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I am trying to model the effect of advertisement on sales in Stata. The data is weekly and there are around 150 observations. I started by applying an ARMAX(1,0,1) model with the following exogenous variables: investment in advertisement, quantities bought by visit and some seasonal dummies (Q1, Q2, Q3).

I would like to have some ideas regarding the model:

  1. Is this the best model to estimate the coefficients accurately?
  2. Should I be worried about endogeneity?
  3. Is there any way to impose (or test) diminishing returns for the investment in advertisement?
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  1. It is difficult to say what the best model is, but you can ask whether the current model is adequate. For that you may check if it has residuals that are close to white noise and whether explanatory power can be considered sufficient. What stands out from your model description is that you use quarterly seasonal dummies although your data is weekly. That does not sound like a sensible way of modelling seasonality. Instead you could try including some Fourier terms to account for weekly seasonality, see e.g. Rob J. Hyndman's blog post and related posts.
  2. Not really if you use lagged (rather than contemporaneous) effects of investment advertisement. Then your regressor will be predetermined, which is often sufficient in time series models.
  3. You may include a transformation of investment in advertisement as another exogenous variable or in place of the level of investment in advertisement. The transformation could be square root or logarithm of the original value, both would yield diminishing effects. If you want to test for diminishing effects, you could include both levels and {square root or logarithm} and test whether the latter is significant.
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  • $\begingroup$ Thank you very much Richard. Regarding the second point, does it mean that I cannot use the contemporaneous effect in the model? Would that always create endogeneity? $\endgroup$ – andrepereira Feb 4 '16 at 10:06
  • $\begingroup$ Generally, if $x_t$ affects $y_t$ and vice versa, having a model of the form $y_t=\beta_0+\beta_1 x_t+\varepsilon_t$ (possibly including other regressors as well) will not work because $x_t$ will be correlated with $\varepsilon_t$ in reality while zero correlation will be mechanically imposed in the model (causing quite a discrepancy). $\endgroup$ – Richard Hardy Feb 4 '16 at 10:10
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  1. Nobody knows what's the best model. One thing I'd be concerned with is that ARMAX is a stationary model if X are stationary. Sales are often modeled as differences, i.e. exponentially growing processes.

  2. You should always be worried, especially, if you're not using lagged ad spending.

  3. There are many ways of doing it. Any power function may work like $a^{\alpha}$, where $\alpha\in(0,1)$.

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