Regression assumption violation: lagged dependent variable as regressor I am studying regression and a bit lost with conceptually understanding why having an independent variable correlated with the error term is a regression assumption violation.
Just to give more context... I am looking into model specifications errors related to Time Series, where a lagged value of the dependent variable is included as an independent variable. And it's said that if we include such lagged dependent variable as independent, then it will be correlated with the error term, thus violating a regression assumption.

In any statistical analysis, such as regressions(inference), there must be randomness entropy that is considered to be not measurable. If not, then what is the point to study already known patterns. The idea is we can discover one.
That being said, we can move towards the Gauss-Markov theorem, which states that if your linear regression model satisfies the first six classical assumptions, then ordinary least squares (OLS) regression produces unbiased estimates that have the smallest variance of all possible linear estimators.
Therefore, the regression model will try to avoid bias on all it's estimators.
Moreover, if you take a look at Jim Frost approach https://statisticsbyjim.com/regression/ols-linear-regression-assumptions/
You will see that a correlation (not causation) between an independent term and the error term, breaks the notion of "unpredictable random error" which can also be called as "random error"(alone because of it randomness...) or "entropic random error".
However, in practice, I can assure you, this happens ... If you run a predictive model on real data or a monte carlo simulation over fake data, you will find correlations between estimates and error all the time.
