DifferencesinDifferences and A Brief Introduction to Panel Data

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Differences-in-Differencesand A Brief
Introduction to Panel Data
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John Snow again
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The Grand Experiment

• Water supplied to households by competing private
  companies
• Sometimes different companies supplied households
  in same street
• In south London two main companies
• Lambeth Company (water supply from Thames Ditton,
  22 miles upstream)
• Southwark and Vauxhall Company (water supply from
  Thames)

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In 1853/54 cholera outbreak

• Death Rates per 10000 people by water company
• Lambeth 10
• Southwark and Vauxhall 150
• Might be water but perhaps other factors
• Snow compared death rates in 1849 epidemic
• Lambeth 150
• Southwark and Vauxhall 125
• In 1852 Lambeth Company had changed supply from
  Hungerford Bridge

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What would be good estimate of effect of clean
water?
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This is basic idea of Differences-in-Differences

• Have already seen idea of using differences to
  estimate causal effects
• Treatment/control groups in experimental data
• Often would like to find treatment and
  control group who can be assumed to be similar
  in every way except receipt of treatment
• This may be very difficult to do

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A Weaker Assumption is..

• Assume that, in absence of treatment, difference
  between treatment and control group is
  constant over time
• With this assumption can use observations on
  treatment and control group pre- and
  post-treatment to estimate causal effect
• Idea
• Difference pre-treatment is normal difference
• Difference pre-treatment is normal difference
  causal effect
• Difference-in-difference is causal effect

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A Graphical Representation
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What is D-in-D estimate?

• Standard differences estimator is AB
• But normal difference estimated as CB
• Hence D-in-D estimate is AC
• Note assumes trends in outcome variables the
  same for treatment and control groups
• This is not testable
• with two periods can get no idea of plausibility
  but can with more periods

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Some Notation

• Define
• µitE(yit)
• Where i0 is control group, i1 is treatment
• Where t0 is pre-period, t1 is post-period
• Standard differences estimate of causal effect
  is estimate of
• µ11-µ01
• Differences-in-Differences estimate of causal
  effect is estimate of
• (µ11-µ01)-(µ10-µ00)

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How to estimate?

• Can write D-in-D estimate as
• (µ11-µ10)-(µ01 -µ00)
• This is simply the difference in the change of
  treatment and control groups so can estimate as

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• This is simply differences estimator applied to
  the difference
• To implement this need to have repeat
  observations on the same individuals
• May not have this individuals observed pre- and
  post-treatment may be different
• What can we do in this case?

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In this case can estimate.

• D-in-D estimate is estimate of ß3 why is this?

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A Comparison of the Two Methods

• Where have repeated observations could use both
  methods
• Will give same parameter estimates
• But will give different standard errors
• levels version will assume residuals are
  independent unlikely to be a good assumption
• Can deal with this by
• Clustering
• Or estimating differences version

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Other Regressors

• Can put in other regressors as before
• Perhaps should think about way in which they
  enter the estimating equation
• E.g. if level of W affects level of y then should
  include ?W in differences version

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Differential Trends in Treatment and Control
Groups

• Key assumption underlying validity of D-in-D
  estimate is that differences between treatment
  and control group would have remained constant in
  absence of treatment
• Can never test this
• With only two periods can get no idea of
  plausibility
• But can with more than two periods

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An ExampleVertical Relationships and
Competition in Retail Gasoline Markets, by
Justine Hastings, American Economic Review, 2004

• Interested in effect of vertical integration on
  retail petrol prices
• Investigates take-over in CA of independent
  Thrifty chain of petrol stations by ARCO (more
  integrated)
• Defines treatment group as petrol stations which
  had a Thrifty within 1 mile
• Control group those that did not
• Lots of reasons why these groups might be
  different so D-in-D approach seems a good idea

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This picture contains relevant information

• Can see D-in-D estimate of 5c per gallon
• Also can see trends before and after change very
  similar D-in-D assumption valid

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A Case which does not look so good..Ashenfelters
Dip

• Interested in effect of government-sponsored
  training (MDTA) on earnings
• Treatment group are those who received training
  in 1964
• Control group are random sample of population as
  a whole

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Earnings for period 1959-69
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Things to Note..

• Earnings for trainees very low in 1964 as
  training not working in that year should ignore
  this year
• Simple D-in-D approach would compare earnings in
  1965 with 1963
• But earnings of trainees in 1963 seem to show a
  dip so D-in-D assumption probably not valid
• Probably because those who enter training are
  those who had a bad shock (e.g. job loss)

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Differences-in-DifferencesSummary

• A very useful and widespread approach
• Validity does depend on assumption that trends
  would have been the same in absence of treatment
• Can use other periods to see if this assumption
  is plausible or not
• Uses 2 observations on same individual most
  rudimentary form of panel data

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A Brief Introduction to Panel Data

• Panel Data has both time-series and cross-section
  dimension N individuals over T periods
• Will restrict attention to balanced panels same
  number of observations on each individuals
• Whole books written about but basics can be
  understood very simply and not very different
  from what we have seen before
• Asymptotics typically done on large N, small T
• Use yit to denote variable for individual i at
  time t

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The Pooled Model

• Can simply ignore panel nature of data and
  estimate
• yitßxiteit
• This will be consistent if E(eitxit)0 or
  plim(X e/N)0
• But computed standard errors will only be
  consistent if errors uncorrelated across
  observations
• This is unlikely
• Correlation between residuals of same individual
  in different time periods
• Correlation between residuals of different
  individuals in same time period (aggregate
  shocks)

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A More Plausible Model

• Should recognise this as model with group-level
  dummies or residuals
• Here, individual is a group

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Three Models

• Fixed Effects Model
• Treats ?i as parameter to be estimated (like ß)
• Consistency does not require anything about
  correlation with xit
• Random Effects Model
• Treats ?i as part of residual (like ?)
• Consistency does require no correlation between
  ?i and xit
• Between-Groups Model
• Runs regression on averages for each individual

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Proposition 5.2The fixed effect estimator of ß
will be consistent if

• E(eitxit)0
• Rank(X,D)NK
• Proof Simple application of what you should know
  about linear regression model

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Intuition

• First condition should be obvious regressors
  uncorrelated with residuals
• Second condition requires regressors to be of
  full rank
• Main way in which this is likely to fail in fixed
  effects model is if some regressors vary only
  across individuals and not over time
• Such a variable perfectly multicollinear with
  individual fixed effect

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Estimating the Fixed Effects Model

• Can estimate by brute force - include separate
  dummy variable for every individual but may be
  a lot of them
• Can also estimate in mean-deviation form

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How does de-meaning work?

• Can do simple OLS on de-meaned variables
• STATA command is like
• . xtreg y x, fe i(id)

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Problems with fixed effect estimator

• Only uses variation within individuals
  sometimes called within-group estimator
• This variation may be small part of total (so low
  precision) and more prone to measurement error
  (so more attenuation bias)
• Cannot use it to estimate effect of regressor
  that is constant for an individual

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Random Effects Estimator

• Treats ?i as part of residual (like ?)
• Consistency does require no correlation between
  ?i and xit
• Should recognise as like model with clustered
  standard errors
• But random effects estimator is feasible GLS
  estimator

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More on RE Estimator

• Will not describe how we compute O-hat see
  Wooldridge
• STATA command
• . xtreg y x, re i(id)

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Proposition 5.3The random effects estimator of
ß will be consistent if

• E(eitxi1,..xit,.. xiT)0
• E(?ixi1,..xit,.. xiT)0
• Rank(XO-1X)k
• Proof RE estimator a special case of the
  feasible GLS estimator so conditions for
  consistency are the same.
• Error has two components so need a. and b.

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Comments

• Assumption about exogeneity of errors is stronger
  than for FE model need to assume eit
  uncorrelated with whole history of x this is
  called strong exogeneity
• Assumption about rank condition weaker than for
  FE model e.g. can estimate effect variables that
  are constant for a given individual

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Another reason why may prefer RE to FE model

• If exogeneity assumptions are satisfied RE
  estimate will be more efficient than FE estimator
• Application of general principle that imposing
  true restriction on data leads to efficiency
  gain.

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Another Useful Result

• Can show that RE estimator can be thought of as
  an OLS regression of
• On
• Where
• This is sometimes called quasi-time demeaning
• See Wooldridge (ch10, pp286-7) if want to know
  more

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Between-Groups Estimator

• This takes individual means and estimates the
  regression by OLS
• Stata command is xtreg y x, be i(id)
• Condition for consistency the same as for RE
  estimator
• But BE estimator less efficient as does not
  exploit variation in regressors for a given
  individual
• And cannot estimate variables like time trends
  whose average values do not vary across
  individuals
• So why would anyone ever use it lets think
  about measurement error

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Measurement Error in Panel Data Models

• Assume true model is
• Where x is one-dimensional
• Assume E(eitxi1,..xit,.. xiT)0 and
  E(?ixi1,..xit,.. xiT)0 so that RE and BE
  estimators are consistent

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Measurement Error Model

• Assume
• where uit is classical measurement error, xi is
  average value of x for individual i and ?it is
  variation around the true value which is assumed
  to be uncorrelated with and uit and iid.
• We know this measurement error is likely to cause
  attenuation bias but this will vary between FE,
  RE and BE estimators.

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Proposition 5.4

• For FE model we have
• For BE model we have
• For RE model we have
• Where

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Proof

• General idea is to write each model as an OLS
  estimator and then use what we know about
  attenuation bias in that model
• Will use those earlier results

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Proof for FE Model

• Can write as OLS estimator on de-meaned data
• We have that
• And that

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• De-meaning we have that
• And that
• Take variances

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• Standard formula for attenuation bias gives us

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Proof for BE estimator

• Using earlier results we have

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Proof for RE estimator (a bit more complicated)

• Use result that can write it as OLS on
  quasi-de-meaned data
• Attenuation bias van be written as

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• Can write these elements as
• Leading to attenuation bias is
• Where

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What should we learn from this?

• All rather complicated dont worry too much
  about details
• But intuition is simple
• Attenuation bias largest for FE estimator
  Var(x) does not appear in denominator FE
  estimator does not use this variation in data

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• Attenuation bias larger for RE than BE estimator
  as Tgt1gt?
• The averaging in the BE estimator reduces the
  importance of measurement error.
• Important to note that these results are
  dependent on the particular assumption about the
  measurement error process and the nature of the
  variation in xit things would be very different
  if measurement error for a given individual did
  not vary over time
• But general point is the measurement error
  considerations could affect choice of model to
  estimate with panel data

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Time Effects

• Have treated time and individual dimensions
  asymmetrically no good reason for this
• Errors likely to be correlated for different
  individuals in same time period most common way
  to deal with this is to include set of time
  dummies

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Estimating Fixed Effects Model in Differences

• Can also get rid of fixed effect by differencing

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Comparison of two methods

• Estimate parameters by OLS on differenced data
• If only 2 observations then get same estimates as
  de-meaning method
• But standard errors different
• Why? assumption about autocorrelation in
  residuals

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What Are these assumptions?

• For de-meaned model

• For differenced model

• These are not consistent

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This leads to time series

• Which is better depends on which assumption is
  right how can we decide this?
• Allow cross-section dimension to wither
• Focus on case where observations are a time
  series for a single unit e.g. a whole economy.