<p>Assume I have quantities and prices for J products across M markets in T periods.
I want to estimate their elasticities.</p>
<p>I construct J regressions of this type:
<span class="math-container">$$\log \left(q_{j m t}\right)=\alpha_{t}+\beta_{m}+\sum_{k=1}^{J} \eta_{j k} \cdot \log \left(p_{k m t}\right)+\epsilon_{j m t}$$</span></p>
<p>I want to instrument the price of <span class="math-container">$p_{jmt}$</span> the price in other markets at the time t (Hausman  IV).
within each regression: do I need to instruments <strong>all</strong> prices (<span class="math-container">$\sum_k{p_{k m t}}$</span>)
or only for <span class="math-container">$p_{jmt}$</span>.</p>
<p>My intuition is that IV for <span class="math-container">$p_{jmt}$</span> is enough, as <span class="math-container">$q_{jmt}$</span> and <span class="math-container">$p_{jmt}$</span> are being set simultanisly, but <span class="math-container">$q_{jmt}$</span> and other product does not.</p>
<p>Am I right?  Is there any reference for such a basic estimation?</p>

CC BY-SA 4.0
2021-01-07T18:18:09.057Z
Do you have more products than markets? Is $J>M$ ? Can you elaborate on this point?
Bertrand
CC BY-SA 4.0
2021-01-07T22:02:37.283Z
Maybe I defined it a bit sloppy, I have 10 products sold by 20 firms. so each firm is a market.
The panel is almost balanced so I have 3 year of data (36 obs)
Number of obs 36*20 per each regression
Number of coefficients is 10 items + 36 periods + 20 retailers
AverageGuy
CC BY-SA 4.0
2021-01-07T22:37:22.993Z
A lot of textbooks have treatments of simultaneous equations models, and the go to example in most if not all is demand and supply estimation. Personally, I learned these methods from Green: Econometric Analysis although im sure most upper-level textbooks will do. I believe your intuition about instrumenting price is correct, although it has been some time since doing this.
Brennan
CC BY-SA 4.0
2021-01-08T07:54:03.797Z
@AverageGuy: If firms have no market power but are price takers, prices are correlated with changes in aggregate shocks in demand and supply, but not with firm specific shocks, so prices could be exogenous. However, it seems that in your case, $m$ denotes a firm, and you have firm specific prices, which is quite interesting. Without more details on your structural model (competitive context), it will be very difficult to give a precise answer to your question. I do not understand why $q$ has an $m$ subscript. If $q$ represents "demand" quantity, it should have a $n$ subscript for the consumer.
Bertrand
<microeconomics><econometrics><demand-system-estimation>
Endogeneity in demand estimation using demand systems estimations
AverageGuy
# Endogeneity in demand estimation using demand systems estimations

(Asked by: username_0 on 2021-01-07)

Assume I have quantities and prices for J products across M markets in T periods.
I want to estimate their elasticities.

I construct J regressions of this type:
$$\log \left(q\_{j m t}\right)=\alpha\_{t}+\beta\_{m}+\sum\_{k=1}^{J} \eta\_{j k} \cdot \log \left(p\_{k m t}\right)+\epsilon\_{j m t}$$

I want to instrument the price of $p\_{jmt}$ the price in other markets at the time t (Hausman IV).
within each regression: do I need to instruments **all** prices ($\sum\_k{p\_{k m t}}$)
or only for $p\_{jmt}$.

My intuition is that IV for $p\_{jmt}$ is enough, as $q\_{jmt}$ and $p\_{jmt}$ are being set simultanisly, but $q\_{jmt}$ and other product does not.

Am I right? Is there any reference for such a basic estimation?

**Tags:** microeconomics, econometrics, demand-system-estimation
> **Comments:**
> **username_1**: Do you have more products than markets? Is $J>M$ ? Can you elaborate on this point?  
> **username_0**: Maybe I defined it a bit sloppy, I have 10 products sold by 20 firms. so each firm is a market. The panel is almost balanced so I have 3 year of data (36 obs) Number of obs 36*20 per each regression Number of coefficients is 10 items + 36 periods + 20 retailers  
> **username_2**: A lot of textbooks have treatments of simultaneous equations models, and the go to example in most if not all is demand and supply estimation. Personally, I learned these methods from Green: Econometric Analysis although im sure most upper-level textbooks will do. I believe your intuition about instrumenting price is correct, although it has been some time since doing this.  
> **username_1**: @AverageGuy: If firms have no market power but are price takers, prices are correlated with changes in aggregate shocks in demand and supply, but not with firm specific shocks, so prices could be exogenous. However, it seems that in your case, $m$ denotes a firm, and you have firm specific prices, which is quite interesting. Without more details on your structural model (competitive context), it will be very difficult to give a precise answer to your question. I do not understand why $q$ has an $m$ subscript. If $q$ represents "demand" quantity, it should have a $n$ subscript for the consumer.  

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## 0 answers

