Is the "within-person" effect estimated by fixed effects regression equivalent to that of a random effects model with mean-centered predictors? I suspect the answer is yes. I explored this with a simulation. The coefficient and standard error are identical in the first model (the fixed effects regression) and the second model (the model with mean centered predictor and random effects). The goal is to estimate the within-"person" effect of x. 
library(lme4)
K <- 50; N <- 5 #50 people, 5 measurements per person. 
ID = rep(1:K,each=N)
x=NULL
for(k in 1:K) x <- c(x, runif(N)+0.1*k)
y=NULL
for(k in 1:K) y <- c(y, runif(N)+0.1*k)
mod2 = lm(y ~ -1 + factor(ID) + x)
xc = NULL
for(k in 1:K)
{
  ix = c(1:5) + (k-1)*5
  xc = c(xc, x[ix]-mean(x[ix]))
}
summary(mod2)
summary(lmer(y~xc+(1|ID)))

If these two models are equivalent for estimating the within-person effect, I don't understand why. And, I don't understand why this doesn't seem to be mentioned by statisticians that have researched the "fixed effects model" in depth, such as Paul Allison. I've always been taught that the random effects model is not sufficient for estimating the within-person effect like this. 
Any thoughts on this are appreciated. 
 A: The models aren't equivalent. The coefficient for x equals the coefficient for xc, but these are different predictors because of the mean-centering; moreover, the coefficient of each group is estimated differently for the two models, which leads to different predictions of y for the same x and group, even when the models are asked to predict y in their own training data.
So why do x and cx end up with the same coefficient? First notice that tail(coef(mod2), 1) and coef(lm(y ~ -1 + xc)) are equal. That is, if you remove group from the OLS model but center x within groups, the coefficient ends up the same. This makes sense considering that the effect of group (according to the first model) is just to add a constant value to the dependent variable for each group. In fact, once you center x within groups, adding groups back into the model, whether as random or fixed effects, won't change the coefficient of xc, because (I believe) xc is uncorrelated with group.
A: They are equivalent in linear models. See reference:
https://academic.oup.com/ije/article/43/1/264/732283 
"An alternative and computationally less demanding way to calculate the linear fixed effects model is the mean-centring approach. In this case the mean (over time) of measurements for each individual is subtracted from all the individual’s measurements. The time-invariant terms (which are not independently identifiable) are eliminated in the mean-centring Equation (2), and only parameters associated with time-varying covariates can be estimated by the model"
