(requires Appendix material) The relationship between the TSLS slope and the corresponding population parameter is:
A) $\left( \hat { \beta } _ { 1 } ^ { \text {TSLS } } - \beta _ { 1 } \right) = \frac { \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left( Z _ { i } - \bar { Z } \right) u _ { i } } { \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left( Z _ { i } - \bar { Z } \right) \left( X _ { i } - \bar { X } \right) } .$
B) $\left( \hat { \beta } _ { 1 } ^ { \text {TSLS } } - \beta _ { 1 } \right) = \frac { \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left( Z _ { i } - \bar { Z } \right) } { \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left( Z _ { i } - \bar { Z } \right) \left( X _ { i } - \bar { X } \right) } .$
C) $\left( \hat { \beta } _ { 1 } ^ { T S L S } - \beta _ { 1 } \right) = \frac { \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left( Z _ { i } - \bar { Z } \right) u _ { i } } { \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left( Z _ { i } - \bar { Z } \right) ^ { 2 } } .$
D) $\left( \hat { \beta } _ { 1 } ^ { \text {TSLS } } - \beta _ { 1 } \right) = \frac { \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left( X _ { i } - \bar { X } \right) u _ { i } } { \frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left( Z _ { i } - \bar { Z } \right) \left( X _ { i } - \bar { X } \right) }$

Question

Quizplus · Accepted Answer

The TSLS slope is a consistent estimator of the population parameter $\beta_1$, meaning that as the sample size increases, the TSLS slope will converge to the true value of $\beta_1$. The formula provided in option A shows the relationship between the TSLS slope and the population parameter, and it includes the term $\frac { 1 } { n } \sum _ { i = 1 } ^ { n } \left( Z _ { i } - \bar { Z } \right) u _ { i }$, which represents the covariance between the instrumental variable and the error term. This term is important because it shows that the TSLS slope will be biased if the instrumental variable is correlated with the error term.

(Requires Appendix Material) the Relationship Between the TSLS Slope and the Corresponding