Error Component Models

1. Error Component Models Methods of Economic Investigation Lecture 8

2. Today’s Lecture • Review of Omitted Variables Bias • Error component models • Fixed Effects • Random Effects

3. Omitted Variable Bias Formula • True Relationship Y = α + βT +γX+ε • We estimate • We get a bias from the omitted variable

4. A specific form of this… • Suppose we had to estimate • Where the error term can be decomposed into: Individual specific factor Going to set this to zero for ease of exposition Random error term uncorrelated with everything else Group specific factor

5. What are the problems with this? • Errors have cross-correlation across terms • OLS is not efficient • Still get consistent estimates, but may be harder to do inference • If X is correlated with γor δ and we cannot observe this factor, then our OLS estimates will be biased

6. Unobserved effects are not correlated with X’s • This can happen if the group effect is ‘random’ • Maybe different geographic areas, not correlated with X • Different within-group correlations, violate OLS assumptions that errors uncorrelated with each other • Don’t have to worry about OVB • Still have problems with standard error • OLS not efficient • Can’t do inference with our estimates

7. OLS standard errors • Usually estimate • Standard way to estimate variance of OLS estimates not valid—won’t necessarily be the same across j’s:

8. Robust Standard Errors • Estimate the residuals: • Then estimate standard errors • Consistent, but may be able to do better…

9. Exploiting the Random Effects Structure • We have more information than we’re using in the robust standard errors • The composite error means there is correlation within groups j but not between groups • Define where

10. FGLS Estimates—2 steps • Estimate OLS and get • Get weighting matrix • Can now adjust the regression for the known correlation

11. FGLS Variance Estimate • With a variance • If our model is correctly specified, this will simplify to:

12. Unobserved effects are correlated with X’s • Simplify, let • So we need to estimate: • When might this happen? • Individuals are part of some “group” and that group has a unique relationship to the outcomes (relative to other groups) • Looking in a time period—that time period is related to the outcome differently than other time periods

13. What does a group specific term do? Y X

14. How might we fix this? • If we could observe multiple individuals in a group j, then we could difference out the group effect • So for example: look at individuals 1 & 2, both of whom are in group 1

15. Differencing out Group Effects • Difference between individuals 1 and 2, gives us the following estimate: • How do we interpret this? • β tells us how a change in the difference in X’s between people 1 and 2, changes the difference in outcomes between people 1 and two • Often don’t really care about difference between 2 people…

16. Average Group Effect • Take the sample average outcome for each group • Crucial assumption: average group effect is the group effect • This means: NO within group variation

17. Estimating Deviation from the Average Group Effect • Difference individual and group average equations: • Define ~ terms as deviation from means so we estimate: • So we can estimate this with our usual OLS (more on the standard errors in a few minutes…)

18. How do ‘fixed effects’ help? • Define δj=1 for group j and zero for all groups • Then we estimate: • And we will get an estimate for β1, our effect of X on Y, and δ our group effect

19. Estimation • To see how, imagine we have two groups: men and women • If we estimated two equations then its: • We won’t recover β0 separate from the δ’s constant, cm constant, cf

20. Multiple Equation Estimation • Let men be the base group, then for men we estimate: • What does this give us? • CEF: E[Y| X, j=m] = δm + β*E[X | j=m] • δmis the intercept for men or E[Y |X=0, j=m] • If X is de-meaned, then this is the average effect for men at the mean of the sample

21. Why estimate a single equation? • Single equation easy to estimate: run a regression for each group separately BUT • Hard to do inference on size of group effect • Lose some power if, conditional on group—the effect of various X’s is the same • For example: • Men and women on average have different starting wages • Conditional on starting level, each year of additional experience, increases wages by some fixed %

22. Single equation estimation • Now estimate for everyone: • For men this estimates • For women Additional “female” effect Constant: E[Y |j=m, X=0]

23. How to interpret your “fixed effect”? • The additional effect of women, relative to men • Do we care about the fixed effects? • Sometimes they have an interpretation e.g. controlling for all observable factors, is there a different level of wages for men and women • Sometimes they are a just differencing out a bunch of things, and don’t have an interpretation, e.g. repeated observations on an individual, difference out an individual effect—not much interpretation for idiosyncratic individual effect

24. When have unobserved group effects can be two issues: Uncorrelated with X’s: OLS not efficient, can fix this with GLS Correlated with X’s: OVB, can include “fixed effects” Fixed effects, within-group differences, and deviation from means differences can all remove bias from unobserved group effect What did we learn today

25. Application: The effect of Schooling on wages Ability Bias Fixing this with “twins” and “siblings” models Next Class