# Multidimensional panel data

In econometrics, panel data is data observed over two dimensions (typically, time and cross-sections). A panel data set is termed "multidimensional" when the phenomenon is observed over three or more dimensions. An example is a data set containing forecasts produced by multiple individuals (the first dimension) forecasting multiple macroeconomic variables (the second dimension) at multiple time horizons (the third dimension) and for multiple target periods (the fourth dimension).

## Analysis of panel data

A multidimensional panel with four dimensions can have the form

$X_{isth}, \; i = 1, \dots, N, \; s = 1, \dots, S, \; t = 1, \dots, T, \; h = 1, \dots, H, \,$

where i is the individual dimension, s is the series dimension, t is the time dimension, and h is the horizon dimension. A general multidimensional panel data regression model is written as

$y_{isth} = \alpha + X_{sith} \beta + u_{sith}.\,$

Complex assumptions can be made on the precise structure of the correlations among errors in this model. For example, serial correlation (error terms correlated across time) has multiple distinct meanings. Error terms can be correlated across time for the same series, individual, and horizon. They can be correlated across time and across series for the same individual and horizon, etc. Similarly, heteroskedasticity can be defined across individuals for the same series, time, and horizon, across individuals and different series for the same time and horizon, etc.

## Data sets which have a multidimensional panel design

• Blue Chip Survey of Professional Forecasters
• Survey of Professional Forecasters (ASA-NBER)