Lesson M03.L04: Omitted Variable Bias

Module: M03 — Multiple Regression Analysis
Level: Graduate (ECMT5001, USYD S1 2027)
Duration: 35 min
Learning Objective: Derive the omitted variable bias formula; determine the direction and magnitude of bias given the sign of the omitted variable's true coefficient and its correlation with the included regressor; apply OVB reasoning to real economic examples.
Provenance: Wooldridge, Introductory Econometrics, Chapter 3 (Section 3.3)

Explanation

What Is Omitted Variable Bias?

Omitted variable bias (OVB) arises when a variable that belongs in the true regression — because it affects $y$ and correlates with at least one included regressor — is left out. The OLS estimate of the included regressor's coefficient absorbs part of the omitted variable's effect and is therefore biased.

OVB is a violation of MLR.4: the omitted variable ends up in the error term $u$, making $E(u \mid x_1) \neq 0$.

The OVB Formula (Two-Variable Case)

True model (population):

\[y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + u, \quad E(u \mid x_1, x_2) = 0\]

Estimated model (researcher omits $x_2$):

\[y = \tilde{\beta}_0 + \tilde{\beta}_1 x_1 + v, \quad v = \beta_2 x_2 + u\]

The simple-regression OLS estimator of $\tilde{\beta}_1$ is:

\[\tilde{\beta}_1 = \frac{\sum_{i=1}^n (x_{i1} - \bar{x}_1)(y_i - \bar{y})}{\sum_{i=1}^n (x_{i1} - \bar{x}_1)^2}\]

Taking expectations (using the true model):

\[E(\tilde{\beta}_1) = \beta_1 + \beta_2 \tilde{\delta}_1\]

where $\tilde{\delta}_1$ is the auxiliary regression coefficient — the slope from regressing $x_2$ on $x_1$:

\[\tilde{\delta}_1 = \frac{\sum_{i=1}^n (x_{i1} - \bar{x}_1)(x_{i2} - \bar{x}_2)}{\sum_{i=1}^n (x_{i1} - \bar{x}_1)^2}\]

Key formula

\[\boxed{E(\tilde{\beta}_1) = \beta_1 + \underbrace{\beta_2 \cdot \tilde{\delta}_1}_{\text{OVB}}}\]

Bias $= \beta_2 \cdot \tilde{\delta}_1$

Direction of Bias

The sign of OVB depends on the signs of $\beta_2$ and $\tilde{\delta}_1$:

$\beta_2 > 0$ (omitted var ↑ $y$)	$\beta_2 < 0$ (omitted var ↓ $y$)
$\tilde{\delta}_1 > 0$ (omitted corr. positively with $x_1$) → Upward bias	$\tilde{\delta}_1 > 0$ → Downward bias
$\tilde{\delta}_1 < 0$ (omitted corr. negatively with $x_1$) → Downward bias	$\tilde{\delta}_1 < 0$ → Upward bias

Mnemonic: Same signs → upward bias; opposite signs → downward bias

Note: "upward bias" means $E(\tilde{\beta}_1) > \beta_1$ (OLS overestimates the true effect).

Classic Example: Ability Bias in Returns to Education

True model:

\[\ln(\text{wage}) = \beta_0 + \beta_1\,\text{educ} + \beta_2\,\text{ability} + u\]

$\beta_2 > 0$: more able workers earn higher wages (ability raises $y$)
$\tilde{\delta}_1 > 0$: more able individuals tend to acquire more education (ability correlates positively with educ)

OVB:

\[E(\tilde{\beta}_1) = \beta_1 + \underbrace{\beta_2}_{>0} \cdot \underbrace{\tilde{\delta}_1}_{>0} > \beta_1\]

Conclusion: Simple OLS of log-wage on education overestimates the true return to education. Part of what we attribute to schooling is really the wage premium for ability.

Australian Example: Family Background and Returns to Education

In Australia, children from wealthier family backgrounds tend to: 1. Achieve higher educational attainment (educ ↑) 2. Earn higher wages due to networks, cultural capital, and inherited advantages (wage ↑)

If a researcher regresses $\ln(\text{wage})$ on educ without controlling for family background (SES index):

$\beta_2 > 0$ (SES raises wages)
$\tilde{\delta}_1 > 0$ (SES correlates positively with education)

→ Upward OVB on $\hat{\beta}_1$. The estimated return to education is inflated.

This has direct policy implications: if we overestimate returns to education, we may over-invest in tertiary subsidies relative to other interventions.

Worked Example: Calculate Direction and Magnitude

Setup: A researcher has data on 500 workers and estimates:

\[\widehat{\ln(\text{wage})} = 0.50 + 0.12\,\text{educ} \qquad (\text{simple regression})\]

An econometrician reveals that the true model is:

\[\ln(\text{wage}) = 0.20 + 0.08\,\text{educ} + 0.04\,\text{ability} + u\]

The auxiliary regression of ability on educ gives:

\[\widehat{\text{ability}} = -1.2 + 0.9\,\text{educ} \implies \tilde{\delta}_1 = 0.9\]

Step 1: Compute predicted OVB

\[\text{OVB} = \beta_2 \cdot \tilde{\delta}_1 = 0.04 \times 0.9 = 0.036\]

Step 2: Check against formula

\[E(\tilde{\beta}_1) = \beta_1 + \text{OVB} = 0.08 + 0.036 = 0.116\]

This is close to the observed $\tilde{\beta}_1 = 0.12$ (small discrepancy due to sampling variation). ✓

Step 3: Direction of bias

$\beta_2 = 0.04 > 0$ and $\tilde{\delta}_1 = 0.9 > 0$ → upward bias. The simple regression overestimates the return to education by roughly $0.036$ log-wage units (≈ 3.6 percentage points).

Policy implication: A policy that raises education by 1 year would produce an approximately 8% wage increase (true $\beta_1 = 0.08$), not the 12% suggested by the biased estimate.

OVB with Multiple Included Regressors

In the general case with multiple included regressors $x_1, \ldots, x_k$ and omitted variable $x_{k+1}$:

\[\text{OVB on }\hat{\beta}_j = \beta_{k+1} \cdot \tilde{\delta}_j\]

where $\tilde{\delta}_j$ is the coefficient on $x_j$ in the auxiliary regression of $x_{k+1}$ on all included $x_1,\ldots,x_k$. Every included coefficient is potentially biased if the omitted variable correlates with it.

Common Misconception

Adding more variables always eliminates OVB

Partially true, but dangerous. Adding the correct omitted variable eliminates OVB. However:

The added variable may itself be endogenous — then you introduce a new form of bias. For example, adding job type to control for ability may be a bad control if job type is partly caused by education.
Proxy variables (e.g., IQ test scores as a proxy for ability) reduce but do not eliminate OVB, because the proxy is an imperfect measure.
Adding irrelevant variables does not reduce any bias but inflates variance and can induce new near-multicollinearity.

Zero correlation means no OVB

If the omitted variable is uncorrelated with $x_1$ ($\tilde{\delta}_1 = 0$), then $\text{OVB} = \beta_2 \cdot 0 = 0$ even if $\beta_2 \neq 0$. A variable that affects $y$ but is uncorrelated with included regressors goes into the error term without biasing any coefficient — it only increases residual variance ($\hat{\sigma}^2$) and hence standard errors.

Practice Prompts

OVB direction. A researcher estimates the effect of the minimum wage (minwage) on employment (emp) without controlling for regional economic conditions (local_gdp). Suppose the true model is:

$$\ln(\text{emp}) = \beta_0 + \beta_1 \ln(\text{minwage}) + \beta_2 \ln(\text{local\_gdp}) + u$$

with $\beta_2 > 0$. Australian states with higher local GDP tend to set higher minimum wages ($\tilde{\delta}_1 > 0$). Without controlling for local_gdp, is the OLS estimate of $\beta_1$ (the employment elasticity of the minimum wage) biased upward or downward? What does this imply about published estimates of minimum wage effects that don't control for regional conditions?

Derive OVB. Start from the true model $y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + u$ and the formula $\tilde{\beta}_1 = \frac{\sum(x_{i1}-\bar{x}_1)(y_i-\bar{y})}{\sum(x_{i1}-\bar{x}_1)^2}$. Substitute the true model for $y_i$ and show that $E(\tilde{\beta}_1) = \beta_1 + \beta_2\tilde{\delta}_1$.
Numerical OVB. True model: $y = 1 + 2x_1 + 3x_2 + u$. Auxiliary regression: $x_2 = 0.5 + 0.4x_1 + \epsilon$. You estimate the misspecified model $y = \tilde{\beta}_0 + \tilde{\beta}_1 x_1 + v$. (a) Compute $E(\tilde{\beta}_1)$. (b) Is OLS upward or downward biased? (c) How much does OLS overstate/understate the effect of $x_1$?

Further Resources

Wooldridge, Ch 3.3 (pp. 89–96)
Angrist & Pischke, Mostly Harmless Econometrics, Ch 3.2 (OVB formula in IV context)
Card (1995) — uses distance to college as IV to address ability bias; classic paper
HILDA longitudinal data — used in Australian studies of education–wage link

\(\beta_2 > 0\) (omitted var ↑ \(y\))	\(\beta_2 < 0\) (omitted var ↓ \(y\))
\(\tilde{\delta}_1 > 0\) (omitted corr. positively with \(x_1\)) → Upward bias	\(\tilde{\delta}_1 > 0\) → Downward bias
\(\tilde{\delta}_1 < 0\) (omitted corr. negatively with \(x_1\)) → Downward bias	\(\tilde{\delta}_1 < 0\) → Upward bias