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Let's say I have two OLS specifications with different IVs (R notation):

lm(outcome ~ IV_1 + I(IV_1^2)) 
lm(outcome ~ IV_2 + I(IV_2^2))

I theorize that the quadratic term on IV_1 will be 'steeper' than IV_2. Comparing the coefficients, I see that the magnitude of I(IV_1^2) > I(IV_2^2), but I want a formal statistical test to demonstrate that this is the case.

What is the right test for comparing the polynomial terms of IVs from two separate specifications?

Parseltongue
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  • https://stats.stackexchange.com/questions/93540/testing-equality-of-coefficients-from-two-different-regressions – user2974951 Jan 23 '19 at 12:15
  • @user2974951 That's a good find--but I believe it's not applicable, and the reason for that is interesting: because the outcome is the same set of observations in the present case, the coefficient estimates are likely to be strongly correlated, whereas all the answers in the linked thread assume the two sets of observations are independent. The basic problem here is that the two implicit regression models are different but not nested. In effect, the stated hypothesis is not a valid statistical hypothesis and therefore cannot be tested with a hypothesis test! – whuber Jan 23 '19 at 13:45
  • @whuber, do you have any suggestions as to how to make this a valid statistical test? – Parseltongue Jan 23 '19 at 20:06
  • I wish I did, but I'm not sure what your hypothesis would be. You can't formally test a pair of non-nested models for the same data, but you can conduct separate goodness of fit tests. In many circumstances people would approach your situation by comparing each of your models to a model of which they are both special cases: namely, a model that includes all four of IV_1, IV_2, IV_1^2, and IV_2^2. – whuber Jan 23 '19 at 20:09
  • Thanks so much for your help! In your stated example involving nesting, how would I get at the underlying question about which relationship is "steeper?"

    Also, what about the possibility of comparing standardized coefficients? (see here: https://www.researchgate.net/post/Can_we_compare_betas_of_two_different_regression_analyses)

     – Parseltongue Jan 23 '19 at 20:15

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