# the conditional expectation of the error term in linear regressions (OLS assumption)

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?

• 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 $Y_i=\beta_0+\beta_1X_i+u_i$? – Henry Jun 10 at 13:48
• Oh, let me rewrite my question in a more general way. – WinnieXi Jun 10 at 14:11
• 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". – stollenm Jun 11 at 9:58