Consider the linear model Yi = B₁ + B₂ dj + Uis where y, is the subjective reported happiness of individual i, d; is a dummy variable which indicates whether individual i owns a dog (d; = 1) or not (d; = 0), and u; is an unobserved error term. Assume that E(u;) = 0 and E(u;d;) = 0. We observe an i.i.d. sample of (yi, di), i = 1,...,n, with sample size n = 100. In that sample we have no = Σ1 (1 - d;) = 80 individuals with d; = 0, and n₁ = ₁=1 di 20 individuals with d; = 1. The average reported happiness in the subpopulation with d; = 0 is no¹ Σ(1 — d;)y; = 3, and in the subpopulation with d; = 1 is n₁¹ Σ=1 d¡y₁ = 8. = (a) Let x₁ = (1, d;) and 3 = (B₁, B₂). Calculate Σ1₁, and Σ1₁, and the OLS estimator Σπιτι (b) Are our assumptions above sufficient to guarantee that is the true causal effect of d; on yi? Explain your answer. i=1 I'; Yi-
Consider the linear model Yi = B₁ + B₂ dj + Uis where y, is the subjective reported happiness of individual i, d; is a dummy variable which indicates whether individual i owns a dog (d; = 1) or not (d; = 0), and u; is an unobserved error term. Assume that E(u;) = 0 and E(u;d;) = 0. We observe an i.i.d. sample of (yi, di), i = 1,...,n, with sample size n = 100. In that sample we have no = Σ1 (1 - d;) = 80 individuals with d; = 0, and n₁ = ₁=1 di 20 individuals with d; = 1. The average reported happiness in the subpopulation with d; = 0 is no¹ Σ(1 — d;)y; = 3, and in the subpopulation with d; = 1 is n₁¹ Σ=1 d¡y₁ = 8. = (a) Let x₁ = (1, d;) and 3 = (B₁, B₂). Calculate Σ1₁, and Σ1₁, and the OLS estimator Σπιτι (b) Are our assumptions above sufficient to guarantee that is the true causal effect of d; on yi? Explain your answer. i=1 I'; Yi-
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter7: Analytic Trigonometry
Section7.6: The Inverse Trigonometric Functions
Problem 92E
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Transcribed Image Text:Consider the linear model
Y: = B1 + B2 d; + u;,
where y; is the subjective reported happiness of individual i, d; is a dummy variable which
indicates whether individual i owns a dog (d; = 1) or not (d; = 0), and u; is an unobserved
error term. Assume that E(u;) = 0 and E(u;d;) = 0. We observe ani.i.d. sample of (yi, d;),
i = 1,...,n, with sample size n = 100. In that sample we have no = E (1 – d;) = 80
individuals with d; = 0, and n1
reported happiness in the subpopulation with d; = 0 is no'E(1 –- d;)y; = 3, and in
the subpopulation with d; = 1 is n¡' E d:y; = 8.
, d; = 20 individuals with d; = 1. The average
(a) Let r; = (1, d;) and 3 = (B1, B2)'. Calculate , ¤; x;, and D, '; Y;, and the OLS
estimator
i=1
(b) Are our assumptions above sufficient to guarantee that B, is the true causal effect
of d; on y;? Explain your answer.
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