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In statistics, a fixed effects model is a statistical model in which the model parameters are fixed or non-random quantities. This is in contrast to random effects models and mixed models in which all or some of the model parameters are random variables. In many applications including econometrics and biostatistics a fixed effects model refers to a regression model in which the group means are fixed (non-random) as opposed to a random effects model in which the group means are a random sample from a population. Generally, data can be grouped according to several observed factors. The group means could be modeled as fixed or random effects for each grouping. In a fixed effects model each group mean is a group-specific fixed quantity. In panel data where longitudinal observations exist for the same subject, fixed effects represent the subject-specific means. In panel data analysis the term fixed effects estimator (also known as the within estimator) is used to refer to an estimator for the coefficients in the regression model including those fixed effects (one time-invariant intercept for each subject). Qualitative description Such models assist in controlling for omitted variable bias due to unobserved heterogeneity when this heterogeneity is constant over time. This heterogeneity can be removed from the data through differencing, for example by subtracting the group-level average over time, or by taking a first difference which will remove any time invariant components of the model. There are two common assumptions made about the individual specific effect: the random effects assumption and the fixed effects assumption. The random effects assumption is that the individual-specific effects are uncorrelated with the independent variables. The fixed effect assumption is that the individual-specific effects are correlated with the independent variables. If the random effects assumption holds, the random effects estimator is more efficient than the fixed effects estimator. However, if this assumption does not hold, the random effects estimator is not consistent. The Durbin–Wu–Hausman test is often used to discriminate between the fixed and the random effects models. Formal model and assumptions Consider the linear unobserved effects model for N {\displaystyle N} observations and T {\displaystyle T} time periods: y i t = X i t β + α i + u i t {\displaystyle y_{it}=X_{it}\mathbf {\beta } +\alpha _{i}+u_{it}} for t = 1 , … , T {\displaystyle t=1,\dots ,T} and i = 1 , … , N {\displaystyle i=1,\dots ,N} Where: y i t {\displaystyle y_{it}} is the dependent variable observed for individual i {\displaystyle i} at time t {\displaystyle t} . X i t {\displaystyle X_{it}} is the time-variant 1 × k {\displaystyle 1\times k} (the number of independent variables) regressor vector. β {\displaystyle \beta } is the k × 1 {\displaystyle k\times 1} matrix of parameters. α i {\displaystyle \alpha _{i}} is the unobserved time-invariant individual effect. For example, the innate ability for individuals or historical and institutional factors for countries. u i t {\displaystyle u_{it}} is the error term. Unlike X i t {\displaystyle X_{it}} , α i {\displaystyle \alpha _{i}} cannot be directly observed. Unlike the random effects model where the unobserved α i {\displaystyle \alpha _{i}} is independent of X i t {\displaystyle X_{it}} for all t = 1 , . . . , T {\displaystyle t=1,...,T} , the fixed effects (FE) model allows α i {\displaystyle \alpha _{i}} to be correlated with the regressor matrix X i t {\displaystyle X_{it}} . Strict exogeneity with respect to the idiosyncratic error term u i t {\displaystyle u_{it}} is still required. Statistical estimation Fixed effects estimator Since α i {\displaystyle \alpha _{i}} is not observable, it cannot be directly controlled for. The FE model eliminates α i {\displaystyle \alpha _{i}} by de-meaning the variables using the within transformation: y i t − y ¯ i = ( X i t − X ¯ i ) β + ( α .... Discover the Chris Pueyo popular books. Find the top 100 most popular Chris Pueyo books.