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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.

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

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