<p>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 $Cov(Z, \mu_1) = 0$.</p>

<p>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!</p>

<p>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.</p>

<p>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).</p>

<p>If the instrument were to be valid, you could form a consistent estimator.</p>

CC BY-SA 3.0
2015-06-25T15:31:27.357Z
2015-06-25T15:31:27.357Z
Pat W.
<p>Given a set of structural equations:</p>

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

<p>Where</p>

<ul>
<li>$P$ = price</li>
<li>$Q$ = quantity</li>
<li>$I$ = income</li>
<li>$R$ = wealth</li>
</ul>

<p>How would you proceed with the Hausman test of simultaneity? Suppose you <em>reject the null</em>, 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?</p>

CC BY-SA 3.0
2015-06-24T11:06:17.550Z
Please include more information in your question: What is the null, what is the model and what are the parameters?
Giskard
CC BY-SA 3.0
2015-06-24T12:42:57.790Z
That's the hard part...you might try finding a valid instrument.
Pat W.
CC BY-SA 3.0
2015-06-24T14:21:58.373Z
This could be a very nice question to have for future references, but as is it doesn't include sufficient information.
cc7768
CC BY-SA 3.0
2015-06-24T14:59:47.787Z
I am sorry  for the lack of full information. I have included more details. Thanks
Christopher
<econometrics><structural-estimation><reduced-form-estimation>
Hausman Test for Simultaneous Equations
Christopher
# Hausman Test for Simultaneous Equations

(Asked by: username_0 on 2015-06-24)

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?

**Tags:** econometrics, structural-estimation, reduced-form-estimation
> **Comments:**
> **username_1**: Please include more information in your question: What is the null, what is the model and what are the parameters?  
> **username_2**: That's the hard part...you might try finding a valid instrument.  
> **username_3**: This could be a very nice question to have for future references, but as is it doesn't include sufficient information.  
> **username_0**: I am sorry  for the lack of full information. I have included more details. Thanks  

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## 1 answers

### Answer by username_2 on 2015-06-25. Score: 3

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.


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