Regression modelling options for time-varying contribution of exogeneous variables

When building a regression model for observations in time $$y_t = f(\beta, x_t, \varepsilon_t)$$, I want the magnitude of $$x_t$$ to have different contribution to $$y_t$$ at different time moments. I consider one option to specify time-varying coefficients $$\beta_t$$, but this puts an extra burden of specifying coefficient dynamics. Another option that I can think of is introducing time $$t$$ into the model or more specifically time interval length $$t_c - t_p$$ between current time moment $$t_c$$ and past time moment $$t_p$$ in addition to observation itself $$x_{t_p}$$. The reason for this is that usually current observations have bigger impact.

Are there any common approaches, econometric models for this situation?

To give an example, suppose a model for product sales explained (among other factors) by daily new Covid-19 cases. We expect that due to increasing number of cases, sales decrease (people visit stores less frequently). But at the beginning, this effect, captured by $$\beta$$, is expected to be stronger for the same value of $$x$$. In general, it is expected to vary over time.

• This is all very abstract. Can you describe your concrete problem in english? Commented Sep 22, 2021 at 21:37
• Added an example case. Commented Sep 23, 2021 at 6:07
• What is the goal? Forecasting? Commented Sep 27, 2021 at 16:12
• Yes, the main goal is forecasting. Commented Sep 28, 2021 at 10:15