Basic Fundamentals question on regression analysis based on economic theory:
You never consider either $r^2$ or significance in choosing a model.  $r^2$ is non-decreasing in the number of parameters.  You can improve $r^2$ quite often by adding a set of unrelated variables, such as the rainfall in Quebec to estimates of the price of diamonds in Singapore.  Significance assumes you believe the null is true.  If you have multiple models, then you obviously do not believe one to be true.

When using Frequentist methods, model selection should be done with something similar to the Akaike Information Criterion or the Bayesian Information Criterion.  Both are summary point estimates of a stylized Bayesian posterior density over the model space.  Only use one, don't use both.  There are also a few other similar methods.  The information criterion are essentially Bayesian solutions with slightly different prior mass functions.

You should write software to go through each combination of variable, excluding variables that are collinear, and run either the AIC or the BIC.  The AIC and the BIC measure how close a model is to the data generating function.  This is not like significance an doesn't imply significance.