Interpreting the intercept in the random-effects model

[Riccardo Marchi]

Consider the model

y_it = a + bX_it + c_i + e_it

Using a fixed effect model, the constant term a cannot be estimated. a is collinear with c_i. To break the collinearity, you need an additional restriction: xtreg, fe adds one restriction, reg another etc.. At the end, there's no way to separate a from c_i.

My question is if this holds true also in a random effect model. In a RE we impose that the mean of all c_i is equal to zero. Is this restriction sufficient to separately estimate a and c_i or not? More in general what is the interpretation of the constant term in a random effect model (xtreg, re)?

Thanks

[Clyde Schechter]

In a random effects model, the colinearity between a and the c_i is simply not an issue as the c_i are not estimated with linear algebra the way they are in a fixed effects model. In fact, the c_i are not estimated at all within the -xtreg, re- or -mixed- commands. (If you want estimates of them, you can get them afterwards with -predict, u- or -predict, reffectrs-, respectively.) Rather, in a random effects model, the c_i are modeled as being sampled independently from a normal distribution with mean 0 and unknown variance, the variance being estimated from the data and reported as part of -xtreg, re- or -mixed- output.

So the interpretation of the constant term in -xtreg, re- is similar to the interpretation of the constant term in -regress-: it is the expected value of the outcome variable when all predictors are 0 and u_i = 0.

[Riccardo Marchi]

Thanks Clyde, clear and instructive! Riccardo

[Carlo Lazzaro]

Riccardo:
in addition to Clyde's as always enlightening explanation, you can find an interesting comparison about the role of -fe- and -re- intercept in linear panel data regression in: https://www.wiley.com/en-us/A+Guide+...-9781405182577, page 282-284.