.915762 0.80778 0.915762
51
Is the beta for the UC Is the beta for the UC Stock Index Fund <1?Stock Index Fund <1?
Step # 1: Formulate the HypothesesStep # 1: Formulate the Hypotheses– HH0 0 : b = 1: b = 1
– HHa a : b < 1: b < 1
Step # 2: choose the test statisticStep # 2: choose the test statistic
Step # 3: If the null hypothesis were Step # 3: If the null hypothesis were true, what is the probability of true, what is the probability of getting a t-statistic this big?getting a t-statistic this big?
4.6027.0/)1862.0(/)]ˆ(ˆ[ ˆ b
bEbstatt
52
Appendix BTable 4p. B-9
5.0 % in the lower tail
t..050
95 1.66
53
-15
-10
-5
0
5
10
-20 -10 0 10
SPNET
UC
ST
OC
KN
ET
Returns Generating ProcessEViews Chart
54
Midterm 2001Midterm 2001
55
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?
Q. 4
d. What is the ratio of the explained mean square to the unexplained mean square?
56
Calfifornia General Fund Expenditures Vs. California Personal Income, Billions of Nominal $
y = 0.066x - 1.1974
R2 = 0.981
0
10
20
30
40
50
60
70
80
90
0 200 400 600 800 1000 1200 1400
Personal Income
Gen
Fu
nd
Exp
end
itu
res
Q 4
Figure 4-1: California General Fund Expenditures Versus California Personal Income, both in Billions of Nominal Dollars
57
Regression StatisticsMultiple R 0.9904673R Square 0.9810255Adjusted R Square 0.9804325Standard Error 2.9988336Observations 34
ANOVA
df SS MS F SignificanceF
Regression 1 14878.68965 14878.69 1654.47398 3.98668E-29Residual 32 287.7761003 8.993003Total 33 15166.46575
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%Intercept -1.197411 0.927956018 -1.29037 0.20616709 -3.08759378 0.6927721X Variable 1 0.0659894 0.001622349 40.67523 3.9867E-29 0.062684796 0.069294
Q 4Table 4-1: Summary Output
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