The two concepts are related but not quite the same. This note demonstrates the difference and discusses when omitted variable bias is a problem and when it is not. For a short answer, see this StackExchange post.
References: - Pearl (2009): Causality - Models, Reasoning, and Inference. - Hayashi (2000): Econometrics.
import statsmodels.api as sm
import statsmodels.formula.api as sm_form
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
from statsmodels.stats.moment_helpers import corr2cov
np.random.seed(5000)
import rpy2
%load_ext rpy2.ipython%%R
library(dagitty)Consider a simple OLS regression. The coefficient estimate for variable \(S\), \(\beta\), can be asymptotically biased if a relevant variable—i.e. a variable that is a) determinant of the dependent variable \(W\), and b) correlated with \(S\)—is omitted from the regression equation. Then we say that estimate \(\hat{\beta}\) suffers from omitted variable bias (OVB). However, this is not always problematic! It all depends on what effect one is interested in.
If one is interested in estimating the direct causal effect of \(S\) on outcome variable \(Y\)—i.e., the causal effect \(S\) has on \(Y\) excluding all causal effects mediated via other variables of the system—then OVB will always pose a problem. To see this, consider a simple regression problem inspired by Hayashi (2000, p. 237; problem slightly modified from the original), where an attempt is made to estimate the effects of schooling and ability on wage:
\[W = \alpha + \beta S + \gamma A \ \ \text{,}\]
where \(W\) denotes wage, \(S\) denotes years of schooling, and \(A\) denotes the ability of the individual. For this simplified demonstration, it is assumed that in the underlying data-generating process (DGP) schooling and ability are indeed the only determinants of wage. Further, we first assume tha,t whereas schooling affects wage directly, ability affects both schooling and wages. We can represent the causal effects by a following directed acyclic graph (DAG), where we draw the three variables (\(S\), \(A\), \(W\)), causal effects between them (arrows) and the magnitude of each effect (denoted by \(E_i\)); for example, the (direct) causal effect of \(A\) on \(S\) is \(E_1\).
%%R -h 300 -w 600
g = dagitty( 'dag {
"S" [pos="0,2"]
"A" [pos="2,0"]
"W" [pos="4,2"]
"E_1" [pos="1,1"]
"E_2" [pos="3,1"]
"E_3" [pos="2,2"]
"S" <- "A" -> "W"
"S" -> "W"
}')
plot(g)
In this case, regression
\[W = \alpha + \beta S + \gamma A\]
would result in estimate \(\hat{\beta}\) being an asymptotically unbiased estimate for effect \(E_3\) and estimate \(\hat{\gamma}\) being an asymptotically unbiased estimate for effect \(E_2\). That is, \(\text{plim } \hat{\beta} = E_3\) and \(\text{plim } \hat{\gamma} = E_2\).
If \(A\) was omitted from the regression, i.e., if we ran regression
\[W = \alpha + \beta S \ \ \text{,}\]
then the estimate \(\hat{\beta}\) would be a biased estimate of the effect \(E_1\). Indeed, we’d have
\[\text{plim } \hat{\beta} = E_3 + E_1 * E_2 \neq E_3\]
due to the direct effect of schooling (\(E_1\)) as well as the “indirect” or “back-door” effect of ability on wages (as \(A\) and \(S\) are correlated).
What if the direction of the causal effect \(E_1\) was changed in the underlying DGP? That is:
%%R -h 300 -w 600
g = dagitty( 'dag {
"S" [pos="0,2"]
"A" [pos="2,0"]
"W" [pos="4,2"]
"E_1" [pos="1,1"]
"E_2" [pos="3,1"]
"E_3" [pos="2,2"]
"S" -> "A" -> "W"
"S" -> "W"
}')
plot(g)
Well, as long as we are interested in estimating the direct causal effect of \(S\) on \(W\) this would not change anything. Regression
\[W = \alpha + \beta S + \gamma A\]
would still yield an unbiased estimate \(\hat{\beta}\) of effect \(E_3\) and an unbiased estimate \(\hat{\gamma}\) of effect \(E_2\). If \(A\) was omitted from the regression, i.e.,
\[W = \alpha + \beta S \ \ \text{,}\]
then again
\[\text{plim } \hat{\beta} = E_3 + E_1 * E_2 \neq E_3 \ \ \text{.}\]
That is, we estimate the effect \(E_3\) with a bias.
Things change drastically if we are interested in the total causal effect of \(S\) on \(W\). By “total effect”, we mean the total causal effect \(S\) has on \(W\), i.e., how would \(W\) react were one to modify \(S\) (e.g., by one year). If we can estimate this total causal effect without bias, the we say our estimate is without confounding bias.
To set the stage, we reproduce the definition of non-confoundness from Pearl (2009, Definition 6.2.1):
Let \(M\) be a causal model of the data generating process – that is, a formal description of how the value of each observed variable is determined. Denote by \(P(y | do(x))\) the probability of the response event \(Y=y\) under the hypothetical intervention \(X=x\), calculated according to \(M\). We say that \(X\) and \(Y\) are not confounded in \(M\) if and only if
\[P(y | do(x)) = P(y | x) \text{, or } P(x | do(y)) = P(x|y) \quad \text{6.10}\]
for all \(x\) and \(y\) in their respective domains, where \(P(y | x)\) is the conditional probability generated by \(M\). If (6.10) holds, we say that \(P(y|x)\) is unbiased.
In this formal definition, \(do()\) denotes the do-calculus operator. The formal definition might seem complicated, and to accompany it Pearl (2009, p. 183; reproduced here with slight modification) also offers an association-based definition that might be more familiar to statisticians and econometricians:
Two variables \(X\) and \(Y\) are not confounded if and only if every variable \(Z\) that is not affected by \(X\) is either 1) unassociated with X or 2) unassociated with \(Y\), conditional on \(X\).
Now we can see what this means in the context of our example. From the two variations above, let’s first consider the situation where in the DGP \(S\) causally affects \(A\):
%%R -h 300 -w 600
g = dagitty( 'dag {
"S" [pos="0,2"]
"A" [pos="2,0"]
"W" [pos="4,2"]
"E_1" [pos="1,1"]
"E_2" [pos="3,1"]
"E_3" [pos="2,2"]
"S" -> "A" -> "W"
"S" -> "W"
}')
plot(g)
Let’s remind ourselves that we are now not interested estimating the direct causal effect \(E_3\), but rather the total causal effect that a change in \(S\) would induce on \(W\). This total effect can be calculated to be the sum of the direct effect \(E_3\) and the mediated effect \(E_1 * E_2\) via \(A\): \(E_3 + E_1 * E_2\). Now observe what happens when we run regression
\[W = \alpha + \beta S + \gamma A\]
From above, we know that this yields asymptotically unbiased estimates \(\text{plim } \hat{\beta} = E_3\) and \(\text{plim } \hat{\gamma} = E_2\). But this is not correct given our aim to estimate the total causal effect \(E_3 + E_1 * E_2\)! Here, controlling for \(A\) introduces a mediator-confounding bias on the causal effect. Note, however, that the estimates are totally fine (i.e., unbiased) for the direct effects; it is just not what we are after!
What about a regression with an omitted variable?
\[W = \alpha + \beta S\]
Again, from above we know that \(\text{plim } \hat{\beta} = E_3 + E_1 * E_2\), which is exactly what we wanted! This means that although this regression suffers from OVB, it does not suffer from confounding bias! We can also see this from the non-formal definition of non-confoundness: \(A\) is the only other variable in the system besides exposure \(S\) and outcome \(W\), and as \(A\) is directly affected by \(S\), then the definition is satisfied and \(W\) and \(S\) are not confounded without any additional adjustments.
What about the second variation where, in the DGP, \(A\) causally affects \(S\):
%%R -h 300 -w 600
g = dagitty( 'dag {
"S" [pos="0,2"]
"A" [pos="2,0"]
"W" [pos="4,2"]
"E_1" [pos="1,1"]
"E_2" [pos="3,1"]
"E_3" [pos="2,2"]
"S" <- "A" -> "W"
"S" -> "W"
}')
plot(g)
In this case, the total and direct effects of \(S\) on \(W\) coincide and both equal \(E_1\). To estimate \(E_1\) in an unbiased fashion (i.e., no confounding bias) it is very important to control for \(A\) in the regression. That is, the correct regression to run is
\[W = \alpha + \beta S + \gamma A\]
This regression suffers neither from OVB nor confounding bias. Again, this can be seen from the non-formal definition of non-confoundness: now \(A\) is not directly affected by \(S\), but it is associated with \(X\), and it is associated with \(W\) conditional on \(S\). Hence, the definition says \(W\) and \(S\) are confounded, and this confounding bias needs to be accounted for.