11.4 Repeated Cross-Sectional Data

Repeated cross-sectional data consists of multiple independent cross-sections collected at different points in time. Unlike panel data, where the same individuals are tracked over time, repeated cross-sections draw a fresh sample in each wave.

This approach allows researchers to analyze aggregate trends over time, but it does not track individual-level changes.

Examples

• General Social Survey (GSS) (U.S.) – Conducted every two years with a new sample of respondents.
• Political Opinion Polls – Monthly voter surveys to track shifts in public sentiment.
• National Health Surveys – Annual studies with fresh samples to monitor population-wide health trends.
• Educational Surveys – Sampling different groups of students each year to assess learning outcomes.

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11.4.1 Key Characteristics

1. Fresh Sample in Each Wave
   • Each survey represents an independent cross-section.
   • No respondent is tracked across waves.
2. Population-Level Trends Over Time
   • Researchers can study how the distribution of characteristics (e.g., income, attitudes, behaviors) changes over time.
   • However, individual trajectories cannot be observed.
3. Sample Design Consistency
   • To ensure comparability across waves, researchers must maintain consistent:
     • Sampling methods
     • Questionnaire design
     • Definitions of key variables

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11.4.2 Statistical Modeling for Repeated Cross-Sections

Since repeated cross-sections do not track the same individuals, specific regression methods are used to analyze changes over time.

1. Pooled Cross-Sectional Regression (Time Fixed Effects)

Combines multiple survey waves into a single dataset while controlling for time effects:

$$y_i = \mathbf{x}_i \beta + \delta_1 y_1 + ... + \delta_T y_T + \epsilon_i$$

where:

• $y_i$ is the outcome for individual $i$,

• $\mathbf{x}_i$ are explanatory variables,

• $y_t$ are time period dummies,

• $\delta_t$ captures the average change in outcomes across time periods.

Key Features:

• Allows for different intercepts across time periods, capturing shifts in baseline outcomes.

• Tracks overall population trends without assuming a constant effect of $\mathbf{x}_i$ over time.

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2. Allowing for Structural Change in Pooled Cross-Sections (Time-Dependent Effects)

To test whether relationships between variables change over time (structural breaks), interactions between time dummies and explanatory variables can be introduced:

$$y_i = \mathbf{x}_i \beta + \mathbf{x}_i y_1 \gamma_1 + ... + \mathbf{x}_i y_T \gamma_T + \delta_1 y_1 + ...+ \delta_T y_T + \epsilon_i$$

• Interacting $x_i$ with time period dummies allows for:
  • Different slopes for each time period.
  • Time-dependent effects of explanatory variables.

Practical Application:

• If $\mathbf{x}_i$ represents education level and $y_t$ represents survey year, an interaction term can test whether the effect of education on income has changed over time.

• Structural break tests help determine whether such time-varying effects are statistically significant.

• Useful for policy analysis, where a policy might impact certain subgroups differently across time.

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3. Difference-in-Means Over Time

A simple approach to comparing aggregate trends:

$$\bar{y}_t - \bar{y}_{t-1}$$

• Measures whether the average outcome has changed over time.
• Common in policy evaluations (e.g., assessing the effect of minimum wage increases on average income).

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4. Synthetic Cohort Analysis

Since repeated cross-sections do not track individuals, a synthetic cohort can be created by grouping observations based on shared characteristics:

• Example: If education levels are collected over multiple waves, we can track average income changes within education groups to approximate trends.

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11.4.3 Advantages of Repeated Cross-Sectional Data

Advantage	Explanation
Tracks population trends	Useful for studying shifts in demographics, attitudes, and economic conditions over time.
Lower cost than panel data	Tracking individuals across multiple waves (as in panel studies) is expensive and prone to attrition.
No attrition bias	Unlike panel surveys, where respondents drop out over time, each wave draws a new representative sample.
Easier implementation	Organizations can design a single survey protocol and repeat it at set intervals without managing panel retention.

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11.4.4 Disadvantages of Repeated Cross-Sectional Data

Disadvantage	Explanation
No individual-level transitions	Cannot track how specific individuals change over time (e.g., income mobility, changes in attitudes).
Limited causal inference	Since we observe different people in each wave, we cannot directly infer individual cause-and-effect relationships.
Comparability issues	Small differences in survey design (e.g., question wording or sampling frame) can make it difficult to compare across waves.

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To ensure valid comparisons across time:

• Consistent Sampling: Each wave should use the same sampling frame and methodology.
• Standardized Questions: Small variations in question wording can introduce inconsistencies.
• Weighting Adjustments: If sampling strategies change, apply survey weights to maintain representativeness.
• Accounting for Structural Changes: Economic, demographic, or social changes may impact comparability.

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