# How can I regress household income on three factors

I have to regress family income (faminc; in dollars) onto husband's educational attainment (he; in years), wife's educational attainment (we; in years), and number of children less than 6 years old in household (kl6) using Stata.

(the file only contains data of 4 above factors)

I use OLS to estimate a model in the form: $$faminc = b_1 + b_2 * he + b_3 * we + b_4 * kl6 + \epsilon$$

      Source |       SS       df       MS              Number of obs =     430
-------------+------------------------------           F(  3,   426) =   28.77
Model |  1.4002e+11     3  4.6673e+10           Prob > F      =  0.0000
Residual |  6.9100e+11   426  1.6221e+09           R-squared     =  0.1685
Total |  8.3102e+11   429  1.9371e+09           Root MSE      =   40275

------------------------------------------------------------------------------
faminc |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
he |   3185.882   795.4493     4.01   0.000     1622.388    4749.376
we |   4637.415   1059.177     4.38   0.000     2555.551    6719.279
kl6 |  -8372.704   4343.059    -1.93   0.055     -16909.2    163.7893
_cons |  -5998.224   11161.51    -0.54   0.591    -27936.72    15940.27


I have some questions:

1) The regression yields $b4<0$. Is this true in fact? I mean that if the family has more children, the less income they gain?

2) Is this model good enough? Should I use natural logarithm or add dummy to make it better?

• The following is the data file: mediafire.com/download/6yug9vqu00m8v32/family_income.dta – user36706 Dec 30 '13 at 16:11
• 1) The p-value for the difference of b4 from 0 is greater than 0.05, so you don't have evidence for a negative coefficient. 2) Answering that requires you to specify the purpose of you analysis. – EdM Dec 30 '13 at 16:13
• Thank you. How can I keep kl6 in this regression? – user36706 Dec 30 '13 at 16:15
• Much depends on what you are trying to accomplish with this analysis: are you trying to make predictions about incomes of other families based on educational attainment and family size, or trying to make inferences about factors that are "important" in determining family income? Some cautions in either case: this doesn't seem to be a very large sample, there are many other variables that are missing (like adult ages, geographic locations), and this particular data set might not be representative. And for inferences, I'm sure you know that correlation doesn't imply causation. – EdM Dec 30 '13 at 16:22
• I have to work on this file and just want to analyze the influences of only 3 above factors on family income, so I try to find a model (not contradicting the fact), in which these 3 factors can explain "family income" as good as possible (although the sample is not large enough, I agree). – user36706 Dec 30 '13 at 16:35

The fact that the p-value is 5,5% only means that the coefficient of kl6 is not statistically significant at 5% level -but it is significant at 6% level, and more so at 10% level. The "5% rule" has no scientific justification whatsoever - it has historical justification and perhaps social justification, but that's another matter and a very large discussion.

Interpretation-wise, the negative coefficient gives us the marginal effect of the number of little children on household income after the educational effect has been controlled for (by the existence of the other two regressors). So what does it say? That more little children tend to reduce household income. This may appear counter-intuitive because one could think "more children provide stronger incentives to earn more income in order to provide for the larger family". Yes, but more children also mean greater demands on the parents time that must be devoted to the children, and so less time available to work and earn income. I would suggest to try a regression where you include in addition the kl6 squared. If this squared regressor obtains a negative coefficient and the plain kl6 obtains a positive coefficient, then you are possibly looking at a non-monotonic relation (i.e. that there is an income-maximizing number of little children below or above which income tends to be lower).

PS: "How can I keep a regressor in a regression?" is the mother-question that leads to data-tampering in those ingenious ways only statistics can offer. I would suggest not to ask yourself again such a question. The regression results are what they are. Statistics should not be the brush with which we paint the world in the colors we want.

• Thanks. You made me realize something. However, I think that a regressor may be not statistically significant in a certain model, but may be so in another by changing the equation. So am I wrong? – user36706 Dec 30 '13 at 17:25
• Indeed such a thing can happen - for example, in the one specification, there may be another regressor which is a common source of influence on the dependent variable and on the statistically insignificant regressor. If in an alternative specification this common source is removed, the other regressor may appear statistically significant. But in such a case you have an "omitted variables" problem -there are variables subsumed in the error term that are correlated with the regressors. – Alecos Papadopoulos Dec 30 '13 at 17:34
1. It's not true or false. It's just saying that more children is correlated with households having lower income. Check the correlation between the two variables. There's certainly no causation here.

2. We have no idea without seeing some residual plots. Are the residuals normally distributed? Post them and I'll update my answer. I would check out some interaction terms and squared terms as well - interaction between education levels and number of children. Overall model fit is weak (low $R^2$) but that's not always the main metric of interest.

Also keep in mind that the difference between 1 and 2 years education may not be the same as 12 and 13 years. Without a log or squared term, OLS is going to force the effect to be constant. A binary indicator for college education or not would be great - could you create one out of your years of education variable?

• Shapiro-Wilk W test for normal data : p-value=0.0000000 < 0.05. So is that mean the residuals are not normally distributed? – user36706 Dec 30 '13 at 17:20
• That is correct. Look at a histogram of the residuals, that's usually more useful. Try logging your Y variable and see if that changes anything. – wcampbell Dec 30 '13 at 18:58