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 , where is the dependent variable, is the independent variable, and are coefficients, and are indices for individuals and time. The error 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, is assumed to vary non-stochastically over or making the fixed effects model analogous to a dummy variable model in one dimension. In a random effects model, is assumed to vary stochastically over or 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. Independently pooled panels Key assumption: There are no unique attributes of individuals within the measurement set, and no universal effects across time. Fixed effect models Key assumption: There are unique attributes of individuals that do not vary over time. That is, the unique attributes for a given individual are time invariant. These attributes may or may not be correlated with the individual dependent variables yi. To test whether fixed effects, rather than random effects, is needed, the Durbin–Wu–Hausman test can be used. Random effect models Key assumption: There are unique, time constant attributes of individuals that are not correlated with the individual regressors. Pooled OLS can be used to derive unbiased and consistent estimates of parameters even when time constant attributes are present, but random effects will be more efficient. Fixed effects is a feasible generalised least squares technique which is asymptotically more efficient than Pooled OLS when time constant attributes are present. Random effects adjusts for the serial correlation which is induced by unobserved time constant attributes. Models with instrumental variables In the standard random effects (RE) and fixed effects (FE) models, independent variables are assumed to be uncorrelated