A two-way fixed effects model. Suppose that the fixed effects model is modified to include a time-specific dummy variable as well as an individual-specific variable. Then y_it= α_i+ γt + x_itβ + ε_it. At every observation, the individual- and time-specific dummy variables sum to 1, so there are some redundant coefficients. The discussion in Section 11.4.4 shows that one way to remove the redundancy is to include an overall constant and drop one of the time specific and one of the time-dummy variables. The model is, thus,

The respective time- or individual-specific variable is zero when t or i equals one. Ordinary least squares estimates of β are then obtained by regression of y̅_it y̅_i. y̅_.t + y̅ on x̅_it x̅_i. x̅_.t + x̅. Then (α_i ?? α₁) and (γ_t ?? γ₁) are estimated using the expressions in (11-25). Using the following data, estimate the full set of coefficients for the least squares dummy variable model:

Test the hypotheses that (1) the ??period?? effects are all zero, (2) the ??group?? effects are all zero, and (3) both period and group effects are zero. Use an F test in each case.

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Ya = µ + (a; – a1) + (y – n1) + Xqß + Eit-