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I understand that the most precise usage of the term independent variable applies it to situations in which the variable is manipulated by the researcher, although the term is used more loosely in other contexts as well.

On the other hand, if a variable is exogenous that means it is outside the system we are modelling. It might cause some other variable in our model, but it is not caused by any variables within our model.

I get that "exogenous variable" implies the modelling of some causal system, but isn't that also true of "independent variable" as well, if we limit it to the experimental scenario I mentioned in the first paragraph?

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An independent variable is defined within the context of a dependent variable. In the context of a model the independent variables are input whereas the dependent variables are the targets (Input vs Output).

An exogenous variable is a variable whose state is independent of the state of other variables in a system.

To my understanding independent variables thus do not need to be exogenous, yet exogenous variables are naturally independent.

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