(Edit: @Peter Flom is right that Vi and Vi2 cannot be assessed in a single model if one is simply a product of the other. But more generally,...) If Vi and Vi2 are both indicators of the incidence of violent crime, to check the relationship of Vi with the dependent variable while controlling for Vi2 will give an erroneous estimate of the former. You will in effect be partialing a portion of the first relationship right out of itself. Statistical control cannot be applied indiscriminately; one needs to assess a relationship X.Y while controlling for some distinct other relationship Z.Y. (Many of us have learned a lot from our own mistakes in this vein; textbooks and courses tend to emphasize the how of statistical control rather than the why or the when.)
Further, to exclude a variable from your model because its correlation with Y (or its partial correlation) falls below 0.4 seems to be a rule of thumb that will not serve you well in the long run. In many cases one wants to study relationships even when they are weaker than that; in selected cases one's cutoff point may need to be stronger. Criteria for inclusion should be dictated by subject-matter knowledge and perhaps also by statistical power. A few people will even recommend that no variable, once tested, should be removed, though that is a controversial position.
Automated variable selection methods are almost sure to cause harm to your research, unless they are applied in a very savvy way, by a researcher fully aware of their drawbacks.
