Therefore, the prior variance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\sigma _{\text {d}}^{2}$ \end{document}σd2 is obtained by subtracting the expected sampling variance from an estimate of the variance of the logarithmic residuals, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $s^{2}_{\text {lr}}$ \end{document}slr2: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\sigma_{\text{d}}^{2} = \operatorname{max}\left\{ s^{2}_{\text{lr}} - \psi_{1}((m-p)/2),\,\,0.25 \right\}. $$ \end{document}σd2=maxslr2−ψ1((m−p)/2),0.25. The prior variance \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\sigma _{\text {d}}^{2}$ \end{document}σd2 is thresholded at a minimal value of 0.25 so that the dispersion estimates are not shrunk entirely to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\alpha _{\text {tr}}(\bar \mu _{i})$ \end{document}αtr(μ¯i) if the variance of the logarithmic residuals is less than the expected sampling variance.
