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2 assumptions for short T

yi, Xi independent over i
parameters in B are common to all i


2 assumptions for short T

yi, Xi independent over i
parameters in B are common to all i


What does predetermined mean?

do not rule out correlations with past v


Pooled OLS

just stack all observations and treat them the same so then just to ols
require x predetermined, uncorrelated with individual effects in the errors to be consistent
Error serially correlated due to individual effects so use cluster-robust standard errors for inference, causes inefficiency


Within Groups (or fixed effects)
Assumption for consistency in short-T
Can it work with lagged dependent variable ie yit-1?
Alternative method of getting same estimator
Degrees of freedom

transform variables by subtracting their mean over time ie sum in t / T
Strict exogeneity: E(xitvis)=0 for all s, t past present and future. If this satisfied, consistency does not require vit to be serially uncorrelated
Does not work with yit-1 as an explanatory variable as necessarily correlated with vit-1
If believe serially correlated errors then report cluster-robust std errors
Can also get same estimator by doing least squares dummy variables (adding dummies for each i=1,...,N)
Hence degrees of freedom=NT-K-N (=N(T-1)-N)


Condition required for consistency
When would classical standard errors be appropriate

First differenced OLS
Lose one equation by doing it (ie 1 t)
E(DxitDvit)=0 where D is 1st diff operator. Ie rules out feedback from vi,t-1 to xit but not from longer lags
if Dvit=white noise (ie vit random walk). Vit serially uncorrelated would lead Dvit to be serially correlated so use cluster-robust


Most efficient estimator if ni endogenous but vit iid?
if vit is a random walk?

Both from Gauss-Markov theorem


When are FDOLS and WG the same?



If one explanatory variable in xit is a step function, not good for which type of estimator?

As could lead to lots of Dxit=0


Long T consistency condition for WG

E(xitvit)=0 ie only need predetermined


Long T consistency condition for WG

Exactly the same as short T:
E(DxitDvit)=0 so not as good as WG in this case


Are FDOLS or WG efficient if ni is uncorrelated with explanatory variables

lose information by differencing away


Between groups estimator
Consistency requirements

regress the time mean of y on the time mean of x (still have variation from different i) Leaves the ni term
error term contains ni and all vi1, vi2,... so requires uncorrelated individual effects and strictly exogenous covariates
Not efficient


Random effects GLS estimator
when to use
How can it be computed using ols on a transformed model?

Compute omega matrix which is E(uu') (NTxNT) which is a diagonal matrix with E(uiui') (TxT) on the diagonal. E(ui,ui')=variance of ni not on the diagonals and the sum of the variances of ni and vit on the diagonal.
Then Bgls=(X'omega^-1X)^-1X'omega^-1y

use when both ni and vit exogenous (but still can have serial correlation as uit=ni+vit), very restrictive assumptions
Yes is efficient under these circumstances
transformed variables yit*=yit-(1-theta) x time mean yi
theta=variance of vit/(variance of vit+Tvariance of ni)


Feasible GLS

Obtain estimates for the variances of n and v from the WG and between groups estimators and then do Random effects GLS


Testing for correlated individual effects

Hausman test comparing WG with GLS, large differences indicate there is an effect
or any which are consistent with fixed effects and not (eg. WG and BG expoloiting the fact they are orthogonal to make variance of difference estimates easier)