Maximum likelihood estimation of the log-normal distribution using R - r

Question

I'm trying to estimate a linear model with a log-normal distributed error term. I already have working code for a linear model with normally distributed errors:
library(Ecdat)
library(assertthat)
library(maxLik)
# Load the data
data(Wages1)
# Check what R says
summary(lm(wage ~ school + exper + sex, data = Wages1))
# Use maxLik from package maxLik
# The likelihood function
my_log_lik_pos <- function(theta, data){
y <- data[, 1]
x <- data[, -1]
beta <- head(theta, -1)
sigma <- tail(theta, 1)
xb <- x%*%beta
are_equal(dim(xb), c(nrow(my_data), 1))
return(sum(log(dnorm(y, mean = xb, sd = sigma))))
}
# Bind the data
my_data <- cbind(Wages1$wage, 1, Wages1$school, Wages1$exper, Wages1$sex)
my_problem <- maxLik(my_log_lik_pos, data = my_data,
start = rep(1,5), method = "BFGS")
summary(my_problem)
I get approximately the same results. Now I try to do the same, but using the log-normal likelihood. For this, I have to first simulate some data:
true_beta <- c(0.1, 0.2, 0.3, 0.4, 0.5)
ys <- my_data[, -1] %*% head(true_beta, -1) +
rlnorm(nrow(my_data), 0, tail(true_beta, 1))
my_data_2 <- cbind(ys, my_data[, -1])
And the log-likelihood function:
my_log_lik_lognorm <- function(theta, data){
y <- data[, 1]
x <- data[, -1]
beta <- head(theta, -1)
sigma <- tail(theta, 1)
xb <- x%*%beta
are_equal(dim(xb), c(nrow(data), 1))
return(sum(log(dlnorm(y, mean = xb, sd = sigma))))
}
my_problem2 <- maxLik(my_log_lik_lognorm, data = my_data_2,
start = rep(0.2,5), method = "BFGS")
summary(my_problem2)
The estimated parameters should be around the values of true_beta, but for some reason I find completely different values. I tried with different methods, different starting values but to no avail. I'm sure that I'm missing something obvious, but I don't see what.
Am I right to assume that the log-likelihood of the log-normal distribution is:
sum(log(dlnorm(y, mean = .., sd = ...))
Unless I'm mistaken, this is the definition of the log-likelihood (sum of the logs of the densities).

Answer 1

I found the issue: it seems the problem is not my log-likelihood function. When I try to estimate the model with glm:
summary(glm(ys ~ school + exper + sex, family=gaussian(link="log"), data=Wages1))
I get the same result as with maxLik and my log-likelihood. It would seem the problem comes from when I tried to simulate some data:
ys <- my_data[, -1] %*% head(true_beta, -1) +
rlnorm(nrow(my_data), 0, tail(true_beta, 1))
The correct way to simulate the data:
ys <- rlnorm(nrow(my_data), my_data[, -1] %*% head(true_beta, -1), tail(true_beta, 1))
Now everything works!