6.4 Sample selection bias
6 min read•august 20, 2024
Sample selection bias is a critical issue in econometrics that can lead to skewed results and faulty conclusions. It occurs when the sample used in a study isn't representative of the population, resulting in and reduced external validity.
Detecting and correcting for sample selection bias is crucial for accurate analysis. Methods like comparing sample characteristics to the population, using the Heckman , and applying techniques such as inverse probability weighting can help address this issue.
Types of sample selection bias
- Sample selection bias occurs when the sample used in a study is not representative of the population of interest, leading to biased and
- Arises due to non-random selection or self-selection of individuals into or out of the sample based on unobserved factors that are correlated with both the dependent variable and the independent variables
- Common types include non-response bias (individuals who refuse to participate in a survey may differ systematically from those who do participate), incidental truncation (sample is truncated based on some variable of interest, such as observing wages only for employed individuals), and (individuals self-select into treatment or control groups based on unobserved characteristics)
Consequences of sample selection bias
Biased parameter estimates
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- Sample selection bias leads to biased and inconsistent estimates of the parameters of interest in a regression model
- The estimated coefficients will be biased because the sample used for estimation is not representative of the true population
- The direction and magnitude of the bias depend on the nature of the selection process and the correlation between the unobserved factors affecting selection and the dependent variable
- For example, if high-ability individuals are more likely to self-select into a training program and also have higher earnings, the estimated effect of the training program on earnings will be upward biased
Incorrect inferences
- Biased parameter estimates due to sample selection can lead to incorrect inferences and conclusions about the relationship between variables
- Hypothesis tests and confidence intervals based on biased estimates will be invalid, potentially leading to Type I (false positive) or Type II (false negative) errors
- The presence of sample selection bias can make it difficult to establish causal relationships between variables, as the observed association may be driven by unobserved factors rather than a true causal effect
Reduced external validity
- Sample selection bias can limit the external validity or generalizability of the study's findings to the broader population of interest
- If the sample is not representative of the population, the estimated relationships and effects may not hold for individuals outside the sample
- This can be particularly problematic when the goal is to make policy recommendations or draw conclusions that are applicable to a wider population
- For instance, if a study on the effectiveness of a job training program only includes individuals who chose to participate, the results may not generalize to the population of all eligible individuals
Detecting sample selection bias
Comparing sample vs population
- One way to detect sample selection bias is to compare the characteristics of the sample used in the study with those of the target population
- Significant differences between the sample and population in terms of observable characteristics (such as demographics or socioeconomic status) may indicate the presence of selection bias
- Statistical tests, such as t-tests or chi-square tests, can be used to assess whether the differences between the sample and population are statistically significant
- For example, if a survey on income has a significantly higher proportion of high-income respondents compared to the population, this may suggest non-response bias
Using Heckman selection model
- The Heckman selection model is a statistical method designed to detect and correct for sample selection bias in regression analysis
- It involves estimating two equations: a selection equation that models the probability of an individual being included in the sample, and an outcome equation that models the relationship between the dependent variable and independent variables for the selected sample
- The Heckman model includes an additional term, the inverse Mills ratio, in the outcome equation to account for the potential correlation between the unobserved factors affecting selection and the dependent variable
- A statistically significant coefficient on the inverse Mills ratio indicates the presence of sample selection bias
Correcting for sample selection bias
Heckman two-step procedure
- The Heckman two-step procedure is a method for correcting sample selection bias in regression analysis
- In the first step, a probit model is estimated to predict the probability of an individual being included in the sample based on observed characteristics (the selection equation)
- In the second step, the inverse Mills ratio is computed from the predicted probabilities and included as an additional regressor in the outcome equation, which is then estimated using ordinary least squares (OLS)
- The coefficient on the inverse Mills ratio captures the effect of sample selection bias, and the remaining coefficients provide consistent estimates of the parameters of interest
Maximum likelihood estimation
- (MLE) is an alternative approach to correcting for sample selection bias
- MLE involves specifying a joint distribution for the selection and outcome equations and estimating the parameters of both equations simultaneously by maximizing the likelihood function
- MLE is more efficient than the two-step procedure and provides consistent estimates of the parameters, but it requires stronger distributional assumptions and can be more computationally intensive
- MLE is often used when the selection and outcome equations are believed to have a specific joint distribution, such as a bivariate normal distribution
Inverse probability weighting
- Inverse probability weighting (IPW) is a method for correcting sample selection bias by reweighting the observed sample to make it representative of the population
- IPW involves estimating the probability of each individual being included in the sample (the propensity score) based on observed characteristics and then weighting each observation by the inverse of its propensity score
- Observations with a low probability of being selected receive higher weights, while observations with a high probability of being selected receive lower weights
- The reweighted sample mimics the distribution of the population, allowing for consistent estimation of the parameters of interest
- IPW is particularly useful when the selection process is based on observable characteristics and does not require specifying a joint distribution for the selection and outcome equations
Examples of sample selection bias
Non-response bias in surveys
- Non-response bias occurs when individuals who refuse to participate in a survey differ systematically from those who do participate
- For example, in a survey on income, high-income individuals may be less likely to respond due to privacy concerns or time constraints
- If non-respondents have systematically different incomes than respondents, the estimated average income from the survey will be biased
- To correct for non-response bias, researchers can use methods such as weighting the sample based on observable characteristics or imputing missing values based on the characteristics of respondents
Incidental truncation in labor economics
- Incidental truncation arises when the sample is truncated based on some variable of interest, such as observing wages only for employed individuals
- In labor economics, incidental truncation can occur when studying the determinants of wages because wages are only observed for individuals who are employed
- If the factors that affect an individual's decision to work are correlated with the factors that affect their wage, the estimated wage equation will suffer from sample selection bias
- The Heckman selection model can be used to correct for incidental truncation by modeling the employment decision and the wage equation simultaneously
Self-selection bias in treatment effects
- Self-selection bias occurs when individuals self-select into treatment or control groups based on unobserved characteristics that are correlated with the outcome of interest
- For example, in a study on the effectiveness of a job training program, individuals who choose to participate may have higher motivation or ability than those who do not participate
- If these unobserved characteristics are positively correlated with employment outcomes, the estimated effect of the training program will be upward biased
- To correct for self-selection bias, researchers can use methods such as instrumental variables (using a variable that affects participation but not the outcome) or propensity score matching (matching treated and control individuals based on observable characteristics)
Key Terms to Review (14)
Biased estimates: Biased estimates are statistical estimates that systematically deviate from the true parameter values they are intended to estimate. This can lead to inaccurate conclusions and decisions based on the analysis, affecting the validity of the model. Biased estimates can arise from several issues, including omitted variables, incorrect model specifications, sample selection problems, and endogeneity, each of which can distort the relationship being analyzed.
Donald Rubin: Donald Rubin is a prominent statistician known for his contributions to causal inference and the development of the Rubin Causal Model (RCM). His work primarily focuses on understanding how to make causal conclusions from observational data, which is crucial in addressing issues like sample selection bias, where the sample studied may not represent the broader population.
Exogeneity: Exogeneity refers to a condition where an explanatory variable is not correlated with the error term in a regression model. When a variable is exogenous, it suggests that any changes in this variable do not arise from the model's error, making it crucial for establishing causal relationships and ensuring valid inference in econometric analysis.
Heckman Correction: The Heckman correction is a statistical technique used to correct for sample selection bias in regression analysis. This method is particularly useful when the data used for estimation is not randomly selected, leading to biased and inconsistent parameter estimates. By accounting for the non-random nature of the sample, the Heckman correction helps improve the validity of the results by estimating the selection process and adjusting the outcome model accordingly.
Identifying Restrictions: Identifying restrictions refer to the specific assumptions or conditions imposed on a model to ensure that the estimates derived from it are valid and interpretable. In the context of econometrics, these restrictions are crucial for addressing issues like sample selection bias, as they help to isolate the effects of variables of interest while controlling for confounding factors.
Inconsistent estimates: Inconsistent estimates occur when the statistical estimates do not converge to the true parameter value as the sample size increases. This means that even with a larger dataset, the estimates can remain off-target, leading to unreliable results. Understanding this concept is crucial because it highlights potential flaws in the model, such as omitted variables or selection biases that could distort the findings.
Instrumental variable: An instrumental variable is a tool used in econometrics to estimate causal relationships when controlled experiments are not feasible and there is potential for confounding. It serves as a proxy for an independent variable that is suspected to be correlated with the error term, thus helping to eliminate bias in the estimation of the causal effect of the independent variable on the dependent variable. This approach is especially important when dealing with issues like sample selection bias or when using two-stage least squares for regression analysis.
James Heckman: James Heckman is a renowned economist known for his work on the theory and application of econometrics, particularly regarding sample selection bias and the Heckman selection model. His research has significantly influenced how economists understand issues of endogeneity, as well as the methods to correct for it, such as two-stage least squares (2SLS). His contributions have advanced empirical research and policy analysis in various fields, including labor economics and education.
Maximum likelihood estimation: Maximum likelihood estimation (MLE) is a statistical method for estimating the parameters of a probability distribution or a statistical model by maximizing the likelihood function. It connects to the concept of fitting models to data by finding the parameter values that make the observed data most probable under the assumed model.
Nonrandom Sampling: Nonrandom sampling is a method of selecting individuals or units for a study where the selection is not based on chance, leading to potential biases in the sample. This approach can skew results because it may not accurately represent the larger population, which is critical in assessing relationships and outcomes. Nonrandom sampling can occur through various means, such as convenience sampling, where researchers choose participants who are readily available, or judgmental sampling, where the researcher selects individuals based on their expertise or characteristics.
Selection Model: A selection model is a statistical technique used to correct for sample selection bias by modeling the process that determines whether an observation is included in the sample. It helps address issues where the outcome of interest may be systematically different between observed and unobserved data, allowing for more accurate inferences about relationships in the data. This model is particularly useful in situations where data is missing not at random, influencing the analysis of causal relationships.
Self-selection bias: Self-selection bias occurs when individuals select themselves into a group, leading to a sample that may not be representative of the population as a whole. This bias can skew the results of a study because those who choose to participate may have different characteristics or behaviors compared to those who do not, affecting the validity of conclusions drawn from the data.
Treatment effect model: A treatment effect model is a statistical framework used to estimate the causal impact of a treatment or intervention on an outcome of interest. This model addresses the problem of sample selection bias by comparing treated and untreated groups, allowing researchers to infer what the effect would have been had the untreated group received the treatment. The goal is to isolate the treatment effect from confounding factors that could skew results.
Two-step estimation: Two-step estimation is a statistical technique used to address issues such as sample selection bias by estimating model parameters in two distinct phases. In the first step, a preliminary model is estimated to predict an outcome or a selection mechanism, and in the second step, the outcome variable is estimated using the predictions from the first step. This method is particularly useful in scenarios where the data may be biased due to missing observations or non-random selection of samples.