Hausman Test for Simultaneous Equations Given a set of structural equations:


Demand Function: $$Q_1 = \alpha_0 + \alpha_1 P_t + \alpha_2 I_t + \alpha_3 R_t + \mu_{1, t}$$
Supply Function: $$Q = \beta_0 + \beta_1 P_t + \mu_{2, t}$$


Where


$P$ = price
$Q$ = quantity
$I$ = income
$R$ = wealth


How would you proceed with the Hausman test of simultaneity? Suppose you reject the null, which states there is no simultaneity between the error term and the endogenous variable; how can you proceed to obtain the parameters in the model?
Generally, price is endogenous in this set of simultaneous equations.  One strategy we can use to overcome the bias is to find a valid instrument for price—call it $Z$.  We’d need something that satisfies $Cov(P, Z) \neq 0$ and $Cov(Z, \mu_1) = 0$.

The trouble with simultaneous equations is that upon observing some $(P, Q)$ pair, all we know is that it lies at the intersection of supply and demand. As we get more pairs (more data), we’re not tracing out supply or demand curves—we’re just getting a bunch of equilibrium points and aren’t
certain whether it’s supply or demand (or both) that moved!

To estimate the demand curve, what we'd like to do conceptually is to hold it fixed while we shift the supply curve around.  In theory, these new equilibrium points would trace out the demand curve itself.

Returning to the instrument, the statement $Cov(P, Z) \neq 0$ means we'd like to find some variable $Z$ that shifts the supply curve.  Maybe this is something like subsidies to tech firms or weather in agricultural regions.  Then you'd hope $Cov(Z, \mu_1) = 0$ so $Z$ doesn't shift demand around, too (then we'd be back where we started).

If the instrument were to be valid, you could form a consistent estimator.