Suppose I have the simple libear regression model:

Y=a0+a1*X+ε where ε~N(0,σ^2)

Suppose that via least squares I estimate a0 (a0_hat) and a1 (a1_hat).

Can anyone tell me what is the asymptotic distribution of the quantity:

sum((y(i)-a0_hat-a1_hat*x(i))^2) ??? (1)

What I am saying is that if I make a code that generates data from a simple model like this, then estimate the parameters, then plug them back in (1), repeat the procedure 10000 and make a histogramm.

1. Generate data from a simple model like the above.

2. Estimate a0 and a1

3. Calculate value of formula (1).

4. repeat the procedure say 10000 times and make the histogramm.

It is Normal I think but what are the parameters? How can I derive this without the use of a computer (mathematically)?

Thanks in advance!