The One-Way Fixed Effects Model

The TSCSREG Procedure

The One-Way Fixed Effects Model

The specification for the one-way fixed effects model is

$u_{it}={\nu}_{i}+{\epsilon}_{it}$

where the ${{\nu}_{i}}$ s are nonrandom. Since including both the intercept and all the ${{\nu}_{i}}$ s induces a redundancy (unless the intercept is suppressed with the NOINT option), the ${{\nu}_{i}}$ estimates are reported under the restriction that ${{\nu}_{N}=0}$ .

Let Q₀ = diag(E_{T_i}), with ${{\bar{J}}_{T_{i}}=J_{T_{i}}/T_{i}}$ and ${E_{T_{i}}=I_{T_{i}}-{\bar J}_{T_{i}}}$ .

The estimators for the intercept and the fixed effects are given by the usual OLS expressions.

If ${\tilde{X}_{s}=Q_{0}X_{s}}$ and ${\tilde{y}=Q_{0}y}$ , the estimator of the slope coefficients is given by

$\tilde{{\beta}}_{s}=( \tilde{X}^{'}_{s} \tilde{X}_{s})^{-1} \tilde{X}^{'}_{s}\tilde{y}$

The estimator of the error variance is

$\hat{{\sigma}}_{{\epsilon}}=\tilde{u}^{'}Q_{0} \tilde{u} / (M-N-(K-1))$

where the residuals ${\tilde{u}}$ are given by ${\tilde{u}}= (I_{M}-J_{M} j{'}_{M}/M)(y -X_{s}{\tilde{\beta}}_{s})$ if there is an intercept and by ${\tilde{u}= (y-X_{s}\tilde{\beta}_{s}) }$ if there is not.

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