Winston Todd

Winston Todd

Answered

2022-10-15

Given this regression model: y i = β 0 + β 1 x i + E i .
All the assumptions are valid except that now: E i N ( 0 , x i σ 2 )
Find Maximum likelihood parameters for β 0 , β 1

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Answer & Explanation

espava8b

espava8b

Expert

2022-10-16Added 12 answers

Let us define the following quantities:
y ¯ i = x i 1 / 2 y i , x ¯ i = x i 1 / 2 , x ¯ i + = x i + 1 / 2 , E ¯ i = x i 1 / 2 E i
Then the equation becomes
y ¯ i = β 0 x ¯ i + β 1 x ¯ i + + E ¯ i , E ¯ i N ( 0 , σ 2 )
What we have now is a standard, unweighted linear regression in two variables.
To solve this we define two matrices
X = [ x ¯ 1 x ¯ 1 + x ¯ n x ¯ n + ] , Y = [ y ¯ 1 y ¯ n ]
What happens if we cannot assume x i > 0? Well, in that case the original relation reduces to y i = β 0 and E i = 0. So that means that β 0 is forced to be equal to y i in that case. If there are multiple values of x i > 0, then we have two scenarios:
- All of the corresponding values of y i are identical, in which case β 0 = y i .
- The corresponding values of y i are not identical, in which case the model is infeasible and cannot be solved.
If the model is feasible, then we fix β 0 and eliminate those ( x i , y i ) pairs from the model, leaving us with a single-variable regression.
z ¯ i = β 1 x ¯ i + + E ¯ i , z ¯ i y ¯ i β 0 x ¯ i β 1 = ( i y i ) n β 0 i x i .

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