Quadratic Regression

[Danish Angela]

i am doing logit regression with workhours and workhours-square as independent variable and depression as dependent variable with several control variables such as age, educ (dummy), marital status, gender, tenure, working benefit, work stress and per capita expenditure as the proxy of household income. So i have one variable with quadratic form in this regression.

I am a bit confused because the peak point of workhours with or without control variable is different. The peak point of workhours is 35 hours when i regress without control variables. However, the peak point of workhours becomes 64 hours if i regress with control variables. Of course the peak point of 64 hours is not consistent with many literature i've ever read.

The command that i input on stata is:

logit depress workhours workhoursq chronic age elementary middle highschool highereduc single female stress tenure reward pce_pctilei2 pce_pctilei3 pce_pctilei4

is the regression model inconsistent or bias? what should i do to fix my model so the peak point of my regression becomes not really that different?

[Maarten Buis]

The first question that needs to be answered is: "why would you expect the shape to remain the same after adding control variables?" The answer is that it would be very unusual for the shape to remain the same. So forcing the peak to remain in one place seems like a very bad idea to me.

Another thing to consider is that the location of the peak can be extremely uncertain. I would not be surprised if the confidence interval would be something like -1000 to +1000, in which case: who cares? Finding the CI is a bit tricky, as we are talking about a ratio of coefficients. So a bootstrap would probably be the most feasible: http://www.maartenbuis.nl/workshops/...l#slide32.smcl

A third thing to consider is that using a quadratic form to find the location of a peak is very sensitive to that functional form. So this is not a very robust way of finding peaks.

[Nick Cox]

A quadratic may have a turning point at 64 but that turning point may be way outside the observed data. Consider e.g.

twoway function 20 * x/32 - 5 * (x/32) ^2, ra(0 108) xli(64)

and then imagine truncating that at say x = 45 or 50.