You should get the "same" estimates no matter which variable you omit; the coefficients may be different, but the estimates of particular quantities or expectations should be the same across all the models.
In a simple case, let $x_i=1$ for men and 0 for women. Then, we have the model:
$$\begin{align*}
E[y_i \mid x_i] &= x_iE[y_i \mid x_i = 1] + (1 - x_i)E[y_i \mid x_i = 0] \\
&= E[y_i \mid x_i=0] + \left[E[y_i \mid x_i= 1] - E[y_i \mid x_i=0]\right]x_i \\
&= \beta_0 + \beta_1 x_i.
\end{align*}$$
Now, let $z_i=1$ for women. Then
$$\begin{align*}
E[y_i \mid z_i] &= z_iE[y_i \mid z_i = 1] + (1 - z_i)E[y_i \mid z_i = 0] \\
&= E[y_i \mid z_i=0] + \left[E[y_i \mid z_i= 1] - E[y_i \mid z_i=0]\right]z_i \\
&= \gamma_0 + \gamma_1 z_i .
\end{align*}$$
The expected value of $y$ for women is $\beta_0$ and also $\gamma_0 + \gamma_1$. For men, it is $\beta_0 + \beta_1$ and $\gamma_0$.
These results show how the coefficients from the two models are related. For example, $\beta_1 = -\gamma_1$. A similar exercise using your data should show that the "different" coefficients that you get are just sums and differences of one another.
The main issue here is the nature of the omitted variable bias. Wikipedia states:
Two conditions must hold true for omitted-variable bias to exist in
linear regression:
- the omitted variable must be a determinant of the dependent variable (i.e., its true regression coefficient is not zero); and
- the omitted variable must be correlated with one or more of the included independent variables (i.e. cov(z,x) is not equal to zero).
It's important to carefully note the second criterion. Your betas will only be biased under certain circumstances. Specifically, if there are two variables that contribute to the response that are correlated with each other, but you only include one of them, then (in essence) the effects of both will be attributed to the included variable, causing bias in the estimation of that parameter. So perhaps only some of your betas are biased, not necessarily all of them.
Another disturbing possibility is that if your sample is not representative of the population (which it rarely really is), and you omit a relevant variable, even if it's uncorrelated with the other variables, this could cause a vertical shift which biases your estimate of the intercept. For example, imagine a variable, $Z$, increases the level of the response, and that your sample is drawn from the upper half of the $Z$ distribution, but $Z$ is not included in your model. Then, your estimate of the population mean response (and the intercept) will be biased high despite the fact that $Z$ is uncorrelated with the other variables. Additionally, there is the possibility that there is an interaction between $Z$ and variables in your model. This can also cause bias without $Z$ being correlated with your variables (I discuss this idea in my answer here.)
Now, given that in its equilibrium state, everything is ultimately correlated with everything in the world, we might find this all very troubling. Indeed, when doing observational research, it is best to always assume that every variable is endogenous.
There are, however, limits to this (c.f., Cornfield's Inequality). First, conducting true experiments breaks the correlation between a focal variable (the treatment) and any otherwise relevant, but unobserved, explanatory variables. There are some statistical techniques that can be used with observational data to account for such unobserved confounds (prototypically: instrumental variables regression, but also others).
Setting these possibilities aside (they probably do represent a minority of modeling approaches), what is the long-run prospect for science? This depends on the magnitude of the bias, and the volume of exploratory research that gets done. Even if the numbers are somewhat off, they may often be in the neighborhood, and sufficiently close that relationships can be discovered. Then, in the long run, researchers can become clearer on which variables are relevant. Indeed, modelers sometimes explicitly trade off increased bias for decreased variance in the sampling distributions of their parameters (c.f., my answer here). In the short run, it's worth always remembering the famous quote from Box:
All models are wrong, but some are useful.
There is also a potentially deeper philosophical question here: What does it mean that the estimate is being biased? What is supposed to be the 'correct' answer? If you gather some observational data about the association between two variables (call them $X$ & $Y$), what you are getting is ultimately the marginal correlation between those two variables. This is only the 'wrong' number if you think you are doing something else, and getting the direct association instead. Likewise, in a study to develop a predictive model, what you care about is whether, in the future, you will be able to accurately guess the value of an unknown $Y$ from a known $X$. If you can, it doesn't matter if that's (in part) because $X$ is correlated with $Z$ which is contributing to the resulting value of $Y$. You wanted to be able to predict $Y$, and you can.
Best Answer
How to use / interpret the coefficients from a regression model with categorical variables to get predicted variables depends on how your variables are coded. There are many different coding schemes (see here for a good overview). It sounds like you used 'reference cell coding', which most people call 'dummy coding'. I gather your race1 category is the reference category. In this case, the intercept is the mean of the race1 group. To compute the predicted value, you would solve the equation using whatever values for other variables apply and omitting the coefficients for the other categories (i.e., race2 & race3). There is some good, relevant info here, and here.
edit: The way the question is phrased made me think about situations in which there is only one factor in the model, however, @Michelle raises the question of the more general case. To keep this relatively simple, imagine a case with just two factors, e.g. race and sex, plus some continuous covariates. Using reference cell coding, we will create a dummy for male. Now, solving the regression equation without including any of the factor coefficients (i.e., just the intercept + continuous covariates) yields the predicted mean of the reference cell, which in this case is the race1 female group. Should you want to know the value for race1 males, you would solve as above, but also include the coefficient for male. If you wanted to ignore sex, or make a prediction for a mixed-sex group, you would calculate a weighted average of the above two predictions. Obviously, this will get more complicated as the number of factors, $J$, increases, but the pattern should be clear enough.