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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 ets 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 eT=0. If you specify the NOINT option to suppress the intercept, only the restriction eT=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 Mt be the number of cross sections observed in year t and let {\sum_{t}M_{t}=M}. Let Dt be the Mt × 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(D1jN,D2 jN, ... ... DTjN). The matrix Z gives the dummy variable structure for the two-way model.


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

( 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

/ (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}
}if there is an intercept in the model and by {\tilde{u}=y_{{\ast}}-X_{{\ast} s}
}if there is no intercept.

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Copyright © 1999 by SAS Institute Inc., Cary, NC, USA. All rights reserved.