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&gt;|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&gt;|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.
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.