Assumptions of random effects for unbiased estimates

I didn't respond to this offline, as it took me a while even to start
to come up to speed on the question.  Random effects are indeed defined
from *very* different points of view in the two communities
([bio]statistical vs. econometric); I'm sure there are points of
contact, but I've been having a hard time getting my head around it all.

Econometric definition:

The wikipedia page <https://en.wikipedia.org/wiki/Random_effects_model>
and CrossValidated question
<http://stats.stackexchange.com/questions/66161/why-do-random-effect-models-require-the-effects-to-be-uncorrelated-with-the-inpu>
were both helpful for me.

 In the (bio)statistical world fixed and random effects are usually
justified practically in terms of shrinkage estimators, or
philosophically in terms of random draws from an exchangeable set of
levels: e.g. see
<http://stats.stackexchange.com/questions/4700/what-is-the-difference-between-fixed-effect-random-effect-and-mixed-effect-mode/>
for links.

  I don't think I can really write an answer yet.  I'm still trying to
understand at an intuitive or heuristic level what it means for
Cov(x_it,c_i)=0, where x_it is a set of explanatory variables over time
for an individual subject and c_i is the conditional mode (=BLUP in
linear mixed-model-land) for the deviation of the individual i from the
population mean ... or more particularly what it means for that
condition to be violated, which is the point at which fixed effects
would become preferred.

  As a side note, some statisticians (Andrew Gelman is the one who
springs to mind) have commented on the possible overemphasis on bias.
(All else being equal unbiased estimators are preferred to biased
estimators but all else is not always equal). Two examples: (1)
penalized estimators such as lasso/ridge regression (closely related to
mixed models) give biased parameter estimates with lower mean squared
error. (2) When estimating variability, one has to choose a particular
scale (variance, standard error, log(standard error), etc.) on which one
would prefer to get an unbiased answer.

Dear all,
Random effects are more efficient estimators ? however they come at the
cost of the assumption that the random effect is not correlated with the
included explanatory variables. Otherwise, using random effects leads to
biased estimates (e.g., as laid out in Woolridge
<https://faculty.fuqua.duke.edu/~moorman/Wooldridge,%20FE%20and%20RE.pdf>'s
Econometrics text). This assumption is a strong one for many
observational datasets, and most analyses in economics do not use random
effects for this reason. *Is there a reason why observational ecological
datasets would be fundamentally different that I am missing? Why is this
important assumption (to have unbiased estimates from random effects)
not emphasized in ecology? *

Thanks!

Laura

-- 
Laura Dee
Post-doctoral Associate
University of Minnesota
ledee at umn.edu <mailto:ledee at umn.edu> 
lauraedee.com <http://lauraedee.com>