Endogeneity with individual fixed effects and time dummies

[Maria Nikolaou]

Hi there,

I am estimating an individual level regression with panel data. i wish to investigate whether immigration shares in a region effects ones like/dislike for their region. to account for unobserved heterogeneity and time-specific effects i am estimating the model using ols fixed effects with time dummies.

i suspect that my immigration share variable may be endogeneous as immigrants may locate into regions on the basis of specific characteristics which may be correlated with those unobservables which affects natives like or dislike for their neighbourhood.

Is it wrong to estimate an IV regression in Stata to account for this? my difficulty is understanding whether this hypothesised bias is in fact captured by the time/individual fixed effects. As i havent included dummies for each region i suspect that this should be ok, however i am a first time user of panel data so i would very much appreciate some verification.

thank you in advance,

Maria

[Andrew Musau]

Maria: I do not think that it is possible to give you a straight answer without knowing the theory and assumptions behind your study. After all, you are the one doing the research, so you should enlighten us on the hidden relationships. Nonetheless, here are some points that you need to consider:

1) It appears that your dependent variable is "ones like/ dislike for their region". If the measurement scale is binary, then linear fixed effects is not suitable. You need a somewhat continuous scale - otherwise xtlogit/ xtprobit, fe is preferable.

2) Ideally, you want to have exogenous regressors, but you state that you think that your main regressor may be endogenous. If you can identify factors correlated with this variable, and get variables to proxy these factors, then you can explicitly include them in your model and your problem is solved.

3) Assuming that such factors in (2) are unobserved, then you need to consider the following: Let us refer to the OLS error term as u_it. Under fixed effects, we can write this error term as
u_it = η_{i +} μ_t + e_it
where η_i is the individual effect, μ_t is the time effect and e_it is the remainder of the stochastic error term. The (two-way) fixed effects estimator treats the individual effect and the time effect as fixed parameters to be estimated. However, e_it always remains, and if your regressor is correlated with e_it, then fixed effects will not give you consistent estimates of the endogenous variable.

In other words, fixed effects is suitable if the unobserved individual effect is time-invariant (and the unobserved time effect is individual invariant). If the unobserved individual effect is time varying, then you will need to consider a different method.

4) It must be that the "immigration share" variable is time varying, since the trade-off for consistency is that FE cannot estimate the effect of time-invariant regressors (hence it is not efficient).

5) If you have a valid instrument, then you will get consistent estimates of the endogenous variable using IV regression.

Now, you need to consider whether the unobserved individual effects are time invariant, and make sure that your main regressors are time-varying inorder to use FE. If you are not sure, my advice is for you to look at the literature (previous papers in the field) and see how other researchers dealt with it.