Media variables without adstock or coefficient constraints

[tjb474]

Hi Sibyl,

Thank you for your efforts in this repo, it's been very helpful.

I would like to get your thoughts on adding additional media variables that
a) don't require an adstock transformation
b) require no restriction on the coefficients

For example, Average SEO Position, which requires no adstock transformation and intuitively has an inverse relationship with Sales.

These could be included in the control model, however it might be useful to see its individual contribution to sales.

In the STAN code (model_code2) i created two input matrices X1 and X2. X1 contains mdip variables and control variables/baseline sales (as in the original example). X2 contains SEO positioning. The model logic is the same as in the control model.

model {
  decay ~ beta(3,3);
  peak ~ uniform(0, ceil(max_lag/2));
  tau ~ normal(0, 5);
  for (i in 1 : num_adstock_media+num_ctrl) {
    beta1[i] ~ normal(0, 1);
  }
  for (i in 1 : num_media) {
    beta2[i] ~ normal(0, 0.5);
  }
  noise_var ~ inv_gamma(0.05, 0.05 * 0.01);
  y ~ normal(tau + X1*beta1 + X2*beta2, sqrt(noise_var));
}

X2 is mean-centred, log1p transformed only:

 // mean-center, log1p transformation for media variables requiring no adstock
  for (nn in 1:N) {
    for (media in 1 : num_media) {
      X2[nn, media] <- log1p(X_media[nn, media]/mu_mdnip[media]);
    }
  }

And there is no lower bound on X2's coefficients:

parameters {
  // residual variance
  real<lower=0> noise_var;
  // the intercept
  real tau;
  // the (positive) coefficients for adstocked media variables and base sales
  vector<lower=0>[num_adstock_media+num_ctrl] beta1;
  // the (positive or negative) coefficients for non-adstocked media variables
  vector[num_media] beta2;
  // the decay and peak parameter for the adstock transformation of
  // each media
  vector<lower=0,upper=1>[num_adstock_media] decay;
  vector<lower=0,upper=ceil(max_lag/2)>[num_adstock_media] peak;
}

Does this seem to be a sensible approach to you?

Thanks

[sibylhe]

The code looks good.
One concern is endogeneity. To me, it seems seo position and seo clicks are not independent. Click is (partially) influenced by position, because top-ranking items get clicked more often. And probably, if an item is more frequently clicked, it will be ranked higher by the search engine. If that endogeneity exists, it should be removed to see their individual contributions to sales. Perhaps use seo position as an instrumental variable. Please research endogeneity/instrumental variable/latent instrumental variable in econometric modeling if you're interested in that approach. (I don't have any experience using instrumental variables, so unable to provide instructions.)