Online Appendix: Preference Identification Under Inconsistent Choice
نویسندگان
چکیده
The proof of 1.1 is provided in the body of the paper. To prove 1.2, let α = p(y i0 = 1; y i1 = 0) denote the fraction of frame defiers and note that now p(c i = 0) = p(y i1 = 1; y i0 = 0) + α. It is straightforward to show that E[y i0 ] = p(c i = 1)E[y * i |c i = 1] + α (1) E[y i1 ] = p(c i = 1)E[y * i |c i = 1] + p(c i = 0) − α (2) Substituting these into the definition of Y c , we have Y c = p(c i = 1)E[y * |c i = 1] + α p(c i = 1) + 2α (3) Subtract 1 2 from both sides of (3) to obtain Y c − 1 2 = p(c i = 1)(E[y * |c i = 1] − 1 2) p(c i = 1) + 2α (4) In addition, subtracting E[y * |c i = 1] from both sides of (3) yields Y c − E[y * i |c i = 1] = α(1 − 2E[y * |c i = 1])) 2α + p(c i = 1) (5) This expression gives the bias in Y c when frame monotonicity fails. To complete the proof, note that by (4), Y c > 1 2 =⇒ E[y * |c i = 1] > 1 2 , and then by (5), E[y * |c i = 1] > 1 2 =⇒ E[y * i |c i = 1] > Y c. This proves 1.2.i. Repeating these two steps with the inequalities reversed proves 1.2.ii. Finally, 1.2.iii follows directly from 4.
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