Econometric Methods 1

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Econometric Methods 1

• Lecture 8 Panel Data and Differences in
  Differences

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Panel Data

• So far in this course have looked at cross
  section data i.e. across individuals, households,
  firms, etc
• And also time series data, over years, quarters,
  days.
• Many surveys of households (individuals, etc)
  return to the same households at regular time
  intervals
• e.g. The British Labour Force Survey interviews
  the same households every quarter for 5 quarters
• This is panel data
• Note we can have pooled cross section-time series
  data where unrelated households are surveyed at
  regular time intervals
• Panel data follows the same cross section units
  over time

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Panel Data

• The main advantage of panel data is that it
  allows us to control for unobservable (fixed)
  effects
• e.g. Previously we had cross section data on
  individual is wages from the LFS and estimated
  the following
• Here we were concerned about omitted
  unobservables that may be correlated with
  Education or Male
• Suppose we now observed the same person in two
  different years
• We now have observations on wagei,t ,Malei,t and
  Education it where wagei,t the wage for person i
  in year t

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Panel Data

• We can now estimate the above equation in the
  following form
• Now that we have more than one observation on
  each individual i, can also add a separate
  intercept for each i.
• So the equation now becomes
• Where µi is an individual (fixed) effect that we
  estimate for each observation

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Unobserved Effects

• The term µi is often referred to as the
  unobserved effect, the fixed effect or unobserved
  heterogeneity
• µi capture all unobserved and time invariant
  factors that have an impact on log(wageit)
• In this example think of µi as capturing the
  impact of unobserved ability or skill, or any
  other unchanging characteristic that may have an
  effect on wages
• i.e. motivation, height,
• Note that the regression can also include
  unchanging but observable factors e.g. sex,
  race.
• The regression also contains a time varying error
  uit

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Estimation of panel models

• How then do we estimate such models
• Suppose we have the general model
• One approach is to pool the years and use OLS
• Where is the composite error
• But for OLS to be unbiased and consistent we
  require vit to be uncorrelated with Xit
• Since vit contains µi we require Xit and µi to be
  uncorrelated which is unlikely

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Estimation of panel models

• In most panels we want to allow for Xit and µi to
  be correlated, e.g. ability is correlated with
  education
• Since µi is fixed we can difference the equation
  above
• Suppose we just have two time periods t1 and
  t2 and allow different intercepts in each year
• Subtracting the top from the bottom equation
  gives
• µi has
  been differenced away!

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First Differenced Estimation

• First differenced model can then be estimated by
  OLS
• With two periods we have a single cross section
  where each variable has been differenced
• We require that be uncorrelated with
  for valid OLS estimates
• We also need to have sufficient variation in
• This rules out time invariant Xs such as sex,
  race, etc which will themselves be differenced
  away
• But even if varies significantly across
  individuals there is no guarantee that
  will

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Potential pitfalls

• Going back to our wage example and taking
  differences
• Although the level of education may vary widely
  across working age adults it is unlikely that the
  change in education will vary much
• For many individuals their education will remain
  unchanged from one year to the next
• Furthermore, panel data is expensive to collect
  since survey has to reach same individuals over
  time
• We can get problems of attrition, where
  individuals drop out of the panel in a non-random
  manner

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Differences in differences

• This is a particular application of panel data
  and is most useful when we have some sort of
  natural experiment
• Lets take an example
• In the 19th Century water was supplied to
  households by competing private companies
• Possible for different companies to supply
  households in same street!
• In 1854 South London had two main companies
• Lambeth Company (water supply from Thames Ditton,
  22 miles upstream)
• Southwark and Vauxhall Company (water supply from
  Thames)
• Lambeth company had switched from Hungerford
  bridge (Thames) in 1852

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Cholera Example

• Death Rates per 10000 people by water company in
  1854
• Lambeth 10
• Southwark and Vauxhall 150
• Difference could be water but perhaps other
  factors
• Death rates in 1849 epidemic
• Lambeth 150
• Southwark and Vauxhall 125
• So death rates fell for those under Lambeth
  company when it changed supply

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Cholera Example
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Cholera Example

• See fall in deaths amongst Lambeth drinkers
• No fall among Southwark and Vauxhall
• Can think of Lambeth as our treatment group and
  Southwark and Vauxhall as our control group
• We want to know what is the effect of switching
  supply to cleaner water
• Change in Lambeth supplied between 1849 and 1854
  tells us what happened
• But we dont know what would have happened in the
  absence of the change in supply
• Use Southwark and Vauxhall as control group to
  estimate what would have happened

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Cholera Example

• Figures from control group suggest that deaths
  would have increased by 25 per 10,000 people
• But actually fell by 140 per 10,000
• So estimated impact of change in supply is a fall
  of 165 per 10,000
• This estimate is valid if two conditions hold
• Change in Southwark and Vauxhall tells us what
  would have happened in absence of supply change
• The supply change doesnt affect what goes on
  among Southwark and Vauxhall supplied households

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Differences in differences

• Ideal use of differences in differences in
  context of a natural experiment e.g. random drug
  trials
• Otherwise we look for a control group who are the
  same in every way from the treatment group other
  than the receipt of the treatment
• This is a strong requirement
• A weaker condition is that, in the absence of
  treatment, the (unobserved) difference between
  treatment and control is constant
• We then compare the pre-treatment difference
  (normal difference) with the post treatment
  difference (normal plus causal effect)

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Graphically
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Differences in differences

• The standard difference estimate is given by A-B.
• But the normal difference between groups is C-B
• So differences in differences estimate is given
  by A-C
• This assumes that the trends would be the same
  for the treatment and control groups
• We are not able to test assumption
• Since we cannot ever observe what would have
  happened in the treatment group in the absence of
  the treatment
• But with data on prior periods with no treatment
  change can test to see if differences in trends

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More formally

• If is the mean of y in period t0 ,1 in
  group g T,C
• Then the differences in differences estimate is
  given by
• Identification of causal effects here is
  dependent on two assumptions
• in the absence of treatment, the (unobserved)
  difference between treatment and control is
  constant
• The control group is unaffected by the treatment
  itself
• The first assumption is likely to hold in
  randomised experiments and is helped by inclusion
  of Xs (below)
• Second assumption may be broken even in
  randomised experiments random job subsidy may
  affect control group if displaced from work by
  treated workers

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Regression approach to DiD

• Can estimate DiD using OLS regression
• Where Di is a dummy 1 if the individual is in
  the treatment group and zero otherwise
• postt is a dummy that is 1 is the time period
  is after the treatment and zero otherwise
• The estimated DiD effect is the coefficient on
  the interaction of the two dummies

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Regression approach to DiD

• The above regression can be extended to include
  observable characteristics X
• The addition of controls can help with the
  assumption above that the treatment and control
  groups have the same trend