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Few notes on panel data (materials by Alan Manning)

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Few notes on panel data (materials by Alan Manning)

Development

Workshop

- 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

- 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)

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

- 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

- 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

- 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:

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

- 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

- 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

- Will not describe how we compute Ω-hat – see Wooldridge
- STATA command: xtreg y x, re i(id)

- 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

- 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.

- 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

- 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

- 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

- 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.

- For FE model we have:
- For BE model we have:
- For RE model we have:
- Where:

- 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

Can also get rid of fixed effect by differencing:

- 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

- For de-meaned model:

- For differenced model:

- These are not consistent:

- Which is ‘better’ depends on which assumption is right – how can we decide this?
- Much of this you have covered in Macroeconometrics course…