Incidental parameters problem results in away-from-zero biased estimates per Greene (2004). Okay. But can this bias result in a directional change as well, e.g., true value is +2.13 but estimate is -1.23 or true value is -4.12 and estimate is +1.53.
Further, Greene (2004) shows downwards biased standard errors and thus inflated test statistics. However, what about if utilizing robust standard errors? Would robust standard errors effectively correct for this inflated test statistics bias?
Lastly, is there a reason NOT to use conditional logit when wanting to do logit with fixed effects? If there is no bias in a setting, would logit with fixed effects give same results as conditional logit (stratified on the fixed effect)?
Best Answer
For your first question, I think anything can happen, an estimator is itself a random variable so there is no reason why it should not have a sign change. In addition, I had simulations about fixed-effect logit models and I did observe sign changes.
For the second question. I don't think the robust standard error would help. I tend to think that the robust standard error (sandwich) itself is inconsistent, if derived from the likelihood function that depends on the estimates, not the true values, of the fixed effects. You can look at Arellano and Hahn (2006) (A Likelihood-Based Approximate Solution to the Incidental...) which leads to that under fixed T, the likelihood is inconsistent (they did not show this but you can derive this from their result easily, I think). So, if the likelihood is already inconsistent, I would tend to think that the robust standard error, which is calculated from likelihood derivatives, are also inconsistent.
For the third question. If T grows faster ($N,T\rightarrow\infty$ but $N/T\rightarrow0$), then there is no problem, if you are doing individual effects (time effects -> you need the opposite, two-way effects - > you need bias correction). I assume the conditional logit you are talking about is the Chamberlain (1980) conditional logit. The result of the usual logit is typically not numerically the same as the conditional logit. But they should be asymptotically equivalent (I haven't proven it but I highly think so). But in practice, I could not think of a case where you should not use conditional logit.