Problem 3. Consider a very large random sample of (Yi, Xi, Zi, Qi), where Yi = Bo+ BıXi+Ui, = 0.9, ởzx = 0.7, and Z₁ and Q₁ are some variables. Suppose o 4,0² = 3, 02 = 2, σχυ ozu = Y, for some number 7, and Q₁ is independent of (Yi, Xi, Zi) (perhaps it was generated randomly on a computer without any connection to the data we have). Let us define a new variable Ri= Zi + Qi. (a) Suppose we run OLS regression of Y; on X, and a constant. Will this result in a consistent estimator of B₁? (b) Suppose y = 0. Is Z; a valid instrumental variable? (c) Suppose y = 0. Is Rį a valid instrumental variable? (d) Suppose y 0. Let 3z denote the TSLS estimator of 3₁ that uses Z; as the instrumental variable. Let bz denote the probability limit of 3z, i.e., BÔz →p bz as n → ∞. Using the information provided above, find bz. Hint: bz could depend on y.
Problem 3. Consider a very large random sample of (Yi, Xi, Zi, Qi), where Yi = Bo+ BıXi+Ui, = 0.9, ởzx = 0.7, and Z₁ and Q₁ are some variables. Suppose o 4,0² = 3, 02 = 2, σχυ ozu = Y, for some number 7, and Q₁ is independent of (Yi, Xi, Zi) (perhaps it was generated randomly on a computer without any connection to the data we have). Let us define a new variable Ri= Zi + Qi. (a) Suppose we run OLS regression of Y; on X, and a constant. Will this result in a consistent estimator of B₁? (b) Suppose y = 0. Is Z; a valid instrumental variable? (c) Suppose y = 0. Is Rį a valid instrumental variable? (d) Suppose y 0. Let 3z denote the TSLS estimator of 3₁ that uses Z; as the instrumental variable. Let bz denote the probability limit of 3z, i.e., BÔz →p bz as n → ∞. Using the information provided above, find bz. Hint: bz could depend on y.
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![**Problem 3.** Consider a very large random sample of \((Y_i, X_i, Z_i, Q_i)\), where
\[ Y_i = \beta_0 + \beta_1 X_i + U_i, \]
and \(Z_i\) and \(Q_i\) are some variables. Suppose \(\sigma_X^2 = 4\), \(\sigma_Z^2 = 3\), \(\sigma_Q^2 = 2\), \(\sigma_{XU} = 0.9\), \(\sigma_{ZX} = 0.7\), \(\sigma_{ZU} = \gamma\), for some number \(\gamma\), and \(Q_i\) is independent of \((Y_i, X_i, Z_i)\) (perhaps it was generated randomly on a computer without any connection to the data we have). Let us define a new variable
\[ R_i = Z_i + Q_i. \]
(a) Suppose we run OLS regression of \(Y_i\) on \(X_i\) and a constant. Will this result in a consistent estimator of \(\beta_1\)?
(b) Suppose \(\gamma = 0\). Is \(Z_i\) a valid instrumental variable?
(c) Suppose \(\gamma = 0\). Is \(R_i\) a valid instrumental variable?
(d) Suppose \(\gamma \neq 0\). Let \(\hat{\beta}_Z\) denote the TSLS estimator of \(\beta_1\) that uses \(Z_i\) as the instrumental variable. Let \(b_Z\) denote the probability limit of \(\hat{\beta}_Z\), i.e., \(\hat{\beta}_Z \rightarrow_p b_Z\) as \(n \rightarrow \infty\). Using the information provided above, find \(b_Z\). *Hint: \(b_Z\) could depend on \(\gamma\).*](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8eeaacd3-f9c7-4223-a8e6-a717e551a98a%2F299b0963-e8a2-4092-8578-eaa3a744e94b%2Fgb1dmhb_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 3.** Consider a very large random sample of \((Y_i, X_i, Z_i, Q_i)\), where
\[ Y_i = \beta_0 + \beta_1 X_i + U_i, \]
and \(Z_i\) and \(Q_i\) are some variables. Suppose \(\sigma_X^2 = 4\), \(\sigma_Z^2 = 3\), \(\sigma_Q^2 = 2\), \(\sigma_{XU} = 0.9\), \(\sigma_{ZX} = 0.7\), \(\sigma_{ZU} = \gamma\), for some number \(\gamma\), and \(Q_i\) is independent of \((Y_i, X_i, Z_i)\) (perhaps it was generated randomly on a computer without any connection to the data we have). Let us define a new variable
\[ R_i = Z_i + Q_i. \]
(a) Suppose we run OLS regression of \(Y_i\) on \(X_i\) and a constant. Will this result in a consistent estimator of \(\beta_1\)?
(b) Suppose \(\gamma = 0\). Is \(Z_i\) a valid instrumental variable?
(c) Suppose \(\gamma = 0\). Is \(R_i\) a valid instrumental variable?
(d) Suppose \(\gamma \neq 0\). Let \(\hat{\beta}_Z\) denote the TSLS estimator of \(\beta_1\) that uses \(Z_i\) as the instrumental variable. Let \(b_Z\) denote the probability limit of \(\hat{\beta}_Z\), i.e., \(\hat{\beta}_Z \rightarrow_p b_Z\) as \(n \rightarrow \infty\). Using the information provided above, find \(b_Z\). *Hint: \(b_Z\) could depend on \(\gamma\).*
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