The Two-Way Fixed Effects Model

The TSCSREG Procedure

The Two-Way Fixed Effects Model

The specification for the two-way fixed effects model is

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

where the ${{\nu}_{i}}$ s and e_ts are nonrandom. If you do not specify the NOINT option, which suppresses the intercept, the estimates for the fixed effects are reported under the restriction that ${{\nu}_{N}=0}$ and e_T=0. If you specify the NOINT option to suppress the intercept, only the restriction e_T=0 is imposed.

Let X_* and y_* be the independent and dependent variables arranged by time and by cross section within each time period. Let M_t be the number of cross sections observed in year t and let ${\sum_{t}M_{t}=M}$ . Let D_t be the M_t × N matrix obtained from the N × N identity matrix from which rows corresponding to cross sections not observed at time t have been omitted. Consider

Z = (Z₁, Z₂)

where Z₁ = ( D'₁, D'₂, ... .. D'_T)'and Z₂ = diag(D₁j_N,D₂ j_N, ... ... D_Tj_N). The matrix Z gives the dummy variable structure for the two-way model.

Let

${\Delta}_{N}= Z^{'}_{1}Z_{1},\hspace*{1em} {\Delta}_{T}= Z^{'}_{2}Z_{2},\hspace*{1em} A= Z^{'}_{2}Z_{1}$

${\bar{Z}}=Z_{2}-Z_{1} {\Delta}^{-1}_{N}A^{'}$

$Q={\Delta}_{T}-A {\Delta}^{-1}_{N} A^{'}$

$P=(I_{M}-Z_{1} {\Delta}^{-1}_{N} Z^{'}_{1}) - {\bar{Z}}Q^{-}{\bar Z}^{'}$

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

The estimate of the regression slope coefficients is given by

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

where ${X_{{\ast} s}}$ is the ${X_{{\ast}}}$ matrix without the vector of 1s.

The estimator of the error variance is

$\hat{{\sigma}}^2_{{\epsilon}}= \tilde{u}^{'}P\tilde{u} / (M-T-N+1-(K-1))$

where the residuals are given by ${\tilde{u}=(I_{M}-j_{M} j^{'}_{M}/ M) (y_{{\ast}}-X_{{\ast} s} \tilde{{\beta}}_{s}) }$ if there is an intercept in the model and by ${\tilde{u}=y_{{\ast}}-X_{{\ast} s} \tilde{{\beta}}_{s} }$ if there is no intercept.

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