# Why does adding a quadratic term to a regression change unrelated coefficients?

I'm in STATA and using 2010 data from Ipums. I'm trying to measure the wage differential between single men, married men, single women, and married women. I ran my first regression and got the following results:

    lwage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
educ |   .1297281   .0003132   414.22   0.000     .1291142    .1303419
age |   .0130395   .0000535   243.56   0.000     .0129346    .0131444
uhrswork |   .0454742   .0000613   741.81   0.000      .045354    .0455943
singlefemale |  -.0749253   .0021686   -34.55   0.000    -.0791756    -.070675
marriedfemale |   .0853371   .0021692    39.34   0.000     .0810856    .0895886
marriedmale |   .3149997   .0021153   148.92   0.000     .3108539    .3191455
_cons |   6.826747    .003847  1774.56   0.000     6.819207    6.834287


Next I added age^2 as an additional explanatory variable. My results changed dramatically:

   lwage |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
educ |   .1258587   .0003049   412.83   0.000     .1252612    .1264563
age |   .0961006   .0003096   310.38   0.000     .0954938    .0967075
agesq |  -.0009427   3.46e-06  -272.14   0.000    -.0009495   -.0009359
uhrswork |   .0406035   .0000622   652.40   0.000     .0404815    .0407255
singlefemale |  -.0865127   .0021091   -41.02   0.000    -.0906465    -.082379
marriedfemale |   -.035098   .0021552   -16.29   0.000    -.0393221    -.030874
marriedmale |   .2403908    .002075   115.85   0.000     .2363239    .2444578
_cons |   5.455941   .0062742   869.58   0.000     5.443643    5.468238


So basically, when I assume age is linearly related to logwage, married women are estimated to earn more than single men, but when I assume a quadratic form I get the reverse. Both are statistically significant. Why is this happening? And how do I choose the better model?

Also, is this common in other applications? I'm surprised that I can change the sign on something just by adding an unrelated quadratic term -- this seems like a source of potential abuse.

## 3 Answers

Your errors aren't the same anymore. For example, instead of writing $Y = \beta_1 + \beta_2 X + U$, you're actually writing $Y = \alpha_1 + \alpha_2 X + \alpha_3 X^2 + V$. There is no expectation that they should be the same.

In other areas, the trouble is that the error term is likely correlated with your regressors.

Fear not: running wages on education is a common specification. Searching for something like

wages education endogeneity

should bring up a whole slew of papers that treat the topic.

This is a well known phenomenon called collinearity. Basically your two independent variables (age and age-squared) are correlated strongly. In the presence of collinearity the coefficient estimates can change significantly. To overcome this problem you may use robust regression, such as ridge regression.

@OccupyGezi 's suggestion is good -you should check for severe collinearity, that may render the estimations unstable and unreliable for purely technical reasons.

As for choosing between the models, there are technical, purely statistical criteria, but there is also the economic essence of the matter, which should not be forgotten: by including the variable "age-squared" you postulate an "inverted U" relationship between the logarithm of wage and age, i.e. that the logarithm of wage peaks at some age, and then starts to fall.

The fact that the coefficient on age-squared is negative while the coefficient on age is positive, provides evidence that support such a relation. But since it may be the case that the coefficient estimates are affected by multicollineaity, it would be good if could you back-up the claim of an inverted-U relationship with economic arguments or out-of-sample information.