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Within a frequentist, deterministic paradigm of multiple linear regression, is there a (standard) method to accomplish "chain modeling for panel data" in a way that avoids formal identity (and/or redundancy) with inclusion of a time variable in one's linear regression model?

E.g., the True ROI on an investment strategy may depend on the same 5 parameters from 1970 - 2010, each having a different, but related, True weight in 1970, 1980, 1990, 2000, 2010? We want to predict those parameters for 2020 under the assumption of a chained (evolving) economy.

$$ y = \beta_0 + \beta X + \epsilon $$

By this phrase ("chain modeling for panel data") I mean the following: you have sample data from times (1, 2,.. n) [I have in mind a panel, if that matters]:

  1. a closed static system might take data captured at different times as useful additions to a (meta-)sample that helps estimate the true population parameters that apply to ($1, 2,.. n = 1$ population)

    1.1. e.g., measuring $x_1$ at each time generates a new $\hat\beta_1$, but they all try to inform us about the one true $\beta_1$

  2. a closed dynamic system might take data captured at different times as separate samples that each estimate the true population parameters of n different populations ($1, 2,.. n = n$ subpopulations)

    2.1. e.g., measuring $x_1$ at each time generates a $\hat\beta_{1.1}$ for time 1, $\hat\beta_{1.2}$ for time 2, ... and each tries to inform us about its time-specific true $\beta_{1.(1,2,.. n)}$

  3. however, a closed chained dynamic system might take data captured at different times as different, but related, samples that each estimate the true population parameters of n different, but related, populations ($1, 2,.. n = n$ chained subpopulations)

    3.1. e.g., similar to 2.1 except we now assume that there is some information from our $n-1, n-2,...$ models to help us estimate the true $\beta_{1.n}$ at time $n$

I ask because it seems useful, assuming we are making predictions (or inferences) within dynamic systems, to utilize more information across time yet still formally recognize that different times may contain distinct, yet related, data generating processes. Can we weight past models coefficients in rank order $n-1, n-2,...$? Or predict the path of the coefficient? Or this assumes too much perfect data? Or perhaps we can only accommodate static or dynamic and not a middle?

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Sure, they're called time-varying parameter models. They try to predict the value of the coefficient beta in the future.

Likely, you are talking about a system that is autoregressive (weight of the past on the present is very high), so I'd probably look at Engles, Lilien, Robins 1987 econometrica paper.

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