Omitted variable bias in linear regression

Question

I have a philosophical question regarding omitted variable bias.

We have the typical regression model (population model) $$ Y= \beta_0 + \beta_1X_1 + ... + \beta_nX_n + \upsilon, $$ where the samples are from $(Y,X_1,...,X_n)$, and then a bunch of conditions by which the OLS estimates behave quite well.

Then we know that, if we omit one of the main variables, $X_k$, this might bias the estimates of $\beta_0, \beta_1, ..., \beta_{k-1}, \beta_{k+1}, ..., \beta_n$. This would affect, at least, the estimated effect of the rest of the variables on $Y$, and also the hypothesis tests about $\beta_1, \beta_2, ...$, as the predicted values are not reliable.

The thing is, we don't know which variables are in the true population model. Instead, we have a bunch of candidates from which we should analyze and find out the most appropriate subset. This process of variable selection uses OLS estimates and hypothesis tests again. Based on that, we reject or include different variables. But since each candidate model is omitting relevant variables (you will never be able to find the true model), wouldn't these decisions be based on biased results? Why then, should we trust them?

(I'm thinking of forward stepwise method, for instance, where you pick one variable then add the rest. You compare the models doing inference, and I'm thinking that omitted variables may be disturbing everything.)

I was never too worried about this topic until I began thinking of it, and I'm sure I'm wrong somewhere.

Answer 1

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.