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Suppose in an economy there are $G$ available goods $X_1,...,X_G$. We can formulate a system of $3G+1$ equations as below:

$$Q^d_g = A_{g1}+A_{g2}P_g+\mathbf{A}_g\mathbf{Z}^d_g+u_g\quad(g = 1,...,G)\quad(1)$$ $$Q^s_g = B_{g1}+B_{g2}P_g+\mathbf{B}_g\mathbf{Z}^s_g+v_g\quad(g = 1,...,G)\quad(2)$$ $$Q^d_g = Q^s_g\quad(g = 1,...,G)\qquad\qquad\qquad\qquad\qquad\quad(3)$$ $$\sum_{g=1}^GP_gQ^d_g = I\quad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad(4)$$

Here $Q^d$ is quantity demanded, $Q^s$ quantity supplied, $P$ price, $\mathbf{Z}^d$ and $\mathbf{Z}^s$ are vectors of other variables (assumed exogenous) relevant to demand and supply respectively, $I$ is available income (also assumed exogenous), the $A$'s and $B$'s are parameters (vectors if in bold), and $u$ and $v$ are error terms.

Suppose we want to estimate demand for good $X_1$. Two methods that might be considered are:

Method 1: Use equations (1), (2) and (3) for $g=1$ only, that is, ignore the equations for the other goods and the income constraint. Use (3) to eliminate $Q^s$. Then estimate (1) within the remaining system of two equations using two-stage least squares (2SLS), on the basis that the equation is overidentified.

Method 2: Use the whole system of $3G+1$ equations. Use (3) to eliminate $Q^s_g$ for $g=1,...,G$. Then estimate, using a suitable estimator, the system consisting of (1) and (2) for $g=1,...,G$, treating (4) as a cross-equation restriction.

Question: What would be the most suitable estimation technique for Method 2, and what advantages might it offer over Method 1?

Method 2 of course is rather impracticable as it is unlikely that all the necessary data could be collected. So the point of the question is to understand what if anything is lost by Method 1 (relative to what might theoretically be obtained by Method 2).

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