# Unstable coefficient in regression without high correlation between variables

I am estimating a linear regression: $Y=f(X_1,X_2,X_3,X_4,X_5)$. My test shows that when the equation includes $X_4$ and $X_5$ only, $X_4$ is not statistically significant ($t$-value=1.26). However, When $X_1$,$X_2$,and $X_3$ are added, its significance is increased to significant level ($t$-value =2.36). In another words, the significance of $X_4$ depends on the presence of $X_1$, $X_2$ and $X_3$ in the model.

I tested the correlation between $X_4$ and $X_1$, $X_2$, and $X_3$. Bivariate correlations between them are all below 0.5. If I understand it correctly, multicollinearity should not be a concern at such correlation level.

My question is: Should I drop X4 from my final model? If no, how should I explain its significance?

• Multicollinearity is not necessarily a function of a single bivariate correlation. In addition to @Greg's answer, you might look at other posts here tagged multicollinearity, including this one. Sep 26, 2013 at 19:38
• Can you clarify what you mean by "unstable coefficient" in the title? The fact that the significance level (ie p-value) changes need not have anything to do with multicollinearity, & I wouldn't use the term "unstable" myself. Is the coefficient estimate similar (just more significant), or is it wildly different? Sep 26, 2013 at 20:11