interpretation of loglog regression

Question

It is common econometric disciplines to model an baseline + incremental quantity due to treatment. E.g baseline sales + incremental sales(due to marketing).

It is common to deploy loglog models to tackle e.g diminishing marginal returns.

Consider e.g the model $y = e^{\text{trend} + \text{seasonality}} \cdot \prod_{k \in \text{channels}} x_k^{b_k}$

where i constrained $0 < b_k < 1$ (to capture diminishing marginal returns)

Which we can linearize into $$\ln(y) = \text{trend} + \text{seasonality} + \sum_{k \in \text{channels}} b_k \ln(x_k)$$

s.t $0 < b_k < 1 $...

Lets assume the trend and seasonality are modeled by some sort of dummies.

Lets assume we estimate this by OLS.

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Now, how can we describe the baseline sales in this scenario thus the sales that would have happened had no marketing activity been taking place?

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As i see it, this type of model assumes that the sole drivers behind sales is the "incremental part" and that there does not exist any "baseline sales". Thus in the case described above, we essentially have timevarying parameters that solely affect our marketing variables. Can someone help me with my intuition here.

Answer 1

The baseline would be the $\exp(\alpha + \tau \cdot t + \gamma_\text{Jan} + \cdots )$ on the original scale. This is sensible since it constrains the baseline to be positive, and the baseline depends on the seasonality and trend. You can imagine adding other variables in there as well. Presumably, there is also a multiplicative error term with a mean of $1.$

However, having the marketing terms enter multiplicatively assumes that when you shut off a single marketing channel, you have zero sales since $0^{\beta_k}=0$. I think this is what many CMOs believe, but that is hard to take seriously for most settings. This sort of thing makes much more sense for production functions where if you have zero labor and/or zero capital, your output would be zero since you need both inputs to make something.

In fitting the log-log model, you can recode 0 to 1 since $\beta \cdot \ln(1) = 0$ to get around the fact that the log of zero is undefined. You may not need to do this if all the marketing variation is on the intensive margin.

To go from the estimated model back to the original scale, you will need to deal with the retransformation bias. Duan's smearing method is a frequently adopted approach.