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I am interested in the analysis of interaction between variables in a regression model.

First, the context : I work in marketing and the explanatory variables corresponds to marketing channels, therefore it is interesting to know what are the combinations of marketing channels that are worth in terms of sales volume (sales volume is the dependent variables $y_t$).

For numerous reason, I work with the following model :

$$ \ln(y_t) = \ln(\beta_0) + \sum_{i=1}^{K}\beta_i \ln(x_{i,t}) + \sum_{j=K+1}^{L}\beta_jx_{j,t} +\ln(\epsilon_t) $$

Now, here is my tentative to analyze these interactions : I have tried, at each period $t$, to take my estimated model $\ln(\hat{y}_t)$ and write it as a function of two explanatory variables of interest (by fixing the others) :

$$ \ln(\hat{y}_t) = f(x_{3,t}, x_{7,t}) $$

and look at what happens when I change the value of $x_{3,t}$ and $x_{7,t}$ which is a fairly poor analysis...

Ideally, after estimating the model I would like to represent the interactions between the $x_k$ in quantitative terms (like numbers or percentage). On the internet, I found nothing like this in the litterature, is this unrealistic ?

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    $\begingroup$ Please explain more clearly what you mean by "quantify" $\endgroup$
    – Gregg H
    Commented May 30, 2023 at 14:46
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    $\begingroup$ You are going to need to provide more context if others are going to be able to help you here. $\endgroup$
    – Gregg H
    Commented May 30, 2023 at 15:42
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    $\begingroup$ Maybe you want to work with a model that actually includes the interactions? I reckon you have a huge database and the number of datapoints is considerably larger than $L$+ the number of interactions that interest you? $\endgroup$
    – Ute
    Commented May 30, 2023 at 22:47
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    $\begingroup$ @Ute Thank you for your comment ! Yes I have a huge data base ! Concerning the interaction, if we take the exponential both side of equality (in the model I wrote in question), we get the following $y_t = \beta_0\Pi_{i=1}^{K}x_{i,t}^{\beta_i}\Pi_{j=K+1}^{L}\beta_{j}x_{j,t}$, doesn't this include interactions? And if not, what kind of model can be proposed for the interaction ? I thought about the classical model $y_t = \beta_0 + \beta_1x_{1,t} + \beta_{2}x_{2,t} + \beta_{3}x_{1,t}x_{2,t}$ but it is clearly a different model from the one I initially have $\endgroup$
    – G2MWF
    Commented May 31, 2023 at 6:59
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    $\begingroup$ Ah, I can see what you mean. When you transform back fom log, the originally additive effects become indeed multiplied. But this is not the same as interaction terms in a linear model. Have a look at this question:(stats.stackexchange.com/questions/615168/…) $\endgroup$
    – Ute
    Commented May 31, 2023 at 8:29

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