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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Few notes on panel data materials by alan manning

Few notes on panel data (materials by Alan Manning)

Development

Workshop


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:

    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)


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


Intuition

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


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.


Comments

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


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


Conclusions

Conclusions

  • 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

Can also get rid of fixed effect by differencing:

Estimating Fixed Effects Model in Differences


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…


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