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I have some questions regarding the least square assumptions for causal and prediction models.

I know that in linear regressions, for the coefficient on the regressors to have causal interpretation, the conditional expectation of the error term $u_i$ on $X_i$'s has to be zero. And I always thought that if we are conducting a regression for prediction instead of causal interpretation, then the error term does not have to be zero.
However, I was studying Autoregression today and the textbook writes "The assumption that the conditional expectation of $u_t$ is zero given past values of $Y_t$––that is, $E(u_t|Y_{t-1},Y_{t-1}...)=0$––has two important implications", and I was confused.
Is this because the way the two regression models are defined are different?

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  • $\begingroup$ If your autoregressive model is $Y_t=\beta_0+\beta_1Y_{t-1}+\beta_2Y_{t-2}+\cdots+u_t$ then is it very different from your linear model $Y_i=\beta_0+\beta_1X_i+u_i$? $\endgroup$
    – Henry
    Commented Jun 10, 2021 at 13:48
  • $\begingroup$ Oh, let me rewrite my question in a more general way. $\endgroup$
    – WinnieXi
    Commented Jun 10, 2021 at 14:11
  • $\begingroup$ Having conditional expectation of the error term equal to zero is not sufficient for a causal interpretation. Using the linear model you should almost always use phrases like "A w-percent increase in x is associated (not causes!) with a y-percent increase in z". $\endgroup$
    – stollenm
    Commented Jun 11, 2021 at 9:58

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