bug in finding error

q5.jl

# Question 5
module Q5
using LinearAlgebra, Distributions, DataFrames

function make_data(T, N, β, σ)
	T = map(1:T*N) do i 
		x = rand(Normal(0, sqrt(σ)))
		u = rand(Normal(0, abs(x)))
		y = x * β + u
		x, y
	end

	first.(T), last.(T)
end

function get_β_FE(y, X, T, N)
	Q = I - kron(I(N), ones(T,T) ./ T)

	β = (X' * Q * X) \ (X' * Q * y)
end

function get_σ_2(β, X, y, T, N)
	u_hat = y - X * β
	X_bars = [mean(X[i:(i+T-1)]) for i in 1:N]
	u_hat_bars = [mean(u_hat[i:(i+T-1)]) for i in 1:N]

	S_xx = let s = 0.0
		for i in 1:N
			sn = 0.0
			X_bar = X_bars[i]
			for t in 1:T
				sn += (X[i+t-1] - X_bar)^2
			end	
			s += sn^2
		end
		s
	end

	diffsum_hat = let s = 0.0
		for i in 1:N
			sn = 0.0
			X_bar = X_bars[i]
			u_hat_bar = u_hat_bars[i]
			for t in 1:T
				sn += (X[i+t-1] - X_bar)*(u_hat[i+t-1] - u_hat_bar)
			end
			s += sn^2
		end
		s
	end


	diffsum_tilde= let s = 0.0 
		for i in 1:N
			sn = 0.0
			X_bar = X_bars[i]
			u_hat_bar = u_hat_bars[i]
			for t in 1:T
				sn += ((X[i+t-1] - X_bar)*(u_hat[i+t-1] - u_hat_bar))^2
			end
			s += sn
		end
		s
	end

	return (σ_2_hat = diffsum_hat / S_xx^2, σ_2_tilde = diffsum_tilde / S_xx^2)
end

function make_estimates(T, N, β, σ_2)
	X, y = make_data(T, N, β, σ_2)
	β_hat = get_β_FE(y, X, T, N)
	σ_2_hat, σ_2_tilde = get_σ_2(β, X, y, T, N)
	σ_2_ols = 1
	#σ_2_ols = FE_error(\beta, X, Y, T, N)
	return (β_hat = β_hat, 
		    σ_2_hat = σ_2_hat, 
		    σ_2_tilde = σ_2_tilde, 
		    σ_2_ols = σ_2_ols)
end

function FE_error(β, X, y, T, N)
	Q = I - kron(I(N), ones(T,T) ./ T)
	X_dot = Q*X
	u_hat = y - X * β
	S_xx_inv = inv(X_dot' * X_dot)

	V = S_xx_inv * X' * u_hat u_hat' * X * S_xx_inv
end
function main()
	S = 1000
	T = 5
	N = 500
	β = 1
	σ_2 = 1

	make_estimates(T, N, β, σ_2)
end

end