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I am looking for a textbook or some other resource where the following multiple linear regression problem is considered.

The model is:

$Y_n = \beta X_n + a+ \epsilon$

Where $\{X_n\}_{n\in\{0,...,T\}}$ are $\mathbb{R}^K$-valued random variables, $\{Y_n\}_{n\in\{0,...,T\}}$ are $\mathbb{R}$-valued random variables. We are therefore looking at the formulation where the regressors are stochastic.

I am interested in the case where the pairs $\{(Y_n,X_n)\}_{n\in\{0,...,T\}}$ are not necessarily i.i.d., but they may satisfy other conditions, for instance stationarity.

The objective is to estimate the parameters $\beta \in \mathbb{R}^K$ and $a\in\mathbb{R}$ given a sample path of the process $\{(Y_n,X_n)\}_{n\in\{0,...,T\}}$.

I have looked into several time-series text books and couldn't find a treatment in this setting.

Winkelried
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    You describe a *model* but you don't state a *problem.* The distinction matters, because for some problems it doesn't matter whether $X_n$ are random. As it stands, it's difficult to see how what you ask for is any different than an account of the theory of time series in general. Could you be more specific? – whuber Aug 04 '18 at 17:28
  • @whuber I have added a clarification stating the goal is to estimate the parameters given a sample path of the process. What the specific problem is will still depend on the assumptions on the processes and what the criterion for the estimation is. I leave this open because I don't know how this is commonly treated in the literature. In time-series analysis textbooks I find AR models where the regressors are the response variable at previous points in time, and MA where the regressors are the variables from a white noise process. I also found something similar to what I... – Winkelried Aug 04 '18 at 19:28
  • ...describe but with fixed regressors. And otherwise the case where the regressors are stochastic but $\{(Y_n,X_n)\}_{n\in\{0,...,T\}}$ are i.i.d. Have I made clear to you what I'm looking for? – Winkelried Aug 04 '18 at 19:32
  • It's clearer, thank you. I believe this is standard material for econometricians--it describes generally what they do--so you're likely to find a great deal of related material on this site searchable from the [tag:econometrics] tag. – whuber Aug 04 '18 at 20:15

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