few notes on panel data materials by alan manning
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Few notes on panel data (materials by Alan Manning). Development Workshop. A Brief Introduction to Panel Data. Panel Data has both time-series and cross-section dimension – N individuals over T periods

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a brief introduction to panel data
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
the pooled model
The Pooled Model
  • Can simply ignore panel nature of data and estimate:


  • 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)
a more plausible model
A More Plausible Model
  • Should recognise this as model with ‘group-level’ dummies or residuals
  • Here, individual is a ‘group’
three models
Three Models
  • Fixed 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
the fixed effect estimator of will be consistent if
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
  • 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
estimating the fixed effects model
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:
how does de meaning work
How does de-meaning work?
  • Can do simple OLS on de-meaned variables
  • STATA command is like:xtreg y x, fe i(id)
problems with fixed effect estimator
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
random effects estimator
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
more on re estimator
More on RE Estimator
  • Will not describe how we compute Ω-hat – see Wooldridge
  • STATA command: xtreg y x, re i(id)
the random effects estimator of will be consistent if
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.
  • 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
another reason why may prefer re to fe model
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.
another useful result
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
between groups estimator
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
measurement error in panel data models
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
measurement error model
Measurement Error Model
  • Assume:
  • where uit is classical measurement error, x*iis 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.
proposition 5 4
Proposition 5.4
  • For FE model we have:
  • For BE model we have:
  • For RE model we have:
  • Where:
what should we learn from this
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
  • 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
comparison of two methods
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
what a re these assumptions
What are these assumptions?
  • For de-meaned model:
  • For differenced model:
  • These are not consistent:
this leads to time series
This leads to time series…
  • Which is ‘better’ depends on which assumption is right – how can we decide this?
  • Much of this you have covered in Macroeconometrics course…