Econometrics

1
Econometrics

• Lecture Notes Hayashi, Chapter 5c
• Unbalanced Panels

2
Missing Observations

• Assuming common coefficients, for each equation m
  (1,,M) and for each observation i (1,,n),yim
  zimd ai him
• Typically, there are missing observations.
• Assumption No Selection Bias
• Whether an observation stays in the sample does
  not depend on the error term.

3
Missing Observations

• Let dim 1 if observation m is in the sample
  dim 0 otherwise. Let Mi be the number of
  observations from i.

4
The Model

• yi Fib di(big) diai hi(i1,,n)

5
The Transformed Model

• Let Qi IMi di(didi)-1di IMi didi/Mi
• Qiyi QiFib Qidi(big) Qidiai Qihi
• Qiyi QiFib Qihi since Qidi 0
• yi Fib hi
• y Fb h
• With zeroing out missing observation, the
  transformed model remains the same as without
  missing observations. Therefore, the formula for
  the fixed-effects estimator remains the same.

6
Fixed-Effects Estimator

• The fixed-effects estimator of b is the pooled
  OLS estimator, which is consistent and
  asymptotically normal
• bFE (FF)1Fy
• bFE- b (FF)1Fh
• Est(Avar(bFE)) 1/n (FF)1(FVF)
  (FF)1 where V (y-FbFE)(y-FbFE)

7
Fixed-Effects Estimator

• If the assumption of spherical distribution is
  assumed E(hihi) s2IMi, so that E(hihi)
  s2Qi
• Est(Avar(bFE)) n s2 (FF)1
• s2 is the estimator of s2

8
No Selection Bias Assumption

• Recall the Orthogonality Assumption for the
  transformed modelE(fim hih) 0 (m,h1,,M)
• A stronger assumption is need for the case with
  missing observations --No Selection Bias E(fim
  hihdi) 0 (m,h1,,M)

9
Large Sample Properties

• If the assumption of no selection bias holds in
  addition to other assumptions required for
  establishing the large sample properties of the
  fixed-effects estimator, the same properties
  carry over for unbalanced panels.