Selection on Unobservables: Solutions with Panel Data

Question 1

What are the main feature of panel data

Answer 1

• observe same subjects at different points in time (at least 2)
• panel called balanced if we have the same time points for all subjects
• balanced panels are easiest to handle
• otherwise potentially attrition bias (e.g. non-random drop out over time)
• important: panel data different from ‘repeated cross-section’
• in RCS different sample at each point in time
• the same individuals may be observed twice, but only by chance

Question 2

When is a panel called balanced? Problems?

Answer 2

same data for all subjects - attrition bias (non-random drop out)

Question 3

How is panel data different from ‘repeated cross-section’?

Answer 3

• in RCS different sample at each point in time - the same individuals may be observed twice, but only by chance

Question 4

advantages of panel data

Answer 4

• generally, more data is better
• specifically, can not only exploit between-subject variation as in cross-section, but also within-subject variation (e.g. change of treatment status)
• this allows to deal with some forms of bias from unobservables

Question 5

Econometric challenges arising with panel data

Answer 5

• serial correlation of errors - need for inclusion of lagged dependent variables? - if so, opens a range of new ‘problems’ (not covered in the course)

Question 6

Bias from unobservables and panel data

Answer 6

Cause of interest (treatment) not always an event with a given timing

• often subjects decide whether to take treatment, and when to take it
• decision often depends on observed factors X and unobserved factors U
  → if these also affect Y,we are back in a familiar world:non-random selection on unobservables
• similarly, if unobservables affect level of Y and X
  → again a familiar problem: omitted variable bias from the unobservables

Question 7

If c is constant across time for each subject (i.e. cit = ci for t = 1, 2, …T ), what are three common situations and models producing consistent estimates?

Answer 7

1. pooled OLS 2. the random effects model 3. the fixed effects model

⇒ Most intuitive to think about these three situations in terms of the intercepts required for unbiased estimation.

Question 8

When is pooled OLS consistent?

Answer 8

• if ci are all the same,that is ci =c
• c is captured by the constant α in
  y =α+X′β+δd +ε
  ⇒ pooled OLS is consistent
  (which means, just for clarification, one can re-index to yk = α + Xk′ β + δdk + εk .

Question 9

When is random effects estimation appropriate?

Answer 9

• if ci differ but are not systematically related to dit and xit
• effects ci can be decomposed into mean component α = c ̄ and individual specific
  component vi = ci − c ̄
• a new model with a composite error term, say uit = vi + εit can be formulated
  y =α+X′β+δd +u it it it it
  ⇒ random effects estimation by means of GLS offers a way to correct for the fact that the vi part in the error is not fully independent from y (explanation is technical)

Question 10

When is a fixed effects estimator needed?

Answer 10

• if ci differ and are related to dit (and potentially also xit )
• RE estimator is biased
  → reason is that the procedure by which RE-GLS estimates the intercept for each i
  picks up some of the effect of interest
• need to either explicitly model the ci or make them ‘disappear’
  ⇒ fixed effects estimators achieve that

Question 11

How does one implement a FE estimator?

Answer 11

a) including a dummy variable for each subject (explicit modelling of intercepts)
b) subtract each subject’s mean values of y , x , d and estimate as pooled OLS (makes ci disappear) (demeaning)
c) take first differences and estimate as pooled OLS (or cross section if T = 2) (also makes ci disappear) (first differencing)

Question 12

Describe the three different FE estimators.

Answer 12

a) including matrix with a dummy variable for each subject y =x′β+δd +I′γ+ε
b) demeaning (yit −y ̄i)=(xit −x ̄i)′β+δ(dit −d ̄i)+(ci −c ̄i)+(εit −ε ̄i) noteherethatci =c ̄i andε ̄i =0 Remark: a and b identical except that a ‘costs’ more degrees of freedom.
c) first differencing (yit − yit−1) = (xit − xit−1)′β + δ(dit − dit−1) + (ci − ci) + ε ̃it Remark:c identical to a and b if T=2, and very similar for T >2. a and b require non-serially correlated errors, whereas c does not.

Question 13

Disadvantages of FE estimators (vs. RE)

Answer 13

• subjects with time-invariant yi are ignored
• cannot obtain coefficient estimates for time-invariant xi
  → intuitive reason:
  FE estimation identifies effect of X on Y solely from within-subject variation.
  → note also: FE estimators very sensitive to bias with short panels (T ∼ 10 or less)

Question 14

WHen are FE & RE estimators consistent?

Answer 14

FE and RE estimators only consistent if effect of unobservables constant for each i

Question 15

How can you test RE assumption?

Answer 15

can be tested using the Hausman-test (the null is that RE is unbiased, if rejected use FE)

Question 16

Advantages of RE vs FE?

Answer 16

RE estimator is superior along these criteria:
- uses within-subject variation and between-subject variation
- can produce coefficients for time-invariant covariates xi
- but requires that unobserved factors ci are unrelated to xit and dit
- often untenable as an assumption
→ can be tested using the Hausman-test
(the null is that RE is unbiased, if rejected use FE)
→ note also: RE model not very suitable for data with few subjects

Question 17

Disadvantages of RE vs FE?

Answer 17

but requires that unobserved factors ci are unrelated to xit and dit note also: RE model not very suitable for data with few subjects

Question 18

Unobserved effects in panel data modelling

Answer 18

Suppose outcome Y is given by
E(Y|X,D) = α + X′β + δD and all X and D are exogenous, then δ can be estimated consistently by a regression y k = α + X k′ β + δ d k + ε k
because Cov(d,ε) = 0. (Note: k = 1, 2, …K indexes observations, not subjects; thus in panel data K = T × N.)
If, however, true model is such that Y is given by
E(Y|X,D) = α + X′β + δD + c,
where c is the effect on Y of unobserved factors, then the model we can estimate is yk =α+Xk′β+δdk +ck +εk.
Coefficients will be biased because ck will end up in the residual (then c + ε) and Cov (d , c + ε) ̸= 0.