Missing at random

What Is Missing at Random?

"Missing at random" (MAR) is a fundamental assumption in quantitative analysis and statistical modeling that describes a specific pattern of incomplete data. Data are considered missing at random when the probability of a value being missing is related to other observed data points in the dataset, but not to the unobserved (missing) value itself.¹⁶, ¹⁷ This means that while there might be systematic reasons for data to be absent, these reasons can be explained by information that is available.¹⁵ Understanding missing at random is crucial for selecting appropriate methods for data imputation and ensuring the validity of statistical inferences in finance and other fields.

History and Origin

The concept of missing data mechanisms, including missing at random, was formally introduced and categorized by statistician Donald Rubin in a seminal 1976 paper.¹¹, ¹², ¹³, ¹⁴ Rubin's work provided a theoretical framework for understanding and addressing the challenges posed by incomplete data in statistical analysis. Prior to this, methods for handling missing data were often ad hoc and could lead to significant statistical bias. Rubin's classification distinguished between three primary mechanisms: "missing completely at random" (MCAR), "missing at random" (MAR), and "missing not at random" (MNAR). This taxonomy became central to the development of more sophisticated and robust imputation techniques, allowing researchers to make more accurate inferences from datasets with missing values.¹⁰

Key Takeaways

• Conditional Missingness: Missing at random implies that the pattern of missingness is predictable based on other observed variables in the dataset.
• Model-Based Imputation: Advanced imputation methods, such as multiple imputation and maximum likelihood estimation, often assume missing at random.
• Bias Reduction: When the MAR assumption holds and appropriate methods are used, it helps to mitigate bias that would otherwise arise from incomplete data.
• Untestability: While the MAR assumption is crucial, it generally cannot be directly tested using the observed data alone.
• Broader than MCAR: MAR is a less restrictive and more frequently plausible assumption than "missing completely at random" (MCAR), where missingness is entirely unsystematic.⁸, ⁹

Formula and Calculation

Missing at random (MAR) is a conceptual assumption about the data generation process, not a statistical formula or calculation that yields a numeric result. Therefore, this section is not applicable.

Interpreting the Missing at Random Assumption

Interpreting the missing at random (MAR) assumption involves considering whether the reasons for data being absent can be fully accounted for by the information already present in the dataset. If a data quality issue leads to missing values, and the missingness depends only on observable characteristics, then the MAR assumption may hold. For example, if older investors are less likely to provide certain survey data about their income, but the missingness of income values does not depend on the actual amount of income they earn (only on their age), then the data is missing at random.⁷

When MAR is a reasonable assumption, it enables the use of more powerful statistical techniques, such as multiple imputation, to create complete datasets for financial modeling and analysis. These methods account for the observed relationships to estimate the missing values, thus preserving statistical power and reducing potential bias compared to simpler approaches like listwise deletion, which can lead to biased estimates if the missingness is not completely random. Analysts must carefully consider the context of their data collection and domain knowledge to assess the plausibility of the MAR assumption.

Hypothetical Example

Consider a hedge fund performing quantitative research on stock returns and company fundamentals. They collect data on daily stock prices, trading volume, company revenue, and employee count. For some smaller, less frequently traded stocks, the employee count data might be missing.

Let's assume the following:

• Stock prices and trading volumes are always observed.
• Company revenue is mostly observed.
• Employee count data is missing for some companies.

If the probability of employee count being missing is higher for companies with lower observed revenue, but not directly related to the actual missing employee count value itself, then the data could be considered missing at random.

Scenario:
The research team has a dataset where:

1. Company A: Revenue = $500M, Employee Count = 10,000
2. Company B: Revenue = $10M, Employee Count = 100
3. Company C: Revenue = $2M, Employee Count = Missing
4. Company D: Revenue = $15M, Employee Count = Missing

If it's found that companies with revenue below $5M tend to have missing employee count data more often, this is a systematic pattern. However, as long as the missingness isn't dependent on the specific number of employees (e.g., companies with exactly 50 employees are more likely to be missing), but rather on their low revenue, then it aligns with the missing at random assumption.

To handle this, the team might use a method like regression analysis to predict the missing employee counts based on observed revenue and other company characteristics, assuming the MAR mechanism holds.

Practical Applications

The missing at random (MAR) assumption has broad practical applications across various financial disciplines where data incompleteness is common:

• Risk Management: In risk management, banks and financial institutions analyze vast datasets for credit risk, market risk, and operational risk. Missing values in client financial statements or transaction records are common. If the missingness of certain variables (e.g., a client's specific debt-to-income ratio) depends on other available information (e.g., their credit score or loan type), but not on the unobserved debt-to-income itself, then MAR-compliant imputation techniques can be used to complete the data, ensuring more accurate risk assessments. The Basel Committee on Banking Supervision (BCBS) emphasizes the need for strong data governance and robust data aggregation capabilities, as outlined in BCBS 239, to ensure data accuracy and completeness, particularly for global systemically important banks (G-SIBs). This regulatory framework implicitly addresses issues of missing data by requiring institutions to strengthen their ability to aggregate and report risk data effectively.
• Credit Scoring: When developing credit scoring models, financial institutions often encounter missing data in applicant information. If an applicant's missing income information is related to their reported employment status or educational background, but not the specific income amount, MAR-based imputation can help build more robust models without discarding valuable partially complete records.
• Economic Research and Surveys: Government agencies, such as the U.S. Census Bureau, frequently deal with missing data in large-scale surveys. They employ sophisticated methods like statistical imputation and weighting adjustments to address both unit nonresponse (entire households not responding) and item nonresponse (specific questions left unanswered). These imputation procedures often rely on the assumption that data are missing at random within subgroups of the population to reduce bias in survey estimates.
• Algorithmic Trading and Predictive Analytics: In these areas, complete and high-quality data are paramount. When market data feeds have gaps or financial news sentiment data is sporadically available, techniques that assume MAR can be used to fill these gaps based on other observed market conditions or related news events, allowing algorithms to maintain continuous operation.
• Portfolio Management: When constructing and rebalancing portfolios, missing data on asset characteristics, such as volatility or correlation, can affect optimal portfolio allocation strategies. Assuming MAR, missing values can be estimated based on observable factors like asset class, market capitalization, or industry sector.

Limitations and Criticisms

Despite its utility, the "missing at random" (MAR) assumption has significant limitations and is a frequent subject of criticism in statistical and financial analysis. The primary challenge is that MAR cannot be directly tested or verified from the observed data alone. This means that while a statistical model might assume MAR, there's no definitive way to prove this assumption holds true in the real world based solely on the data available. Researchers must rely on subject matter expertise and logical reasoning to assess its plausibility.

One key critique highlights that when dealing with missingness across multiple variables, the plausibility of the MAR assumption becomes even more complex and stringent than often appreciated. If the underlying reason for data being missing is related to the unobserved value itself, even after accounting for all observed variables, then the data are "missing not at random" (MNAR), and methods relying on MAR will produce biased results. For instance, if high-income individuals are systematically less likely to report their full income because they have high income (and prefer privacy), then the data is MNAR.

Furthermore, incorrectly assuming MAR when the true mechanism is MNAR can lead to inaccurate parameter estimates, underestimated variance, and potentially flawed conclusions, impacting everything from hypothesis testing to capital allocation decisions. Critics also point out that the precise definition of MAR, as outlined by Rubin, has been inconsistently understood and applied, leading to confusion about when it is truly valid to ignore the missingness mechanism for inference. This ambiguity underscores the importance of conducting sensitivity analysis to understand how different assumptions about missing data might affect the final results.

Missing at Random vs. Missing Not at Random

"Missing at random" (MAR) and "missing not at random" (MNAR) are two distinct mechanisms for incomplete data, with MNAR posing a greater challenge for data analysis. The key difference lies in whether the missingness is explained by the observed data or by the unobserved (missing) data itself.

Essentially, if the missingness can be predicted from the information you do have, it's MAR. If the missingness itself provides unique information about the missing value that isn't captured elsewhere, it's missing not at random, which is generally more difficult to address without introducing significant bias or requiring strong, untestable assumptions.

FAQs

Q: Why is understanding "missing at random" important in finance?

A: Understanding missing at random is crucial in finance because financial datasets frequently contain incomplete information due to various factors like reporting errors, data collection issues, or non-responses. Recognizing when data is missing at random allows financial professionals to apply appropriate statistical techniques, such as multiple imputation, to estimate missing values. This helps maintain the integrity of the data, reduces sampling error, and ensures that analyses, models, and investment decisions are based on the most complete and accurate information possible, preventing biased outcomes.

Q: Can you always assume data is missing at random?

A: No, you cannot always assume data is missing at random. While it's a common and often convenient assumption for applying advanced imputation techniques, the missing at random condition is fundamentally untestable using the observed data alone. Analysts must rely on a deep understanding of the data collection process, domain knowledge, and logical reasoning to assess the plausibility of the MAR assumption. If the assumption is incorrect and the data are "missing not at random," applying MAR-based methods can lead to biased estimates and misleading conclusions.

Q: What happens if I treat missing not at random data as missing at random?

A: If you treat data that is "missing not at random" (MNAR) as "missing at random" (MAR), the results of your analysis are likely to be biased. This is because MNAR implies that the reason for the data being missing is related to the unobserved value itself, even after accounting for other observed variables. By incorrectly assuming MAR, your imputation methods will fail to capture this underlying systematic pattern, leading to inaccurate estimates, incorrect probability distribution assumptions, and potentially flawed conclusions about financial trends, risks, or performance. It can distort relationships between variables and reduce the statistical power of your analysis.