How to determine the likelihood function for $Y_i=u+(1+\phi*x_i)e_i$, $e_i \sim N(0,1)$

Question

The way I see it there's two way to define my Likelihood function: $L(Y_1,Y_2,...,Y_n;u,\phi)=Y_1*Y_2*...*Y_n$

or

$L(Y_1,Y_2,...,Y_n;u,\phi)=f(Y_1)*f(Y_2)*...*f(Y_n)$, where $f(Y_i)=N(u,(1+\phi*x_i)^2*1)$

Are any of those the right to determine the likelihood function?

Answer 1

The second one is right.

Not sure where you got the idea that the first one might be from, since it doesn't depend on the parameters at all while the densities clearly do...