In statistics and econometrics, panel data and longitudinal data are both multi-dimensional data involving measurements over time. Panel data is a subset of longitudinal data where observations are for the same subjects each time. Time series and cross-sectional data can be thought of as special cases of panel data that are in one dimension only (one panel member or individual for the former, one time point for the latter). A literature search often involves time series, cross-sectional, or panel data. A study that uses panel data is called a longitudinal study or panel study.
Example
In the multiple response permutation procedure (MRPP) example above, two datasets with a panel structure are shown and the objective is to test whether there's a significant difference between people in the sample data. Individual characteristics (income, age, sex) are collected for different persons and different years. In the first dataset, two persons (1, 2) are observed every year for three years (2016, 2017, 2018). In the second dataset, three persons (1, 2, 3) are observed two times (person 1), three times (person 2), and one time (person 3), respectively, over three years (2016, 2017, 2018); in particular, person 1 is not observed in year 2018 and person 3 is not observed in 2016 or 2018. A balanced panel (e.g., the first dataset above) is a dataset in which each panel member (i.e., person) is observed every year. Consequently, if a balanced panel contains N {\displaystyle N} panel members and T {\displaystyle T} periods, the number of observations ( n {\displaystyle n} ) in the dataset is necessarily n = N ⋅ T {\displaystyle n=N\cdot T} . An unbalanced panel (e.g., the second dataset above) is a dataset in which at least one panel member is not observed every period. Therefore, if an unbalanced panel contains N {\displaystyle N} panel members and T {\displaystyle T} periods, then the following strict inequality holds for the number of observations ( n {\displaystyle n} ) in the dataset: n < N ⋅ T {\displaystyle n<N\cdot T} . Both datasets above are structured in the long format, which is where one row holds one observation per time. Another way to structure panel data would be the wide format where one row represents one observational unit for all points in time (for the example, the wide format would have only two (first example) or three (second example) rows of data with additional columns for each time-varying variable (income, age).
Analysis
A panel has the form
X i t , i = 1 , … , N , t = 1 , … , T , {\displaystyle X_{it},\quad i=1,\dots ,N,\quad t=1,\dots ,T,}
where i {\displaystyle i} is the individual dimension and t {\displaystyle t} is the time dimension. A general panel data regression model is written as y i t = α + β ′ X i t + u i t {\displaystyle y_{it}=\alpha +\beta 'X_{it}+u_{it}} . Different assumptions can be made on the precise structure of this general model. Two important models are the fixed effects model and the random effects model. Consider a generic panel data model:
y i t = α + β ′ X i t + u i t , {\displaystyle y_{it}=\alpha +\beta 'X_{it}+u_{it},}
u i t = μ i + v i t . {\displaystyle u_{it}=\mu _{i}+v_{it}.}
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