Download Power 17 - UCSB Economics

Econ 240A
Power 17
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Outline
• Review
• Projects
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Review: Big Picture 1
• #1 Descriptive Statistics
– Numerical
central tendency: mean, median, mode
dispersion: std. dev., IQR, max-min
skewness
kurtosis
– Graphical
•
•
•
•
Bar plots
Histograms
Scatter plots: y vs. x
Plots of a series against time (traces)
Question: Is (are) the variable (s) normal?
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Review: Big Picture 2
• # 2 Exploratory Data Analysis
– Graphical
• Stem and leaf diagrams
• Box plots
• 3-D plots
4
Review: Big Picture 3
• #3 Inferential statistics
– Random variables
– Probability
– Distributions
• Discrete: Equi-probable (uniform), binomial, Poisson
– Probability density, Cumulative Distribution Function
• Continuous: normal, uniform, exponential
– Density, CDF
• Standardized Normal, z~N(0,1)
– Density and CDF are tabulated
• Bivariate normal
– Joint density, marginal distributions, conditional distributions
– Pearson correlation coefficient, iso-probability contours
– Applications: sample proportions from polls
pˆ  successes / n  x / n, where : x ~ B( p, n)
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Review: Big Picture 4
• Inferential Statistics, Cont.
– The distribution of the sample mean is different than the
distribution of the random variable
• Central limit theorem
z  [ x  Ex ] /  x  [ x   ] /  / n
– Confidence intervals for the unknown population mean
p[ x  1.96 / n    x  1.96 / n ]  0.95
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Review: Big Picture 5
• Inferential Statistics
– If population variance is unknown, use sample standard
deviation s, and Student’s t-distribution
p[ x  t0.025s / n    x  t0.025s / n ]  0.95
– Hypothesis tests
H 0 :   0, H A :   0, t  [ x  Ex ] /( s / n )
– Decision theory: minimize the expected costs of errors
• Type I error, Type II error
– Non-parametric statistics
• techniques of inference if variable is not normally distributed
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Review: Big Picture 6
• Regression, Bivariate and Multivariate
– Time series
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•
•
•
•
•
Linear trend: y(t) = a + b*t +e(t)
Exponential trend: ln y(t) = a +b*t +e(t)
Quadratic trend: y(t) = a + b*t +c*t2 + e(t)
Elasticity estimation: lny(t) = a + b*lnx(t) +e(t)
Returns Generating Process: ri(t) = c + b*rM(t) + e(t)
Problem: autocorrelation
–
–
–
–
Diagnostic: Durbin-Watson statistic
Diagnostic: inertial pattern in plot(trace) of residual
Fix-up: Cochran-Orcutt
Fix-up: First difference equation
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Review: Big Picture 7
• Regression, Bivariate and Multivariate
– Cross-section
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•
•
•
•
•
•
•
Linear: y(i) = a + b*x(i) + e(i), i=1,n ; b=dy/dx
Elasticity or log-log: lny(i) = a + b*lnx(i) + e(i); b=(dy/dx)/(y/x)
Linear probability model: y=1 for yes, y=0 for no; y =a + b*x +e
Probit or Logit probability model
Problem: heteroskedasticity
Diagnostic: pattern of residual(or residual squared) with y and/or x
Diagnostic: White heteroskedasticity test
Fix-up: transform equation, for example, divide by x
– Table of ANOVA
• Source of variation: explained, unexplained, total
• Sum of squares, degrees of freedom, mean square, F test
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Review: Big Picture 8
• Questions: quantitative dependent, qualitative
explanatory variables
– Null: No difference in means between two or more
populations (groups), One Factor
• Graph
• Table of ANOVA
• Regression Using Dummies
– Null: No difference in means between two or more
populations (groups), Two Factors
• Graph
• Table of ANOVA
• Comparing Regressions Using Dummies
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Review: Big Picture 9
• Cross-classification: nominal categories, e.g.
male or female, ordinal categories e.g. better
or worse, or quantitative intervals e.g. 13-19,
20-29
– Two Factors mxn; (m-1)x(n-1) degrees of freedom
– Null: independence between factors; expected
number in cell (i,j) = p(i)*p(j)*n
– Pearson Chi- square statistic = sum over all i, j of
[observed(i, j) – expected(i, j)]2 /expected(i, j)
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Summary
• Is there any relationship between 2 or
more variables
– quantitative y and x: graphs and regression
– Qualitative binary y and quantitative x:
probability model, linear or non-linear
– Quantitative y and qualitative x: graphs and
Tables of ANOVA, and regressions with
indicator variables
– Qualitative y and x: Contingency Tables
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Projects
• Learning by doing
• Learning from one another
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Control of Social Problems
• HIV/AIDS
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HIV/AIDS
What can we do to
prevent it?!
Group 4:
Pinar Sahin
Darren Egan
David White
Yuan Yuan
Miguel Delgado Helleseter
David Rhodes
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The Problem is Controllable
Control Curve
Morbidity
Per capita
Abatement Expenditure
Per Capita
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Is there a relationship?
regression of HIV infection and CDC expenditure
Dependent Variable: INFECT
Method: Least Squares
Date: 11/21/03 Time: 17:10
Sample: 1 9
Included observations: 9
#of HIV vs CDC expenditure
number of morbidity
80000
70000
60000
Variable
Coefficient Std. Error t-Statistic
Prob.
50000
40000
30000
20000
y = -78.53x + 104053
10000
2
R = 0.61
0
500
600
700
800
CDC expenditure ($million)
900
1000
CDCMONEY
C
-78.5302 23.73235 -3.3089951 0.012959
104053.4 17180.64 6.05643533 0.000513
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.610016
0.554304
9233.366
5.97E+08
-93.8147
0.822936
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
48122.44
13830.59
21.29216
21.33599
10.94945
0.012959
both of t and F statistic are significant. R-squared is 0.61, which is also fine.
• Both the t and F statistics are significant
Group 4
• R^2 is .61, which is decent
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HIV/AIDS cases vs. per capita
funding per state
Dependent Variable: INFECT
Method: Least Squares
Date: 11/21/03 Time: 18:07
Sample: 1 50
Included observations: 50
160
Coefficient Std. Error t-Statistic
Prob.
PERCAPFUND
C
5.605922 2.044885 2.74143582 0.008568
4.614507 2.447325 1.88553113 0.065418
R-squared
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
Durbin-Watson stat
0.135376
0.117363
8.222325
3245.118
-175.269
1.952071
Mean dependent var
S.D. dependent var
Akaike info criterion
Schwarz criterion
F-statistic
Prob(F-statistic)
10.518
8.751926
7.090761
7.167242
7.51547
0.008568
# of cases per 100,000
people
Variable
# of cases VS per Capital funding per state
140
120
100
80
60
40
20
0
0
2
4
6
8
10
12
per Capita funding per state
Group 4
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Controlling Social Problems
• This same analytical framework works for
various social ills
– Morbidity per capita
– Offenses per capita
– Pollution per capita
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The Problem is Controllable
Control Curve
Morbidity
Per capita
Offenses
Per Capita
Pollution
Per Capita
Abatement Expenditure
Per Capita
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Source: Report to the Nation on Crime and Justice
Causal
factors
control
Source: Report to the Nation on Crime and Justice
The Problem is Not Controllable
Morbidity
Per capita
Controllability is an empirical
question that we want to answer
Control Curve
Abatement Expenditure
Per Capita
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Optimizing Behavior
• Cost Curve:
– Cost = Damages from Morbidity + Abatement
Expenditures
– C = p*M + Exp
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Cost Curve
C = p*M + Exp
Exp=0, M=C/p
Morbidity M
M=0, Exp=C
Abatement Exp
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Family of Cost Curves
Higher Cost
Morbidity M
Lower Cost
Abatement Exp
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The Problem is Not Controllable: Don’t Throw Money At It
Morbidity
Per capita
Control Curve
Optimum: Zero
Abatement
Higher Cost
Lowest Cost
Abatement Expenditure
Per Capita
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The Problem is Controllable: Optimum Expenditures
Control Curve
Morbidity
Per capita
Higher Cost Curve
Optimum
Lowest Attainable Cost
Abatement Expenditure
Per Capita
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The Problem is Controllable: Optimum Expenditures
Control Curve
Morbidity
Per capita
Higher Cost Curve
Optimum
Spend too Much
But Morbidity Is Low
Lowest Attainable Cost
Abatement Expenditure
Per Capita
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The Problem is Controllable: Optimum Expenditures
Morbidity
Per capita
Spend Too Little, Morbidity Is Too High
Higher Cost Curve
Optimum
Control Curve
Lowest Attainable Cost
Abatement Expenditure
Per Capita
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Economic Paradigm
• Step One: Describe the feasible
alternatives
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The Problem is Controllable
Control Curve
Morbidity
Per capita
Abatement Expenditure
Per Capita
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Economic Paradigm
• Step One: Describe the feasible
alternatives
• Step Two: Value the alternatives
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Cost Curve
C = p*M + Exp
Exp=0, M=C/p
Morbidity M
M=0, Exp=C
Abatement Exp
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Economic Paradigm
• Step One: Describe the feasible
alternatives
• Step Two: Value the alternatives
• Step Three: Optimize, pick the lowest cost
alternative
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The Problem is Controllable: Optimum Expenditures
Control Curve
Morbidity
Per capita
Higher Cost Curve
Optimum
Lowest Attainable Cost
Abatement Expenditure
Per Capita
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The Problem is Controllable: Family of Control Curves
Control Curve
Morbidity
Per capita
Control Curve
Another Time
Or Another Place
Abatement Expenditure
Per Capita
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Behind the Control Curve
• Morbidity Generation
– M = f(sex-ed, risky behavior)
– M = f(sex-ed, RB)
• Producing Morbidity Abatement
– Sex-ed = g(labor)
– Sex-ed = g(L)
• Abatement Expendtiture
– Exp = wage*labor = w*L
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Morbidity Generation
Morbidity, M
M = f(Sex-ed, RB)
Sex-ed
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Morbidity Generation
Riskier behavior
Morbidity, M
M = f(Sex-ed, RB)
Sex-ed
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Production Function
Sex-ed
Labor, L
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Expenditure On Wage Bill
(Abatement)
Exp
Exp = w*L
Labor, L
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Control Curve
Morbidity Generation
Morbidity, M
M = f(Sex-ed, RB)
Sex-ed
Exp
Sex-ed = g(L)
Exp = w*L
Production function
Labor,L
Expenditure
function
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Control Curve
Morbidity, M
M = f(Sex-ed, RB)
Sex-ed
Exp
Sex-ed = g(L)
Exp = w*L
Labor,L
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Control Curve
Morbidity, M
M = f(Sex-ed, RB)
Sex-ed
Exp
Sex-ed = g(L)
Exp = w*L
Labor,L
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Control Curve
Morbidity, M
M = f(Sex-ed, RB)
Sex-ed
Exp
Sex-ed = g(L)
Exp = w*L
Labor,L
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Control Curve
Morbidity, M
Higher Risky Behavior
M = f(Sex-ed, RB)
Sex-ed
Exp
Sex-ed = g(L)
Exp = w*L
Labor,L
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Exercise
• Derive the control curve for the jurisdiction
with more risky behavior
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Expansion Path
• Assume the family of control curves is
nested, i.e. have the same slope along
any ray from the origin
• Assume all jurisdictions place the same
value, p, on morbidity
• Assume all jurisdictions are optimizing
• Then the expansion path is a ray from the
origin
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The Problem is Controllable: Family of Control Curves
Control Curve
Morbidity
Per capita
Control Curve
Another Time
Or Another Place
Expansion path
Abatement Expenditure
Per Capita
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The Problem is Controllable: Family of Control Curves
Control Curve
Morbidity
Per capita
Control Curve
Another Time
Or Another Place
M
Expansion path
Exp
Abatement Expenditure
Per Capita
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Econometric Issues
• Two Relationships
– Control curve: M = h(exp, RB)
– Expansion path: M/EXP = k
• Variation in risky behavior from one
jurisdiction to the next shifts the control
curve and traces out (identifies) the
expansion path
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• Unless price, technology, or optimizing
behavior changes from jurisdiction to
jurisdiction, there will not be enough
movement in the expansion path to trace
out(identify) the control curve
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The Problem is Controllable: Family of Control Curves
Control Curve
Morbidity
Per capita
Control Curve
Another Time
Or Another Place
M
Expansion path
Exp
Abatement Expenditure
Per Capita
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California Expenditure VS.
Immigration
By: Daniel Jiang, Keith Cochran,
Justin Adams, Hung Lam, Steven
Carlson, Gregory Wiefel
Fall 2003
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Immigration VS Expenditure
Immigration VS Expenditure
90000
80000
Expenditure
70000
60000
50000
y = 0.2363x + 814.96
R2 = 0.3733
40000
30000
20000
100000
150000
200000
250000
Immigration
300000
350000
Simultaneity
Immigration
Function
CA EXP
Expenditure
function
Immigration
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Simultaneity Concepts
• Jointly determined: Morbidity and abatement
expenditure are jointly determined by the
control curve and the cost curve
• Morbidity and abatement expenditure are
referred to as endogenous variables
• Risky behavior is an exogenous variable
• For a 2-equation simultaneous system, at
least one exogenous variable must be
excluded from a behavioral (structural)
equation to identify it
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Theory
• Minimize Cost, C = p*M + Exp
• Subject to the control curve, M = h(Exp, RB)
• Lagrangian, La = p*M + Exp + l[M-h(RB, Exp]
La / M  p  l  0
La / Exp  1  lh / Exp  0
 h / Exp  1 / l  1 / p
• Slope of the control curve = slope of cost curve
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Model
• Production Function: Cobb-Douglas
– Sex-ed = a*Lb *eu b>0
• Abatement Expenditure
– Exp = w*L
• Morbidity Abatement
– M = d*sex-edm *RBn *ev m<0, n>0
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Model Cont.
• Combine production function, expenditure and
morbidity abatement functions to obtain control
function
–
–
–
–
M = d*[a*Lb *eu ]m *RBn *ev
M = d*[a*(exp/w)b *eu ]m *RBn *ev
M = d* am * expb*m * w-b*m *RBn *eu*m *ev
lnM = ln(d*am) + b*m lnexp –b*m lnw + n* lnRB +
(u*m + v)
– Or assuming w is constant:
y1 = constant1 + b*m y2 + n x + error1
– We would like to show that b*m is negative, i.e. that
morbidity is controllable
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Model Cont.
• Expansion Path
– M/exp = k*ez
– lnM = -lnexp + lnk + z
– Or y1 = constant2 – y2 + error2
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Reduced Form
• Solve for y1 and y2, the two endogenous
variables
• y1 = [constant1 + constant2]/(1-b*m) + n/(1-b*m)
x + (error1 + b*m error2)/(1-b*m)
• y2 ={ -[constant1 + constant2]/(1-b*m) +
constant2} - n/(1-b*m) x + {-(error1 + b*m
error2)/(1-b*m) + error2}
• There is no way to get from the estimated
parameter on x, n/(1 – b*m) to n or b*m, the
parameters of interest for the control function
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