Few notes on panel data (materials by Alan Manning)

1. Few notes on panel data (materials by Alan Manning) Development Workshop

2. 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

3. 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)

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

5. 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

6. 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

7. 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

8. 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:

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

10. 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

11. 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

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

13. 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.

14. 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

15. 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.

16. 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

17. 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

18. 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

19. 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.

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

21. 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

22. 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

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

24. 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

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

26. 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…