Optim with MLE R - r

Question

suppose you have this simulated data:
N = 1000; alpha_bar = 0.7; gamma_h = 16; gamma_c = 50
mu_hw = 4; sigma_hw = 0.1; mu_i = 4; sigma_i = 0.5; sigma_eps = 0.01
set.seed(1)
wage <- exp(rnorm(N, mean = mu_hw, sd = sigma_hw))
set.seed(1)
nlincome <- exp(rnorm(N, mu_i, sigma_i))
set.seed(1)
e_ <- rnorm(N, 0, sigma_eps)
obs_data <- data.frame(w = wage,
I = nlincome,
eps = e_) %>%
mutate(alpha = alpha_bar + eps,
h = alpha*gamma_h - ((1-alpha)*(I-gamma_c))/w) %>%
select(w,I,h)
And now assume that you've forgotten these parameters (alpha_bar, gamma_h, gamma_c, sigma_eps). So you want to estimate those by maximum likelihood using the obs_data dataframe.
From the model I am working, I know that epsilon has this form:
Furthermore, I have derived the Log-Likelihood function like this: (I wrote this)
So I know I have to minimize the negative log-likelihood function, so I have tried to do it like this:
normal_loglik <- function(par){
g_h <- par[1]; g_c <- par[2]; a_bar <- par[3]; sigma_e <- par[4]
d <- mutate(obs_data,
num = h + ((I-g_c)/w),
den = g_h + ((I-g_c)/w),
eps = (num/den)-a_bar,
arg_1 = 0.5*(eps/sigma_e)^2,
arg_2 = log(abs(den)),
opt = log(sigma_e * sqrt(2*pi)) + arg_1 + arg_2)
LL <- - sum(d$opt)
return(-LL)
}
opt_par <- optim(rep(1e-4,4),
normal_loglik,
method = "SANN")$par
real_param <- c(gamma_h, gamma_c, alpha_bar, sigma_eps)
# Comparison
print(real_param); print(opt_par)
It does run, but I get these estimates that are way too far away from the real ones: