5 edited tags
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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.):

$Y_1 = X_1^' \beta + \epsilon$ epsilon_1Y_2 = X_2^' \beta + \epsilon$epsilon_2$

$Y_3 = X_3^' \beta + \epsilon$ epsilon_3$where,$X_3 = x_{3,1}^'$x_{3,1}$ if $y_1$ $y_2$ > 0,

OR

$X_3 = x_{3,2}^'$ x_{3,2}$if$y_1y_2$< 0,$\epsilon$is \epsilon_i$, ($i$ = 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: {$Y_1$ = $y_1$,$Y_2$ = $y_2$, $X_{31}$ 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 $) Thanks EDIT: Fixed some typos and notation in light of comments by Bjørn. 3 added 5 characters in body Context Consider the following sequential data generating process for$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 + \epsilonY_2 = X_2^' \beta + \epsilonY_3 = X_3^' \beta + \epsilon$where,$X_3 = x_{3,1}^'$if$y_1y_2$> 0, OR$X_3 = x_{3,2}^'$if$y_1y_2$< 0,$\epsilon ~\epsilon$is$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_{31}$=$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

2 fixed minor typos in notation
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