Consider the model $\hat{y}_t = e^{\text{trend} + \text{seasonality}} \prod_k^K x_{k, t}^{b_k}$

where $K$ denotes different investment alternatives. You can think that trend and seasonality are modelled a linear trend and dummies.

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

We can linearize the multiplicative model to $$\ln(y) = \text{trend} + \text{seasonality} + \sum_{k \in \text{channels}} b_k \ln(x_k)$$

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

Now comes what bugs me out;

Imagine that we estimated the linearized version with OLS and retrieved estimates for our $\text{trend}, \text{seasonality}, b_k$ parameters.

Imagine that we want to optimally plan a given budget $B$ over our investments $K$ such that we maximize $y_t$ over a horizon $H$.

There must be some issue with $log$ not commuting with $\sum$ since the two following reward functions are vastly different and will yield different solutions.

$$\max_{x_k}\sum_t^H e^{\hat{\text{trend}} + \hat{\text{seasonality}}} \prod_k^K x_{k, t}^{\hat{b}_k}$$

$$\max_{x_k} \sum_t^H \hat{\text{trend}} + \hat{\text{seasonality}} + \sum_k^K x_{k, t}^{\hat{b}_k}$$

....s.t $\sum_k^K \sum_t^T x_{k, t} \le B$

Notice how the seasonality and trend interact with our spend variables in the multiplicative model and do so in the linearized model. This means that we may ease in on spending when the season is bad and spend more when the season is good in the first model. This is not accounted for in the second model.

How would one deal with this scenario, is it up to interpretation and are there any guidelines?



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