Econometric fixed effects (control for between variation to estimate within variation)

Interfaces brms
1

Hi there,

I’m wondering if there is an easy way to implement the econometric idea of fixed effects modeling via brms?

This blog post summarises the issue succinctly, basically that the partial pooling happens via the brms setup between subjects as well as within subjects, and when unobservable subject-specific traits are correlated with a parameter to be estimated, there is a confounding that happens that leads to a biased population-level estimate.

Of course, I can go into the stan code, but in combination with trying to fit more complex models I think such an implementation would also facilitate a lot!

Thank you!

  • Operating System: Windows 7
  • brms Version: 2.19
2

Hi,
are you interested in the fixed effects model or the multilevel model?

If it’s the multilevel model, the trick is to include group means of x into your model. The tidyverse way, that would be:

library(dplyr)

d <- d %>% 
  mutate(x_mean = mean(x),
         .by = firm_id)

And then fit the model:

library(brms)

m1 <- brm(y ~ x + x_mean + (1|firm_id),
          data = d,
          backend = "cmdstanr",
          cores = 4,
          seed = 123)
#          Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#Intercept     0.06      0.19    -0.32     0.44 1.00     4058     3137
#x             1.03      0.04     0.94     1.11 1.00     3346     2973
#x_mean        0.65      0.05     0.56     0.75 1.00     3096     2926

x corresponds to the within-group-variation (a1 parameter in the blog post) and x_mean is the between-group-variation (b in the blogpost).

In econometrics, this is called Mundlak model. Snijders and Bosker explain it in detail in chapter 4 of their book Multilevel analysis: An introduction to Basic and Advanced Multilevel Modeling.

If you are asking about the fixed effects model, then for a single fixed effect, getting the point estimate is easy. You just need to mean center x.

d$x_centered <- d$x - d$x_mean

m2 <- brm(y ~ x_centered,
          data = d,
          backend = "cmdstanr",
          cores = 4,
          seed = 123)
#           Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
#Intercept      5.58      0.41     4.78     6.39 1.00     4003     2919
#x_centered     1.03      0.11     0.82     1.24 1.00     3608     2950

Unfortunately, this model makes errors nonindependent (because observation from the same group are correlated with each other). With frequentist/maximum likelihood models, this issue can be solved using clustered standard errors. But I don’t think clustered/robust standard error exist in bayesian context? So there is no easy way to replicate the results of fixest package. Also, with more than one fixed effect, the centering becomes much more complex (using so-called alternating projections) and I don’t think there is a way to do it in brms. So the multilevel approach is your best best.

2 Likes
3

Hi @AlesVomacka ,

Thanks very much for the detailed reply and references!

To be more specific in my case, I’m trying to fit a non-linear model with group memberships, as well as time and subject fixed effects.

Without the FE, it’s written out like this, but will try including the group means for one FE first, I guess it should work despite the nonlinear structure.

model <- brm(
 bf(y ~  a*(b^x),
     a ~  (1|group1|group2),
     b ~  (1|group1),  nl = TRUE), init = 0, 
data = data , family = student(), ... )

I see that the Mundlak model lets us estimate the within- and between-group-variation given one group (or fixed effect). I see that there has been recent work on a two-way Mundlak that is equivalent to (or improves) the two-way FE regression by including both group-means as well as all the possible interactions effects of the groups. ( Woolridge 2021 and Baltagi 2023).

Seems like some more complex work include two-way FE through either the Mundlak or more typical two-way FE regressions, either via brms or through STAN.

Happy to hear if you have some further ideas :+)