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Few notes on panel data (materials by Alan Manning) - PowerPoint PPT Presentation

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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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

• 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

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

• 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)

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

• 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

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

• 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

Estimating Fixed Effects Model in Differences

• 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 are these assumptions?

• 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…