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I am trying to estimate the impact on a politician's re-election of bills dealing with immigration. Suppose that the number of bills on immigration represents a high share of total bills and the two counts are correlated (correlation is around 0.7). If I run a logit model like this

$$ Reelection = \beta_{1} immigration $$

$\beta_{1}$ is significant. If I run a model including both the counts (immigration and total)

$$ Reelection = \beta_{1} immigration + \beta_{2} total $$

$\beta_{1}$ becomes insignificant and $\beta_{2}$ is significant. Is this evidence that the effect is driven by the total number of bills rather than immigration bills?

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I would advise building a model with just the total and observing the significance of that model. Your situation often happens when one predictor explains your response similar to the other predictor's ability to explain the response.

Variance Inflation Factors are generally a better metric than correlations, especially if you plan on adding more predictors (any VIF over 10 is considered dangerous). If significance tests are not helping, you could compare models and attempt to minimize AIC/BIC. You can then check if adding an interaction term (even if a main effect is left out) improves the fit.

After this, you will have some more clarity on what is driving reelection in your data.

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