# Multiplicative linear model

I am considering the model:

$$y_t = \beta_0\left(\Pi_{i=0}^{K}x_{i,t}^{\beta_i}\right)\left(\Pi_{j = K+1}^{L}e^{\beta_{j}x_{j,t}}\right)$$

where we want to have multiplicative effect between variables and linear for the others. My question is the following: in this model, how can we interprate and estimate the coefficients ? I mean we cannot keep the interpretation from the simple linear regression model where the coefficient $$\beta_i$$ represents the impact of the explanatory variable $$x_{i,t}$$ on the explained variable $$y_t$$.

The advantage of a such model is to separate explanatory variables in two sets in order to have something realistic about the model when we should have $$y_t = 0$$ because in the set of fundamental explanatory variables we have one variable $$x_{i,t} = 0$$ so there is a kind of ''interaction" between variables.

• If you take the logs of both sides, it becomes a linear model - linear in $\log x_i$ for the first $K$ terms and linear in the $x_i$ themselves for the remaining $L-K$ terms. Commented Apr 6, 2023 at 15:00
• since you have x^beta, it is not a linear model. IN a linear model, if you want to have multiplicative effect of a variable on the outocme you can just log-transform it. Commented Apr 6, 2023 at 15:02
• This model is incompletely specified. How it is fit and interpreted depends on your probability model for the deviations between the left and right sides. What is that model?
– whuber
Commented Apr 6, 2023 at 15:50
• The coefficients are exactly the same as in the model formulation you've written, so the interpretation is too. Note also @whuber's comment. Commented Apr 6, 2023 at 16:50
• Start with the log-log model (which is readily fit using ordinary least squares regression) and study the residuals. But before doing that, please explain what you mean about the zero values: that raises many red flags but it's unclear what you're trying to say.
– whuber
Commented Apr 6, 2023 at 21:45