Update: @FrankHarrell advises this situation requires multiple imputation. See hbiostat.org/rmsc/missing for how to do multiple imputation. The rest is my original answer.
Before getting to the question of missing covariate values, here are a few points to consider. Some of them might not be relevant in your area of study.
- You have four regression models: a regression for graduate hourly earnings and for non-graduate hourly earnings, with and without covariates. Why do you model graduates and non-graduates separately? One model for hourly earnings doesn't necessarily mean that graduates and non-graduates share effects. (For example, we can add a
graduate_status term to give each group its own intercept.) However, with two different models the two groups definitely share no effects.
- You have multiple observations for most individuals, so how does your model handle that? It's incorrect to assume that observations for the same individual are independent. For example, it's reasonable to expect that the earnings of the same person in 2010 and 2011 are more similar than the earnings of two different persons in 2010. This can be represented with individual random effects.
- You model the time effect with one dummy variable per year even though clearly the time effect increases with time (at least in the regressions without covariates). Have you considered modeling the functional relationship between time and hourly earnings, eg. as a line? See plot below.
- Some of the covariates likely vary with time, eg., number of family members and number of children in the household. For many people these numbers would have changed over the course of 10 years. Have you thought about when these numbers are recorded and how you interpret their effects?
- Time (as either categorical or continuous variable) might be confounding two effects: with each year, people gain more experience and they start getting paid more. Concurrently, average salaries might be increasing with inflation and cost of living. Maybe you already take this into account with Experience?
As you note yourself, you don't have data on at least one covariate for 83% of graduates (observations drop from 7476 to 1269) and 82% of non-graduates (10940 → 1948). There are methods to deal with missing values; otherwise the regression is fitted on the subset of observations without missing values.
Yes, this is allowed but as it's often the case in statistics, it means making (strong) assumptions. In this case the required assumption is that the patterns of missingness are not associated with the outcome, hourly earnings: either values are missing completely at random (think about flipping a coin that determine whether a value is observed or not) or values are missing at random (the other observed variables explain the missingness).
So broadly you have (at least) three options:
- Proceed with regression without covariates. However, you cannot unsee that covariates seem to be associated with the outcome. So it might be hard to justify the conclusions you draw from this analysis. [Regression analysis assumes validity, ie. the model includes all relevant predictors and the sample is representative of the population of interest.]
- Make strong assumptions about missingness and proceed with regression with covariates. You can also use your domain knowledge to decide that it's reasonable to ignore some covariates altogether.
- Impute missing values.
Finally, I'd like to show what I mean by functional relationship between time and outcome. Since I don't have the raw data, I plot estimated yearly effects from the regressions without covariates. You can see that the relationship seems to be linear with a different intercept. A straight line is the simplest function; it's possible to allow for a smooth relationship as well.
