LSDV is identical to the Within estimator. Prove that given in (2.7) can be obtained from

Question:

LSDV is identical to the Within estimator. Prove that
β given in (2.7) can be obtained from OLS on (2.5) using results on partitioned inverse. This can be easily obtained using the Frisch–Waugh–Lovell theorem of Davidson and MacKinnon (1993, p. 19). Hint: this theorem states that the OLS estimate of β
from (2.5) will be identical to the OLS estimate of β from (2.6). Also, the least squares residuals will be the same. See Chap. 1 of Baltagi (2009).