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The price of a product has signficant impact on the total sales. Hence modeling sales would give the incentive to include price as a regressor (amongst other variables). Suppose we would estimate this using OLS, this gives something like

$sales_t=\alpha_0+\beta_1\times price_t+...+\epsilon_t\;\;\;\;\;\;\;\;\;(1)$.

In addition, we know that

$price=\frac{sales}{volume} \iff sales=price\times volume \;\;\;\;\;\;\;(2) $.

Estimating (1) results in a large Variance Inflation Factor (VIF) for price, which is comes from the relation (2).

It seems that one can never include price in a sales model, simply because of relation (2). Equation (1) shows a linear relationship, and under ceteris paribus, whenever price increase by one unit, sales should increase with $\beta_1$. But whenever price increases, the sales increases as well because of (2). And price and volume have a non-linear relationship: a price increase of \$0.05 would have a different impact on volume than an increase of \$50000.-

Hence simultaneity occurs and the OLS estimator $\beta_1$ seems to be invalid.

One could reason to use an instrument, but equation (2) implies than any instrument correlated to price is automatically correlated to sales which suggests that there exists no valid instrument.

What is your opinion about modeling sales including the price as a regressor? Am I making a mistake in reasoning here?

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    $\begingroup$ For instrumental variables it is required they do not correlate with regression error, not with the response variable. In general practically all econometric textbooks with chapter on simulataneous equations give an example with supply and demand models, where you model the volume and the price. The sales is then simply volumes times the price. $\endgroup$ – mpiktas Jan 31 '14 at 9:36
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Eq. (2) is accounting identity, you cannot form economic theory from it.

Here would be economic theory based on few assumptions and this accounting identity:

(1) price = sales /volume, where we assume volume=f(income,price)

(2) log(sales)=alpha + beeta_one *price + beeta_two*income + epsilon

EDIT:

log(price)=log(sales)-log(volume), and by re-arranging we get
log(sales)=log(volume)-log(price)

Above is accounting identity. Now we invoke economic assumption and believe income is given outside to our model. So that peoples higher income will lead more goods to be bought at given price level.

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  • $\begingroup$ Eq. (2) holds, simply because revenues will always equal price*volume. Your stated assumption "volume=f(income,price)" is a relation, but volume is not caused by income. Your suggested model will suffer from simultaneity and endogeneity. $\endgroup$ – Sweetbabyjesus Jan 31 '14 at 12:22
  • $\begingroup$ @LongLaw it is theory which we assume to be able to identify parameters from the joint density of prices and sales. My model will not suffer from endogeneity bias since income is given outside to my model. I do not have a separate equation for the determination of prices as function of volume since income is exogenous and my parameters are combination of structural disturbances and of course exogenous income from the two equations. $\endgroup$ – Analyst Jan 31 '14 at 12:52

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