1 Econ 240A Power 17

2 Outline Review Projects

3 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?

4 Review: Big Picture 2 # 2 Exploratory Data Analysis –Graphical Stem and leaf diagrams Box plots 3-D plots

5 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

6 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 –Confidence intervals for the unknown population mean

7 Review: 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

8 Review: 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*t 2 + e(t) Elasticity estimation: lny(t) = a + b*lnx(t) +e(t) Returns Generating Process: r i (t) = c +  r M (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

9 Review: 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

10 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

11 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 , –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)

12 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

13 Projects Learning by doing Learning from one another

14 Control of Social Problems HIV/AIDS

15 HIV/AIDS What can we do to prevent it?! Group 4: Pinar Sahin Darren Egan David White Yuan Yuan Miguel Delgado Helleseter David Rhodes

16 Morbidity Per capita Abatement Expenditure Per Capita Control Curve The Problem is Controllable

17 Is there a relationship? Both the t and F statistics are significant R^2 is.61, which is decent Group 4

18 HIV/AIDS cases vs. per capita funding per state Group 4

19 Controlling Social Problems This same analytical framework works for various social ills –Morbidity per capita –Offenses per capita –Pollution per capita

20 Morbidity Per capita Abatement Expenditure Per Capita Control Curve The Problem is Controllable Offenses Per Capita Pollution Per Capita

Source: Report to the Nation on Crime and Justice

control Causal factors

23 Morbidity Per capita Abatement Expenditure Per Capita Control Curve The Problem is Not Controllable Controllability is an empirical question that we want to answer

24 Optimizing Behavior Cost Curve: –Cost = Damages from Morbidity + Abatement Expenditures –C = p*M + Exp

25 Cost Curve Abatement Exp Morbidity M C = p*M + Exp Exp=0, M=C/p M=0, Exp=C

26 Family of Cost Curves Abatement Exp Morbidity M Higher Cost Lower Cost

27 Morbidity Per capita Abatement Expenditure Per Capita Control Curve The Problem is Not Controllable: Don’t Throw Money At It Higher Cost Lowest Cost Optimum: Zero Abatement

28 Morbidity Per capita Abatement Expenditure Per Capita Control Curve The Problem is Controllable: Optimum Expenditures Lowest Attainable Cost Optimum Higher Cost Curve

29 Morbidity Per capita Abatement Expenditure Per Capita Control Curve The Problem is Controllable: Optimum Expenditures Lowest Attainable Cost Optimum Higher Cost Curve Spend too Much But Morbidity Is Low

30 Morbidity Per capita Abatement Expenditure Per Capita Control Curve The Problem is Controllable: Optimum Expenditures Lowest Attainable Cost Optimum Higher Cost Curve Spend Too Little, Morbidity Is Too High

31 Economic Paradigm Step One: Describe the feasible alternatives

32 Morbidity Per capita Abatement Expenditure Per Capita Control Curve The Problem is Controllable

33 Economic Paradigm Step One: Describe the feasible alternatives Step Two: Value the alternatives

34 Cost Curve Abatement Exp Morbidity M C = p*M + Exp Exp=0, M=C/p M=0, Exp=C

35 Economic Paradigm Step One: Describe the feasible alternatives Step Two: Value the alternatives Step Three: Optimize, pick the lowest cost alternative

36 Morbidity Per capita Abatement Expenditure Per Capita Control Curve The Problem is Controllable: Optimum Expenditures Lowest Attainable Cost Optimum Higher Cost Curve

37 Morbidity Per capita Abatement Expenditure Per Capita Control Curve The Problem is Controllable: Family of Control Curves Control Curve Another Time Or Another Place

38 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

39 Morbidity Generation Morbidity, M Sex-ed M = f(Sex-ed, RB)

40 Morbidity Generation Morbidity, M Sex-ed M = f(Sex-ed, RB) Riskier behavior

41 Production Function Sex-ed Labor, L

42 Expenditure On Wage Bill (Abatement) Labor, L Exp Exp = w*L

43 Control Curve Labor,L Exp Exp = w*L Sex-ed Sex-ed = g(L) Morbidity, M M = f(Sex-ed, RB) Expenditure function Production function Morbidity Generation

44 Control Curve Labor,L Exp Exp = w*L Sex-ed Sex-ed = g(L) Morbidity, M M = f(Sex-ed, RB)

45 Control Curve Labor,L Exp Exp = w*L Sex-ed Sex-ed = g(L) Morbidity, M M = f(Sex-ed, RB)

46 Control Curve Labor,L Exp Exp = w*L Sex-ed Sex-ed = g(L) Morbidity, M M = f(Sex-ed, RB)

47 Control Curve Labor,L Exp Exp = w*L Sex-ed Sex-ed = g(L) Morbidity, M M = f(Sex-ed, RB) Higher Risky Behavior

48 Exercise Derive the control curve for the jurisdiction with more risky behavior

49 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

50 Morbidity Per capita Abatement Expenditure Per Capita Control Curve The Problem is Controllable: Family of Control Curves Control Curve Another Time Or Another Place Expansion path

51 Morbidity Per capita Abatement Expenditure Per Capita Control Curve The Problem is Controllable: Family of Control Curves Control Curve Another Time Or Another Place Expansion path M Exp

52 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

53 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

54 Morbidity Per capita Abatement Expenditure Per Capita Control Curve The Problem is Controllable: Family of Control Curves Control Curve Another Time Or Another Place Expansion path M Exp

55 California Expenditure VS. Immigration By: Daniel Jiang, Keith Cochran, Justin Adams, Hung Lam, Steven Carlson, Gregory Wiefel Fall 2003

Immigration VS Expenditure

57 Simultaneity Immigration CA EXP Expenditure function Immigration Function

58 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

59 Theory 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

60 Model Production Function: Cobb-Douglas –Sex-ed = a*L b *e u b>0 Abatement Expenditure –Exp = w*L Morbidity Abatement –M = d*sex-ed m *RB n *e v m 0

61 Model Cont. Combine production function, expenditure and morbidity abatement functions to obtain control function –M = d*[a*L b *e u ] m *RB n *e v –M = d*[a*(exp/w) b *e u ] m *RB n *e v –M = d* a m * exp b*m * w -b*m *RB n *e u*m *e v –lnM = ln(d*a m ) + 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

62 Model Cont. Expansion Path –M/exp = k*e z –lnM = -lnexp + lnk + z –Or y1 = constant2 – y2 + error2

63 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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