Measurement

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Measurement

• Denis Cogneau
• Delphine Roy

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Constructing social facts (1)

• Fact should not require an inter-subjective
  agreement (or a poll) to be true truth is
  conditional to a method or procedure of data
  collection and aggregation this procedure can be
  liked or not, anyway the procedural link between
  the data and the fact remains
• Ex. Food price ? Tasty character of a given
  food product
• Social we are not talking of facts that are
  particular to an individual or to very small
  groups of individuals
• (even if there can be only one or zero poor,
  everybody can afford being poor likewise
  Durkheim will not talk about suicide as a
  personal distress but as a widespread and regular
  phenomenon and as a social symptom)

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Constructing social facts (2)

• Social facts are counts, even in anthropology
  is prohibition of incest general, or even
  this group of people has a very peculiar
  mythology (as ? from other groups)
• (even most psychological facts, that must occur
  many times)

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Constructing social facts (3)

• A social fact is a construct depending on a
  methodology of accounting that has 4 components
• What is to be measured? Definition
• What is to be computed? Axiomatics
• How to measure variables? Measure
• How to count individuals? Sampling

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1 Definition (a)

• Concepts come from theory
• - of growth, international economics
• - of development
• - of justice
• - of social integration
• - etc.
• Before to be measured, concepts should be made
  precise in terms of population covered and
  dimensions concerned

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1- Definition (b)

• Theory should define the concepts to be measured
• national wealth non-market activities, waste of
  natural resources?
• access to goods is it income or consumption?
  full time income? income per adult equivalent?
• poverty do differences between the poor matter?
• income inequality relative or absolute income
  differences?
• health inequality does it matter per se or only
  correlation with other inequalities?
• unemployment or migration voluntary vs.
  involuntary, transient vs. permanent
• trade openness trade intra-firms ? extra-firms?
• education market returns or cognitive
  achievements?

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2 - Axiomatics

• Quantities observed in nature price of Big
  Mac, quantity of rice, height stature, number of
  people in a place at a time Not so natural but
  elementary skin color, mark at an exam
• Measuring concepts require an aggregation of
  natural quantities
• - in the space of variables cost of living,
  agricultural production, individual income
  literacy
• - in the space of individuals mean income,
  stunting / overweight, poverty, inequality

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2- Ex. Basic axioms of inequality

• Identical individuals (in needs) and heterogenous
  income
• Axiom 1 Inequality should decrease when
  transferring income from a rich to a poor
• (Varlog no Gini yes)
• Axiom 2a Inequality is unchanged if all incomes
  are doubled (Gini)
• Axiom 2b Inequality is unchanged if a given
  amount is added to all incomes (Kolm Gini index
  not divided by the mean)

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• At date t n individuals indexed by i (or j)
  income yi(t) mean income µy(t)
• Traditional relative Gini
• G 1/(2n²µy(t)) Si Sj yi(t) yj(t)
• Absolute Gini
• Abs-G 1/(2n²) Si Sj yi(t) yj(t)
• In the figure divided by 1992 median, i.e. a
  time-invariant factor, just for normalization

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3 Measure

• How to ask questions in a census or a survey
  formulation, order, items
• Should one rely on self-declarations? On
  interviewer observation?
• When to use direct measurements literacy,
  health?
• How to mimic real life buying products to know
  prices, making real exams?
• How to reach the underground informal trade
  flows, unregistered economy, capital flight?
• How to avoid refusals to answer?

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3- Measurement errors

• Classical white noise
• Y Y u E(u)0 E(uY)Cov(u,Y)0
• Increases variance and decreases correlations
• V(Y)V(Y)V(u)V(Y)(1?), ?gt0
• Corr(Y,X)Corr(Y,X).1/(1?)
• Hence, enhances the impression of inequality,
  mobility, or multidimensionality.
• Computations of the sensitivity of indexes to
  measurement errors may help (assume ?20 and see
  what comes out)
• Non-classical correlated with the true value
• Bounded variables like dummies negative
  correlation
• Y and Y0 or 1 u-1, 0 or 1
• Comparisons between direct data and self-declared
  data on income ? around 20-30 on US data
  (Handbook of Econometrics, vol.5, measurement
  errors in survey data)

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• V(Y)V(Y)1V(u)/V(Y) V(Y) (1?)
• X correlated with Y but not with u
• Cov(X,Y)Cov(X,Y)Cov(X,u)Cov(X,Y)0
• Corr(X,Y)Cov(X,Y)/(V(X)V(Y))1/2Corr(X,Y)/(1?
  )
• If Y and Y are dummies
• E(u)0 imposes same number
• for u-1 and u1, i.e. n01
• Cov(Y,u)- n01/n lt0
• Var(u)2n01/n Var(Y)(n01n10)/n
• Etc. (same thing when Y Y bounded)
• Let ?-corr(Y,u)gt0, V(Y)V(Y)1
  ?-2?V(u)gtV(Y) iff V(Y)lt1/2?
• Etc.

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4 Sampling

• How to obtain an unbiased and precise image of
  the desired population (of products, of firms, of
  people)?
• - avoiding selection or attrition
• minimizing confidence intervals
• with a given hierarchical structure clusters,
  networks, biographies
• ? Sample theory as a branch of statistics

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4 The revolution of probabilistic samples

• Law of large numbers
• n independent and identically distributed
  (i.i.d.) random variables Xn, with finite
  expected value E(Xn)µ and finite variance
  V(Xn)s². Let MnSnXn/n. (empirical mean). X can
  be either discrete (frequencies) or continuous.
• Weak version
• For all egt0, limn?8 P( Mn-µ e) 0 i.e.
  the distribution of Mn concentrates around µ.
• Comes from Bienaymé-Chebyshev inequality
  P(Mn-µ e)lt s²/ne²
• Coin tossing you may draw samples with many
  heads or tails but the relative share of these
  samples decreases as n increases
• Strong version the probability of samples ? that
  are systematically drawn away from µ becomes
  negligible P(? limn?8 Mn(?)µ)1
• Coin tossing drawing samples whose empirical
  mean does not converge to ½ become less and less
  probable as n increases

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Bienaymé-Chebyshev Proof

• Case X discrete random variable, m(x)
  distribution function
• P(X-µ e) Sx-µ e m(x)
• V(X) Sx (x-µ)²m(x) Sx-µ e (x-µ)²m(x)
• Sx-µ e (x-µ)²m(x) Sx-µ e e²m(x)
• e² Sx-µ e m(x)
• e² P(X-µ e)
• So P(X-µ e) lt V(X)/ e²
• X continuous same proof with density function
  f(x) instead of m(x) and integrals ? instead of
  sums S

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Weak law of large numbers

• Xn i.i.d. V(X1Xn) ns² ? V(Mn)s²/n E(Mn)µ
• Then, applying Bienaymé-Chebyshev to Mn
• P(Mn-µ e)lt s²/ne²
• Without replacement, a little bit more
  complicated calculations give
• V(Mn)(N-n)/(N-1)s²/n

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• Poll survey n 1000 respondents Obama Mn(?)
  54 true p, unknown
• P(Mn-p t) p(1-p)/nt²
• ? P(Mn-tltpltMnt) 1-1/4nt²
• Mn-t lt p lt Mnt confidence interval at level
  a1-1/4nt²
• To have P a0.95 the minimum t is t0.07, so
  47 lt p lt 61
• Does not depend much on the sampling rate modify
  formulas of variances by a (N-n)/(N-1) factor
  (sampling without replacement)

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• Samples without replacement are the real world
  samples. Formulas should hence be modified a
  little to take into account the violation of the
  independence hypothesis
• The change is very simple multiply variances by
  (1-n/N), and usually n/N is very small so that
  nothing changes
• The sample size n is (usually) more important
  than the sample rate n/N
• However precision only increases with vn (root-n
  samples) So that to double precision you have
  to increase sample size four times

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Back to concepts

• Regarding society (or nature), a concept that can
  not be measured is like a theoretical proposition
  that can not be empirically identified through a
  statistical analysis they are empty
• How long should a concept or a theory survive
  without empirical contents?

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Facts and counterfactuals (1)

• A social fact is an aggregation of natural
  quantities grounded on theory. Many questions are
  factual. Other questions are counterfactual.
• The difference can be large
• What are the differences between Côte dIvoire
  and Ghana, in terms of economic and social
  structures?
• How would have looked Côte dIvoire if it had
  been colonized by the British instead of the
  French?

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Facts and counterfactuals (2)

• Or the difference can look thinner
• Do richer people have taller children? (Not much)
• When income increases, do children grow taller as
  a consequence? (Yes indeed)
• - Do income differentials between social origins
  represent a large share of total income
  inequality in Brazil?
• What income inequality would be observed if
  social mobility was maximal?

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Facts and counterfactuals (3)

• Facts are often too quickly interpreted as
  counterfactuals, especially when they produce a
  rich description, for instance through
  decomposition techniques (for growth, inequality,
  mobility, etc.)
• Facts rule out some theories that can not account
  for them but they are compatible with many
  others.

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A very brief history of stat(e)-istics

• Land property (writing), then cadastre
• Fiscal income based on population (censuses),
  trade (customs), production (agricultural
  censuses in Rome)
• (Control of) Prices wages (Diocletian edict)
• National accounts (Quesnay)
• Probability theory and sample surveys
• Internet data

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Where to get some data on the Web?

• More and more data, macro and micro, becomes
  freely available on the Web. Suggestions will be
  given within each session.
• UN, IMF, WB, WTO, ILO, Eurostat
• Pop. censuses IPUMS initiative
• Survey series DHS, LSMS (WB), General Values
  Surveys, Rand Corporation surveys

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Some free micro data on the Web

• IPUMS international (census data)
• https//international.ipums.org/international/
• Demographic and Health Surveys
• http//www.measuredhs.com/
• World Values Surveys
• http//www.worldvaluessurvey.org/
• Rand Corporation surveys
• http//www.rand.org/labor/FLS/
• Some LSMS surveys (World Bank)
• http//iresearch.worldbank.org/lsms/lsmssurveyFind
  er.htm
• African Poverty Databank (World Bank)
• http//www4.worldbank.org/afr/poverty/databank/cdr
  oms/default.cfm
• Other free data on WebPages of academics who
  produced data