Fixed Effects Regression
Comparing and Contrasting With Multilevel Modeling.
1 Acknowledgement
This presentation of these ideas draws heavily upon the Stata documentation, although I have changed the notation slightly, and drawn out a few steps.
2 Derivation
2.1 A Regression Model With Person Specific Effects
We start with our regression equation.
\[y_{it} = \beta_0 + \beta_1 x_{it} + u_{0i} + e_{it} \tag{1}\]
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2.1.1 Intermediate Step 1
We can sum both sides over the \(t\) time points.
\[\sum_{t} y_{it} = \sum_{t} \beta_0 + \sum_{t} \beta_1 x_{it} + \sum_{t} u_{0i} + \sum_{t} e_{it}\]
2.1.2 Intermediate Step 2
We can then divide by the number of time points (\(T_i\)).
\[\sum_{t} y_{it} / T_i = \sum_{t} \beta_0 / T_i + \sum_{t} \beta_1 x_{it} / T_i + \sum_{t} u_{0i} / T_i + \sum_{t} e_{it} / T_i\]
2.2 The Between Estimator
If Equation 1 is true, then the below must also be true:
\[\bar y_i = \beta_0 + \beta_1 \bar x_i + u_{0i} + \bar e_i \tag{2}\]
This is sometimes called the between estimator.
2.3 The Fixed Effects Estimator
We can subtract Equation 2 from Equation 1.
\[y_{it} - \bar y_i = \beta_1 (x_{it} - \bar x_i) + (e_{it} - \bar e_i) \tag{3}\]
Equation 3 is the fixed effects (or within) estimator.