▁and ▁corresponding ▁parameters ▁in ▁the ▁regression ▁can ▁be ▁log ically ▁split ▁into ▁two ▁groups , ▁so ▁that ▁the ▁regression ▁takes ▁form ▁▁ ▁where ▁X 1 ▁and ▁X 2 ▁have ▁dimensions ▁n × p 1 , ▁n × p 2 , ▁and ▁ β 1 , ▁ β 2 ▁are ▁p 1 × 1 ▁and ▁p 2 × 1 ▁vectors , ▁with ▁. ▁ ▁The ▁Fr isch – W augh – L ov ell ▁theorem ▁states ▁that ▁in ▁this ▁regression ▁the ▁resid uals ▁ ▁and ▁the ▁O LS ▁estimate ▁ ▁will ▁be ▁numer ically ▁identical ▁to ▁the ▁resid uals ▁and ▁the ▁O LS ▁estimate ▁for ▁ β 2 ▁in ▁the ▁following ▁regression : ▁▁ ▁where ▁M 1 ▁is ▁the ▁anni h il ator ▁matrix ▁for ▁re gress ors ▁X 1 . ▁ ▁The ▁theorem ▁can ▁be ▁used ▁to ▁establish ▁a ▁number ▁of ▁theoretical ▁results . ▁For ▁example , ▁having ▁a ▁regression ▁with ▁a ▁constant ▁and ▁another ▁re gress or ▁is ▁equivalent ▁to ▁subtract ing ▁the ▁means ▁from ▁the ▁dependent ▁variable ▁and ▁the ▁re gress or ▁and ▁then ▁running ▁the ▁regression ▁for ▁the ▁de - mean ed ▁variables ▁but ▁without ▁the ▁constant ▁term . ▁▁ ▁Con stra ined ▁estimation ▁▁ ▁Suppose ▁it ▁is ▁known ▁that ▁the ▁coefficients ▁in ▁the ▁regression ▁satisfy ▁a ▁system ▁of ▁linear ▁equations ▁▁ ▁where ▁Q ▁is ▁a ▁p × q ▁matrix ▁of ▁full ▁rank , ▁and ▁c ▁is ▁a ▁q × 1 ▁vector ▁of ▁known ▁constants , ▁where ▁. ▁In ▁this ▁case ▁least ▁squares ▁estimation ▁is ▁equivalent ▁to ▁minim izing ▁the