In most areas (social sciences, medicine) there is no "standard" for statistical significance. In economics, people try to deal with the issue by looking at the robustness of a result. If a result remains when you make various changes to the specification of your model, then it is less likely to be a statistical artifact.
Another step in the right direction is greater reproducibility, so that at least people can play with your analysis and see if what you did was the most direct and natural analysis, or if there are clear signs of playing with the parameters until you get the result you want.
Another step in the right direction is greater reproducibility, so that at least people can play with your analysis and see if what you did was the most direct and natural analysis, or if there are clear signs of playing with the parameters until you get the result you want.