There exists an example at Optim.jl’s tutorial, I copy the code here.

```
using Optim, NLSolversBase, Random
using LinearAlgebra: diag
Random.seed!(0); # Fix random seed generator for reproducibility
n = 500 # Number of observations
nvar = 2 # Number of variables
β = ones(nvar) * 3.0 # True coefficients
x = [ones(n) randn(n, nvar - 1)] # X matrix of explanatory variables plus constant
ε = randn(n) * 0.5 # Error variance
y = x * β + ε; # Generate Data
function Log_Likelihood(X, Y, β, log_σ)
σ = exp(log_σ)
llike = -n/2*log(2π) - n/2* log(σ^2) - (sum((Y - X * β).^2) / (2σ^2))
llike = -llike
end
func = TwiceDifferentiable(vars -> Log_Likelihood(x, y, vars[1:nvar], vars[nvar + 1]),
ones(nvar+1); autodiff=:forward);
opt = optimize(func, ones(nvar+1))
parameters = Optim.minimizer(opt)
parameters[nvar+1] = exp(parameters[nvar+1])
numerical_hessian = hessian!(func,parameters)
var_cov_matrix = inv(numerical_hessian)
β = parameters[1:nvar]
temp = diag(var_cov_matrix)
temp1 = temp[1:nvar]
t_stats = β./sqrt.(temp1)
# This file was generated using Literate.jl, https://github.com/fredrikekre/Literate.jl
```

as you see, `Log_Likelihood`

function accepts log\_\sigma as one of parameters, so the estimated parameters should also be log\_\sigma, however, if you change log\_\sigma to \sigma by `parameters[nvar+1] = exp(parameters[nvar+1])`

, then the hessian matrix is not right, as `Log_Likelihood`

's input is log\_\sigma, am I right?

I can’t figure out why to take exponential to log\_\sigma before calculating hessian matrix.