# Baseline adjustment in a change model given bidirectional causation

I come from a non-statistical background and am trying to wrap my head around whether baseline adjustment is necessary in a change model when analysed using OLS regression. I am considering different OLS models used in analyses of longitudinal epidemiology papers examining the associations between exercise as a predictor, and either follow-up weight or weight change as outcomes. Four models are commonly used (apologies that I've written them in simple terms):

1) Follow-up weight = baseline exercise + baseline weight + covariates

2) Follow-up - baseline weight = baseline exercise + covariates

3) Follow-up weight = change exercise + baseline weight + covariates

4) Follow-up weight - baseline weight = change exercise + covariates

When comparing models 1 and 2, and comparing models 3 and 4, I am aware that the coefficients for exercise will differ due to Lord's paradox, and that depending on the situation, each result is valid.

However, given that baseline weight is the strongest predictor of future weight, and that there is bi-directional causation between exercise and weight, I am wondering whether both models 2) and 4) shouldn't also be adjusted for baseline weight? Without such an adjustment, the exercise coefficient will only reflect the influence of exercise on weight change, not accounting for the influence of baseline weight on baseline exercise.

Equally, shouldn’t model 4 also be adjusted for baseline weight given that baseline weight status will likely influence whether somebody changes their exercise level over time?

Thanks in advance for helping me understand this.