Limited Dependent Variables
13.1 The Linear Probability Model

a. Let к і = Pr[y; = 1], then y; = 1 when u; = 1 — x0" with probability к; as shown in the table above. Similarly, y; = 0 when u; = —x0" with probability 1 — к ;. Hence, E(u;) = к; (1 — x[") + (1 — к 😉 (—x0").
For this to equal zero, we get, к; — к ;xi" + к ;xi" — x0" = 0 which gives к ; = xi" as required.
b. var(u;) = E(u2) = (1 — xi")2 к ; + (—x0")2 (1 — к 😉
1 — 2×0" + (x0")2 к; + (xi")2 (1 — к i)
= к ; — 2×0" к ; + (x0")2 = к ; — к 2 = к ;(1 — к 😉 = x0" (1 — xi") using the fact that к ; = xi".
13.2 a. Since there are no slopes and only a constant, x0" = a and (13.16) becomes
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