documentclass[english]{scrartcl}
usepackage[T1]{fontenc}
usepackage[latin9]{inputenc}
usepackage{textcomp}
usepackage{amstext}
usepackage{babel}
begin{document}
As an application of these results, consider the case in which $mathbf{X}_{1}$
is $mathbf{i}$, a constant term that is a column of 1s in the first
column of $mathbf{X}$. The solution for $b_{2}$ in this case will
then be the slopes in a regression that contains a constant term.
Using Theorem 3.2 the vector of residuals for any variable in $mathbf{X}_{2}$
in this case will be

$mathbf{xtext{textasteriskcentered}}=mathbf{x}-mathbf{X}_{1}(mathbf{X}_{1}'mathbf{X}_{1})^{-1}mathbf{X}_{1}mathbf{x}$

$textrm{xtext{textasteriskcentered}}=textrm{x}-textrm{X}_{1}(textrm{X}_{1}'textrm{X}_{1})^{-1}textrm{X}_{1}textrm{x}$
end{document}

documentclass{article}
usepackage{amsmath}

begin{document}

$mathbf{x}text{textasteriskcentered}=mathbf{x}-mathbf{X}_{1}(mathbf{X}_{1}^{'}mathbf{X}_{2}')^{-1}mathbf{X}'_{1}mathbf{x}$


$mathrm{x}text{textasteriskcentered}=mathrm{x}-mathrm{X}_{1}(mathrm{X}_{1}^{'}mathrm{X}_{2}')^{-1}mathrm{X}'_{1}mathrm{x}$

end{document}

documentclass[english]{scrartcl}
usepackage[T1]{fontenc}
usepackage[latin9]{inputenc}
usepackage{textcomp}
usepackage{amstext}
usepackage{babel}
%%%%%
usepackage{scalerel,lmodern}
input pdf-trans
newboxqbox
defusecolor#1{csnamestringcolor@#1endcsnamespace}
newcommandoutline[1]{leavevmode%
  defmaltext{#1}%
  setboxqbox=hbox{maltext}%
  boxgs{Q q 2 Tr thicknessspace w 0 0 0 rg 0 G}{}%
  copyqbox%
}
newcommandmathbbf[2][.2]{%
  defthickness{#1}%
  ThisStyle{outline{$mathbf{SavedStyle#2}$}}%
}
%%%%%
begin{document}
$mathbbf{xtext{textasteriskcentered}}=mathbbf{x}-mathbbf{X}_{1}(mathbbf{X}_{1}'mathbbf{X}_{1})^{-1}mathbbf{X}_{1}mathbbf{x}$

$mathbf{xtext{textasteriskcentered}}=mathbf{x}-mathbf{X}_{1}(mathbf{X}_{1}'mathbf{X}_{1})^{-1}mathbf{X}_{1}mathbf{x}$

$textrm{xtext{textasteriskcentered}}=textrm{x}-textrm{X}_{1}(textrm{X}_{1}'textrm{X}_{1})^{-1}textrm{X}_{1}textrm{x}$
end{document}