1 Econ 240A Power 17. 2 Outline Review Projects 3 Review: Big Picture 1 #1 Descriptive Statistics...

transcript

Econ 240AEcon 240A

Power 17Power 17

OutlineOutline

• Review

• Projects

Review: Big Picture 1Review: Big Picture 1• #1 Descriptive Statistics

– Numericalcentral tendency: mean, median, modedispersion: std. dev., IQR, max-minskewnesskurtosis

– Graphical• Bar plots• Histograms• Scatter plots: y vs. x• Plots of a series against time (traces)

Question: Is (are) the variable (s) normal?

Review: Big Picture 2Review: Big Picture 2

• # 2 Exploratory Data Analysis– Graphical

• Stem and leaf diagrams• Box plots• 3-D plots

Review: Big Picture 3Review: 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

),(~:,//ˆ npBxwherenxnsuccessesp

Review: Big Picture 4Review: Big Picture 4• Inferential Statistics, Cont.

– The distribution of the sample mean is different than the distribution of the random variable

• Central limit theorem

– Confidence intervals for the unknown population mean

nxxExz x //][/][

95.0]/96.1/96.1[ nxnxp

Review: Big Picture 5Review: Big Picture 5• Inferential Statistics

– If population variance is unknown, use sample standard deviation s, and Student’s t-distribution

– Hypothesis tests

– 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

95.0]//[ 025.0025.0 nstxnstxp

)//(][,0:,0:0 nsxExtHH A

Review: Big Picture 6Review: Big Picture 6• Regression, Bivariate and Multivariate

– Time series• 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 + 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

Review: Big Picture 7Review: Big Picture 7• Regression, Bivariate and Multivariate

– Cross-section• 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

Review: Big Picture 8Review: 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

Review: Big Picture 9Review: 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)

SummarySummary

• 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

ProjectsProjects

• Learning by doing

• Learning from one another

Control of Social ProblemsControl of Social Problems

• HIV/AIDS

HIV/AIDS HIV/AIDS What can we do to What can we do to

prevent it?!prevent it?!Group 4:Group 4:

Pinar SahinPinar SahinDarren EganDarren EganDavid WhiteDavid WhiteYuan YuanYuan Yuan

Miguel Delgado HelleseterMiguel Delgado HelleseterDavid RhodesDavid Rhodes

MorbidityPer capita

Abatement ExpenditurePer Capita

Control Curve

The Problem is Controllable

Is there a relationship?Is there a relationship?

#of HIV vs CDC expenditure

y = -78.53x + 104053

R2 = 0.610

500 600 700 800 900 1000

CDC expenditure ($million)

regression of HIV infection and CDC expenditureDependent Variable: INFECTMethod: Least SquaresDate: 11/21/03 Time: 17:10Sample: 1 9Included observations: 9

Variable Coefficient Std. Error t-Statistic Prob.

CDCMONEY -78.5302 23.73235 -3.3089951 0.012959C 104053.4 17180.64 6.05643533 0.000513

R-squared 0.610016 Mean dependent var 48122.44Adjusted R-squared 0.554304 S.D. dependent var 13830.59S.E. of regression 9233.366 Akaike info criterion 21.29216Sum squared resid 5.97E+08 Schwarz criterion 21.33599Log likelihood -93.8147 F-statistic 10.94945Durbin-Watson stat 0.822936 Prob(F-statistic) 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• R^2 is .61, which is decent Group 4

HIV/AIDS cases vs. per capita HIV/AIDS cases vs. per capita funding per statefunding per state

Dependent Variable: INFECTMethod: Least SquaresDate: 11/21/03 Time: 18:07Sample: 1 50Included observations: 50

Variable Coefficient Std. Error t-Statistic Prob.

PERCAPFUND 5.605922 2.044885 2.74143582 0.008568C 4.614507 2.447325 1.88553113 0.065418

R-squared 0.135376 Mean dependent var 10.518Adjusted R-squared 0.117363 S.D. dependent var 8.751926S.E. of regression 8.222325 Akaike info criterion 7.090761Sum squared resid 3245.118 Schwarz criterion 7.167242Log likelihood -175.269 F-statistic 7.51547Durbin-Watson stat 1.952071 Prob(F-statistic) 0.008568

# of cases VS per Capital funding per state

0 2 4 6 8 10 12

per Capita funding per state

Group 4

Controlling Social ProblemsControlling Social Problems

• This same analytical framework works for various social ills– Morbidity per capita– Offenses per capita– Pollution per capita

MorbidityPer capita

Control Curve

Offenses Per Capita

PollutionPer Capita

Source: Report to the Nation on Crime and Justice

control

Causalfactors

MorbidityPer capita

Control Curve

The Problem is Not Controllable

Controllability is an empiricalquestion that we want to answer

Optimizing BehaviorOptimizing Behavior

• Cost Curve: – Cost = Damages from Morbidity + Abatement

Expenditures– C = p*M + Exp

Cost CurveCost Curve

Abatement Exp

Morbidity M

C = p*M + Exp

Exp=0, M=C/p

M=0, Exp=C

Family of Cost CurvesFamily of Cost Curves

Abatement Exp

Morbidity M

Higher Cost

Lower Cost

MorbidityPer capita

Control Curve

The Problem is Not Controllable: Don’t Throw Money At It

Higher Cost

Lowest Cost

Optimum: ZeroAbatement

MorbidityPer capita

Control Curve

The Problem is Controllable: Optimum Expenditures

Lowest Attainable Cost

Optimum

Higher Cost Curve

MorbidityPer capita

Control Curve

Optimum

Higher Cost Curve

Spend too MuchBut Morbidity Is Low

MorbidityPer capita

Control Curve

Optimum

Higher Cost Curve

Spend Too Little, Morbidity Is Too High

Economic ParadigmEconomic Paradigm

• Step One: Describe the feasible alternatives

MorbidityPer capita

Control Curve

• Step Two: Value the alternatives

Cost CurveCost Curve

Abatement Exp

Morbidity M

C = p*M + Exp

Exp=0, M=C/p

M=0, Exp=C

• Step Two: Value the alternatives

• Step Three: Optimize, pick the lowest cost alternative

MorbidityPer capita

Control Curve

Optimum

Higher Cost Curve

MorbidityPer capita

Control Curve

The Problem is Controllable: Family of Control Curves

Control CurveAnother TimeOr Another Place

Behind the Control CurveBehind 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

Morbidity GenerationMorbidity Generation

Morbidity, M

Sex-ed

M = f(Sex-ed, RB)

Morbidity GenerationMorbidity Generation

Morbidity, M

Sex-ed

M = f(Sex-ed, RB)

Riskier behavior

Production FunctionProduction Function

Sex-ed

Labor, L

Expenditure On Wage Bill Expenditure On Wage Bill (Abatement)(Abatement)

Labor, L

Exp = w*L

Control CurveControl Curve

Labor,L

Exp = w*L

Sex-ed

Sex-ed = g(L)

Morbidity, M

M = f(Sex-ed, RB)

ExpenditurefunctionProduction function

Morbidity Generation

Labor,L

Exp = w*L

Sex-ed

Sex-ed = g(L)

Morbidity, M

M = f(Sex-ed, RB)

Labor,L

Exp = w*L

Sex-ed

Sex-ed = g(L)

Morbidity, M

M = f(Sex-ed, RB)

Labor,L

Exp = w*L

Sex-ed

Sex-ed = g(L)

Morbidity, M

M = f(Sex-ed, RB)

Labor,L

Exp = w*L

Sex-ed

Sex-ed = g(L)

Morbidity, M

M = f(Sex-ed, RB)

Higher Risky Behavior

ExerciseExercise

• Derive the control curve for the jurisdiction with more risky behavior

Expansion PathExpansion 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

MorbidityPer capita

Control Curve

Expansion path

MorbidityPer capita

Control Curve

Expansion path

Econometric IssuesEconometric 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

• 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

MorbidityPer capita

Control Curve

Expansion path

California Expenditure VS. California Expenditure VS. ImmigrationImmigration

By: Daniel Jiang, Keith Cochran, By: Daniel Jiang, Keith Cochran, Justin Adams, Hung Lam, Steven Justin Adams, Hung Lam, Steven

Carlson, Gregory WiefelCarlson, Gregory Wiefel

Fall 2003Fall 2003

Immigration VS ExpenditureImmigration VS Expenditure

Immigration VS Expenditure

y = 0.2363x + 814.96

R2 = 0.3733

100000 150000 200000 250000 300000 350000

Immigration

SimultaneitySimultaneity

Immigration

CA EXP

Expenditurefunction

ImmigrationFunction

Simultaneity ConceptsSimultaneity 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

TheoryTheory

• Minimize Cost, C = p*M + Exp• Subject to the control curve, M = h(Exp, RB)• Lagrangian, La = p*M + Exp + [M-h(RB, Exp]

• Slope of the control curve = slope of cost curve

ExphExpLa

ModelModel

• 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

Model Cont.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

Model Cont.Model Cont.

• Expansion Path– M/exp = k*ez

– lnM = -lnexp + lnk + z– Or y1 = constant2 – y2 + error2

Reduced FormReduced 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

1 Econ 240A Power 17. 2 Outline Review Projects 3 Review: Big Picture 1 #1 Descriptive Statistics...

Documents