做静态与动态面板数据模型的很好

– finite sample ≠ asymptotic
Users should understand motivation and limits of estimator
Linear GMM in one slide
• Instrument vector z such that E[z ] 0
βˆ2SLS

X'Z
Z'Z
1 Z'X
1
X'Z
Z'Z
1 Z'Y
• If errors i.i.d., efficient GMM is 2SLS • If not, 2SLS inefficient
Linear GMM in another slide
(Holtz-Eakin, Newey, and Rosen 1988)
• Arellano and Bond calculate its standard deviation
• z test for AR()
• More general than other AR() tests in Stata.
• abar: post-estimation command for regress, ivreg, ivreg2
•A problem of short panels •Individual dummies (=Within Groups) don’t help
o Transformed yi,t1 endogenous, as are deeper lags
Partial solution: OLS in differences
. Z'ΩZ 1 Rerun. Two-step is feasible, theoretically efficient.
Linear GMM and 2SLS
βˆEGMM

X'Z
Z'ΩZ
1 Z' X
1
X'Z
Z'ΩZ
1 Z'Y
• If Ω 2I, reduces to
• E.g., compare eit ei,t1 and ei,t2 ei,t3 to detect ei,t1 ~ ei,t2
• Test statistic for AR(l) in differences:
eit ei,tl
i,t
• Normal under null of no AR(l)
Windmeijer correction, revamped syntax Estimators designed for
– Small-T, large-N panels – One dependent variable – Dynamic – Linear – Regressors endogenous and predetermined – Fixed individual effects – Arbitrary autocorrelation and het. within panels – General application
endogenous in same way o Still other variables may be endogenous
from the start •For general application, assume no perfect instruments waiting in the wings
• If we find AR(l) in it , use lags l + 1 and deeper
Arellano-Bond AR() test
• Expect AR() in it i it
• To check for AR(1) in it , test for AR(2) in eit
How to Do xtabond2
David Roodman Research Fellow Center for Global Development
xtabond2 in a nutshell
First ado version in 11/03, Mata version in 11/05. Extends built-in xtabond, to do system GMM,
•
#
instruments
>
#
parameters
so
can’t
have
EN[z ]
1 N
Z'Eˆ 0
•
Want to “minimize”
1 N
Z'Eˆ
in
some
sense
• In what sense? By a pos-semi-def. quad. form given by A:
OLS efficient Var E* I
OLS on (3) = GLS on (2) = GMM on (1)
GMM = GLS on Z-moments
Difference and system GMM
Basic model:
yit
yi,t1

x
' it
β
it
it i it
Solution: Instrument with lags (2SLS) (Anderson and Hsiao 1981)
• Assuming no AR() init, natural
instruments
for
are yi ,t 1
and yi ,t 2
yi ,t 2
• Both mathematically related to yi,t1
Problem: Autocorrelation
• E.g., if it are AR(1), then yi,t2 ~ i,t2 ~ i,t1 ~ it
• Must assume yi,t2 is invalid instrument in i,t
Solution: Restrict to deeper lags
Left-multiply
by
Z'ΩZ
:
1 2
Z'ΩZ

1 2
Z'Y

Z'ΩZ

1 2
Z'Xβ

Z'ΩZ

1 2
Z'E
Let
X*
Z'ΩZ

1 2
Z'
X,
Y*

Z'ΩZ

1 2
Z
'Y,
E*
Z'ΩZ

1 2
Z'E
(3) Y* X* β E*
(1) Y Xβ E
OLS inconsistent: E X'E 0
(2) Take Z-moments: Z'Y Z'Xβ Z'E
OLS consistent EX'ZZ'E 0
but inefficient Var Z'E Z'ΩZ not scalar
moments
z
' i
E
inversely
with
their
variances
and
covariances:
AEGMM Var Z'E X, Z 1 Z' Var E X, Z Z 1 Z'ΩZ 1
• To make feasible, choose arbitrary proxy for Ω, call it H. Do GMM (one-step). Use residuals to make robust sandwich estimator of
yit yi,t1 xi't β it
•Purges fixed effects, doesn’t spread endogeneity much •Transformed yi,t1 still becomes endogenous since the
yi,t1 in yi,t1 yi,t1 yi,t2 correlates with the i,t1 in
it it i,t1
•But deeper lags exogenous if no AR(), offering instruments
Problem: Other endogeneity
•Differencing eliminates endogeneity to fixed effects error component. But o yi,t1 now endogenous to it o Other predetermined variables become
EN z A
1 Z'Eˆ
N
1
Z'Eˆ
'
A
1
Z'Eˆ
1
Eˆ 'ZAZ'Eˆ
N A N N N
• Given A, minimizing leads to βˆA X'ZAZ'X 1 X'ZAZ'Y
• Always unbiased, but which A is efficient? Answer: A should weight