As usual Richard Williams words are wise.
To my students I tell them, be careful about doing individual tests of significance on the different coefficients of a polynomial form. The reason is that you're interested in the partial effect of the variable being considered, not the significance of each coefficient. The appropriate test is an F test of joint significance for the actual parameters. But if you're looking into the actual partial effect, you will have to look at it at many different values of the independent variable. It may be insignificantly different from zero for some and significantly different from zero for others.
If you want to analyze the significance of the individual terms of a quadratic function, you would have to compare the partial effect of the quadratic form at many different values of the independent variable to that of a linear function (to see the significance of the squared term), or those of the model where you just include the squared term (to analyze the significance of the coefficient on the level term of the quadratic function). I also tell them that decisions of entering a quadratic functional form should be made on the basis of data exploration if there is evidence of a nonlinear relationship between the dependent and independent variable of interest.
To my students I tell them, be careful about doing individual tests of significance on the different coefficients of a polynomial form. The reason is that you're interested in the partial effect of the variable being considered, not the significance of each coefficient. The appropriate test is an F test of joint significance for the actual parameters. But if you're looking into the actual partial effect, you will have to look at it at many different values of the independent variable. It may be insignificantly different from zero for some and significantly different from zero for others.
If you want to analyze the significance of the individual terms of a quadratic function, you would have to compare the partial effect of the quadratic form at many different values of the independent variable to that of a linear function (to see the significance of the squared term), or those of the model where you just include the squared term (to analyze the significance of the coefficient on the level term of the quadratic function). I also tell them that decisions of entering a quadratic functional form should be made on the basis of data exploration if there is evidence of a nonlinear relationship between the dependent and independent variable of interest.
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