endogneity in a multilevel model

[Romuald Landry]

Hello everyone, I'm trying to test endogeneity on two levels of hierarchical data (student, school). To do this, I'm running a hausman test with the following commands

  xtset id_ecole,
xtreg Nrdtlectf  diflcev1ir avoirfaim rvauxdome11ique langue1arlé Nbelvdanslécole nbsalledeclasse agedirecteur, fe
est store fixe
xtreg Nrdtlectf  diflcev1ir avoirfaim rvauxdome11ique langue1arlé Nbelvdanslécole nbsalledeclasse agedirecteur, re
hausman fixe

here are the results in the following screenshot

There's no endogenity, the problem is that the empirical literature agrees on the link between the characteristics of the school and those of the pupils, so I'm thinking that maybe I've misjudged Hausman's test, do you need special precision when it comes to hierarchical data? Thank you for helping me to understand

[Carlo Lazzaro]

Romuald:
1) the -hausman- outcome simply tells you that -re- is the way to go. It gives no clue about the evidence of endogeneity;
2) I'm not clear with the type of potential endogenity you're concerned about: latent variable driven endogeneity?
In addition:
a) a continuous regressand and a nested design call for -mixed-;
b) I'd cluster standard errors at -id_ecole- level;
c) I'd add a square term for -agedirecteur- via -fvvarlist- notation:

c.agedirecteur##c.agedirecteur

[Jeff Wooldridge]

A few comments. First, you probably want to use the robust form of the Hausman test that allows for cluster correlation, since it is likely the explanatory variables are correlated within school. Second, there is no practical difference between RE and FE. People are somewhat turned off by RE because, nominally, it assumes that the covariates are uncorrelated with the unobserved effect -- in this case, the school effect. That's the sense in which RE assumes "exogeneity." FE allows full correlation. Given your findings, you should clearly go with FE, say you included school fixed effects, and cluster your standard errors at the school level.