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.
