Why including variable square in the regression change the sign of main variable

[Maryam Bidgoli]

Dear Stata users,

I am estimating a simultaneous equation model of two equations using _cmp_ command.

cmp (rinlbrf = rshlt_#  age rabplace  i.seperated##i.kid18_  i.married##i.kid18_ lesshighschool college collegemore whitecollar2_ bluecollar wealth midwest northeast west hispanic black rsmoke rdrink|| hhidpn:) (rshlt_ =rinlbrf#  age rabplace married seperated  hispanic black midwest northeast west lesshighschool college collegemore whitecollar2_ bluecollar wealth rsmoke rdrink rconde || hhidpn:) if (sex==1 & age<=65) ,ind($cmp_probit $cmp_oprobit) nolr

where
rshlt is health categorical variable, and rinlbrf stands for participation in the labor market

The result shows that the effect of age on both health and labor participation is negative as expected. However, when "age2" variable [age2=age*age] is included in the regression, the age effect is positive which doesn't make sense. why?

Thanks,
Maryam

[William Lisowski]

You have a misunderstanding.

It does not make sense to think about the effect of changing age alone: you cannot increase age without increasing age². I would guess that if you calculate the predictions for health and labor participation for two different values of age (with age2 set to age²) in the range of your data, holding all the variables other that age and age2 constant, you will see a negative effect on both for the larger value of age.

Without being familiar with the cmp command, I will say that in general it is not a good idea to create a separate age2 variable as you suggest you have done. Instead, replace each occurence of

age age2

in your cmp command with

c.age##c.age

See help factor variables for an explanation of that syntax. The cmp command apparently supports the margins postestimation command (with the reduced form estimates), which can help you better explore the marginal effects of your independent variables, but only if the model is constructed in a way that doesn't hide the dependency of age² on age from the model by using a second variable.