Can we remove significant variables in a regression? I first ran a regression with all the variables and then ran the regression again with only significant variables (or the variables of interest).
One of the variables in the 2nd regression is highly correlated with another variable of interest, though both of them are significant but are highly correlated. And this is causing the sign of the coefficient of other variables to go opposite way (not as per expectation or theory).
My question is that can we still remove one of those (correlated) significant variables? It makes the whole result come as per the expectations.
Thanks

Please do not choose variables for a regression model based on p-values.
Also please do not choose variables so that you achieve results that "meet expectations".
If you include variables in your model to begin with, this was presumably because they were identified as possible confounders or as variables that were not associated with your main exposure but associated with the outcome (competing exposures). These are GOOD reasons, and p-values are not part of the decision making.
Of course it is important not to over-adjust for possible confounding, and to avoid things like conditioning on a collider. The best way to do this in my opinion is by considering the possible causal relations between the variables using a causal diagram or DAG, which will then inform you of the minimally sufficient set of variables to condition on. A good free online too for doing this is http://www.dagitty.net/
Also, this answer may help you understand the pitfalls of not choosing variables in a principled manner.
How do DAGs help to reduce bias in causal inference?
