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**What is panel data, here is the short intro!**

**Panel data**** **are multi-dimension with time. Panel data contain phenomena obtained over time across firms or individuals.

Below is the example of Panel data

MRPP balanced panel: | MRPP unbalanced panel: | |
---|---|---|

The example above is two datasets with a panel structure. Individual characteristics (income, age, sex) are collected for different persons and years. In the left dataset, two persons (1, 2) are observed every year for three years (2016, 2017, 2018). In the right dataset, three persons, two times (person 1), three times (person 2), and one time (person 3), respectively, over three years, in particular.

(e.g., the left-hand dataset above) is a dataset in which *each* panel member (i.e., person) is observed *every* year. Consequently, if a balanced panel contains *N* panel members and *T* periods, the number of observations (*n*) in the dataset is necessarily *n* = *N*×*T*.

(e.g., the right-hand dataset above) is a dataset in which *at least one panel* member is not observed every period. Therefore, if an unbalanced panel contains *N* panel members and *T* periods, then the following strict inequality holds for the number of observations (*n*) in the dataset: *n* < *N*×*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 full format would have only two (left model) or three (right example) rows of data with additional columns for each time-varying variable (income, age).

A panel has the form

\( X_{it},\quad i=1,\dots ,N,\quad t=1,\dots ,T, \)

where \( {\displaystyle i}\) is the individual dimension and \({\displaystyle t}\) 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.

Consider a generic panel data model:

\( y_{it}=\alpha +\beta 'X_{it}+u_{it},\)

\(u_{it}=\mu _{i}+v_{it}. \)

\( \mu_{i}\) are individual-specific, time-invariant effects (for example in a panel of countries this could include geography, climate etc.) which are fixed over time., whereas \({\displaystyle v_{it}}\) is a time-varying random component.

If {\displaystyle \mu _{i}}\) is unobserved and correlated with at least one of the independent variables, then it will cause omitted variable bias in a standard OLS regression. However, panel data methods, such as the fixed effects estimator or the First-difference estimator, can be used to control for it.

If\( \mu _{i}\) 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. However, because \({\displaystyle \mu _{i}}\) is fixed over time, it will induce serial correlation in the error term of the regression. This means that more efficient estimation techniques are available. Random effects are one such method: it is a special case of feasible generalized least squares which controls for the structure of the serial correlation induced by \({\displaystyle \mu _{i}}\) .

Dynamic panel data describes the case where a of the dependent variable is used as a regressor:

\(y_{it}=\alpha +\beta 'X_{it}+\gamma y_{it-1}+u_{it},}\)

The presence of the lagged dependent variable violates strict exogeneity, that is, endogeneity may occur. The fixed effect estimator and the first differences estimator both rely on the assumption of strict exogeneity. Hence, if \({\displaystyle u_{i}}\) 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.

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