Question

Consider the following simultaneous equations model: \[ Y_t=\beta_1+\beta_2 X_t+\beta_3 X_{t-1}+\beta_4 Z_t+\mu_{1t} (1),\qquad Z_t=\delta_1+\delta_2 Y_t+\delta_3 W_t+\mu_{2t} (2) \] Before estimating, identification tests (order and rank) show that equation (2) is {overidentified. Which method is appropriate to estimate equation (2)?}

• A. Two-Stage Least Squares
• B. Indirect Least Squares
• C. Weighted Least Squares
• D. Ordinary Least Squares

Hint:

Rule of thumb: {Endogenous regressor + overidentified} $\Rightarrow$ use an \textbf{IV} method—most commonly \textbf{2SLS}. Exactly identified $\Rightarrow$ ILS/2SLS coincide; underidentified $\Rightarrow$ not estimable.

Answer 1

The Correct Option is A

Step 1: Nature of equation (2).
Equation (2) contains the endogenous regressor $Y_t$; OLS would be inconsistent because $Y_t$ is correlated with the structural error $\mu_{2t}$.
Step 2: Identification status.
“Overidentified” means there are {more} valid instruments available than the number of endogenous regressors in (2) (here, at least one more instrument than $Y_t$), e.g., excluded exogenous variables like $X_t$, $X_{t-1}$, $W_t$ as applicable.
Step 3: Appropriate estimator.
For overidentified structural equations, Two-Stage Least Squares (2SLS) (or other IV estimators like LIML/3SLS) provides consistent estimates by first projecting $Y_t$ on the instrument set and then regressing $Z_t$ on the fitted values.
Step 4: Eliminate distractors.
- Indirect LS is suited to exactly identified systems (via reduced form) but not necessary here.
- WLS addresses heteroskedasticity, not endogeneity/identification.
- OLS is biased/inconsistent due to simultaneity.
Final Answer: (A) Two-Stage Least Squares.