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I have been trying to estimate cross-price elasticity of demand on market optimum data by simple regression such as:

$$ln(x_i) = \beta_0 + \beta_i ln(P_i) + \sum_j^J \beta_j \cdot ln(P_j) + \epsilon$$

Where $P_j$ is the price of $j$-th product and $x_i$ is the quantity of $i$-th product bought that day with $\beta_j$ indicating the cross-price elasticity between products $i$ and $j$.

Now, I know that these results should be biased such that $\beta_j = \beta_j^* + BIAS(\beta_j)$, mostly because of simultaneity, requiring me to explain the supply side by instrumental variable to get the proper results for demand. The funny thing is that comparing results obtained for $\beta_j$ with basic intuition and other methods for determining complements/substitutes shows as if the $\beta_j$ was correctly estimated in terms of sign!

To me, this seems as if the bias was self-reinforcing. We expect $\beta_j >0$ for substitutes and $\beta_j <0$ for complements, while my results indicate as if:

$$ BIAS(\beta_j) \approx \left\{ \begin{array}{ll} BIAS(\beta_j) > 0 & \{ \text{if $\beta_j > 0$} \} \\ BIAS(\beta_j) = 0 & \{ \text{if $\beta_j = 0$} \} \\ BIAS(\beta_j) < 0 & \{ \text{if $\beta_j < 0$} \} \end{array}\right. $$

This can be denoted as a multiplicative instead of additive bias such that $\beta_j = \beta_j^* \cdot BIAS(\beta_j)$.

If two goods are independent, the simple method of cross-price elasticity estimation shows them as independent, and if they have certain relation between each other, the simple estimation identifies this relation appropriately. I have tried to simulate it (the simulation is not good yet; however, preliminary results illustrate similar phenomenon).

The question:

  • Does this principle hold in general (i.e., cross-price elasticity having self-reinforcing bias)?
  • Does simple biased estimation of cross-price elasticity provide a good estimate of relations between products (i.e., the bias can be there in any form but is generally just insignificant)?
  • Most importantly, is there a literature describing this phenomenon?
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Does this principle hold in general (i.e., cross-price elasticity having self-reinforcing bias)?

No this is not something that holds. If you did get correct signs, you just got lucky, and its a special case that has little to do with what you describe in question.

Structurally the correct model looks like:

$(1) \quad \ln x_i = \beta_0 + \beta_1 \ln P_i + \beta_2 \ln P_j + \epsilon$

$(2) \quad \ln P_i = \alpha_0 + \alpha_1 \ln x_i + \eta$

$(3) \quad \ln x_j = \omega_0 + \omega_1 \ln P_i + \omega_2 \ln P_j + u$

$(4) \quad \ln P_j = \gamma_0 + \gamma_1 \ln x_j + \xi$

Now solve the system above for $\ln P_j$, you get:

$$ \ln P_j = \frac{1}{\left((1-\frac{\gamma_1\omega_1 \alpha_1\beta_2}{(1- \alpha_1\beta_1 )} -\gamma_1\omega_2 )\right)} \left( \gamma_0 + \gamma_1\omega_0 + \frac{\gamma_1\omega_1(\alpha_0 + \alpha_1 \beta_0) }{(1- \alpha_1\beta_1 )} + \frac{\gamma_1\omega_1 \alpha_1}{(1- \alpha_1\beta_1 )}\epsilon + \frac{\gamma_1\omega_1 }{(1- \alpha_1\beta_1 )}\eta+ \gamma_1u + \xi \right)$$

The simultaneity bias comes from $cov(\ln P_j, \epsilon) \neq 0$. The equation above shows that unless we have special cases such as $\gamma_1=0$ etc, there will be bias because $P_j$ is function of $\epsilon$.

Hence what you are claiming is not correct:

  • bias of a estimated coefficient is not 0 if any of the $\beta$'s are zero, or even when they both together are zero, i.e. even if $\beta_1=\beta_2= 0$ there will still be bias. $\ln P_j$ will still be function of $\frac{\gamma_1 \omega_1 \alpha_1}{1-\gamma_1 \omega_2}$.

To get bias of $\beta_2$ or you call it $\beta_j$ to zero you in essence need the price of $j$ that is independent of quantity ( $\gamma_1 = 0$). However, this is not revolutionary observation. In that case the price is not endogenous. Hence this is nothing new just econometrics 101.

  • signs of $\beta_1$ or $\beta_2$ do not tell you the sign of bias. Depending on sings of other parameters the overall sign can flip.

Does simple biased estimation of cross-price elasticity provide a good estimate of relations between products (i.e., the bias can be there in any form but is generally just insignificant)?

As the math in the previous section says, no.

  • naïve OLS does not give you correct sign unless $P_j$ is independent of output, e.g. maybe there is some price fixed by government and government does not respond to shortages or surpluses of the commodity. If that is the case then you can trivially use OLS because then $P_j$ is exogenous

  • the size of bias cannot be known without first estimating other parameters from other equations, if you have data to do that you might as well just run some structural model and get the correct estimates directly instead of trying to laboriously correct biased estimates.

Most importantly, is there a literature describing this phenomenon?

Since the phenomenon does not really generally exist there shouldn't be a literature on it.

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  • $\begingroup$ Your equations (1) and (2) are incompatible, eq (2) should depend on $x_j$ (if you do not want to have it in the random term). The same is true for eq. (4). If the goods $i$ and $j$ are substitutes (or complements) the quantity $x_j$ produced has an impact on $P_i$. $\endgroup$
    – Bertrand
    Feb 8 at 10:33
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    $\begingroup$ @Bertrand but that is the supply, supply of a competitive firm producing $x_i$, and $x_i$ only, does not necessarily depend on supply of complements. Eq 2 and 4 are supply side of the system, the prices have effect on quantities just through the demand side (eq 1 and 3) $\endgroup$
    – 1muflon1
    Feb 8 at 12:26
  • $\begingroup$ oh sorry, it is obvious now... I thought is was the inverse demand formulation $\endgroup$
    – Bertrand
    Feb 8 at 14:47

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