Hi Jesse,
Your response here set my head of spinning, i seen you responding with a lot of very informational content teaching us beginners w.r.t timeseries and causal inference etc.
The scenario i depicted above was a very specific one in which our regressors were the sole IV’s affecting our output.
However, your response set me of on a spiral of thoughts where you may be able to bring some clarity, maybe this deserve a separate thread and maybe this is not even applicable to this forum. If i should create a separate thread to this matter and relate it to bayesian analysis just let me know.
Consider the following high-level time series model:
y_t = \text{seasonality}_t + \text{trend}_t + \text{cyclical}_t + \text{regressor_effects}_t
My primary focus lies in the causal analysis of the influence exerted by the regressors.
Let us consider y_t as indicative of total sales, with our regressors signifying distinct investment opportunities.
Acknowledging that these investment opportunities are not the sole determinants of sales, we have chosen to estimate the baseline sales by incorporating \text{seasonality}_t,\text{trend}_t and \text{cyclical}_t.
Herein lies my concern: given that the effects of our regressors are affected by an underlying demand that concurrently affects our baseline sales, and bearing in mind that my ultimate objective is to make informed decisions about allocation of investments across these varied opportunities, the fluctuating cofounder demand could potentially result in misleading estimates.
Understanding that the impacts of our investments are likely to oscillate in correlation with seasonal variations due to this underlying demand,
I have chosen to employ time-varying parameters that interact with these investment inputs. This composite notion is encapsulated by the variable \text{regressor_effects}_t.
I have refrained from explicitly delineating a low-level model, as I believe that an appropriate response can encompass any suitable model of choice to address my query.
To facilitate a more insightful response to the ensuing question, envision the following scenarios:
- Our expenditure demonstrates a perfect correlation with the seasonality.
- Our expenditure is equally split between random and perfect correlation with the seasonality.
- Our expenditure is entirely random.
Question: How can we gain an intuitive comprehension of the manner in which a time-series model would differentiate the baseline-sales seasonality from the regressor-effect seasonality given the different expenditure scenarios?
any related resources would be of great appreciation.
Thank you in advance Jesse, i seen that you have done a lot for this forum and i really appreciate it.