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.
Cross-panel data (CPD) is an innovative yet underappreciated source of information in the mathematical and statistical sciences.
CPD stands out from other research methods because it vividly illustrates how independent and dependent variables may shift between countries.
This panel data collection allows researchers to examine the connection between variables across several cross-sections and time periods and analyze the results of policy actions in other nations.
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.
periods, then the following strict inequality holds for the number of observations (
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).
A general panel data regression model is written as
Different assumptions can be made on the precise structure of this general model.
are individual-specific, time-invariant effects (e.g., in a panel of countries this could include geography, climate, etc.)
is unobserved, and correlated with at least one of the independent variables, then it will cause omitted variable bias in a standard OLS regression.
is not correlated with any of the independent variables, ordinary least squares linear regression methods can be used to yield unbiased and consistent estimates of the regression parameters.
is fixed over time, it will induce serial correlation in the error term of the regression.
Random effects is one such method: it is a special case of feasible generalized least squares which controls for the structure of the serial correlation induced by
Dynamic panel data describes the case where a lag of the dependent variable is used as regressor: The presence of the lagged dependent variable violates strict exogeneity, that is, endogeneity may occur.
is believed to be correlated with one of the independent variables, an alternative estimation technique must be used.
Instrumental variables or GMM techniques are commonly used in this situation, such as the Arellano–Bond estimator.
While estimating this we should have the proper information about the instrumental variables.