Insignificant Coefficient: Useless or Perfect variable?

There is one query question whose response I needed to confirm..

Consider if an independent variable is insignificant with a valid model. How do we confirm that it is one of two cases:

  1. The variable was not related to the dependent variable as it is showing marginal impact = 0
  2. The variable is optimally effecting dependent variable as it is showing first derivative = 0

Possible un-confirmed answer

This is what I had thought to confirm which option to be selected.

  1. See if the insignificant variable has no relation, then if we make a scatter graph with estimated dependent on Y axis and insignificant variable on X axis it will be horizontal line, means that even if we add the square of that variable, it will also be insignificant.
  2. But if is the case of optimum relation then we would experience significant squared term if it is added, so that the graph formed will be U-shaped or inverted U-shaped.

Note that this solution might only work if the model is linear not logarithmic because it will not show the quadratic effect in logarithmic so there will be no way to confirm using the solution above.

Your comments are welcome…

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