pr.probability - Likelihood function for sequential random variables

Context

Consider the following sequential data generating process for $Y_1$, $Y_2$, $Y_3$. (By sequential I mean that we generate $Y_1$, $Y_2$, $Y_3$ in sequence.):

$Y_1 = X_1^' beta + epsilon_1$

$Y_2 = X_2^' beta + epsilon_2$

$Y_3 = X_3^' beta + epsilon_3$

where,

$X_3 = x_{3,1}$ if $y_1$ $y_2$ > 0,

OR

$X_3 = x_{3,2}$ if $y_1$ $y_2$ < 0,

$epsilon_i$, ($i$ = 1, 2, 3) are i.i.d $N(0,sigma^2)$,

$beta$ is a $p x 1$ vector and

$X_1$, $X_2$, and $X_3$ are vectors of appropriate dimensions.

Question

Suppose we observe the following sequence: {$Y_1$ = $y_1$,$Y_2$ = $y_2$, $X_3$ = $x_{31}$, $Y_3$ = $y_3$} and wish to estimate the parameters $beta$ and $sigma$. Is the likelihood function given below the correct function?

L( $beta$, $sigma$ | $y_1$, $y_2$, $y_3$, $x_1$, $x_2$, $x_{31}$ ) = ( $f(y_1|x_1,beta, sigma)$ $f(y_2|x_2,beta, sigma)$ $f(y_3|x_{31},beta, sigma)$ ) / Prob( $Y_1 Y_2 >0 $ )

Thanks

EDIT: Fixed some typos and notation in light of comments by Bjørn.