Measuring and Valuing Mobility
|
|
- Thomasina Porter
- 6 years ago
- Views:
Transcription
1 Measuring and Valuing Aggregate Frank London School of Economics January 2011
2 Select bibliography Aggregate, F. A. and E. Flachaire, (2011) Measuring, PEP Paper 8, STICERD, LSE D Agostino, M. and V. Dardanoni (2009). The measurement of rank. Journal of Economic 144, Demuynck, T. and D. Van de gaer (2010). Rank dependent relative measures. Ghent University, Working Paper, 10/628, Fields, G. S. (2007).. Cornell University, I.L.R School Working Paper. Gottschalk, P. and E. Spolaore (2002). On the evaluation of economic. Review of Economic Studies 69, Van Kerm, P. (2004).What lies behind income? Reranking and distributional change in Belgium, Western Germany and the USA. Economica 71,
3 Approaches to Aggregate Model of often depends on application: income or wealth wage educational in terms of social status Measurement addressed from different standpoints in relation to a specific dynamic model as an abstract distributional concept Focus on the measures in the abstract covers income or wealth covers also rank where the underlying data are categorical separates out the fundamental components of the -measurement problem
4 modelling Aggregate Basic information is the temporal pair z i = (x i,t 1, x i,t ) Bivariate distribution distribution function F (x t 1, x t ) marginal distributions F t 1 and F t give income distribution in each period Time-aggregated income derived from (x i,t 1, x i,t ) using weights w t 1, w t x i := w t 1 x i,t 1 + w t x i,t Distribution F w derived from F
5 measures in practice Aggregate Stability indices: 1 I(F w ) w t 1 I(F t 1 )+w t I(F t ) Hart (1976): 1 r(log x t 1, log x t ) where r is the correlation coefficient " R R x t e γr(f,(x t 1,x k t)) df(xt 1,x t ) King (1983): 1 µ k (F t ) k 1, k 6= 0, γ 0 where r(f; (x t 1, x t )) is a rank indicator: µ 1 (F t ) 1 jx t Q(F t ; F t 1 (x t 1 ))j Q(G; q) := inffx : G(x) qg Fields-Ok (1999): c R R jlog x t 1 log x t j df(x t 1, x t ) # 1 k
6 Fundamentals Aggregate How to characterise in terms of individual income? in terms of social position? Ingredients for a theory of measurement: 1 time frame of two or more periods; 2 measure of individual status within society 3 aggregation of changes in status over the time frame.
7 Ingredients of the problem: classes Aggregate Income as a generic term any cardinally measurable, comparable quantity cardinality turns out not to be crucial for our approach Ordered set of K income classes class k is associated with income level x k where x k < x k+1, k = 1, 2,..., K 1 p k 2 R + is the size of class k, k = 1, 2,..., K and K k=1 p k = n, the size of the population k 0 (i) and k 1 (i): income class occupied by person i in periods 0 and 1 respectively characterised by x k0 (1),..., x k0 (n) and x k1 (1),..., x k1 (n)
8 Ingredients of the problem: valuation Aggregate Don t have to use simple aggregation of the x k to compute Could carry out a relabelling of the income classes For example use n 0 (x k ) := k h=1 p h, k = 1,..., K number of persons in, or below, each income class according to the distribution in period 0 Suppose that the class sizes (p 1,..., p K ) in period 0 change to (q 1,..., q K ) in period 1 Relabelling the income classes:n 1 (x k ) := k h=1 q h, k = 1,..., K
9 Ingredients of the problem: status Aggregate u i, v i denote individual i s status in the 0-distribution, the 1-distribution personal history: z i := (u i, v i ) Distribution-independent, static (1). z i = x k0 (i), x k1 (i) Distribution-independent, static (2). z i = ϕ x k0 (i), ϕ x k1 (i) ϕ essentially arbitrary (utility of x?) independent of ϕ? Distribution-dependent, static. z i = n 0 x k0 (i), n 0 x k1 (i) cumulative numbers in class value the class Distribution-dependent, dynamic. z i = n 0 x k0 (i), n 1 x k1 (i).
10 Example Aggregate Consider the following example: period 0 period 1 x 1 A _ x 2 B A x 3 C B x 4 _ C x 5 Different definitions of status will produce different evaluations of such a change.
11 Basic axioms Aggregate Let m be individual, increasing in ju i Continuity is continuous on Z n Monotonicity. If z, z 0 2 Z n differ only in their ith component then m (u i, v i ) > m u 0 i, v0 i () z z 0. Independence. For z, z 0 2 Z n such that: z s z 0 and z i = z 0 i for some i then z (ζ, i) s z0 (ζ, i) for all ζ 2 [z i 1, z i+1 ] \ z 0 i 1, z0 i+1. Local im. Let z, z 0 2 Z n be such that, for some i and j, u i = v i, u j = v j, u 0 i = u i + δ, v 0 i = v i + δ, u 0 j = u j δ, v 0 j = v j δ and, for all h 6= i, j, u 0 h = u h, v 0 h = v h. Then z s z 0. v i j
12 Representation results (1) Aggregate Theorem. Given basic axioms: is representable by the continuous function given by n i=1 φ i (z i), 8z 2 Z n φ i : Z! R is a continuous function that is strictly decreasing in ju i v i j φ i (u, u) = a i + b i u. Corollary. is also representable by φ ( n i=1 φ i (z i)) φ : R! R is continuous and strictly monotonic increasing.
13 Representation results (2) Aggregate Status scale irrelevance. For any z, z 0 2 Z n such that z s z 0, tz s tz 0 for all t > 0.: Theorem. Given Basic axioms and scale irrelevance: is representable by φ n i=1 u ui ih i v i where H i is a real-valued function.
14 Representation results (3) Aggregate This suggests we can compare the (u, v) vectors in different parts of the distribution in terms of proportional differences m (z i ) = max ui v, v i i u i scale irrelevance. Suppose there are z 0, z0 0 2 Zn such that z 0 s z0 0. Then for all t > 0 and z, z0 such that m (z) = tm (z 0 ) and m (z 0 ) = tm (z0 0 ): z s z0. Theorem. Given our axioms is representable by Φ (z) = φ n i=1 uα i v1 α where α 6= 1 is a constant. i
15 Generating an aggregate Aggregate Consider a subset of Z : Z (ū, v) := fz 2Zj n i=1 z i = (ū, v)g. From theorem 3, that the must take the form: Φ (z) = φ α ; ū, v. n i=1 uα i v1 i Φ (z) should be zero when there is no using the standard interpretation of φ ( n i=1 u i; ū, ū) = 0, i.e. φ (ū; ū, ū) = 0.
16 Using a broader interpretation of zero Aggregate Scaling up everyone s status should not matter v i = λu i, i = 1,.., n (where λ = v/ū) φ λ 1 α n i=1 u i; ū, v = 0 φ ū α v 1 α ; ū, v = 0. This requires φ and φ are equivalent to: h i ψ n ui α h i vi 1 α i=1 µ u µ. v A suitable cardinalisation of ψ(.) gives M. h i M α := 1 ui α h i vi 1 α α[α 1]n n i=1 µ u µ 1. v
17 Limiting cases Aggregate Two limiting cases α = 0: α = 1 M 0 = 1 n n i=1 M 1 = 1 n n i=1. v i µ log ui vi v µ u µ, v. u i µ log ui vi u µ u µ. v We have a class of aggregate measures high α > 0: M sensitive to downward movements α < 0: M sensitive to upward movements
18 1 Aggregate Concerned with ranks not income levels? Then make status an ordinal concept (Chakravarty 1984) Variety of ways to define status ordinally: tables or transition matrices. However, these approaches are sensitive to the adjustment of class boundaries: Consider the case where in the original set of classes p k = 0 and p k+1 > 0; if the is sensitive to small values of p and the income boundary between classes k and k + 1 is adjusted there could be a big jump in the. This will not happen if the is defined in terms of u i and v i
19 2 Aggregate The derivation is value free. Can we introduce a social valuation of? Could construct an explicit welfare approach something analagous to Atkinson inequality? (Gottschalk-Spolaore 2002) but you must go beyond simplistic welfare models (Markandya 1982, 1983) Can also introduce normative elements in the above framework definition of status value range of α
20 Aggregate Simplest case: status before and after i.e. distribution-independent, static Movements of incomes: u i = x 0i and v i = x 1i. Define moment µ g(u,v) = n 1 n i=1 g(u i, v i ) g(.) is a specific function consider three cases: M α, M 0 and M 1
21 General case Aggregate Rewrite the as M α = 1 α(1 α) In terms of moments: M α = 1 α(α 1) n 1 ui αv1 α i µ u α µ1 v α h µu α v 1 α µ u αµ1 v α i Central Limit Theorem implies asymptotic normality under standard regularity conditions. h dvar(m α ) = D ˆΣD > with D = Mα M µ ; α u µ ; v D in terms of sample moments: D = h µu α v 1 α µ u α 1 µ v α 1 ; µ u α v 1 α µ α (α 1) α u µα v 2 ; µ u αµα v 1 α(α 1) i M α µ u α v 1 α i.
22 Covariance matrix Aggregate ˆΣ is the estimator of the covariance matrix of µ u, µ v and µ u α v 1 α We have:
23 Limiting case (1) Aggregate M 0 as a function of four moments: M 0 = µ v log v µ v log u µu µ + log v µ. v dvar(m 0 ) = D 0 ˆΣ 0 D 0 > h M0 M D 0 = µ ; 0 u µ ; v ˆΣ 0 : M 0 µ v log v ; i M 0 µ v log u
24 Limiting case (2) Aggregate M 1 as a function of four moments: M 1 = µ u log u µ u log v µv µ + log u µ u dvar(m 1 ) = D 1 ˆΣ 1 D > 1 with D 1 = D 1 = ˆΣ 1 : h M1 M µ ; 1 u µ ; v h µu log u +µ u log v µ 2 u M 1 µ u log u ; µ u i M 1 µ u log v i ; µ ; v µ ; u µ u
25 Aggregate we now consider the distribution-dependent, dynamic status. Scale independence means we can define status using proportions u i = ˆF 0 (x 0i ) and v i = ˆF 1 (x 1i ) ˆF 0 (.) and ˆF 1 (.) are the empirical distribution functions ˆF k (x) = 1 n n j=1 I(x kj x) Method of moments does not apply as the values in u and v are non i.i.d.
26 Establishing the asymptotic distribution Aggregate Ruymgaart and van Zuijlen (1978): asymptotic normality for the multivariate rank statistic T n = 1 n n i=1 c inφ 1 (u i )φ 2 (v i ). c in are given real constants, φ 1 and φ 2 are (scores) functions defined on (0,1), these are allowed to tend to infinity near 0 and 1 but not too quickly. need to assume the existence of K 1, a 1 and a 2, s.t. φ 1 (t) K 1 [t(1 t)] a 1 and φ 2 (t) K 1 [t(1 t)] a 2 with a 1 + a 2 < 1 2 for t 2 (0, 1). Then, φ 1 (t) and φ 2 (t) tend to infinity near 0 at a rate slower than the functions t a 1 and t a 2. Variance of T n is finite, even if not analytically tractable.
27 Applying the results - general case Aggregate M α can be written as a function of T n Note that µ u = µ v = n 1 n i=1 n i = n+1 2n. M α as a function of one moment: M α = 1 2n α(α 1) n+1 µ u α v 1. 1 α Hence, M α = 1 α(α 1) [T n 1].
28 Applying the results - limiting cases Aggregate M 0 as a function of T n M 0 = 2n n+1 (k µ v log u ) = l T n. M 1 as a function of T n M 1 = 2n n+1 (k µ u log v ) = l T n It can be shown that the relevant conditions are met for 0.5 < α < 1.5. M α is asymptotically normal asymptotic justification for the bootstrap
29 Aggregate coverage error rate of a confidence interval probability that CI does not include the true value of a parameter should be close to the nominal rate e.g. 5% for a 95% CI use Monte-Carlo simulations to approximate coverage error rates for different methods: asymptotic percentile bootstrap and studentized bootstrap
30 Asymptotic confidence intervals Aggregate CI asym = [M α c dvar(m α ) 1/2 ; M α + c dvar(m α ) 1/2 ] c is a critical value from the Student distribution T(n 1). finite sample often poor bootstrap confidence intervals can be expected to perform better
31 Percentile bootstrap method Aggregate does not require the (asymptotic) standard error Method: generate B bootstrap samples by resampling in the original data for each resample, we compute the. obtain B bootstrap statistics, M b α, b = 1,..., B. The percentile bootstrap confidence interval is equal to CI perc = [c b ; cb ] c b and cb are the 2.5 and 97.5 percentiles of the EDF of the bootstrap statistics.
32 Studentized bootstrap method Aggregate uses the asymptotic standard error Method: generate B bootstrap samples by resampling in the original data for each resample, compute a t-statistic. obtain B bootstrap t-statistics t b α = (M b α M α )/dvar(m b α) 1/2, b = 1,..., B, where M α is the original CI stud = [M α c d Var(M α ) 1/2 ; M α c d Var(M α ) 1/2 ] c0.025 and c are percentiles of the EDF of the bootstrap t-statistics.
33 Comparison Aggregate studentized bootstrap based on asymptotically pivotal statistic t-statistics follow (asymptotically) a known distribution superior of the bootstrap over asymptotic confidence intervals both bootstrap intervals asymmetric, asymptotic confidence interval is symmetric bootstrap CIs more accurate if the underlying distribution is asymmetric.
34 Experiments Aggregate Bivariate Lognormal distribution: (x 0, x 1 ) LN(µ, Σ) 1 ρ with µ = (0, 0) and Σ = ρ 1 increases as ρ decreases we try different indices, different sample sizes and different levels for each combination: draw 10,000 samples (from the bivariate lognormal distribution) compute M α compute confidence interval at 95% How often does the interval does not include the true parameter?
35 Asymptotic confidence intervals Aggregate α n = 100, ρ = n = 100, ρ = n = 100, ρ = n = 100, ρ = n = 100, ρ = n = 100, ρ = n = 100, ρ = n = 200, ρ = n = 500, ρ = n = 1000, ρ = n = 5000, ρ = n = 10000, ρ = Table: Coverage error rate of asymptotic confidence intervals at 95% of income measures. The nominal error rate is 0.05, replications
36 Results Aggregate Distribution-independent, static. Recall: coverage error rate should be close to 5% asymptotic intervals perform poorly for α = 1, 2 coverage error rate is stable as ρ varies (for α = 0, 0.5, 1 and n = 100) coverage error rate decreases as n increases coverage error rate close to 0.05 for n and α = 0, 0.5, 1. asymptotic confidence intervals perform well in very large sample, with α 2 [0, 1]. dismal of asymptotic confidence intervals for small and moderate samples poorest results for ρ = 0.8 try other two methods for this value of ρ
37 Other methods Aggregate α n = 100, ρ = 0.8 Asymptotic Boot-perc Boot-stud n = 200, ρ = 0.8 Asymptotic Boot-perc Boot-stud n = 500, ρ = 0.8 Asymptotic Boot-perc Boot-stud n = 1.000, ρ = 0.8 Asymptotic Boot-perc Boot-stud Table: Coverage error rate of asymptotic and bootstrap confidence intervals at 95% of income measures replications, 199 bootstraps
38 Results Aggregate percentile bootstrap and asymptotic confidence intervals perform similarly studentized bootstrap confidence intervals outperform other methods siginificant improvement over asymptotic confidence intervals
39 Aggregate distribution-dependent, dynamic status variance of M α is not analytically tractable cannot use asymptotic and studentized bootstrap confidence intervals use the percentile bootstrap method
40 Percentile bootstrap method (n=100) Aggregate α ρ = ρ = ρ = ρ = ρ = ρ = ρ = Table: Coverage error rate of percentile bootstrap confidence intervals at 95% of rank- measures replications, 199 bootstraps and 100 observations.
41 Results Aggregate the coverage error rate can be very different for different values of ρ and α, it decreases as ρ increases, except for the case of nearly zero (ρ = 0.99). the coverage error rate is close to 0.05 for ρ = 0.8, 0.9 and α = 0, 0.5, 1. What happens as the sample size increases?
42 Percentile bootstrap method (variable n) Aggregate α n = 100, ρ = n = n = n = n = 100, ρ = n = n = n = n = 100, ρ = n = n = n = Table: Coverage error rate of percentile bootstrap confidence intervals at 95% of rank- measures replications, 199 bootstraps.
43 Results Aggregate the coverage error rate gets closer to 0.05 as the sample size increases, the coverage error rate is smaller when α = 0, 0.5, 1. better statistical properties as the sample size increases
44 Asking about Aggregate A questionnaire study use same type of methods as for inequality? focus on whether people value contrast with preferences for equality? Study using 356 students from three countries: Italy (120) UK (89) Israel (147) problem obviously more complex is a from-to concept not a snapshot graphics representation is tricky Method: bus queue pictures combine equality and intergenerational get personal characteristics
45 Q1 Full Mixing v Rigidity Aggregate
46 Q2 Full Mixing and Widening Aggregate
47 Q3 Rigidity v Full Mixing+Widening Aggregate
48 Q4 Partial mixing v Rigidity Aggregate
49 Q5 Partial Mixing and Widening Aggregate
50 Q6 Rigidity v Partial Mixing+Widening Aggregate
51 Q7 Full v Partial Mixing Aggregate
52 Q8 Rigidity v Simple Widening Aggregate
53 Asking about Aggregate Yes if A chosen more often than B in Q1 (Full mixing v rigidity) Q4 (Partial mixing v rigidity) Q7 (Full v partial mixing) A responses B responses Q1 Q4 Q7 Q1 Q4 Q7 Italy UK Israel All
54 Asking about equality Aggregate Yes if A chosen more often than B in Q2 (Full mixing and widening) Q5 (Partial mixing and widening) Q8 (Rigidity v Simple widening) A responses B responses Q2 Q5 Q8 Q2 Q5 Q8 Italy UK Israel All
55 Categorical variables Aggregate Check for each person the answers to Q1,Q4,Q7 () Q2,Q5,Q8 (equality) Percentages in each category 0A 1A 2A 3A Italy UK Israel TOTAL Equality Italy UK Israel TOTAL
56 Regression model Aggregate Seek to explain 1 attitudes to 2 attitudes to equality Use categorical variables 1 preferences 0A, 1A, 2A, 3A 2 equality preferences 0A, 1A, 2A, 3A Variety of personal characteristics Standard ordered probit
57 Personal characteristics 1 Aggregate
58 Personal characteristics 2 Aggregate
59 Personal characteristics 3 Aggregate
60 Baseline model Aggregate equality age gender familyincome livingstandards prospects perspectivn independendencea independendenceb governmentsrolee
61 Model with country dummies Aggregate equality age gender familyincome livingstandards prospects perspectivn independendencea independendenceb governmentsrole italy uk
62 Model with nationality dummy Aggregate equality age gender nationality familyincome livingstandards prospects perspectivn independendencea independendenceb governmentsrole
63 : Measurement Aggregate Key step involves a logical separation of fundamental concepts measure of individual status aggregation of changes in status Status concept derived directly from the information in the marginals Apply standard principles to movements in status get a superclass of measures generally applicable to wide variety of status concepts parameter α that determines type of measure Principal status types yield statistically tractable indices
64 : Values Aggregate key factors: The more independent are children s and parents economic positions in a society... I am from around here Italy country dummy (Italians don t value....!) Equality key factors: family income role of government prospective social position
Measuring Mobility. Frank A. Cowell. and. Emmanuel Flachaire. April 2011
Measuring Mobility by Frank A. Cowell and Emmanuel Flachaire April 2011 The Suntory Centre Suntory and Toyota International Centres for Economics and Related Disciplines London School of Economics and
More informationWinter School, Canazei. Frank A. Cowell. January 2010
Winter School, Canazei Frank A. STICERD, London School of Economics January 2010 Meaning and motivation entropy uncertainty and entropy and inequality Entropy: "dynamic" aspects Motivation What do we mean
More informationInequality with Ordinal Data
Inequality with Ordinal Data by Frank A. Cowell STICERD London School of Economics Houghton Street London, WC2A 2AE, UK email: f.cowell@lse.ac.uk and Emmanuel Flachaire GREQAM, Aix-Marseille University
More informationSTAT Section 2.1: Basic Inference. Basic Definitions
STAT 518 --- Section 2.1: Basic Inference Basic Definitions Population: The collection of all the individuals of interest. This collection may be or even. Sample: A collection of elements of the population.
More informationThe Nonparametric Bootstrap
The Nonparametric Bootstrap The nonparametric bootstrap may involve inferences about a parameter, but we use a nonparametric procedure in approximating the parametric distribution using the ECDF. We use
More information11. Bootstrap Methods
11. Bootstrap Methods c A. Colin Cameron & Pravin K. Trivedi 2006 These transparencies were prepared in 20043. They can be used as an adjunct to Chapter 11 of our subsequent book Microeconometrics: Methods
More informationRegression Analysis. BUS 735: Business Decision Making and Research. Learn how to detect relationships between ordinal and categorical variables.
Regression Analysis BUS 735: Business Decision Making and Research 1 Goals of this section Specific goals Learn how to detect relationships between ordinal and categorical variables. Learn how to estimate
More informationGROWTH AND INCOME MOBILITY: AN AXIOMATIC ANALYSIS
XXIII CONFERENZA CRISI ECONOMICA, WELFARE E CRESCITA Pavia, Aule Storiche dell Università, 19-20 settembre 2011 GROWTH AND INCOME MOBILITY: AN AXIOMATIC ANALYSIS FLAVIANA PALMISANO società italiana di
More informationFall 2017 STAT 532 Homework Peter Hoff. 1. Let P be a probability measure on a collection of sets A.
1. Let P be a probability measure on a collection of sets A. (a) For each n N, let H n be a set in A such that H n H n+1. Show that P (H n ) monotonically converges to P ( k=1 H k) as n. (b) For each n
More informationBivariate Relationships Between Variables
Bivariate Relationships Between Variables BUS 735: Business Decision Making and Research 1 Goals Specific goals: Detect relationships between variables. Be able to prescribe appropriate statistical methods
More informationCIRPÉE Centre interuniversitaire sur le risque, les politiques économiques et l emploi
CIRPÉE Centre interuniversitaire sur le risque, les politiques économiques et l emploi Cahier de recherche/working Paper 06-34 Poverty, Inequality and Stochastic Dominance, Theory and Practice: Illustration
More informationCasuality and Programme Evaluation
Casuality and Programme Evaluation Lecture V: Difference-in-Differences II Dr Martin Karlsson University of Duisburg-Essen Summer Semester 2017 M Karlsson (University of Duisburg-Essen) Casuality and Programme
More information9. Linear Regression and Correlation
9. Linear Regression and Correlation Data: y a quantitative response variable x a quantitative explanatory variable (Chap. 8: Recall that both variables were categorical) For example, y = annual income,
More informationThe Bootstrap: Theory and Applications. Biing-Shen Kuo National Chengchi University
The Bootstrap: Theory and Applications Biing-Shen Kuo National Chengchi University Motivation: Poor Asymptotic Approximation Most of statistical inference relies on asymptotic theory. Motivation: Poor
More informationMulti-dimensional Human Development Measures : Trade-offs and Inequality
Multi-dimensional Human Development Measures : Trade-offs and Inequality presented by Jaya Krishnakumar University of Geneva UNDP Workshop on Measuring Human Development June 14, 2013 GIZ, Eschborn, Frankfurt
More information36. Multisample U-statistics and jointly distributed U-statistics Lehmann 6.1
36. Multisample U-statistics jointly distributed U-statistics Lehmann 6.1 In this topic, we generalize the idea of U-statistics in two different directions. First, we consider single U-statistics for situations
More informationØ Set of mutually exclusive categories. Ø Classify or categorize subject. Ø No meaningful order to categorization.
Statistical Tools in Evaluation HPS 41 Fall 213 Dr. Joe G. Schmalfeldt Types of Scores Continuous Scores scores with a potentially infinite number of values. Discrete Scores scores limited to a specific
More informationA better way to bootstrap pairs
A better way to bootstrap pairs Emmanuel Flachaire GREQAM - Université de la Méditerranée CORE - Université Catholique de Louvain April 999 Abstract In this paper we are interested in heteroskedastic regression
More informationConfidence intervals for kernel density estimation
Stata User Group - 9th UK meeting - 19/20 May 2003 Confidence intervals for kernel density estimation Carlo Fiorio c.fiorio@lse.ac.uk London School of Economics and STICERD Stata User Group - 9th UK meeting
More informationGoodness of Fit: an axiomatic approach
Goodness of Fit: an axiomatic approach by Frank A. Cowell STICERD London School of Economics Houghton Street London, WC2A 2AE, UK email: f.cowell@lse.ac.uk Russell Davidson AMSE-GREQAM Department of Economics
More informationLecture I. What is Quantitative Macroeconomics?
Lecture I What is Quantitative Macroeconomics? Gianluca Violante New York University Quantitative Macroeconomics G. Violante, What is Quantitative Macro? p. 1 /11 Qualitative economics Qualitative analysis:
More informationFinding Relationships Among Variables
Finding Relationships Among Variables BUS 230: Business and Economic Research and Communication 1 Goals Specific goals: Re-familiarize ourselves with basic statistics ideas: sampling distributions, hypothesis
More informationØ Set of mutually exclusive categories. Ø Classify or categorize subject. Ø No meaningful order to categorization.
Statistical Tools in Evaluation HPS 41 Dr. Joe G. Schmalfeldt Types of Scores Continuous Scores scores with a potentially infinite number of values. Discrete Scores scores limited to a specific number
More informationBootstrap (Part 3) Christof Seiler. Stanford University, Spring 2016, Stats 205
Bootstrap (Part 3) Christof Seiler Stanford University, Spring 2016, Stats 205 Overview So far we used three different bootstraps: Nonparametric bootstrap on the rows (e.g. regression, PCA with random
More informationInequality Measures and the Median:
Inequality Measures and the Median: Why inequality increased more than we thought by Frank A. Cowell STICERD London School of Economics Houghton Street London, W2CA 2AE, UK f.cowell@lse.ac.uk and Emmanuel
More informationIntroduction There is a folk wisdom about inequality measures concerning their empirical performance. Some indices are commonly supposed to be particu
Sensitivity of Inequality Measures to Extreme Values by Frank A. Cowell Sticerd, London School of Economics and Emmanuel Flachaire Eurequa, University Paris I Panthéon-Sorbonne February 22 Abstract We
More informationRobust Outcome Analysis for Observational Studies Designed Using Propensity Score Matching
The work of Kosten and McKean was partially supported by NIAAA Grant 1R21AA017906-01A1 Robust Outcome Analysis for Observational Studies Designed Using Propensity Score Matching Bradley E. Huitema Western
More informationAdvanced Statistics II: Non Parametric Tests
Advanced Statistics II: Non Parametric Tests Aurélien Garivier ParisTech February 27, 2011 Outline Fitting a distribution Rank Tests for the comparison of two samples Two unrelated samples: Mann-Whitney
More informationMATH 1150 Chapter 2 Notation and Terminology
MATH 1150 Chapter 2 Notation and Terminology Categorical Data The following is a dataset for 30 randomly selected adults in the U.S., showing the values of two categorical variables: whether or not the
More informationThe bootstrap. Patrick Breheny. December 6. The empirical distribution function The bootstrap
Patrick Breheny December 6 Patrick Breheny BST 764: Applied Statistical Modeling 1/21 The empirical distribution function Suppose X F, where F (x) = Pr(X x) is a distribution function, and we wish to estimate
More informationDistributive Justice and Economic Inequality
Distributive Justice and Economic Inequality P. G. Piacquadio UiO, January 17th, 2017 Outline of the course Paolo: Introductory lecture [Today!] Rolf: Theory of income (and wealth) inequality Geir: Distributive
More informationTable B1. Full Sample Results OLS/Probit
Table B1. Full Sample Results OLS/Probit School Propensity Score Fixed Effects Matching (1) (2) (3) (4) I. BMI: Levels School 0.351* 0.196* 0.180* 0.392* Breakfast (0.088) (0.054) (0.060) (0.119) School
More informationChapter 1:Descriptive statistics
Slide 1.1 Chapter 1:Descriptive statistics Descriptive statistics summarises a mass of information. We may use graphical and/or numerical methods Examples of the former are the bar chart and XY chart,
More informationStatistical methods for Education Economics
Statistical methods for Education Economics Massimiliano Bratti http://www.economia.unimi.it/bratti Course of Education Economics Faculty of Political Sciences, University of Milan Academic Year 2007-08
More informationConfidence Intervals in Ridge Regression using Jackknife and Bootstrap Methods
Chapter 4 Confidence Intervals in Ridge Regression using Jackknife and Bootstrap Methods 4.1 Introduction It is now explicable that ridge regression estimator (here we take ordinary ridge estimator (ORE)
More informationGoodness-of-Fit: An Economic Approach. and. Sanghamitra Bandyopadhyay ±
Goodness-of-Fit: An Economic Approach Frank A. Cowell, Emmanuel Flachaire and Sanghamitra Bandyopadhyay ± April 2009 DARP 0 April 2009 The Toyota Centre Suntory and Toyota International Centres for Economics
More informationLong-Run Covariability
Long-Run Covariability Ulrich K. Müller and Mark W. Watson Princeton University October 2016 Motivation Study the long-run covariability/relationship between economic variables great ratios, long-run Phillips
More informationEXPLICIT NONPARAMETRIC CONFIDENCE INTERVALS FOR THE VARIANCE WITH GUARANTEED COVERAGE
EXPLICIT NONPARAMETRIC CONFIDENCE INTERVALS FOR THE VARIANCE WITH GUARANTEED COVERAGE Joseph P. Romano Department of Statistics Stanford University Stanford, California 94305 romano@stat.stanford.edu Michael
More informationRegression Analysis. BUS 735: Business Decision Making and Research
Regression Analysis BUS 735: Business Decision Making and Research 1 Goals and Agenda Goals of this section Specific goals Learn how to detect relationships between ordinal and categorical variables. Learn
More informationSupplement to Quantile-Based Nonparametric Inference for First-Price Auctions
Supplement to Quantile-Based Nonparametric Inference for First-Price Auctions Vadim Marmer University of British Columbia Artyom Shneyerov CIRANO, CIREQ, and Concordia University August 30, 2010 Abstract
More informationStatistics - Lecture One. Outline. Charlotte Wickham 1. Basic ideas about estimation
Statistics - Lecture One Charlotte Wickham wickham@stat.berkeley.edu http://www.stat.berkeley.edu/~wickham/ Outline 1. Basic ideas about estimation 2. Method of Moments 3. Maximum Likelihood 4. Confidence
More informationEstimation and sample size calculations for correlated binary error rates of biometric identification devices
Estimation and sample size calculations for correlated binary error rates of biometric identification devices Michael E. Schuckers,11 Valentine Hall, Department of Mathematics Saint Lawrence University,
More informationEconometric Modelling Prof. Rudra P. Pradhan Department of Management Indian Institute of Technology, Kharagpur
Econometric Modelling Prof. Rudra P. Pradhan Department of Management Indian Institute of Technology, Kharagpur Module No. # 01 Lecture No. # 28 LOGIT and PROBIT Model Good afternoon, this is doctor Pradhan
More informationCh 7: Dummy (binary, indicator) variables
Ch 7: Dummy (binary, indicator) variables :Examples Dummy variable are used to indicate the presence or absence of a characteristic. For example, define female i 1 if obs i is female 0 otherwise or male
More informationIncome distribution and inequality measurement: The problem of extreme values
Income distribution and inequality measurement: The problem of extreme values Frank A. Cowell, Emmanuel Flachaire To cite this version: Frank A. Cowell, Emmanuel Flachaire. Income distribution and inequality
More informationPRESENTATION ON THE TOPIC ON INEQUALITY COMPARISONS PRESENTED BY MAG.SYED ZAFAR SAEED
PRESENTATION ON THE TOPIC ON INEQUALITY COMPARISONS PRESENTED BY MAG.SYED ZAFAR SAEED Throughout this paper, I shall talk in terms of income distributions among families. There are two points of contention.
More informationLecture 6: Linear Regression
Lecture 6: Linear Regression Reading: Sections 3.1-3 STATS 202: Data mining and analysis Jonathan Taylor, 10/5 Slide credits: Sergio Bacallado 1 / 30 Simple linear regression Model: y i = β 0 + β 1 x i
More informationEconometrics I KS. Module 2: Multivariate Linear Regression. Alexander Ahammer. This version: April 16, 2018
Econometrics I KS Module 2: Multivariate Linear Regression Alexander Ahammer Department of Economics Johannes Kepler University of Linz This version: April 16, 2018 Alexander Ahammer (JKU) Module 2: Multivariate
More informationBinary Logistic Regression
The coefficients of the multiple regression model are estimated using sample data with k independent variables Estimated (or predicted) value of Y Estimated intercept Estimated slope coefficients Ŷ = b
More informationFinal Exam - Solutions
Ecn 102 - Analysis of Economic Data University of California - Davis March 19, 2010 Instructor: John Parman Final Exam - Solutions You have until 5:30pm to complete this exam. Please remember to put your
More informationLecture I. What is Quantitative Macroeconomics?
Lecture I What is Quantitative Macroeconomics? Gianluca Violante New York University Quantitative Macroeconomics G. Violante, What is Quantitative Macro? p. 1 /11 Qualitative economics Qualitative analysis:
More informationDear Author. Here are the proofs of your article.
Dear Author Here are the proofs of your article. You can submit your corrections online, via e-mail or by fax. For online submission please insert your corrections in the online correction form. Always
More informationHANDBOOK OF APPLICABLE MATHEMATICS
HANDBOOK OF APPLICABLE MATHEMATICS Chief Editor: Walter Ledermann Volume VI: Statistics PART A Edited by Emlyn Lloyd University of Lancaster A Wiley-Interscience Publication JOHN WILEY & SONS Chichester
More informationBootstrap Confidence Intervals
Bootstrap Confidence Intervals Patrick Breheny September 18 Patrick Breheny STA 621: Nonparametric Statistics 1/22 Introduction Bootstrap confidence intervals So far, we have discussed the idea behind
More informationOne-Sample Numerical Data
One-Sample Numerical Data quantiles, boxplot, histogram, bootstrap confidence intervals, goodness-of-fit tests University of California, San Diego Instructor: Ery Arias-Castro http://math.ucsd.edu/~eariasca/teaching.html
More informationA multivariate multilevel model for the analysis of TIMMS & PIRLS data
A multivariate multilevel model for the analysis of TIMMS & PIRLS data European Congress of Methodology July 23-25, 2014 - Utrecht Leonardo Grilli 1, Fulvia Pennoni 2, Carla Rampichini 1, Isabella Romeo
More informationUniversity of California San Diego and Stanford University and
First International Workshop on Functional and Operatorial Statistics. Toulouse, June 19-21, 2008 K-sample Subsampling Dimitris N. olitis andjoseph.romano University of California San Diego and Stanford
More informationBootstrapping Heteroskedasticity Consistent Covariance Matrix Estimator
Bootstrapping Heteroskedasticity Consistent Covariance Matrix Estimator by Emmanuel Flachaire Eurequa, University Paris I Panthéon-Sorbonne December 2001 Abstract Recent results of Cribari-Neto and Zarkos
More informationMeasuring rank mobility with variable population size
Measuring rank mobility with variable population size Walter Bossert Department of Economics and CIREQ, University of Montreal, Canada walter.bossert@videotron.ca Burak Can Department of Economics, Maastricht
More informationReliable Inference in Conditions of Extreme Events. Adriana Cornea
Reliable Inference in Conditions of Extreme Events by Adriana Cornea University of Exeter Business School Department of Economics ExISta Early Career Event October 17, 2012 Outline of the talk Extreme
More informationCSSS/STAT/SOC 321 Case-Based Social Statistics I. Levels of Measurement
CSSS/STAT/SOC 321 Case-Based Social Statistics I Levels of Measurement Christopher Adolph Department of Political Science and Center for Statistics and the Social Sciences University of Washington, Seattle
More informationOnline Appendix: Misclassification Errors and the Underestimation of the U.S. Unemployment Rate
Online Appendix: Misclassification Errors and the Underestimation of the U.S. Unemployment Rate Shuaizhang Feng Yingyao Hu January 28, 2012 Abstract This online appendix accompanies the paper Misclassification
More informationFactor Analysis. Robert L. Wolpert Department of Statistical Science Duke University, Durham, NC, USA
Factor Analysis Robert L. Wolpert Department of Statistical Science Duke University, Durham, NC, USA 1 Factor Models The multivariate regression model Y = XB +U expresses each row Y i R p as a linear combination
More informationPredicting the Treatment Status
Predicting the Treatment Status Nikolay Doudchenko 1 Introduction Many studies in social sciences deal with treatment effect models. 1 Usually there is a treatment variable which determines whether a particular
More informationWELFARE: THE SOCIAL- WELFARE FUNCTION
Prerequisites Almost essential Welfare: Basics Welfare: Efficiency WELFARE: THE SOCIAL- WELFARE FUNCTION MICROECONOMICS Principles and Analysis Frank Cowell July 2017 1 Social Welfare Function Limitations
More informationPolitical Science 236 Hypothesis Testing: Review and Bootstrapping
Political Science 236 Hypothesis Testing: Review and Bootstrapping Rocío Titiunik Fall 2007 1 Hypothesis Testing Definition 1.1 Hypothesis. A hypothesis is a statement about a population parameter The
More informationStat 5101 Lecture Notes
Stat 5101 Lecture Notes Charles J. Geyer Copyright 1998, 1999, 2000, 2001 by Charles J. Geyer May 7, 2001 ii Stat 5101 (Geyer) Course Notes Contents 1 Random Variables and Change of Variables 1 1.1 Random
More informationSources of Inequality: Additive Decomposition of the Gini Coefficient.
Sources of Inequality: Additive Decomposition of the Gini Coefficient. Carlos Hurtado Econometrics Seminar Department of Economics University of Illinois at Urbana-Champaign hrtdmrt2@illinois.edu Feb 24th,
More informationStochastic Dominance
GREQAM Centre de la Vieille Charité 2 rue de la Charité 13236 Marseille cedex 2, France Stochastic Dominance by Russell Davidson email: Russell.Davidson@mcgill.ca Department of Economics McGill University
More informationAP Statistics Cumulative AP Exam Study Guide
AP Statistics Cumulative AP Eam Study Guide Chapters & 3 - Graphs Statistics the science of collecting, analyzing, and drawing conclusions from data. Descriptive methods of organizing and summarizing statistics
More informationThe measurement of welfare change
The measurement of welfare change Walter Bossert Department of Economics and CIREQ, University of Montreal walter.bossert@videotron.ca Bhaskar Dutta Department of Economics, University of Warwick B.Dutta@warwick.ac.uk
More informationNon-parametric bootstrap and small area estimation to mitigate bias in crowdsourced data Simulation study and application to perceived safety
Non-parametric bootstrap and small area estimation to mitigate bias in crowdsourced data Simulation study and application to perceived safety David Buil-Gil, Reka Solymosi Centre for Criminology and Criminal
More informationAN EMPIRICAL LIKELIHOOD RATIO TEST FOR NORMALITY
Econometrics Working Paper EWP0401 ISSN 1485-6441 Department of Economics AN EMPIRICAL LIKELIHOOD RATIO TEST FOR NORMALITY Lauren Bin Dong & David E. A. Giles Department of Economics, University of Victoria
More informationSemiparametric Regression
Semiparametric Regression Patrick Breheny October 22 Patrick Breheny Survival Data Analysis (BIOS 7210) 1/23 Introduction Over the past few weeks, we ve introduced a variety of regression models under
More informationCIRPÉE Centre interuniversitaire sur le risque, les politiques économiques et l emploi
CIRPÉE Centre interuniversitaire sur le risque, les politiques économiques et l emploi Cahier de recherche/working Paper 03-41 Bi-Polarization Comparisons Jean-Yves Duclos Damien Échevin Octobre/October
More informationApplied Statistics and Econometrics
Applied Statistics and Econometrics Lecture 5 Saul Lach September 2017 Saul Lach () Applied Statistics and Econometrics September 2017 1 / 44 Outline of Lecture 5 Now that we know the sampling distribution
More informationDynamic Fairness: Mobility, Inequality, and the Distribution of Prospects
Dynamic Fairness: Mobility, Inequality, and the Distribution of Prospects Baochun Peng 1 The Hong Kong Polytechnic University Haidong Yuan The Chinese University of Hong Kong Abstract This paper clarifies
More informationDynamic Fairness: Mobility, Inequality, and the Distribution of Prospects
Dynamic Fairness: Mobility, Inequality, and the Distribution of Prospects Baochun Peng 1 The Hong Kong Polytechnic University Haidong Yuan The Chinese University of Hong Kong Abstract This paper clarifies
More informationIdentify the scale of measurement most appropriate for each of the following variables. (Use A = nominal, B = ordinal, C = interval, D = ratio.
Answers to Items from Problem Set 1 Item 1 Identify the scale of measurement most appropriate for each of the following variables. (Use A = nominal, B = ordinal, C = interval, D = ratio.) a. response latency
More informationSupplemental Material for KERNEL-BASED INFERENCE IN TIME-VARYING COEFFICIENT COINTEGRATING REGRESSION. September 2017
Supplemental Material for KERNEL-BASED INFERENCE IN TIME-VARYING COEFFICIENT COINTEGRATING REGRESSION By Degui Li, Peter C. B. Phillips, and Jiti Gao September 017 COWLES FOUNDATION DISCUSSION PAPER NO.
More informationIEOR E4703: Monte-Carlo Simulation
IEOR E4703: Monte-Carlo Simulation Output Analysis for Monte-Carlo Martin Haugh Department of Industrial Engineering and Operations Research Columbia University Email: martin.b.haugh@gmail.com Output Analysis
More informationStatistical Inference: Estimation and Confidence Intervals Hypothesis Testing
Statistical Inference: Estimation and Confidence Intervals Hypothesis Testing 1 In most statistics problems, we assume that the data have been generated from some unknown probability distribution. We desire
More informationPareto Efficiency (also called Pareto Optimality)
Pareto Efficiency (also called Pareto Optimality) 1 Definitions and notation Recall some of our definitions and notation for preference orderings. Let X be a set (the set of alternatives); we have the
More informationSimulation. Where real stuff starts
1 Simulation Where real stuff starts ToC 1. What is a simulation? 2. Accuracy of output 3. Random Number Generators 4. How to sample 5. Monte Carlo 6. Bootstrap 2 1. What is a simulation? 3 What is a simulation?
More informationHarvard University. Rigorous Research in Engineering Education
Statistical Inference Kari Lock Harvard University Department of Statistics Rigorous Research in Engineering Education 12/3/09 Statistical Inference You have a sample and want to use the data collected
More informationThe Role of Inequality in Poverty Measurement
The Role of Inequality in Poverty Measurement Sabina Alkire Director, OPHI, Oxford James E. Foster Carr Professor, George Washington Research Associate, OPHI, Oxford WIDER Development Conference Helsinki,
More informationBootstrap, Jackknife and other resampling methods
Bootstrap, Jackknife and other resampling methods Part III: Parametric Bootstrap Rozenn Dahyot Room 128, Department of Statistics Trinity College Dublin, Ireland dahyot@mee.tcd.ie 2005 R. Dahyot (TCD)
More informationConsistent multidimensional poverty comparisons
Preliminary version - Please do not distribute Consistent multidimensional poverty comparisons Kristof Bosmans a Luc Lauwers b Erwin Ooghe b a Department of Economics, Maastricht University, Tongersestraat
More information11. Further Issues in Using OLS with TS Data
11. Further Issues in Using OLS with TS Data With TS, including lags of the dependent variable often allow us to fit much better the variation in y Exact distribution theory is rarely available in TS applications,
More informationBootstrap Approach to Comparison of Alternative Methods of Parameter Estimation of a Simultaneous Equation Model
Bootstrap Approach to Comparison of Alternative Methods of Parameter Estimation of a Simultaneous Equation Model Olubusoye, O. E., J. O. Olaomi, and O. O. Odetunde Abstract A bootstrap simulation approach
More informationMultidimensional inequality comparisons
Multidimensional inequality comparisons Jean-Yves Duclos, David E. Sahn and Stephen D. Younger February 5, 29 Abstract Keywords:. JEL Classification:. Institut d Anàlisi Econòmica (CSIC), Barcelona, Spain,
More informationLinear Regression With Special Variables
Linear Regression With Special Variables Junhui Qian December 21, 2014 Outline Standardized Scores Quadratic Terms Interaction Terms Binary Explanatory Variables Binary Choice Models Standardized Scores:
More informationRegression models for multivariate ordered responses via the Plackett distribution
Journal of Multivariate Analysis 99 (2008) 2472 2478 www.elsevier.com/locate/jmva Regression models for multivariate ordered responses via the Plackett distribution A. Forcina a,, V. Dardanoni b a Dipartimento
More informationRethinking Inequality Decomposition: Comment. *London School of Economics. University of Milan and Econpubblica
Rethinking Inequality Decomposition: Comment Frank A. Cowell and Carlo V. Fiorio *London School of Economics University of Milan and Econpubblica DARP 82 February 2006 The Toyota Centre Suntory and Toyota
More informationThe assumptions are needed to give us... valid standard errors valid confidence intervals valid hypothesis tests and p-values
Statistical Consulting Topics The Bootstrap... The bootstrap is a computer-based method for assigning measures of accuracy to statistical estimates. (Efron and Tibshrani, 1998.) What do we do when our
More informationConfidence Estimation Methods for Neural Networks: A Practical Comparison
, 6-8 000, Confidence Estimation Methods for : A Practical Comparison G. Papadopoulos, P.J. Edwards, A.F. Murray Department of Electronics and Electrical Engineering, University of Edinburgh Abstract.
More informationInteractions among Continuous Predictors
Interactions among Continuous Predictors Today s Class: Simple main effects within two-way interactions Conquering TEST/ESTIMATE/LINCOM statements Regions of significance Three-way interactions (and beyond
More informationDouble Bootstrap Confidence Intervals in the Two Stage DEA approach. Essex Business School University of Essex
Double Bootstrap Confidence Intervals in the Two Stage DEA approach D.K. Chronopoulos, C. Girardone and J.C. Nankervis Essex Business School University of Essex 1 Determinants of efficiency DEA can be
More informationDescriptive Univariate Statistics and Bivariate Correlation
ESC 100 Exploring Engineering Descriptive Univariate Statistics and Bivariate Correlation Instructor: Sudhir Khetan, Ph.D. Wednesday/Friday, October 17/19, 2012 The Central Dogma of Statistics used to
More information104 Business Research Methods - MCQs
104 Business Research Methods - MCQs 1) Process of obtaining a numerical description of the extent to which a person or object possesses some characteristics a) Measurement b) Scaling c) Questionnaire
More information