Consider the following model with one covariate: Y = x + ui u~ N(0, 1) (a)...

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Consider the following model with one covariate: Y₁ = x + ui u~ N(0, 1) (a) Show that BoLs is a consistent estimator for 3. Be clear on any assumptions that you make. (b) Derive the asymptotic distribution of √n (BOLS-B). Now suppose that you do not observe the variable z. Rather, you observe: x₁ = x + v₁ U₂ ~ N(0,0²) In addition, assume that , is independent of u. (c) Derive the plim of BoLs and an expression for the attenuation bias in terms of Var (v.) and Var(a). Explain how this bias would affect your interpretation of the estimate of BOLS. (d) Suppose that there is an instrument z, that satisfies E[zu] = 0, E[zv] = 0, and E[za] # 0. You estimate the following first-stage regression: I₁ = π²₁ + 1, and the reduced-form equation: Yi = 7²₁ + €2. Show how your estimates of foLs and fols can be used to construct a consistent esti- mator for B. Consider the following model with one covariate: Y₁ = x + ui u~ N(0, 1) (a) Show that BoLs is a consistent estimator for 3. Be clear on any assumptions that you make. (b) Derive the asymptotic distribution of √n (BOLS-B). Now suppose that you do not observe the variable z. Rather, you observe: x₁ = x + v₁ U₂ ~ N(0,0²) In addition, assume that , is independent of u. (c) Derive the plim of BoLs and an expression for the attenuation bias in terms of Var (v.) and Var(a). Explain how this bias would affect your interpretation of the estimate of BOLS. (d) Suppose that there is an instrument z, that satisfies E[zu] = 0, E[zv] = 0, and E[za] # 0. You estimate the following first-stage regression: I₁ = π²₁ + 1, and the reduced-form equation: Yi = 7²₁ + €2. Show how your estimates of foLs and fols can be used to construct a consistent esti- mator for B.

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a To show that the ordinary least squares OLS estimator BOLS is consistent for B we need to demonstrate that it converges in probability to the true parameter B as the sample size n increases Assumpti... View the full answer