# Panel data

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In statistics the term panel data refers to two-dimensional data.

Data is broadly classified according to the number of dimensions. A data set containing observations on a single phenomenon observed over multiple time periods is called time series. In time series data, both the values and the ordering of the data points have meaning. A data set containing observations on multiple phenomena observed at a single point in time is called cross-sectional. In cross-sectional data sets, the values of the data points have meaning, but the ordering of the data points does not. A data set containing observations on multiple phenomena observed over multiple time periods is called panel data. Panel Data aggregates all the individuals, and analyzes them in a period of time. Alternatively, the second dimension of data may be some entity other than time. For example, when there is a sample of groups, such as siblings or families, and several observations from every group, the data are panel data. Whereas time series and cross-sectional data are both one-dimensional, panel data sets are two-dimensional.

Data sets with more than two dimensions are typically called multidimensional panel data.

## ExampleEdit

balanced panel: unbalanced panel:
$\begin{matrix} \mathrm{person} & \mathrm{year} & \mathrm{income} & \mathrm{age} & \mathrm{sex}\\ 1 & 2003 & 1500 & 27 & 1 \\ 1 & 2004 & 1700 & 28 & 1 \\ 1 & 2005 & 2000 & 29 & 1 \\ 2 & 2003 & 2100 & 41 & 2 \\ 2 & 2004 & 2100 & 42 & 2 \\ 2 & 2005 & 2200 & 43 & 2 \end{matrix}$ $\begin{matrix} \mathrm{person} & \mathrm{year} & \mathrm{income} & \mathrm{age} & \mathrm{sex}\\ 1 & 2003 & 1500 & 27 & 1 \\ 1 & 2004 & 1700 & 28 & 1 \\ 2 & 2003 & 2100 & 41 & 2 \\ 2 & 2004 & 2100 & 42 & 2 \\ 2 & 2005 & 2200 & 43 & 2 \\ 3 & 2004 & 3000 & 35 & 1 \end{matrix}$

In the example above, two data sets with a two-dimensional panel structure are shown. Individual characteristics (income, age, sex) are collected for different persons and different years. In the left data set two persons (1, 2) are observed over three years (2003, 2004, 2005). Because each person is observed every year, the left-hand data set is called a balanced panel, whereas the data set on the right hand is called an unbalanced panel, since Person 1 is not observed in year 2005 and Person 3 is not observed in 2003 or 2005.

## Analysis of panel dataEdit

A panel has the form

$X_{it}, \; i = 1, \dots, N \; t = 1, \dots, T,$
where $i$ is the individual dimension and $t$ is the time dimension. A general panel data regression model is written as $y_{it} = \alpha + \beta' X_{it} + u_{it}.$ 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. The fixed effects model is denoted as

$y_{it} = \alpha + \beta' X_{it} + u_{it},$
$u_{it} = \mu_i + \nu_{it}.$

$\mu_i$ are individual-specific, time-invariant effects (for example in a panel of countries this could include geography, climate etc.) and because we assume they are fixed over time, this is called the fixed-effects model. The random effects model assumes in addition that

$\mu_i \sim \text{i.i.d.} N(0, \sigma^2_{\mu})$

and

$\nu_{it} \sim \text{i.i.d.} N(0, \sigma^2_{\nu}),$

that is, the two error components are independent from each other.

## ReferencesEdit

Arellano, Manuel. Panel Data Econometrics, Oxford University Press 2003.

Hsiao, Cheng, 2003. Analysis of Panel Data, Cambridge University Press.

Davies, A. and Lahiri, K., 2000. "Re-examining the Rational Expectations Hypothesis Using Panel Data on Multi-Period Forecasts," Analysis of Panels and Limited Dependent Variable Models, Cambridge University Press.

Davies, A. and Lahiri, K., 1995. "A New Framework for Testing Rationality and Measuring Aggregate Shocks Using Panel Data," Journal of Econometrics 68: 205-227.

Frees, E., 2004. Longitudinal and Panel Data, Cambridge University Press.