What is Omitted Variable Bias & How to Avoid it

In statistical analysis, the accuracy of conclusions hinges on the quality of the models we use. Even the smallest oversight can lead to significantly misleading outcomes. This leads to confirmation bias and affects the source evaluation method. One of the most prevalent issues statisticians and researchers face is the “Omitted Variable Bias” (OVB). 

Understanding OVB is fundamental for anyone seeking to interpret or conduct empirical research. Let’s look into the omitted variable bias definition in detail. 

What is an Omitted Variable?

An omitted variable, in the context of statistical modelling and econometrics, refers to a variable that is not included in a regression model but should be. When significant variables are left out of a model, it can lead to biased or inconsistent estimates of other parameters. This omission can also distort the relationship between the independent and dependent variables.

What is Omitted Variable Bias?

Omitted Variable Bias (OVB) refers to the bias that appears in the coefficients’ estimates of a regression model due to the omission of a relevant variable. This bias occurs when:

• The omitted variable is a determinant of the dependent variable.
• The omitted variable is correlated with one or more of the independent variables already included in the model.

When these two conditions hold, the effect of the omitted variable can get mistakenly attributed to the included independent variables, thus biasing their coefficient estimates.

To illustrate with an example, it is similar to the actor-observer bias in social psychology, 

let’s consider a simple linear regression, where we are trying to estimate the effect of years of education (Edu) on income (Income). Now, imagine that work experience (Experience) is also a determinant of Income, but we fail to include it in our regression. If Experience is also correlated with Edu, then our regression coefficient for Edu will capture not just the effect of education on income but also some of the effect of experience on income. This leads to a biased estimate of the true effect of education on income.

Mathematically, consider the true but unobserved model:

Income = β0 + β1 x Edu+ β2 x Experience + ϵ

​If we incorrectly estimate:

Income = α0 + α1 × Edu + u 

The coefficient α1 will be biased because it tries to capture the effect of education and experience on income, as long as education and experience are correlated.

Why is Omitted Variable Bias a Problem?

Omitted Variable Bias (OVB) is a significant issue in statistical analysis and econometrics because it can lead to incorrect conclusions about the relationships between variables. Just as cognitive bias can distort one’s judgment, OVB can distort statistical interpretations. Specifically, when an important variable is omitted from a regression model, the coefficients on the included variables can be biased. This can lead researchers to make incorrect inferences about the strength and direction of relationships. 

Here is a more detailed look at why OVB is problematic:

Incorrect Coefficient Estimates

If an omitted variable is correlated with both the independent variable(s) and the dependent variable in a regression model, then the coefficient estimates of the included independent variables can be biased. This means that the estimated effect of the independent variable on the dependent variable is not accurate.

Misleading Conclusions

Due to the incorrect coefficient estimates, researchers may draw wrong conclusions about the relationships between variables. For instance, an independent variable might seem to have a significant effect on the dependent variable when, in reality, the effect is due to the omitted variable.

Loss of Efficiency

Even if the omitted variable only correlates with the dependent variable and not with the independent variables, its omission will lead to inefficiency. The standard errors of the coefficients will be larger than they would be if the omitted variable were included, reducing the efficiency of the estimates.

Model Specification Errors

OVB is essentially a type of model misspecification. Relying on misspecified models can lead to poor prediction and unreliable conclusions.

Difficulty in Remediation

Detecting OVB can be challenging, especially when researchers are unaware of all the relevant variables that should be included in the model. Once detected, finding the necessary data to include the omitted variable can also be a hurdle.

Complications in Policy and Decision-Making

Research findings often guide policy and decision-making in fields like economics, public policy, and social sciences. If there is OVB in the analyses, the policies or decisions based on those results could be ineffective or counterproductive.

How to Avoid Omitted Variable Bias

In order to avoid omitted variable bias, consider the following steps:

Theoretical Understanding

Begin with a solid theoretical framework of the issue under investigation. This will help you to identify potentially relevant variables. Sometimes, domain-specific knowledge is essential to ensuring that all crucial variables are considered.

Examine Previous Research

Review the literature on your topic to see which variables other researchers have considered important. Always ensure that your references come from a scholarly source, distinguishing between primary source and secondary source information. This ensures accuracy and minimises publication bias.

Include Relevant Variables

Once you identify potential omitted variables, include them in your regression if you have the data. However, be careful not to confuse adding more variables with better specifications. Overfitting, where too many variables are added to the model, can be as problematic as OVB.

Instrumental Variables (IV)

Consider using instrumental variables when there is concern about omitted variables and you suspect some independent variables might correlate with the error term. An instrument correlates with the potentially endogenous independent variable, but not with the error term. The IV approach can help adjust for the endogeneity caused by OVB.

Fixed Effects

If omitted variables are time-invariant in panel data, using fixed effects can control for them, even if you can not directly measure them.

Random Effects

A random effects model can be a viable solution if the omitted variables are uncorrelated with the included independent variables.

Sensitivity Analysis

After selecting your variables, conduct a sensitivity analysis by adding and subtracting different combinations of variables to see how robust your results are to specification changes.

Proxy Variables

If you suspect an omitted variable but don’t have data on it, see if there is a related variable (a proxy) that you can include. While not perfect, a proxy can help mitigate some of the bias. This could help when there is a ceiling effect on the data you can gather. However, while trying to cater to OVB, ensure you don’t fall into the trap of bias for action. However, be cautious because proxy variables can introduce measurement errors if not chosen wisely.

Look for Natural Experiments

Natural experiments can provide variation in your independent variable of interest that is uncorrelated with the error term, helping to mitigate OVB.

Limit the Scope of Your Analysis

If you are uncertain about the potential for OVB, be clear about the limitations of your analysis. Do not overstate the causal implications of your findings if there’s potential for OVB.

Post-Estimation Diagnostics

After estimating your model, use diagnostic tests for detecting potential specification errors, such as Ramsey’s RESET test.

Collinearity Checks

If you add a variable and another previously significant variable becomes insignificant, it could indicate multicollinearity. This is not omitted variable bias per se, but it’s related in that adding omitted variables can reveal multicollinearity issues.

Regular Communication with Experts

Engage in discussions with subject experts to continuously refine your model specification. Doing so can also help combat explicit bias and affinity bias, ensuring that your research is as objective as possible.

External Validity Checks

If possible, validate your model with other datasets or in different settings to check its robustness.

How to Estimate Omitted Variable Bias

Estimating Omitted Variable Bias (OVB) requires some assumptions and a bit of algebra. OVB refers to the bias that can arise in the coefficient estimates of a regression model when a relevant variable is left out of the model.

To illustrate how to estimate OVB, consider the following simple linear regression model:

Y = β0 + β1X1 + β2X2 + u 

Suppose X2 is the omitted variable. We are interested in understanding how the omission of X2 ​might bias our estimate of β1 when we wrongly estimate the model:

Y = α0 + α1X1 + v

The bias in our estimate of  β1 (when we omit X2) can be represented as:

Bias (α1) = E [α1] – β1 

Under classical linear model assumptions:

Bias (α1) = β2 x (Cov X1X2/ Var X1)

Where:

• β2 is the coefficient on the omitted variable in the correctly specified model.
• Cov (X1, X2) is the covariance between the included and omitted variables.
• Var (X1) is the variance of the included variable.

So, to estimate the omitted variable bias, you would:

1. Estimate the relationship between X2 (the omitted variable) and X1 (the included variable). This gives you the term (Cov X1X2/ Var X1)
2. Estimate the relationship between X2 and Y when X1 is also in the model. This will give you an estimate of β2. 
3. Multiply the results from 1 and 2 to estimate the bias.

This formula shows that if either:

• β2 = 0 (meaning the omitted variable has no direct effect on Y), or 
• Cov (X1, X2) = 0 (meaning the omitted variable is not correlated with the included variable), then the OVB will be zero.

It is essential to understand that while this method can estimate the bias due to omitting a specific variable, there might be other omitted variables or sources of endogeneity that can further bias results. The best way to avoid omitted variable bias is to carefully specify the model based on theory and prior evidence, and use techniques like instrumental variables or fixed effects if feasible and appropriate.

Frequently Asked Questions