Fixed and random effects models

• When you have repeated observations per individual this is a problem and an advantage:
  • the observations are not independent
  • we can use the repetition to get better parameter estimates
• If we pooled the observations and used e.g., OLS we would have biased estimates
• If we fit fixed-effect or random-effect models which take account of the repetition we can control for fixed or random individual differences.

• In the econometrics literature these models are called `cross-sectional time-series' models, because we have time-series of observations at individual rather than aggregate level.
• If we have a small number of individuals, we can simply fit a dummy for the individual:
  
  $$y_{it} = \alpha + \delta_i + \beta \mathbf{x_{it}} + \epsilon_t$$
  
  

• This can be considered a `fixed-effects' model because the regression line is raised or lowered by a fixed amount for each individual
• If there are many individuals this cannot be done directly, but there are mathematically equivalent models which achieve the same effect
• This model is appropriate where we consider each individual to have a fixed effect shifting the $y_{it}$ up or down
• We may prefer to consider the individual differences as random disturbances drawn from some specified distribution:
  
  $$y_{it} = \alpha + \beta \mathbf{x_{it}} + \nu_i + \epsilon_t$$
  
  

• This has the advantage of using fewer degrees of freedom, and that individual differences are considered random rather than fixed and estimable.
• It has the disadvantage of requiring no correlation between the regressors (the $x_{it}$s) and the $\nu_i$: there are tests for this assumption (Hausman test).

  The xt series of commands provide tools for analyzing cross-sectional time-
  series (panel) datasets:
          help xtdes      Describe pattern of xt data
          help xtsum      Summarize xt data
          help xttab      Tabulate xt data
          help xtreg      Fixed-, between- and random-effects, and population-
                              averaged linear models
          help xtdata     Faster specification searches with xt data
          help xtlogit    Fixed-effects, random-effects, & population-averaged
                              logit models
          help xtprobit   Random-effects and population-averaged probit models
          help xttobit    Random-effects tobit models
          help xtpois     Fixed-effects, random-effects, & population-averaged
                              Poisson models
          help xtnbreg    Fixed-effects, random-effects, & population-averaged
                              negative binomial models
          help xtclog     Random-effects and population-averaged cloglog models
          help xtintreg   Random-effects interval data regression models
          help xtrchh     Hildreth-Houck random coefficients models
          help xtgls      Panel-data models using GLS
          help xtgee      Population-averaged panel-data models using GEE
  

• Fitting these models in Stata is easy:
  • With data in long format, one record per individual per wave
  • . xtreg yvar x1 x2, fe i(pid)
  • . xtreg yvar x1 x2, re i(pid)
  • . xtlogit yvar x1 x2, re i(pid)
• Reference: William Greene, Econometric Analysis, Maxwell Macmillan 1991, Ch 16 section 4.