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We have data for 3 goods (equilibrium prices and quantities) and we want to know the cross-price elasticities among them.

I think, this could be estimated as the following system of regressions:

$$ \ln(x_1) = \beta_{1,1} \ln(P_1) + \beta_{1,2} \ln(P_2) + \beta_{1,3} \ln(P_3) + \epsilon_1$$

$$ \ln(x_2) = \beta_{2,1} \ln(P_1) + \beta_{2,2} \ln(P_2) + \beta_{2,3} \ln(P_3) + \epsilon_2$$

$$ \ln(x_3) = \beta_{3,1} \ln(P_1) + \beta_{3,2} \ln(P_2) + \beta_{3,3} \ln(P_3) + \epsilon_3$$

Where $\beta_{i,j}$ denotes the cross-price elasticity between $i$ and $j$. While information about all prices is known, each product has its own dimension, meaning $dim(x_1) \neq dim(x_2)$. The problem is, that after estimation, we have two estimates of this cross-price elasticity, that is $\beta_{i,j}$ and $\beta_{j,i}$. Now, when I would like to say, the cross-price elasticity between $i$ and $j$ is this and that...

...Is the correct approach to average the both estimates?

$$\beta_{i,j}^* = \frac{\beta_{i,j} + \beta_{j,i}}{2}$$

...Or can I report just one-way effects? Or do something else entirely?

I know, that both estimates will differ as I am trying to estimate the cross-price elasticity on Marshallian demand, but they differ significantly. I guess, this may be caused by different dimensionality of products. Additionally, is there a good reference to this?

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  • $\begingroup$ You could estimate them simultaneously while imposing the symmetry conditions $\beta_{i,j} = \beta_{j,i}$ using (for example) restricted least squares. $\endgroup$
    – tdm
    Mar 5 at 8:56
  • $\begingroup$ Is this possible even when the amount of observations between $x_1$ and $x_2$ differ and LASSO is a necessary approach? Is there a way how to legitimately argue that averaging is not necessary? $\endgroup$
    – Athaeneus
    Mar 5 at 13:03

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