Claudio LUCIFORA

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Università Cattolica del Sacro Cuore Istituto di
Economia dell'Impresa e del Lavoro
APPLIED ECONOMETRICS Module 2 - Multiple
Regression Analysis with dummy variables

• Claudio LUCIFORA

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Main references

• - Brucchi Luchino (2000) Manuale di economia del
  lavoro, CH 19-20, Il Mulino.
• Wooldridge (2000), J.M., Introductory
  Econometrics A Modern Approach, Second edition,
  CH.19, MIT Press.
• J. Altonji and R. Blank (1999), "Race and Gender
  in the Labor Market", in O. Ashenfelter and Card,
  D. Handbook of Labour Economics, vol.3(c),
  North-Holland

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Structure of the Presentation

• Multiple Regression Analysis with dummy variables
• Defining the dummies
• Interpreting coefficients
• Dummy variable trap
• Dummy Interactions
• Spline functions
• An empirical application gender pay
  differentials (Oaxaca decomposition)

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Structure of the Presentation (continued)

• Problems
• Some caveats (selection, endogeneity,
  heterogeneity)
• Self selection
• The index number (or discrimination free)
• Differences in distributions the Juhn, Murphy
  and Pierce decomposition

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Using dummy variables

• The purpose of this module is to show how to
  define, construct, specify, estimate and
  interpret dummy variables in empirical analysis.
• Dummies are the most common variables found in
  the empirical analysis of Survey Data.
• We use dummies to account for qualitative
  factors, such as membership in a group (ie
  gender), selected time period (ie 2001), specific
  threshold (ie highest level of education
  achieved), etc.
• The use of dummies can produce an impressive
  variety of models

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Dummy variable/1

• A dummy variable is a variable that takes on the
  value 1 or 0
• it can be the recoding of a binary attribute such
  as gender (male1 female0)
• The recoding of a multilevel character such as
  region (1-20) (i.e. north1 if region 1-5, 0
  otherwise south1 if region 15-20, 0
  otherwise,), etc.
• The recoding of a continuous varible such as age
  (i.e. young1 if age 1-15, 0 otherwise old1 if
  age 55-over, 0 otherwise,), etc.

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A Dummy Independent Variable

• Consider a simple model with one continuous
  variable (x) and one dummy (d)
• y b0 d0d b1x u
• This can be interpreted as an intercept shift
• If d 0, then y b0 b1x u
• If d 1, then y (b0 d0) b1x u
• The case of d 0 is the base group

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Comparing two means

• If for example y is income and d is whether or
  not the individual attended college (ie ignoring
  the continuous variable x)
• Eincomedid not attend collegeb0
• Eincomeattended college(b0 d0)

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Example of d0 gt 0
y (b0 d0) b1x
y
d 1
slope b1

d0
d 0

y b0 b1x
b0
x
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Dummies for Multiple Categories

• We can use dummy variables to control for
  something with multiple categories
• Suppose you want to analyse education, and
  everyone in your data is either a HS dropout (1),
  HS grad only (2), or college grad (3)
• To compare HS grad and college grads to HS
  dropouts, you need 2 dummy variables
• hsgrad 1 if HS grad only, 0 otherwise and
  colgrad 1 if college grad, 0 otherwise

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Dummy variable trap

• Because the base group is represented by the
  intercept, if there are n categories there should
  be n 1 dummy variables
• Industry differentials (20 industries) include
  (20 1)19 industry dummies
• If there are a lot of categories (or continuous
  variables), it may make sense to group some
  together (such as, regions 1-20 into north,
  centre, south)

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Interactions Among Dummies

• Interacting dummy variables is like subdividing
  the group
• Example have dummies for gender, as well as
  hsgrad and colgrad
• Add genderhsgrad and gendercolgrad, for a
  total of 5 dummy variables gt 6 categories
• Base group is female HS dropouts
• hsgrad is for female HS grads, colgrad is for
  female college grads
• The interactions reflect male HS grads and
  male college grads

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Dummy Interactions/1

• Formally, the model is
• y b0 d1gender d2hsgrad d3colgrad
  d4genderhsgrad d5gendercolgrad b1x u
• If gender0 and hsgrad0 and colgrad0
• y b0 b1x u (base group female, HS
  dropouts)
• If gender0 and hsgrad1 and colgrad0
• y b0 d2hsgrad b1x u (female, HS grads)
• If gender1 and hsgrad0 and colgrad1
• y b0 d1gender d3colgrad d5gendercolgrad
  b1x u (male, college grads)

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Dummy Interactions/2

• Can also consider interacting a dummy variable,
  d, with a continuous variable, x
• y b0 d1d b1x d2dx u
• If d 0, then y b0 b1x u
• If d 1, then y (b0 d1) (b1 d2) x u
• N.B. This is interpreted as a change in the slope

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Example of d0 gt 0 and d1 lt 0
y
y b0 b1x
d 0
d 1
y (b0 d0) (b1 d1) x
x
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Spline regression

• The function we wish to estimate
• y b0 d0age u if agelt18
• y b1 d1age u if agegt18 and lt22
• y b2 d2age u if agegt22
• Specify using dummy variables
• d11 if agegt18
• d21 if agegt22
• y b d age ? d1 ? d1age t d2 s d2age
  u
• (1.) d (2.) (d ?) (3.) (d ? s)
• In this way it is discontinuous

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It is discontinuous
y
x
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Spline regression/2

• To make it continuos we require the segments to
  join at the knots
• b d 18 (b ? ) (d ?)18
• (b ? ) (d ?)22 (d ? t ) (d ? s )22
  
• These are linear restrictions on the parameters
• y b dage ? d1(age 18) s d2 (age 22)
  u
• estimate with constrained least squares
• X1age X2(age 18) if agegt18, other 0
• X3(age 22) if agegt22, other 0

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Testing for Differences

• Testing whether a regression function is
  different for one group versus another can be
  thought of as simply testing for the joint
  significance of the dummy and its interactions
  with all other x variables
• can estimate a model with all the interactions
  and without and form an F statistic
• Alternatively, you can perform a CHOW TEST (a
  simple F test for exclusion restrictions)

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Caveats on Program Evaluation

• A typical use of a dummy variable is when we are
  looking for a treatment effect
• For example, we may have individuals that
  received job training, or welfare, etc
• We need to remember that usually individuals
  choose whether to participate in a program, which
  may lead to a self-selection problem

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(Self-)selection Problems

• If we can control for everything that is
  correlated with both participation and the
  outcome of interest then its not a problem
• Often, though, there are unobservables that are
  correlated with participation
• In this case, the estimate of the program effect
  is biased, and we dont want to set policy based
  on it!

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An empirical application

• The analysis of the gender pay differential

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Pay gap/1

• The crudest approach consists in including a
  gender dummy in a single wage regression for
  women and men.
• y b0 d0gender b1x u
• The coefficient d0 is often interpreted as an
  estimate of the standardised gender pay gap
• The underlying assumption here is that female and
  male wages differ by a fixed amount (shift
  parameter), but that human capital
  characteristics and other explanatory variables
  (x) have the same impact on womens and mens
  wages

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Pay gap/2

• The assumption of similar returns might not be
  true as gender differences in pay may go through
  several explanatory variables (x).
• Use gender (1,0) to split the sample
• yM bM0 bM1xM uM iff gender1, male
• yF bF0 bF1xF uF iff gender0, female
• In this case, differences in pay depend on the
  intercept, as well as beta coefficients and error
  terms

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Pay gap/3

• Wage differential between two groups of people
  defined by gender, race, ethnicity etc. can be
  decomposed into two parts
• The first is explained by differences in human
  capital endowments of both groups,
• The second reflects differences in prices, that
  is the remuneration of these endowments. (i.e.
  wage discrimination).

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Decomposition of the gender wage gap

• Male-female wage differential as the difference
  in logarithmic mean wages
• Decompose into an explained part to reflect
  productivity differences endowment eff. and an
  unexplained part to reflect differences in the
  remuneration of those characteristics
  remuneration eff. often referred to as a
  measure of discrimination

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Pay gap two equations

• The wage equations for men and women are
  specified as follows
• where i indexes individuals within the male and
  female samples. lnWi is the log wage. Vector Xi
  contains all explanatory variables. The error
  term represents an iid idiosyncratic error term
  with mean zero and constant variance s2

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Pay gap two equations/2

• The estimated price vector ß and the average
  human capital and job characteristics Xs for
  males and females are used to compute weighted
  differences in mean characteristics.

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Counterfactuals

• To decompose the raw wage gap it is further
  necessary to make assumptions on a competitive
  price vector which operates as standard in
  valuing the different characteristics. This price
  vector should reflect the remuneration of human
  capital characteristics in absence of
  discrimination.
• The predicted mean wage for women is computed
  with coefficient estimates from the male wage
  regression and average characteristics of
  females

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Oaxaca decomposition
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non-discriminatory wage structure

• there is a vector ß which reflects the rates of
  return on human capital characteristics in the
  absence of discrimination.

(1)
(2)
(3)
1. differences in endowments
2. discrimination in favour of males
3.discrimination against females
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Some caveats/1

• This simple wage model is often estimated by
  ordinary least squares. Yet this method only
  provides consistent estimates if the following
  orthogonality conditions are fulfilled
• where Ii denotes a latent index variable which
  is positive if an individual i is employed and
  non-positive otherwise.

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Some caveats/2

• Sample selection is a source of violation of the
  orthogonality condition. The sample of working
  people excludes, by definition, those who do not
  participate in the labour market and therefore
  may not be a random selection of the overall
  population. If the participation decision is
  correlated with the earnings function, the
  expected value of the error term of the latter
  may not be zero.

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Overview of methodological problems wage
equations
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Pay gap refinements

• More complex decomposition methods have been
  developed
• taking also the residual wage distribution into
  account (Juhn, Murphy and Pierce 1993)
• treating occupational or sector segregation as
  endogenous (Brown, Moon and Zoloth 1980).

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Decomposition by Juhn, Murphy and Pierce (JMP)

• It extends the approach of Oaxaca and Blinder
  (O-B) by decomposing the pay gap not only at the
  mean but over the whole wage distribution,
  thereby accounting for the residual (unexplained)
  wage distribution.
• While O-B can be applied with cross-section data,
  JMP needs a two time periods, two countries,
  etc.
• Predicted wages are then used to derive
  hypothetical wage distributions that serve to
  extend the decomposition of the unadjusted wage
  gap by a wage structure effect.
• The decomposition technique is employed to
  distinguish the effects of gender specific
  factors from those associated with the underlying
  wage structures of both economies (i.e. wage
  inequality).

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• The decomposition of the raw wage gap then
  includes three components related to differences
  in endowments, in estimated coeffcients and
  in the residual wage distribution

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• Define where is
  country js residual standard deviation of wages,
  and
• is the standardised unobservable
  productivity.
• The wage equation for a male worker from country
  j is
• The male-female log wage gap for country j is
  then given by
• NB. we assume ßM ßF ß (i.e. non
  discriminatory returns)

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JMP Decomposition

• Gender pay difference between two countries j and
  k
• 1. and 2. as usual (but by country)
• 3. effect of cross country differences in the
  relative wage position of males and females after
  controlling for observed human capital
  characteristics differences in the level of
  unobservables.
• 4. differences in the returns to unobservable
  skills inequality.
• 1 and 3 are gender specific 2 and 4 reflect
  inter-country differences in the underlying wage
  structure

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An empirical application/2

• The analysis of the public-private
• pay differential

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