Usually, you would not care about both of them simultaneously. Depending on the goal of your analysis (say, description vs. prediction vs. causal inference), you would only care about at most one of them.
- For description, multicollinearity is just a fact to be mentioned, just one of the characteristics of the data.
- For prediction, multicollinearity and omitted variable bias are largely irrelevant as you are not interested in model's coefficients per se, only in predictions.
Description$\color{red}{^*}$
Multicollinearity (MC) is just a fact to be mentioned, just one of the characteristics of the data to report.
The notion of omitted variable bias (OVB) does not apply to descriptive modelling. (See the definition of OVB in the Wikipedia quote provided below.) In contrast to causal modelling, the causal notion of relevance of variables does not apply. You can freely choose the variables you are interested in describing probabilistically, e.g. in the form of a regression. You evaluate your model w.r.t the chosen set of variables, not variables not chosen.
Prediction
MC and OVB are largely irrelevant as you are not interested in model coefficients per se, only in predictions.
Causal modelling / causal inference
You may care about both of themMC and OVB at once when attempting to do causal inference. I will argue that you should actually worry about the omitted variable biasOVB but not multicollinearityMC. Omitted variable biasOVB results from a faulty model (a cause in contrast to, not from the characteristics of the underlying phenomenon). You can remedy it by changing the model. Meanwhile, impecrfect multicollinearityimperfect MC can very well arise in a well specified model as a characteristic of the underlying phenomenon. Given the well specified model and the data that you have, there is no sound escape from multicollinearity MC. In that sense you should just acknowledge it and the resulting uncertainty in your parameter estimates and inference.
$\color{red}{^*}$I am not 100% sure about the definition of description / descriptive modelling. In this answer, I take description to constitute probabilistic modelling of data, e.g. joint, conditional and marginal distributions and their specific features. In contrast to causal modelling, description focuses on probabilistic but not causal relationships between variables.
In defence of my statement that omitted variable bias (OVB)OVB is largely irrelevant for prediction, let us first see what OVB is. According to Wikipedia,
- This is tangential to my points because I did not discuss the effect of omitting a variable on prediction. I only discussed the relevance of omitted variable bias for prediction. The two are not the same.
- I agree that omitting a variable does affect prediction under imperfect multicollinearityMC. While this would not be called OVB (see the Wikipedia quote above for what OVB typically means), this is a real issue. The question is, how important is that under multicollinearityMC? I will argue, not so much.
- Under multicollinearityMC, the information set of all the regressors vs. the reduced set without one regressor are close. As a consequence, the loss of predictive accuracy from omitting a regressor is small, and the loss shrinks with the degree of multicollinearityMC. This should come as no surprise. We are routinely omitting regressors in predictive models so as to exploit the bias-variance trade-off.
- Also, the linear prediction is unbiased w.r.t. the reduced information set, and as I mentioned above, that information set is close to the full information set under multicollinearityMC. The coefficient estimators are also predictively consistent; see "T-consistency vs P-consistency" for a related point.
