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I am running an OLS model where my dependent variable Y is continous and among the explanatory vars I have a count variable X. I want to test if the effect of X on Y changes sing. To do so I would include X squared:

Y_i = b0 + b1X_i + b2X^2_i + e_i

I get b1 > 0 and b2 <0, signaling a possible parabolic shape of my relation. Of course I have to check if the quadratic relation holds: calculation of the threshold value and its confidence intervals by means of Delta Method.

Besides the common tests for a quadratic regression, do you see any problem in regressing using the square of a count variable?

Indeed, because of its nature, my count variable does not really fit into semi-parametric models. The graphical output would be impossible to be interpreted due to natural jumps in the variable.

Moreover, my suspect is that the jumps may affect also my parametric estimates and what is detected as a parabolic shape by the coeff b1 and b2 may be due to a sudden jump in the variable for which the slope change.

Any suggestion on how to handle quadratic regressions with explanatory count variables?

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