Omitted Variable Bias (OVB)

For OVB to exist, an omitted variable Z must simultaneously satisfy two strict criteria:

• Correlation with an Included Regressor: Z must be correlated with at least one independent variable X already in the model. If Z is uncorrelated with X, its omission does not bias the coefficient on X.
• Direct Effect on the Dependent Variable: Z must be a determinant of the outcome Y. If Z does not causally affect Y, omitting it only inflates the residual variance without introducing bias.

If either condition fails, OVB is zero. The severity of the bias is a product of both relationships.

The asymptotic bias in the OLS estimator for a simple regression with one included regressor X and one omitted variable Z is given by:

Bias(β̂₁) = β₂ × δ̃₁

Where:

• β₂ is the true coefficient of the omitted variable Z in the full model (the direct effect on Y).
• δ̃₁ is the slope coefficient from an auxiliary regression of the omitted Z on the included X.

This formula generalizes to the multivariate case using the Frisch-Waugh-Lovell theorem, where the bias vector equals the product of the coefficients on omitted variables and the linear projection coefficients linking omitted to included regressors.

The sign of the bias is determined by the signs of the two component relationships:

• Positive Bias (Upward): Occurs when β₂ and δ̃₁ share the same sign. The estimated coefficient overstates the true effect.
• Negative Bias (Downward): Occurs when β₂ and δ̃₁ have opposite signs. The estimated coefficient understates the true effect.
• Attenuation Toward Zero: A specific case where the bias pulls the estimate toward zero, potentially masking a real effect.
• Sign Reversal: In extreme cases, the bias can be so large and opposite in sign that the estimated coefficient takes the wrong sign entirely, reversing the apparent causal direction.

In quantitative finance, OVB is pervasive and dangerous:

• Factor Model Misspecification: Omitting a relevant risk factor (e.g., momentum) from an asset pricing regression biases the estimated alpha and factor loadings on included factors like value or size.
• Market Impact Models: Failing to include volatility regime or order flow imbalance when estimating the price impact of trade size leads to biased execution cost predictions.
• Causal Inference in Corporate Finance: Studying the effect of board diversity on firm performance without controlling for firm culture or industry dynamics yields spurious conclusions.
• Algorithmic Trading Signal Evaluation: Backtesting a signal derived from sentiment analysis without controlling for market-wide liquidity conditions can attribute returns to sentiment when liquidity is the true driver.

While OVB is fundamentally untestable without the omitted data, several diagnostic approaches can raise red flags:

• Sensitivity Analysis: Bounding exercises that estimate how strong the omitted confounder would need to be to nullify the observed result, often visualized with sensitivity contour plots.
• Coefficient Stability Tests: Comparing coefficient estimates across nested specifications. Large movements when adding controls suggest potential OVB in the simpler model.
• Auxiliary Regressions: Testing whether candidate omitted variables are correlated with included regressors and the outcome, even if the variable cannot be added to the main model.
• Instrumental Variables: If a valid instrument exists, comparing OLS and IV estimates provides a formal Hausman test for endogeneity, of which OVB is one source.

Addressing OVB requires structural, not just statistical, solutions:

• Directed Acyclic Graphs (DAGs): Explicitly encoding causal assumptions forces the analyst to identify potential confounders before estimation, making the identification strategy transparent.
• Fixed Effects: Including entity and time fixed effects absorbs all time-invariant or entity-invariant unobserved heterogeneity, eliminating OVB from these dimensions.
• Randomized Controlled Trials: Random assignment breaks the correlation between treatment and all confounders—observed and unobserved—by construction, eliminating OVB.
• Proxy Variables: When the true confounder is unmeasurable, including a correlated proxy reduces, though does not eliminate, the bias. The residual bias depends on the proxy's measurement error.

The condition where an explanatory variable is correlated with the error term, making OLS estimates biased and inconsistent. OVB is a primary cause of endogeneity, occurring when a relevant variable is excluded and absorbed into the error structure.

• Simultaneity: X causes Y, but Y also causes X (bid-ask bounce)
• Measurement Error: Noisy proxies for true latent variables
• Consequence: Coefficients no longer represent causal effects

In algorithmic trading, endogeneity invalidates backtests by attributing predictive power to features that are merely correlated with unobserved confounders.

A two-stage estimation method that rescues causal inference when OVB is unavoidable. An instrument must satisfy two conditions:

• Relevance: Strongly correlated with the endogenous regressor
• Exogeneity: Affects the outcome only through the endogenous variable

In market microstructure, trade size is often instrumented with lagged order book depth to isolate its causal effect on price impact, bypassing unobserved trader intent.

An extraneous variable that influences both the treatment and outcome, creating a spurious association. When omitted, it becomes the root cause of OVB.

Classic finance example: A study finds that firms with larger boards have lower profitability. The confounder is firm size—larger firms have larger boards and different margin structures. Without controlling for size, board composition appears causally detrimental.

• Direction of bias: Depends on the sign of correlations between confounder-treatment and confounder-outcome
• Detection: Use Directed Acyclic Graphs (DAGs) to map assumed causal structure before estimation

A graphical rule from Judea Pearl's causal framework that identifies the minimal set of variables to condition on to block all spurious paths between treatment and outcome.

Application to OVB:

• Draw a DAG of your assumed market relationships
• Identify all backdoor paths (non-causal associations)
• Condition on variables that block these paths

This formalizes variable selection, preventing both OVB (from under-controlling) and collider bias (from over-controlling on common effects like post-earnings volatility).

A modern causal estimation technique that removes regularization bias when using ML models to control for confounders in high-dimensional settings.

How it works:

• Orthogonalization: Residualize both treatment and outcome on controls using flexible ML (gradient boosting, neural nets)
• Cross-fitting: Split data into folds to avoid overfitting bias
• Final regression: Regress residualized outcome on residualized treatment

DML is particularly valuable in alternative data alpha research, where thousands of potential confounders exist but the true causal structure is unknown.

A regression that suggests a statistically significant relationship between independent non-stationary variables when no economic link exists. This is a special case of OVB where the omitted variable is time itself.

Warning signs:

• High R-squared with low Durbin-Watson statistic
• Both series are I(1) but not cointegrated
• Classic example: Cumulative S&P 500 returns vs. cumulative rainfall in Nepal

Solution: First-difference non-stationary series or test for cointegration before interpreting regression coefficients as meaningful relationships.