1. Differences-in-Differences and A Brief Introduction to Panel Data

2. John Snow again…

3. 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)

4. In 1853/54 cholera outbreak
Death Rates per 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

5. What would be good estimate of effect of clean water?
1849
1853/54
Difference
Lambeth
150 10
-140
Vauxhall and Southwark
125 25
-25
140
-165

6. 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

7. 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

8. A Graphical Representationy
Time
Treatment
Control
Pre-
Post- A B C

9. 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

10. Some Notation Define: μit=E(yit)
Where i=0 is control group, i=1 is treatment
Where t=0 is pre-period, t=1 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)

11. 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:

12. 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?

13. In this case can estimate….
D-in-D estimate is estimate of β3 – why is this?

14. 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

15. 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 ΔW in differences version

16. 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

17. An Example: “Vertical 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

18. 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

19. A Case which does not look so good…..Ashenfelter’s 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

20. Earnings for period

21. 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)

22. Differences-in-Differences: Summary
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

23. 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

24. The Pooled Model Can simply ignore panel nature of data and estimate:
yit=β’xit+εit
This will be consistent if E(εit|xit)=0 or plim(X’ ε/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)

25. A More Plausible Model
Should recognise this as model with ‘group-level’ dummies or residuals
Here, individual is a ‘group’

26. Three Models Fixed Effects Model Random Effects Model
Treats θi as parameter to be estimated (like β)
Consistency does not require anything about correlation with xit
Random Effects Model
Treats θi as part of residual (like θ)
Consistency does require no correlation between θi and xit
Between-Groups Model
Runs regression on averages for each individual

27. Proposition 5.2 The fixed effect estimator of β will be consistent if:
E(εit|xit)=0
Rank(X,D)=N+K
Proof: Simple application of what you should know about linear regression model

28. 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

29. 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:

30. How does de-meaning work?
Can do simple OLS on de-meaned variables
STATA command is like:
. xtreg y x, fe i(id)

31. 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

32. Random Effects Estimator
Treats θi as part of residual (like θ)
Consistency does require no correlation between θi and xit
Should recognise as like model with clustered standard errors
But random effects estimator is feasible GLS estimator

33. More on RE Estimator
Will not describe how we compute Ω-hat – see Wooldridge
STATA command
. xtreg y x, re i(id)

34. Proposition 5.3 The random effects estimator of β will be consistent if:
E(εit|xi1,..xit,.. xiT)=0
E(θi|xi1,..xit,.. xiT)=0
Rank(X’Ω-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.

35. Comments
Assumption about exogeneity of errors is stronger than for FE model – need to assume εit 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

36. 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.

37. 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

38. 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

39. Measurement Error in Panel Data Models
Assume true model is:
Where x is one-dimensional
Assume E(εit|xi1,..xit,.. xiT)=0 and E(θi|xi1,..xit,.. xiT)=0 so that RE and BE estimators are consistent

40. Measurement Error Model
Assume:
where uit is classical measurement error, x*i is average value of x* for individual i and ηit 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.

41. Proposition 5.4 For FE model we have: For BE model we have:
For RE model we have:
Where:

42. What should we learn from this?
All rather complicated – don’t 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

43. Attenuation bias larger for RE than BE estimator as T>1>κ
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

44. 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:

45. Estimating Fixed Effects Model in Differences
Can also get rid of fixed effect by differencing:

46. 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

47. What Are these assumptions?
For de-meaned model:
For differenced model:
These are not consistent:

48. This leads to time series…
Which is ‘better’ depends on which assumption is right – how can we decide this?
We are not going to cover this in this course.