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?

• Please include more information in your question: What is the null, what is the model and what are the parameters? Jun 24 '15 at 11:06
• That's the hard part...you might try finding a valid instrument. Jun 24 '15 at 12:42
• This could be a very nice question to have for future references, but as is it doesn't include sufficient information. Jun 24 '15 at 14:21
• I am sorry for the lack of full information. I have included more details. Thanks Jun 24 '15 at 14:59

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!
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).