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well. First of all its depend the intuition behind comparing X with X^2. Suppose if you want to check the nonlinear relationship between two variables, not only the value of x^2 will b greater but will the sign will be change. Check page # 724(15) for further understanding. Thanx

i am not sure what you want.. May b this will help you.. https://www.statalist.org/forums/forum/general-stata-discussion/general/1292877-calculate-market-betas-for-stocks

well its quite later however may be helpful for other. suppose you have two variables say Y and X. Y is the dependent variables and X is the explanatory variable. The relationship between Y and X either be negative or positive.. if Y increase X decrease ( Negative) or when Y increase X also...