I have difficulties understanding the concept of Almost Ideal Demand System or, most probably, I have a problem of how to perform microeconomic estimates from data, in general...

So my data look like this:

ID Good1 Good2 Good3 Good4 Price1 Price2 Price3 Price4 date
ID1 2 0 0 0 4 2 3 2 day1
ID2 2 3 2 4 4 2 3 2 day1
ID3 0 1 2 0 3 5 4 2 day2

I have the ID of a transaction, how many goods were bought during that transaction, prices of all the goods, and the date. And what I would like to know is how the quantity demanded depends on prices and whether goods are complements or substitutes, therefore approximating elasticity of substitution or (rather) price elasticities, cross and self...

In short, there are multiple possible approaches how to solve this question and I would like to ask you what should I do if I want to measure it: What equation to estimate and by which estimator (OLS?)

OPTION A: Simple regression

$$ ln(Good_i) = \sum_j \beta_{ij} * ln(Price_{ij}) + \epsilon_i $$

Where $\beta_j$ is price elasticity of $Good_i$ with respect to $Price_j$, right? Is this good estimate or not?

Should this be used in panel or some form of time series?

OPTION B: Elasticity equation

$$\ln \left( \frac{Good_q}{Good_r} \right) = \sigma_{qr} \ln \left( \frac{Price_q}{Price_r} \right) + \epsilon$$

Where $\sigma_{qr} $ would be elasticity of substitution between

In which case I would have problem with those abundant zeros. So I don't think I could use it. I was thinking, maybe I could only work with ratios without zero in denominator so, for example, in ID1 consider the ratio $Good_2 / Good_1$ and within ID3 consider the ratio $Good_1 / Good_3$... But I don't know...

OPTION C: Demand system - the hardest

Here I understand I have to prepare a new variable: the budget ratio, meaning $M_i = \sum P_i x_i $ and then I would divide each $Good$ by this $M$, therefore, obtaining $s_i = Good_i / M_i$. Is that so?

Then I could estimate the following equation:

$$s_i = \alpha_i + \sum_j \gamma_{ij} * ln(P_j) + \psi_i * ln(M / INDEX) $$

But... Is the $M$ here the same as $M_i$ previously? Is $\gamma_{ij}$ interpretation-wise the same as $\beta_{ij}$ from OPTION A, meaning it tells me the price elasticity? What is $\psi_i$?

OPTION D: Slutsky equation

I have found out a video from @EconJohn about estimating Slutsky equation from data: Video

He uses the following equation (in translation to my notation):

$$ Good_i = \sum_j \beta_{ij} * ln(Price_{ij}) + \gamma_i * M_i + \epsilon_i $$

And then he plugs the results into Slutsky equation:

$$\frac{\partial x_i^*}{\partial P_j} = \frac{\partial \widetilde{x_i}}{\partial P_j} - \frac{\partial x_i^*}{\partial M_i}* \widetilde{x_j}$$

From this he could technically get Slutsky matrix since he can make this eq. for each good and price... The question, however, is: Am I able to construct $M_i$ from my data as $\sum Good_i Price_i$ per transaction?

OPTION E: Hurdle model

Those zeroes seem to be really problematic, because I cannot then distinguish if a good was perfect substitute for a consumer of just an economic bad... Is there some possibility to filter them out, probably using hurdle model? Or just omitting them at all? Could I go for option B when I have dealt with zeroes by this way?

Thank you very much for help. I am completely lost on this. I have tried to read Varian but it did not help me much.



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