I1k n 2 i∈I C = 1 (n I1k |I1k | − | | − |I1k |) | | n = 1 |I1k | − | | + n = , ) R2 : β1 {(β0 ∈ When I1k (x) = the solution set of the least squares problem is infinite, i.e. argmin ∅ 55 x − β0 2 = (1/n) Ln (x) as undefined in this case, then the bias from using the conditional relative β∈R2 β0 i k x If we treat }. β i=1 frequency is equal to the finite sample bias in the coefficients of the associated linear probability model. If instead we define (x) = 0, then the bias in the coefficients of the associated linear probability model will be less than the β1k bias in the conditional relative frequency.
i∈I C n I1k 2 = |I1k | 1 − | = |I1k | 1 − | | n I1k | n + (n − |I1k |) | | 55 When I1k (x) = ∅ the solution set of the least squares problem is infinite, i.e. argmin β∈R2 x − β0 2 = {(β0 , β1 ) ∈ R2 : β0 = (1/n) Ln i k (x) as undefined in this case, then the bias from using the conditional relative i=1 x }. If we treat β frequency is equal to the finite sample bias in the coefficients of the associated linear probability model. If instead we define β1k (x) = 0, then the bias in the coefficients of the associated linear probability model will be less than the bias in the conditional relative frequency.
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