Panel analysis

Panel (data) analysis is a statistical method, widely used in social science, epidemiology, and econometrics to analyze two-dimensional (typically cross sectional and longitudinal) panel data. The data are usually collected over time and over the same individuals and then a regression is run over these two dimensions. Multidimensional analysis is an econometric method in which data are collected over more than two dimensions (typically, time, individuals, and some third dimension). A common panel data regression model looks like y_{it}=a+bx_{it}+\varepsilon_{it} , where y is the dependent variable, x is the independent variable, a and b are coefficients, i and t are indices for individuals and time. The error \varepsilon_{it} is very important in this analysis. Assumptions about the error term determine whether we speak of fixed effects or random effects. In a fixed effects model, \varepsilon_{it} is assumed to vary non-stochastically over i or t making the fixed effects model analogous to a dummy variable model in one dimension. In a random effects model, \varepsilon_{it} is assumed to vary stochastically over i or t requiring special treatment of the error variance matrix. Panel data analysis has three more-or-less independent approaches: *independently pooled panels; *random effects models; *fixed effects models or first differenced models.…

Panel (data) analysis is a statistical method, widely used in social science, epidemiology, and econometrics to analyze two-dimensional (typically cross sectional and longitudinal) panel data. The data are usually collected over time and over the same individuals and then a regression is run over these two dimensions. Multidimensional analysis is an econometric method in which data are collected over more than two dimensions (typically, time, individuals, and some third dimension). A common panel data regression model looks like y_{it}=a+bx_{it}+\varepsilon_{it} , where y is the dependent variable, x is the independent variable, a and b are coefficients, i and t are indices for individuals and time. The error \varepsilon_{it} is very important in this analysis. Assumptions about the error term determine whether we speak of fixed effects or random effects. In a fixed effects model, \varepsilon_{it} is assumed to vary non-stochastically over i or t making the fixed effects model analogous to a dummy variable model in one dimension. In a random effects model, \varepsilon_{it} is assumed to vary stochastically over i or t requiring special treatment of the error variance matrix. Panel data analysis has three more-or-less independent approaches: independently pooled panels; random effects models; fixed effects models or first differenced models. The selection between these methods depends upon the objective of the analysis, and the problems concerning the exogeneity of the explanatory variables. y_{it}=x_{it}\beta+c_i+u_{it} where c_i is unobserved unit-specific time-invariant effect (call it unobserved effect) and x_{it} can be correlated with u_{is} for s possibly different from t. Suppose there exists a set of valid instruments z_i=(z_{i1},\ldots,z_{it}) . In REIV setting, key assumptions include that z_i is uncorrelated with c_i as well as u_{it} for t=1,\ldots,T . In fact, for REIV estimator to be efficient, conditions stronger than uncorrelatedness between instruments and unobserved effect are necessary. On the other hand, FEIV estimator only requires that instruments be exogenous with error terms after conditioning on unobserved effect i.e. E[u_{it} \mid z_i,c_i]=0[1] . The FEIV condition allows for arbitrary correlation between instruments and unobserved effect. However, this generality does not come for free: time-invariant explanatory and instrumental variables are not allowed. As in the usual FE method, the estimator uses time-demeaned variables to remove unobserved effect. Therefore, FEIV estimator would be of limited use if variables of interest include time-invariant ones. The above discussion has parallel to the exogenous case of RE and FE models. In the exogenous case, RE assumes uncorrelatedness between explanatory variables and unobserved effect, and FE allows for arbitrary correlation between the two. Similar to the standard case, REIV tends to be more efficient than FEIV provided that appropriate assumptions hold. y_{it}=a+bx_{it}+\rho y_{it-1}+\varepsilon_{it} The assumptions of the fixed effect and random effect models are violated in this setting. Instead, practitioners use a technique like the Arellano–Bond estimator.

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