12.3 Repeated Cross Sections

For each time point (day, month, year, etc.), a set of data is sampled. This set of data can be different among different time points.

For example, you can sample different groups of students each time you survey.

Allowing structural change in pooled cross section

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

Dummy variables for all but one time period

• allows different intercept for each time period
• allows outcome to change on average for each time period

Allowing for structural change in pooled cross section

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

Interact \(x_i\) with time period dummy variables

• allows different slopes for each time period
• allows effects to change based on time period (structural break)
• Interacting all time period dummies with \(x_i\) can produce many variables - use hypothesis testing to determine which structural breaks are needed.

12.3.1 Pooled Cross Section

\[ y_i=\mathbf{x_i\beta +x_i \times y1\gamma_1 + ...+ x_i \times yT\gamma_T + \delta_1y_1+...+ \delta_Ty_T + \epsilon_i} \]

Interact \(x_i\) with time period dummy variables

• allows different slopes for each time period

• allows effect to change based on time period (structural break)

  • interacting all time period dummies with \(x_i\) can produce many variables - use hypothesis testing to determine which structural breaks are needed.