%\VignetteEngine{knitr::knitr}
%\VignetteIndexEntry{Asymptotic Distribution of the Markowitz Portfolio}
%\VignetteKeyword{Finance}
%\VignetteKeyword{Markowitz}
%\VignettePackage{MarkowitzR}
\documentclass[10pt,a4paper,english]{article}
% front matter%FOLDUP
\usepackage[hyphens]{url}
\usepackage{amsmath}
\usepackage{amsfonts}
% for therefore
\usepackage{amssymb}
% for theorems?
\usepackage{amsthm}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{example}{Example}[section]
\theoremstyle{remark}
\newtheorem*{remark}{Remark}
\newtheorem*{caution}{Caution}
\newtheorem*{note}{Note}
% see http://tex.stackexchange.com/a/3034/2530
\PassOptionsToPackage{hyphens}{url}\usepackage{hyperref}
\usepackage{hyperref}
\usepackage[square,numbers]{natbib}
%\usepackage[authoryear]{natbib}
%\usepackage[iso]{datetime}
%\usepackage{datetime}
%compactitem and such:
\usepackage[newitem,newenum,increaseonly]{paralist}
\makeatletter
\makeatother
%\input{sr_defs.tex}
\usepackage{MarkowitzR}
\providecommand{\sideWarning}[1][0.5]{\marginpar{\hfill\includegraphics[width=#1\marginparwidth]{warning}}}
% knitr setup%FOLDUP
<<'preamble', include=FALSE, warning=FALSE, message=FALSE>>=
library(knitr)
# set the knitr options ... for everyone!
# if you unset this, then vignette build bonks. oh, joy.
#opts_knit$set(progress=TRUE)
opts_knit$set(eval.after='fig.cap')
# for a package vignette, you do want to echo.
# opts_chunk$set(echo=FALSE,warning=FALSE,message=FALSE)
opts_chunk$set(warning=FALSE,message=FALSE)
#opts_chunk$set(results="asis")
opts_chunk$set(cache=TRUE,cache.path="cache/asymarko")
#opts_chunk$set(fig.path="figure/",dev=c("pdf","cairo_ps"))
opts_chunk$set(fig.path="figure/asymarko",dev=c("pdf"))
opts_chunk$set(fig.width=5,fig.height=4,dpi=64)
# doing this means that png files are made of figures;
# the savings is small, and it looks like shit:
#opts_chunk$set(fig.path="figure/",dev=c("png","pdf","cairo_ps"))
#opts_chunk$set(fig.width=4,fig.height=4)
# for figures? this is sweave-specific?
#opts_knit$set(eps=TRUE)
# this would be for figures:
#opts_chunk$set(out.width='.8\\textwidth')
# for text wrapping:
options(width=64,digits=2)
opts_chunk$set(size="small")
opts_chunk$set(tidy=TRUE,tidy.opts=list(width.cutoff=50,keep.blank.line=TRUE))
compile.time <- Sys.time()
# from the environment
# only recompute if FORCE_RECOMPUTE=True w/out case match.
FORCE_RECOMPUTE <-
(toupper(Sys.getenv('FORCE_RECOMPUTE',unset='False')) == "TRUE")
# compiler flags!
# not used yet
LONG.FORM <- FALSE
mc.resolution <- ifelse(LONG.FORM,1000,200)
mc.resolution <- max(mc.resolution,100)
# library(quantmod)
# options("getSymbols.warning4.0"=FALSE)
library(MarkowitzR)
gen_norm <- rnorm
lseq <- function(from,to,length.out) {
exp(seq(log(from),log(to),length.out = length.out))
}
@
%UNFOLD
% SYMPY preamble%FOLDUP
%\usepackage{graphicx} % Used to insert images
%\usepackage{adjustbox} % Used to constrain images to a maximum size
\usepackage{color} % Allow colors to be defined
\usepackage{enumerate} % Needed for markdown enumerations to work
%\usepackage{geometry} % Used to adjust the document margins
\usepackage{amsmath} % Equations
\usepackage{amssymb} % Equations
%\usepackage[utf8]{inputenc} % Allow utf-8 characters in the tex document
%\usepackage[mathletters]{ucs} % Extended unicode (utf-8) support
\usepackage{fancyvrb} % verbatim replacement that allows latex
%\usepackage{grffile} % extends the file name processing of package graphics
% to support a larger range
% The hyperref package gives us a pdf with properly built
% internal navigation ('pdf bookmarks' for the table of contents,
% internal cross-reference links, web links for URLs, etc.)
\usepackage{hyperref}
%\usepackage{longtable} % longtable support required by pandoc >1.10
\definecolor{orange}{cmyk}{0,0.4,0.8,0.2}
\definecolor{darkorange}{rgb}{.71,0.21,0.01}
\definecolor{darkgreen}{rgb}{.12,.54,.11}
\definecolor{myteal}{rgb}{.26, .44, .56}
\definecolor{gray}{gray}{0.45}
\definecolor{lightgray}{gray}{.95}
\definecolor{mediumgray}{gray}{.8}
\definecolor{inputbackground}{rgb}{.95, .95, .85}
\definecolor{outputbackground}{rgb}{.95, .95, .95}
\definecolor{traceback}{rgb}{1, .95, .95}
% ansi colors
\definecolor{red}{rgb}{.6,0,0}
\definecolor{green}{rgb}{0,.65,0}
\definecolor{brown}{rgb}{0.6,0.6,0}
\definecolor{blue}{rgb}{0,.145,.698}
\definecolor{purple}{rgb}{.698,.145,.698}
\definecolor{cyan}{rgb}{0,.698,.698}
\definecolor{lightgray}{gray}{0.5}
% bright ansi colors
\definecolor{darkgray}{gray}{0.25}
\definecolor{lightred}{rgb}{1.0,0.39,0.28}
\definecolor{lightgreen}{rgb}{0.48,0.99,0.0}
\definecolor{lightblue}{rgb}{0.53,0.81,0.92}
\definecolor{lightpurple}{rgb}{0.87,0.63,0.87}
\definecolor{lightcyan}{rgb}{0.5,1.0,0.83}
% commands and environments needed by pandoc snippets
% extracted from the output of `pandoc -s`
\DefineShortVerb[commandchars=\\\{\}]{\|}
\DefineVerbatimEnvironment{Highlighting}{Verbatim}{commandchars=\\\{\}}
% Add ',fontsize=\small' for more characters per line
\newenvironment{Shaded}{}{}
\newcommand{\KeywordTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{\textbf{{#1}}}}
\newcommand{\DataTypeTok}[1]{\textcolor[rgb]{0.56,0.13,0.00}{{#1}}}
\newcommand{\DecValTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
\newcommand{\BaseNTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
\newcommand{\FloatTok}[1]{\textcolor[rgb]{0.25,0.63,0.44}{{#1}}}
\newcommand{\CharTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
\newcommand{\StringTok}[1]{\textcolor[rgb]{0.25,0.44,0.63}{{#1}}}
\newcommand{\CommentTok}[1]{\textcolor[rgb]{0.38,0.63,0.69}{\textit{{#1}}}}
\newcommand{\OtherTok}[1]{\textcolor[rgb]{0.00,0.44,0.13}{{#1}}}
\newcommand{\AlertTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}}
\newcommand{\FunctionTok}[1]{\textcolor[rgb]{0.02,0.16,0.49}{{#1}}}
\newcommand{\RegionMarkerTok}[1]{{#1}}
\newcommand{\ErrorTok}[1]{\textcolor[rgb]{1.00,0.00,0.00}{\textbf{{#1}}}}
\newcommand{\NormalTok}[1]{{#1}}
% Define a nice break command that doesn't care if a line doesn't already
% exist.
\def\br{\hspace*{\fill} \\* }
% Math Jax compatability definitions
\def\gt{>}
\def\lt{<}
% Pygments definitions
\makeatletter
\def\PY@reset{\let\PY@it=\relax \let\PY@bf=\relax%
\let\PY@ul=\relax \let\PY@tc=\relax%
\let\PY@bc=\relax \let\PY@ff=\relax}
\def\PY@tok#1{\csname PY@tok@#1\endcsname}
\def\PY@toks#1+{\ifx\relax#1\empty\else%
\PY@tok{#1}\expandafter\PY@toks\fi}
\def\PY@do#1{\PY@bc{\PY@tc{\PY@ul{%
\PY@it{\PY@bf{\PY@ff{#1}}}}}}}
\def\PY#1#2{\PY@reset\PY@toks#1+\relax+\PY@do{#2}}
\expandafter\def\csname PY@tok@gd\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.63,0.00,0.00}{##1}}}
\expandafter\def\csname PY@tok@gu\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.50,0.00,0.50}{##1}}}
\expandafter\def\csname PY@tok@gt\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.27,0.87}{##1}}}
\expandafter\def\csname PY@tok@gs\endcsname{\let\PY@bf=\textbf}
\expandafter\def\csname PY@tok@gr\endcsname{\def\PY@tc##1{\textcolor[rgb]{1.00,0.00,0.00}{##1}}}
\expandafter\def\csname PY@tok@cm\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
\expandafter\def\csname PY@tok@vg\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}}
\expandafter\def\csname PY@tok@m\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
\expandafter\def\csname PY@tok@mh\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
\expandafter\def\csname PY@tok@go\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.53,0.53,0.53}{##1}}}
\expandafter\def\csname PY@tok@ge\endcsname{\let\PY@it=\textit}
\expandafter\def\csname PY@tok@vc\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}}
\expandafter\def\csname PY@tok@il\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
\expandafter\def\csname PY@tok@cs\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
\expandafter\def\csname PY@tok@cp\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.74,0.48,0.00}{##1}}}
\expandafter\def\csname PY@tok@gi\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.63,0.00}{##1}}}
\expandafter\def\csname PY@tok@gh\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,0.50}{##1}}}
\expandafter\def\csname PY@tok@ni\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.60,0.60,0.60}{##1}}}
\expandafter\def\csname PY@tok@nl\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.63,0.63,0.00}{##1}}}
\expandafter\def\csname PY@tok@nn\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}}
\expandafter\def\csname PY@tok@no\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.53,0.00,0.00}{##1}}}
\expandafter\def\csname PY@tok@na\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.49,0.56,0.16}{##1}}}
\expandafter\def\csname PY@tok@nb\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
\expandafter\def\csname PY@tok@nc\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}}
\expandafter\def\csname PY@tok@nd\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.67,0.13,1.00}{##1}}}
\expandafter\def\csname PY@tok@ne\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.82,0.25,0.23}{##1}}}
\expandafter\def\csname PY@tok@nf\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,1.00}{##1}}}
\expandafter\def\csname PY@tok@si\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.73,0.40,0.53}{##1}}}
\expandafter\def\csname PY@tok@s2\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
\expandafter\def\csname PY@tok@vi\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}}
\expandafter\def\csname PY@tok@nt\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
\expandafter\def\csname PY@tok@nv\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}}
\expandafter\def\csname PY@tok@s1\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
\expandafter\def\csname PY@tok@sh\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
\expandafter\def\csname PY@tok@sc\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
\expandafter\def\csname PY@tok@sx\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
\expandafter\def\csname PY@tok@bp\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
\expandafter\def\csname PY@tok@c1\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
\expandafter\def\csname PY@tok@kc\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
\expandafter\def\csname PY@tok@c\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.25,0.50,0.50}{##1}}}
\expandafter\def\csname PY@tok@mf\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
\expandafter\def\csname PY@tok@err\endcsname{\def\PY@bc##1{\setlength{\fboxsep}{0pt}\fcolorbox[rgb]{1.00,0.00,0.00}{1,1,1}{\strut ##1}}}
\expandafter\def\csname PY@tok@kd\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
\expandafter\def\csname PY@tok@ss\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.10,0.09,0.49}{##1}}}
\expandafter\def\csname PY@tok@sr\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.40,0.53}{##1}}}
\expandafter\def\csname PY@tok@mo\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
\expandafter\def\csname PY@tok@kn\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
\expandafter\def\csname PY@tok@mi\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
\expandafter\def\csname PY@tok@gp\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.00,0.50}{##1}}}
\expandafter\def\csname PY@tok@o\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.40,0.40,0.40}{##1}}}
\expandafter\def\csname PY@tok@kr\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
\expandafter\def\csname PY@tok@s\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
\expandafter\def\csname PY@tok@kp\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
\expandafter\def\csname PY@tok@w\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.73,0.73}{##1}}}
\expandafter\def\csname PY@tok@kt\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.69,0.00,0.25}{##1}}}
\expandafter\def\csname PY@tok@ow\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.67,0.13,1.00}{##1}}}
\expandafter\def\csname PY@tok@sb\endcsname{\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
\expandafter\def\csname PY@tok@k\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.00,0.50,0.00}{##1}}}
\expandafter\def\csname PY@tok@se\endcsname{\let\PY@bf=\textbf\def\PY@tc##1{\textcolor[rgb]{0.73,0.40,0.13}{##1}}}
\expandafter\def\csname PY@tok@sd\endcsname{\let\PY@it=\textit\def\PY@tc##1{\textcolor[rgb]{0.73,0.13,0.13}{##1}}}
\def\PYZbs{\char`\\}
\def\PYZus{\char`\_}
\def\PYZob{\char`\{}
\def\PYZcb{\char`\}}
\def\PYZca{\char`\^}
\def\PYZam{\char`\&}
\def\PYZlt{\char`\<}
\def\PYZgt{\char`\>}
\def\PYZsh{\char`\#}
\def\PYZpc{\char`\%}
\def\PYZdl{\char`\$}
\def\PYZhy{\char`\-}
\def\PYZsq{\char`\'}
\def\PYZdq{\char`\"}
\def\PYZti{\char`\~}
% for compatibility with earlier versions
\def\PYZat{@}
\def\PYZlb{[}
\def\PYZrb{]}
\makeatother
% Exact colors from NB
\definecolor{incolor}{rgb}{0.0, 0.0, 0.5}
\definecolor{outcolor}{rgb}{0.545, 0.0, 0.0}
% Prevent overflowing lines due to hard-to-break entities
\sloppy
% Setup hyperref package
\hypersetup{
breaklinks=true, % so long urls are correctly broken across lines
colorlinks=true,
urlcolor=blue,
linkcolor=darkorange,
citecolor=darkgreen,
}
% Slightly bigger margins than the latex defaults
%\geometry{verbose,tmargin=1in,bmargin=1in,lmargin=1in,rmargin=1in}
%UNFOLD
%UNFOLD
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% document incantations%FOLDUP
\begin{document}
\title{Asymptotic Distribution of the \txtMwtz Portfolio}
\author{Steven E. Pav \thanks{\email{spav@alumni.cmu.edu}}}
%\date{\today, \currenttime}
\maketitle
%UNFOLD
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{abstract}%FOLDUP
The asymptotic distribution of the \txtMP, \minvAB{\svsig}{\svmu},
is derived, for the general case (assuming fourth moments of returns exist),
and for the case of multivariate normal returns.
The derivation allows for inference which is robust to heteroskedasticity
and autocorrelation of moments up to order four. As a side effect, one
can estimate the proportion of error in the \txtMP due to
mis-estimation of the covariance matrix. A likelihood ratio test is given
which generalizes Dempster's Covariance Selection test to allow inference
on linear combinations of the precision matrix and the
\txtMP. \cite{dempster1972} Extensions of the main method to deal with
hedged portfolios, conditional heteroskedasticity, and conditional
expectation are given.
\end{abstract}%UNFOLD
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}%FOLDUP
Given \nlatf assets with expected return \pvmu and covariance of return \pvsig,
the portfolio defined as
\begin{equation}
\pportwopt \defeq \lambda \minvAB{\pvsig}{\pvmu}
\end{equation}
plays a special role in modern portfolio
theory. \cite{markowitz1952portfolio,brandt2009portfolio,GVK322224764}
It is known as the `efficient portfolio', the `tangency portfolio',
and, somewhat informally, the `\txtMP'.
It appears, for various $\lambda$, in the solution to numerous
portfolio optimization problems.
Besides the classic mean-variance formulation,
it solves the (population) \txtSR maximization problem:
\begin{equation}
\max_{\pportw : \qform{\pvsig}{\pportw} \le \Rbuj^2}
\frac{\trAB{\pportw}{\pvmu} - \rfr}{\sqrt{\qform{\pvsig}{\pportw}}},
\label{eqn:opt_port_I}
\end{equation}
where $\rfr\ge 0$ is the risk-free, or `disastrous', rate of return, and
$\Rbuj > 0$ is some given `risk budget'.
The solution to this optimization problem is $\lambda \minvAB{\pvsig}{\pvmu}$,
where $\lambda = \fracc{\Rbuj}{\sqrt{\qiform{\pvsig}{\pvmu}}}.$
\nocite{markowitz1952portfolio} % does not actually mention this portfolio?
In practice, the \txtMP has a somewhat checkered history.
The population parameters \pvmu and \pvsig are not known and must
be estimated from samples. Estimation error results in a feasible
portfolio, \sportwopt, of dubious value. Michaud went so far as to call
mean-variance optimization,
``error maximization.'' \cite{michaud1989markowitz}
It has been suggested that simple portfolio heuristics outperform the
\txtMP in practice. \cite{demiguel2009optimal}
This paper focuses on the asymptotic distribution of the sample \txtMP.
By formulating the problem as a linear regression, Britten-Jones
very cleverly devised hypothesis tests on elements of \pportwopt,
assuming multivariate Gaussian returns. \cite{BrittenJones1999}
In a remarkable series of papers, Okhrin and Schmid, and
Bodnar and Okhrin give the (univariate) density of
the dot product of \pportwopt and a deterministic vector, again for the
case of Gaussian returns. \cite{okhrin2006distributional,SJOS:SJOS729}
Okhrin and Schmid also show that all moments of
$\fracc{\sportwopt}{\trAB{\vone}{\sportwopt}}$ of order greater than or
equal to one do not exist. \cite{okhrin2006distributional}
Here I derive asymptotic normality of \sportwopt, the sample
analogue of \pportwopt, assuming only that the first four moments
exist. Feasible estimation of the variance of \sportwopt is amenable
to heteroskedasticity and autocorrelation robust
inference. \cite{Zeileis:2004:JSSOBK:v11i10}
The asymptotic distribution under Gaussian returns is also derived.
After estimating the covariance of \sportwopt, one can compute Wald
test statistics for the elements of \sportwopt, possibly leading one
to drop some assets from consideration (`sparsification'). Having
an estimate of the covariance can also allow portfolio shrinkage.
\cite{demiguel2013size,kinkawa2010estimation}
The derivations in this paper actually solve a more general problem
than the distribution of the sample \txtMP. The covariance of
\sportwopt and the `precision matrix,' \minv{\svsig} are derived.
This allows one, for example, to estimate the proportion of error
in the \txtMP attributable to mis-estimation of the covariance
matrix. According to lore, the error in portfolio weights is
mostly attributable to mis-estimation of \pvmu,
not of \pvsig. \cite{chopra1993effect,NBERw0444}
Finally, assuming Gaussian returns, a likelihood ratio test for
performing inference on linear combinations of elements of
the \txtMP and the precision matrix is derived. This test
generalizes a procedure by Dempster for inference on
the precision matrix alone. \cite{dempster1972}
% \cite{tu2011markowitz} % is this needed? really. talmud ...
%UNFOLD
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The augmented second moment}%FOLDUP
Let \vreti be an array of returns of \nlatf assets, with mean \pvmu, and
covariance \pvsig. Let \avreti be \vreti prepended with a 1:
$\avreti = \asvec{1,\tr{\vreti}}$. Consider the second
moment of \avreti:
\begin{equation}
\pvsm \defeq \E{\ogram{\avreti}} =
\twobytwo{1}{\tr{\pvmu}}{\pvmu}{\pvsig + \ogram{\pvmu}}.
\label{eqn:pvsm_def}
\end{equation}
By inspection one can confirm that the
inverse of \pvsm is
\begin{equation}
\minv{\pvsm}
= \twobytwo{1 + \qiform{\pvsig}{\pvmu}}{-\tr{\pvmu}\minv{\pvsig}}{-\minv{\pvsig}\pvmu}{\minv{\pvsig}}
= \twobytwo{1 + \psnrsqopt}{-\tr{\pportwopt}}{-\pportwopt}{\minv{\pvsig}},
\label{eqn:trick_inversion}
\end{equation}
where $\pportwopt=\minvAB{\pvsig}{\svmu}$ is the \txtMP,
and $\psnropt=\sqrt{\qiform{\pvsig}{\pvmu}}$ is the \txtSR of
that portfolio. The matrix \pvsm contains the first and second
moment of \vreti, but is also the uncentered second moment of
\avreti, a fact which makes it amenable to analysis via the
central limit theorem.
The relationships above are merely facts of linear algebra, and so
hold for sample estimates as well:
\begin{equation*}
\minv{\twobytwo{1}{\tr{\svmu}}{\svmu}{\svsig + \ogram{\svmu}}}
= {\twobytwo{1 +
\ssrsqopt}{-\tr{\sportwopt}}{-\sportwopt}{\minv{\svsig}}},
\end{equation*}
where \svmu, \svsig are some sample estimates of \pvmu and \pvsig, and
$\sportwopt = \minvAB{\svsig}{\svmu}, \ssrsqopt = \qiform{\svsig}{\svmu}$.
Given \ssiz \iid observations \vreti[i], let \amreti be the matrix
whose rows are the vectors \tr{\avreti[i]}. The na\"{i}ve sample estimator
\begin{equation}
\svsm \defeq \oneby{\ssiz}\gram{\amreti}
\end{equation}
is an unbiased estimator since $\pvsm = \E{\gram{\avreti}}$.
\subsection{Matrix derivatives}%FOLDUP
\label{subsec:matrix_derivatives}
Some notation and technical results concerning matrices are required.
\begin{definition}[Matrix operations]%FOLDUP
For matrix \Mtx{A},
let \fvec{\Mtx{A}}, and \fvech{\Mtx{A}} be the vector and half-space
vector operators. The former turns an $\nlatf\times\nlatf$ matrix into
an $\nlatf^2$ vector of its columns stacked on top of each other; the
latter vectorizes a symmetric (or lower triangular) matrix into a vector
of the non-redundant elements.
Let \Elim be the `Elimination Matrix,' a matrix of zeros and ones with the
property that
$\fvech{\Mtx{A}} = \Elim\fvec{\Mtx{A}}.$ The `Duplication Matrix,' \Dupp,
is the matrix of zeros and ones that reverses this operation:
$\Dupp \fvech{\Mtx{A}} = \fvec{\Mtx{A}}.$ \cite{magnus1980elimination}
Note that this implies that
$$\Elim\Dupp = \eye \wrapParens{\ne \Dupp\Elim}.$$
%This implies that \Dupp is the Moore-Penrose pseudoinverse of \Elim.
Let \Unun be the `remove first' matrix, whose size should be inferred
in context. It is a matrix of all rows but the first of the identity
matrix. It exists to remove the first element of a vector.
%Let \diag{\Mtx{A}} be the diagonal matrix with
%the same diagonal as \Mtx{A}.
%Define
%\begin{equation}
%\fsymd{\Mtx{A}} \defeq \Mtx{A} + \tr{\Mtx{A}} - \diag{\Mtx{A}}.
%\end{equation}
\end{definition}%UNFOLD
\begin{definition}[Derivatives]%FOLDUP
For $m$-vector \vect{x}, and $n$-vector \vect{y}, let the derivative
\dbyd{\vect{y}}{\vect{x}} be the $n\times m$ matrix whose first column
is the partial derivative of \vect{y} with respect to $x_1$.
This follows the so-called `numerator layout' convention.
For matrices \Mtx{Y} and \Mtx{X}, define
\begin{equation*}
\dbyd{\Mtx{Y}}{\Mtx{X}} \defeq \dbyd{\fvec{\Mtx{Y}}}{\fvec{\Mtx{X}}}.
\end{equation*}
\end{definition}%UNFOLD
\begin{lemma}[Miscellaneous Derivatives]%FOLDUP
\label{lemma:misc_derivs}
For symmetric matrices \Mtx{Y} and \Mtx{X},
%\begin{align}
%\dbyd{\fvech{\Mtx{Y}}}{\fvec{\Mtx{X}}} &= \Elim \dbyd{\Mtx{Y}}{\Mtx{X}},\\
%\dbyd{\fvec{\Mtx{Y}}}{\fvech{\Mtx{X}}} &= \dbyd{\Mtx{Y}}{\Mtx{X}}\Dupp,\\
%\dbyd{\fvech{\Mtx{Y}}}{\fvech{\Mtx{X}}} &= \EXD{\dbyd{\Mtx{Y}}{\Mtx{X}}}.
%\end{align}
\begin{equation}
\dbyd{\fvech{\Mtx{Y}}}{\fvec{\Mtx{X}}} = \Elim \dbyd{\Mtx{Y}}{\Mtx{X}},\quad
\dbyd{\fvec{\Mtx{Y}}}{\fvech{\Mtx{X}}} = \dbyd{\Mtx{Y}}{\Mtx{X}}\Dupp,\quad
\dbyd{\fvech{\Mtx{Y}}}{\fvech{\Mtx{X}}} = \EXD{\dbyd{\Mtx{Y}}{\Mtx{X}}}.
\end{equation}
\end{lemma}%UNFOLD
\begin{proof}%FOLDUP
For the first equation, note that
$\fvech{\Mtx{Y}} = \Elim\fvec{\Mtx{Y}}$, thus by the chain rule:
$$
\dbyd{\fvech{\Mtx{Y}}}{\fvec{\Mtx{X}}} =
\dbyd{\Elim \fvec{\Mtx{Y}}}{\fvec{\Mtx{Y}}} = \Elim \dbyd{\Mtx{Y}}{\Mtx{X}},
$$
by linearity of the derivative. The other identities follow similarly.
\end{proof}%UNFOLD
\begin{lemma}[Derivative of matrix inverse]%FOLDUP
For invertible matrix \Mtx{A},
\begin{equation}
\dbyd{\minv{\Mtx{A}}}{\Mtx{A}}
%= \dvecbydvec{\minv{\Mtx{A}}}{\Mtx{A}}
= - \wrapParens{\trminv{\Mtx{A}}\kron\minv{\Mtx{A}}}
= - \minv{\wrapParens{\tr{\Mtx{A}}\kron\Mtx{A}}}.
\label{eqn:deriv_vec_matrix_inverse}
\end{equation}
For \emph{symmetric} \Mtx{A}, the derivative with respect to the
non-redundant part is
\begin{equation}
\dbyd{\fvech{\minv{\Mtx{A}}}}{\fvech{\Mtx{A}}}
= - \EXD{\wrapParens{\minv{\Mtx{A}}\kron\minv{\Mtx{A}}}}.
\label{eqn:deriv_vech_matrix_inverse}
\end{equation}
\label{lemma:deriv_vech_matrix_inverse}
\end{lemma}%UNFOLD
Note how this result generalizes the scalar derivative:
$\dbyd{x^{-1}}{x} = - \wrapParens{x^{-1} x^{-1}}.$
\begin{proof}%FOLDUP
\eqnref{deriv_vec_matrix_inverse} is a known
result. \cite{facklernotes,magnus1999matrix}
\eqnref{deriv_vech_matrix_inverse} then follows using \lemmaref{misc_derivs}.
\end{proof}%UNFOLD
%UNFOLD
\subsection{Asymptotic distribution of the \txtMP}%FOLDUP
\label{subsec:dist_markoport}
\nocite{BrittenJones1999}
Collecting the mean and covariance into the second moment matrix
gives the asymptotic distribution of the sample \txtMP
without much work. In some sense, this computation
generalizes the `standard' asymptotic analysis of Sharpe ratio of
multiple assets. \cite{jobsonkorkie1981,lo2002,Ledoit2008850,Leung2008}
%\cite{jobsonkorkie1981,lo2002,mertens2002comments,Ledoit2008850,Leung2008,Wright2012}
%The asymptotic distribution of \minv{\pvsm} then follows from standard
%techniques.
\begin{theorem}%FOLDUP
\label{theorem:inv_distribution}
Let \svsm be the unbiased sample estimate of
\pvsm, based on \ssiz \iid samples of \vreti.
Let \pvvar be the variance of $\fvech{\ogram{\avreti}}$.
Then, asymptotically in \ssiz,
%\begin{multline}
%\sqrt{\ssiz}\wrapParens{\fvech{\minv{\svsm}} - \fvech{\minv{\pvsm}}}
%\rightsquigarrow \\
%\normlaw{0,\qoform{\pvvar}{\wrapBracks{\EXD{\wrapParens{\AkronA{\minv{\pvsm}}}}}}}.
%\label{eqn:mvclt_isvsm}
%\end{multline}
%\begin{equation}
%\begin{split}
%\sqrt{\ssiz}\wrapParens{\fvech{\minv{\svsm}} - \fvech{\minv{\pvsm}}}
%&\rightsquigarrow
%\normlaw{0,\qoform{\pvvar}{\Mtx{H}}},\\
%\mbox{where}\quad\Mtx{H} &= -\EXD{\wrapParens{\AkronA{\minv{\pvsm}}}}.
%\label{eqn:mvclt_isvsm}
%\end{split}
%\end{equation}
\begin{equation}
\sqrt{\ssiz}\wrapParens{\fvech{\minv{\svsm}} - \fvech{\minv{\pvsm}}}
\rightsquigarrow \normlaw{0,\qoform{\pvvar}{\Mtx{H}}},
\label{eqn:mvclt_isvsm}
\end{equation}
where
\begin{equation}
\Mtx{H} = -\EXD{\wrapParens{\AkronA{\minv{\pvsm}}}}.
\end{equation}
Furthermore, we may replace \pvvar in this equation with an asymptotically
consistent estimator, \svvar.
\end{theorem}%UNFOLD
\begin{proof}%FOLDUP
Under the multivariate central limit theorem \cite{wasserman2004all}
\begin{equation}
\sqrt{\ssiz}\wrapParens{\fvech{\svsm} - \fvech{\pvsm}}
\rightsquigarrow
\normlaw{0,\pvvar},
\label{eqn:mvclt_svsm}
\end{equation}
where \pvvar is the variance of $\fvech{\ogram{\avreti}}$, which, in general,
is unknown.
%(For the case where \vreti is multivariate Gaussian,
%\pvvar is known; see \theoremref{ ...
By the delta method \cite{wasserman2004all},
\begin{equation*}
\sqrt{\ssiz}\wrapParens{\fvech{\minv{\svsm}} - \fvech{\minv{\pvsm}}}
\rightsquigarrow
\normlaw{0,\qoform{\pvvar}{\wrapBracks{\dbyd{\fvech{\minv{\pvsm}}}{\fvech{\pvsm}}}}}.
\end{equation*}
The derivative is given by \lemmaref{deriv_vech_matrix_inverse}, and
the result follows.
\end{proof}%UNFOLD
%To estimate the covariance of $\fvech{\minv{\svsm}} - \fvech{\minv{\pvsm}}$,
To estimate the covariance of $\fvech{\minv{\svsm}}$,
plug in \svsm for \pvsm in the covariance computation, and use some
consistent estimator for \pvvar, call
it \svvar.
%Rather, one must estimate $\qform{\pvvar}{\Elim}$.
One way to compute \svvar is to via the sample covariance of the
vectors $\fvech{\ogram{\avreti[i]}} =
\asvec{1,\tr{\vreti[i]},\tr{\fvech{\ogram{\vreti[i]}}}}$.
More elaborate covariance estimators can be used, for example, to deal with
violations of the \iid assumptions. \cite{Zeileis:2004:JSSOBK:v11i10}
\nocite{magnus1999matrix,magnus1980elimination}
\nocite{BrittenJones1999}
Note that because the first element of
$\fvech{\ogram{\avreti[i]}}$ is a deterministic $1$, the first row and
column of \pvvar is all zeros, and we need not estimate it.
%UNFOLD
\subsection{The \txtSR optimal portfolio}%FOLDUP
\begin{lemma}[\txtSR optimal portfolio]%FOLDUP
\label{lemma:sr_optimal_portfolio}
Assuming $\pvmu \ne \vzero$, and \pvsig is invertible,
the portfolio optimization problem
\begin{equation}
\argmax_{\pportw :\, \qform{\pvsig}{\pportw} \le \Rbuj^2}
\frac{\trAB{\pportw}{\pvmu} - \rfr}{\sqrt{\qform{\pvsig}{\pportw}}},
\label{eqn:sr_optimal_portfolio_problem}
\end{equation}
for $\rfr \ge 0, \Rbuj > 0$ is solved by
\begin{equation}
\pportwoptR \defeq \frac{\Rbuj}{\sqrt{\qiform{\pvsig}{\pvmu}}}
\minvAB{\pvsig}{\pvmu}.
\end{equation}
Moreover, this is the unique solution whenever $\rfr > 0$.
The maximal objective achieved by this portfolio is
$\sqrt{\qiform{\pvsig}{\pvmu}} - \fracc{\rfr}{\Rbuj} =
\psnropt - \fracc{\rfr}{\Rbuj}$.
\end{lemma}%UNFOLD
\begin{proof}%FOLDUP
By the Lagrange multiplier technique, the optimal portfolio
solves the following equations:
\begin{equation*}
\begin{split}
0 &= c_1 \pvmu - c_2 \pvsig \pportw - \gamma \pvsig \pportw,\\
\qform{\pvsig}{\pportw} &\le \Rbuj^2,
\end{split}
\end{equation*}
where $\gamma$ is the Lagrange multiplier, and $c_1, c_2$ are
scalar constants.
Solving the first equation gives us
$$
\pportw = c\,\minvAB{\pvsig}{\pvmu}.
$$
This reduces the problem to the univariate optimization
\begin{equation}
\max_{c :\, c^2 \le \fracc{\Rbuj^2}{\psnrsqopt}}
\sign{c} \psnropt - \frac{\rfr}{\abs{c}\psnropt},
\end{equation}
where $\psnrsqopt = \qiform{\pvsig}{\pvmu}.$ The optimum
occurs for $c = \fracc{\Rbuj}{\psnropt}$, moreover the optimum
is unique when $\rfr > 0$.
\end{proof}%UNFOLD
Note that the first element of \fvech{\minv{\pvsm}} is $1 +
\qiform{\pvsig}{\pvmu}$, and elements 2 through $\nlatf+1$ are
$-\pportwopt$. Thus, \pportwoptR, the portfolio that maximizes the \txtSR,
is some transformation of \fvech{\minv{\pvsm}}, and another
application of the delta method gives its asymptotic distribution,
as in the following corollary to \theoremref{inv_distribution}.
\begin{corollary}%FOLDUP
\label{corollary:portwoptR_dist}
Let
\begin{equation}
\pportwoptR = \frac{\Rbuj}{\sqrt{\qiform{\pvsig}{\pvmu}}}
\minvAB{\pvsig}{\pvmu},
\end{equation}
and similarly, let \sportwoptR be the sample analogue, where \Rbuj is
some risk budget. Then
%\begin{multline}
%\sqrt{\ssiz}\wrapParens{\sportwoptR - \pportwoptR}
%\rightsquigarrow \\
%\normlaw{0,\qform{\qoform{\pvvar}{\wrapBracks{\EXD{\wrapParens{\AkronA{\minv{\pvsm}}}}}}}{\Mtx{K}}},
%\label{eqn:mvclt_portfolio}
%\end{multline}
%where
%$$
%\Mtx{K} = - \Rbuj \asvec{\half \frac{\pportwoptR}{\psnrsqopt},
%\oneby{\psnropt}\eye[\nlatf],\mzero},
%$$
%and $\psnrsqopt \defeq \qiform{\pvsig}{\pvmu}.$
\begin{equation}
\sqrt{\ssiz}\wrapParens{\sportwoptR - \pportwoptR}
\rightsquigarrow
\normlaw{0,\qoform{\pvvar}{\Mtx{H}}},
\label{eqn:mvclt_portfolio}
\end{equation}
where
\begin{equation}
\begin{split}
\Mtx{H} &= \wrapParens{- \asrowvec{\oneby{2\psnrsqopt} \pportwoptR,
\frac{\Rbuj}{\psnropt}\eye[\nlatf],\mzero}}
\wrapParens{-\EXD{\wrapParens{\AkronA{\minv{\pvsm}}}}},\\
\psnrsqopt &\defeq \qiform{\pvsig}{\pvmu}.
\end{split}
\end{equation}
\end{corollary}%UNFOLD
\begin{proof}%FOLDUP
By the delta method, and \theoremref{inv_distribution}, it suffices
to show that
%$$\tr{\Mtx{K}} = \dbyd{\pportwoptR}{\fvech{\minv{\pvsm}}}.$$
$$\dbyd{\pportwoptR}{\fvech{\minv{\pvsm}}}
= - \asrowvec{\oneby{2\psnrsqopt} \pportwoptR,
\frac{\Rbuj}{\psnropt}\eye[\nlatf],\mzero}.
$$
To show this, note that \pportwoptR is $-\Rbuj$ times elements
2 through $\nlatf+1$ of \fvech{\minv{\pvsm}} divided by
$\psnropt = \sqrt{\trAB{\basev[1]}{\fvech{\minv{\pvsm}}} - 1}$, where
$\basev[i]$ is the \kth{i} column of the identity matrix. The result
follows from basic calculus.
\end{proof}%UNFOLD
The sample statistic \ssrsqopt is, up to scaling involving \ssiz, just
Hotelling's \Tstat statistic. \cite{anderson2003introduction}
One can perform inference on \psnrsqopt via this statistic, at least under
Gaussian returns, where the distribution of \Tstat takes a (noncentral)
\Fstat{}-distribution. Note, however, that \psnropt is the maximal
population \txtSR of \emph{any} portfolio, so it is an upper bound of the
\txtSR of the sample portfolio \sportwoptR. It is of little comfort
to have an estimate of \psnropt when the sample portfolio may have
a small, or even negative, \txtSR.
Because \psnropt is an upper bound on the \txtSR of a portfolio, it
seems odd to claim that the \txtSR of the sample portfolio might be
asymptotically normal with mean \psnropt. In fact, the delta method
will fail because the gradient of \psnropt with respect to the portfolio
is zero at \pportwoptR. One solution to this puzzle
is to estimate the `\txtSNR,' incorporating a strictly positive \rfr.
In this case a portfolio may achieve a higher value than
$\psnropt - \fracc{\rfr}{\Rbuj}$, which is achieved by \pportwoptR,
by violating the risk budget. This leads to the following corollary.
\begin{corollary}%FOLDUP
\label{corollary:portwoptR_hc_dist}
Suppose $\rfr > 0, \Rbuj > 0$.
Define the \txtSNR as
\begin{equation}
\pSNR{\sportw} \defeq \frac{\trAB{\sportw}{\pvmu} -
\rfr}{\sqrt{\qform{\pvsig}{\sportw}}}.
\label{eqn:snr_def}
\end{equation}
Let \pportwoptR and \sportwoptR be defined as in
\corollaryref{portwoptR_dist}. As per \lemmaref{sr_optimal_portfolio},
$\pSNR{\pportwoptR} = \psnropt - \fracc{\rfr}{\Rbuj}$.
Let \pvvar be the variance of $\fvech{\ogram{\avreti}}$.
Then, asymptotically in \ssiz,
\begin{equation}
\pSNR{\sportwoptR}
%\sqrt{\ssiz}\wrapParens{\pSNR{\sportwoptR} - \pSNR{\pportwoptR}}
\rightsquigarrow
%\normlaw{0,\qoform{\pvvar}{\Mtx{H}}},
\normlaw{\pSNR{\pportwoptR},\oneby{\ssiz}\qform{\pvvar}{\vect{h}}},
\label{eqn:mvclt_snr}
\end{equation}
where
%\begin{equation}
%\begin{split}
%\Mtx{H} &= \wrapParens{- \asrowvec{\oneby{2\psnrsqopt} \pportwoptR,
%\frac{\Rbuj}{\psnropt}\eye[\nlatf],\mzero}}
%\wrapParens{-\EXD{\wrapParens{\AkronA{\minv{\pvsm}}}}},\\
%\tr{\vect{z}}
%&= \wrapParens{ \frac{\tr{\pvmu}}{\sqrt{\qform{\pvsig}{\pportwoptR}}}
%- \frac{\pSNR{\pportwoptR}}{\qform{\pvsig}{\pportwoptR}}
%\trAB{\pportwoptR}{\pvsig}},\\
%\psnrsqopt &\defeq \qiform{\pvsig}{\pvmu}.
%\end{split}
%\end{equation}
%\begin{equation}
%\begin{split}
%\Mtx{H} &=
%\wrapParens{-\EXD{\wrapParens{\AkronA{\minv{\pvsm}}}}},\\
%\tr{\vect{z}} &= -\frac{1}{\psnrsqopt}
%\asrowvec{\frac{\rfr}{2\Rbuj}, \Rbuj\wrapParens{\tr{\pvmu} - \wrapBracks{1 -
%\frac{\rfr}{\Rbuj\psnropt}} \trAB{\pportwoptR}{\pvsig}}, \mzero}.
%\end{split}
%\end{equation}
%\begin{equation}
%\begin{split}
%\Mtx{H} &=
%\wrapParens{-\EXD{\wrapParens{\AkronA{\minv{\pvsm}}}}},\\
%\tr{\vect{z}} &= -\frac{1}{\psnrsqopt}
%\asrowvec{\frac{\rfr}{2\Rbuj}, \frac{\Rbuj}{\psnrsqopt}\wrapBracks{\psnrsqopt -
%\Rbuj\psnropt + \rfr}\tr{\pvmu}, \mzero}.
%\end{split}
%\end{equation}
\begin{equation}
\tr{\vect{h}}
= - \frac{\rfr}{\Rbuj\psnrsqopt}
\asrowvec{\oneby{2},\tr{\pvmu},\vzero}
\wrapParens{-\EXD{\wrapParens{\AkronA{\minv{\pvsm}}}}}.
\end{equation}
\end{corollary}%UNFOLD
\begin{proof}%FOLDUP
By the delta method,
$$
\pSNR{\sportwoptR}
\rightsquigarrow
\normlaw{\pSNR{\pportwoptR},\oneby{\ssiz}\qform{\pvvar}{\vect{h}}},
\,\mbox{with}\,
\tr{\vect{h}} = \tr{\dbyd{\pSNR{\pportwoptR}}{\pportwoptR}}
\dbyd{\pportwoptR}{\fvech{\pvsm}}.
$$
Then, via \corollaryref{portwoptR_dist},
$$
\tr{\vect{h}} = {\tr{\dbyd{\pSNR{\pportwoptR}}{\pportwoptR}}
{\asrowvec{-\oneby{2\psnrsqopt} \pportwoptR,
-\frac{\Rbuj}{\psnropt}\eye[\nlatf],\mzero}}}
\wrapParens{-\EXD{\wrapParens{\AkronA{\minv{\pvsm}}}}}.
$$
By simple calculus,
\begin{equation}
\dbyd{\pSNR{\pportw}}{\pportw}
= \frac{\sqrt{\qform{\pvsig}{\pportw}}\pvmu
- \frac{\trAB{\pportw}{\pvmu} -
\rfr}{\sqrt{\qform{\pvsig}{\pportw}}}\pvsig\pportw}{\qform{\pvsig}{\pportw}}
= \frac{\sqrt{\qform{\pvsig}{\pportw}}\pvmu
- \pSNR{\pportw}\pvsig\pportw}{\qform{\pvsig}{\pportw}}
\end{equation}
Since, by definition,
$\pportwoptR = \frac{\Rbuj}{\psnropt}\minvAB{\pvsig}{\pvmu},$ and
$\pSNR{\pportwoptR} = \psnropt - \fracc{\rfr}{\Rbuj}$, plugging
in gives
\begin{equation}
\left.\dbyd{\pSNR{\pportw}}{\pportw} \right\vert_{\pportw = \pportwoptR}
= \frac{\Rbuj\pvmu - \wrapParens{\psnropt -
\fracc{\rfr}{\Rbuj}}\frac{\Rbuj}{\psnropt}\pvmu}{\Rbuj^2}
= \frac{\rfr}{\Rbuj^2\psnropt}\pvmu.
\end{equation}
Then we have
\begin{equation}
\begin{split}
{\tr{\dbyd{\pSNR{\pportwoptR}}{\pportwoptR}}
{\asrowvec{-\oneby{2\psnrsqopt} \pportwoptR,
-\frac{\Rbuj}{\psnropt}\eye[\nlatf],\mzero}}}
&= - \frac{\rfr}{\Rbuj^2\psnropt}\tr{\pvmu}
\asrowvec{\oneby{2\psnrsqopt} \pportwoptR,
\frac{\Rbuj}{\psnropt}\eye[\nlatf],\mzero},\\
&= - \frac{\rfr}{\Rbuj\psnrsqopt}\tr{\pvmu}
\asrowvec{\oneby{2\psnrsqopt} \minvAB{\pvsig}{\pvmu},
\eye[\nlatf],\mzero},\\
&= - \frac{\rfr}{\Rbuj\psnrsqopt}
\asrowvec{\oneby{2},\tr{\pvmu},\vzero},
\end{split}
\end{equation}
completing the proof.
\end{proof}%UNFOLD
\begin{caution}%FOLDUP
Since \pvmu and \pvsig are population parameters, \pSNR{\sportwoptR} is
an unobserved quantity. Nevertheless, we can estimate the variance of
\pSNR{\sportwoptR}, and possibly construct confidence intervals on it
using sample statistics.
\end{caution}%UNFOLD
%UNFOLD
%UNFOLD
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Distribution under Gaussian returns}%FOLDUP
The goal of this section is to derive a variant of
\theoremref{inv_distribution} for the case where \vreti follows a
multivariate Gaussian distribution. First,
assuming $\vreti\sim\normlaw{\pvmu,\pvsig}$, we can express the density
of \vreti, and of \svsm, in terms of \nlatf, \ssiz, and \pvsm.
\begin{lemma}[Gaussian sample density]%FOLDUP
Suppose $\vreti\sim\normlaw{\pvmu,\pvsig}$. Letting
$\avreti = \asvec{1,\tr{\vreti}}$, and $\pvsm = \E{\ogram{\avreti}}$,
then the negative log likelihood of \vreti is
\begin{equation}
- \log\normpdf{\vreti}{\pvmu,\pvsig} =
c_{\nlatf}
+ \half \logdet{\pvsm}
+ \half \trace{\minv{\pvsm}\ogram{\avreti}},
\end{equation}
for the constant
$c_{\nlatf} = -\half + \half[\nlatf]\log\wrapParens{2\pi}.$
\label{lemma:x_dist_gaussian}
\end{lemma}%UNFOLD
\begin{proof}%FOLDUP
By the block determinant formula,
\begin{equation*}
\det{\pvsm}
= \det{1}\det{\wrapParens{\pvsig + \ogram{\pvmu}} - \pvmu 1^{-1} \tr{\pvmu}}
= \det{\pvsig}.
\end{equation*}
Note also that
\begin{equation*}
\qiform{\pvsig}{\wrapParens{\vreti - \pvmu}} =
\qiform{\pvsm}{\avreti} - 1.
\end{equation*}
These relationships hold without assuming a particular distribution for
\vreti.
The density of \vreti is then
\begin{equation*}
\begin{split}
\normpdf{\vreti}{\pvmu,\pvsig} &= \frac{1}{\sqrt{\wrapParens{2\pi}^{\nlatf}\det{\pvsig}}}
\longexp{-\half \qiform{\pvsig}{\wrapParens{\vreti - \pvmu}}},\\
&= \frac{\detpow{\pvsig}{-\half}}{\wrapParens{2\pi}^{\nlatf/2}}
\longexp{-\half \wrapParens{\qiform{\pvsm}{\avreti} - 1}},\\
&= \wrapParens{2\pi}^{-\nlatf/2} \detpow{\pvsm}{-\half}
\longexp{-\half \wrapParens{\qiform{\pvsm}{\avreti} - 1}},\\
&= \wrapParens{2\pi}^{-\nlatf/2}
\longexp{\half - \half \logdet{\pvsm} - \half \trace{\minv{\pvsm}\ogram{\avreti}}},
%\therefore - \log\normpdf{\vreti}{\pvmu,\pvsig} &=
%- \half + \half[\nlatf]\log\wrapParens{2\pi}
%+ \half \logdet{\pvsm}
%+ \half \trace{\minv{\pvsm}\ogram{\avreti}}.
\end{split}
\end{equation*}
and the result follows.
\end{proof}
%UNFOLD
\begin{lemma}[Gaussian second moment matrix density]%FOLDUP
Let $\vreti\sim\normlaw{\pvmu,\pvsig}$,
$\avreti = \asvec{1,\tr{\vreti}}$,
and $\pvsm = \E{\ogram{\avreti}}$.
Given \ssiz \iid samples \vreti[i], let
Let $\svsm = \oneby{\ssiz}\sum_i \ogram{\avreti[i]}$.
Then the density of \svsm is
\begin{equation}
\FOOpdf{}{\svsm}{\pvsm} =
\longexp{c'_{\ssiz,\nlatf}}\frac{\det{\svsm}^{\half[\ssiz-\nlatf-2]}}{\det{\pvsm}^{\half[\ssiz]}}
\longexp{-\half[\ssiz]\trace{\minv{\pvsm}\svsm}},
\label{eqn:theta_dist_gaussian}
\end{equation}
for some $c'_{\ssiz,\nlatf}.$
\label{lemma:theta_dist_gaussian}
\end{lemma}%UNFOLD
\begin{proof}%FOLDUP
Let \amreti be the matrix
whose rows are the vectors \tr{\vreti[i]}.
From \lemmaref{x_dist_gaussian}, and using linearity of the trace,
the negative log density of \amreti is
\begin{equation*}
\begin{split}
- \log\normpdf{\amreti}{\pvsm} &=
\ssiz c_{\nlatf}
+ \half[\ssiz] \logdet{\pvsm}
+ \half \trace{\minv{\pvsm}\gram{\amreti}},\\
\therefore
\frac{- 2\log\normpdf{\amreti}{\pvsm}}{\ssiz} &=
2 c_{\nlatf}
+ \logdet{\pvsm}
+ \trace{\minv{\pvsm}\svsm}.
\end{split}
\end{equation*}
By Lemma (5.1.1) of Press \cite{press2012applied}, this can be expressed
as a density on \svsm:
\begin{equation*}
\begin{split}
\frac{- 2\log\FOOpdf{}{\svsm}{\pvsm}}{\ssiz}
&= \frac{- 2\log\normpdf{\amreti}{\pvsm}}{\ssiz}
-\frac{2}{\ssiz}\wrapParens{\half[\ssiz-\nlatf-2]\logdet{\svsm}}\\
&\phantom{=}\,
-\frac{2}{\ssiz}\wrapParens{\half[\nlatf+1]\wrapParens{\ssiz - \half[\nlatf]} \log\pi -
\sum_{j=1}^{\nlatf+1} \log\funcit{\Gamma}{\half[\ssiz +1-j]}},\\
&= \wrapBracks{2c_{\nlatf} - \frac{\nlatf+1}{\ssiz}\wrapParens{\ssiz - \half[\nlatf]} \log\pi
- \frac{2}{\ssiz} \sum_{j=1}^{\nlatf+1}
\log\funcit{\Gamma}{\half[\ssiz +1-j]}}\\
&\phantom{=}\,
+ \logdet{\pvsm} - \frac{\ssiz-\nlatf-2}{\ssiz}\logdet{\svsm}
+ \trace{\minv{\pvsm}\svsm},\\
&= c'_{\ssiz,\nlatf}
- \log\frac{\det{\svsm}^{\frac{\ssiz-\nlatf-2}{\ssiz}}}{\det{\pvsm}}
+ \trace{\minv{\pvsm}\svsm},
\end{split}
\end{equation*}
where $c'_{\ssiz,\nlatf}$ is the term in brackets on the third line.
Factoring out $\fracc{-2}{\ssiz}$ and taking an exponent gives the
result.
\end{proof}%UNFOLD
\begin{corollary}%FOLDUP
The random variable $\ssiz\svsm$ has the same density, up to a constant
in \nlatf and \ssiz, as a
$\nlatf+1$-dimensional Wishart random variable with \ssiz degrees of freedom
and scale matrix \pvsm. Thus $\ssiz\svsm$ is a \emph{conditional} Wishart,
conditional on $\svsm_{1,1} = 1$. \cite{press2012applied,anderson2003introduction}
%conditional on $\svsm_{\nlatf+1,\nlatf+1} = 1$. % for the other form.
\end{corollary}%UNFOLD
\begin{corollary}%FOLDUP
The derivatives of log likelihood are given by
\begin{equation}
\begin{split}
\drbydr{\log\FOOpdf{}{\svsm}{\pvsm}}{\fvec{\pvsm}}
&= - \half[\ssiz]\tr{\wrapBracks{\fvec{\minv{\pvsm} -
\minv{\pvsm}\svsm\minv{\pvsm}}}},\\
\drbydr{\log\FOOpdf{}{\svsm}{\pvsm}}{\fvec{\minv{\pvsm}}}
&= -\half[\ssiz]\tr{\wrapBracks{\fvec{\pvsm - \svsm}}}.
\end{split}
\label{eqn:deriv_gauss_loglik}
\end{equation}
\end{corollary}%UNFOLD
\begin{proof}%FOLDUP
Plugging in the log likelihood gives
\begin{equation*}
\drbydr{\log\FOOpdf{}{\svsm}{\pvsm}}{\fvec{\pvsm}}
= - \half[\ssiz]\wrapBracks{\drbydr{\logdet{\pvsm}}{\fvec{\pvsm}} +
\drbydr{\trace{\minv{\pvsm}\svsm}}{\fvec{\pvsm}}},
\end{equation*}
and then standard matrix calculus gives the first
result. \cite{magnus1999matrix,petersen2012matrix}
Proceeding similarly gives the second.
\end{proof}%UNFOLD
This immediately gives us the Maximum Likelihood Estimator.
\begin{corollary}[MLE]%FOLDUP
\svsm is the maximum likelihood estimator of \pvsm. \label{corollary:theta_mle}
\end{corollary}%UNFOLD
To compute the covariance of \fvech{\pvsm}, \pvvar, in the Gaussian case, one
can compute the Fisher Information, then appeal to the fact that
\pvsm is the MLE. However, because the first element of \fvech{\pvsm}
is a deterministic $1$, the first row and column of \pvvar are all
zeros. This is an unfortunate wrinkle. The solution is to compute the
Fisher Information with respect to the nonredundant variables,
$\Unun\fvech{\pvsm}$, as follows.
%Since $\pvsm[\txtMLE] = \svsm$, the log likelihood of the MLE is
%\begin{equation}
%\begin{split}
%\log\FOOlik{}{\svsm}{\pvsm[\txtMLE]}
%&= - \half[\ssiz] c'_{\ssiz,\nlatf} - \half[\ssiz] \logdet{\pvsm[\txtMLE]} +
%\half[\ssiz-\nlatf-2]\logdet{\svsm}\\
%&\phantom{=}\,+ \trace{\minv{\pvsm[\txtMLE]}\svsm},\\
%&= -\half[\ssiz] c'_{\ssiz,\nlatf}
%- \half[\nlatf+2]\logdet{\svsm} + \wrapParens{\nlatf+1}.
%\end{split}
%\end{equation}
%\begin{lemma}%FOLDUP
%The Fisher Information of $\fvec{\minv{\pvsm}}$ is
%\begin{equation}
%\FishIf[\ssiz]{\fvec{\minv{\pvsm}}}
%= \half[\ssiz] {\AkronA{\pvsm}},
%\label{eqn:itheta_fish_inf_eqn}
%\end{equation}
%and, by change of variables, the Fisher Information
%of $\fvech{\minv{\pvsm}}$ is
%\begin{equation}
%\FishIf[\ssiz]{\fvech{\minv{\pvsm}}}
%= \half[\ssiz] \qform{\wrapParens{\AkronA{\pvsm}}}{\Dupp}.
%\label{eqn:itheta_vech_fish_inf_eqn}
%\end{equation}
%\label{lemma:itheta_fish_inf}
%\end{lemma}%UNFOLD
\begin{lemma}[Fisher Information]%FOLDUP
The Fisher Information of $\Unun\fvech{\pvsm}$ is
\begin{multline}
\FishIf[\ssiz]{\Unun\fvech{\pvsm}}
=\\ \half[\ssiz]
\qoform{ \qform{ \qform{\wrapParens{\AkronA{\pvsm}}}{\Dupp}
}{\wrapBracks{\EXD{\wrapParens{\minv{\pvsm}\kron\minv{\pvsm}}}}} }{\Unun}.
\label{eqn:theta_fish_inf_eqn}
\end{multline}
\label{lemma:theta_fish_inf}
\end{lemma}%UNFOLD
\begin{proof}%FOLDUP
First compute the Hessian of $\log\FOOpdf{}{\svsm}{\pvsm}$ with
respect to $\fvec{\minv{\pvsm}}$. The Hessian is defined as
\begin{equation*}
\drbydr[2]{\log\FOOpdf{}{\svsm}{\pvsm}}{\wrapParens{\fvec{\minv{\pvsm}}}}
\defeq
\drbydr{\tr{\wrapBracks{\drbydr{\log\FOOpdf{}{\svsm}{\pvsm}}{\fvec{\minv{\pvsm}}}}}}{\fvec{\minv{\pvsm}}}.
\end{equation*}
Then, from \eqnref{deriv_gauss_loglik},
\begin{equation*}
\begin{split}
\drbydr[2]{\log\FOOpdf{}{\svsm}{\pvsm}}{\wrapParens{\fvec{\minv{\pvsm}}}}
&= - \half[\ssiz] \dbyd{\wrapBracks{\pvsm - \svsm}}{\fvec{\minv{\pvsm}}},\\
&= - \half[\ssiz] \wrapParens{\AkronA{\pvsm}},
\end{split}
\end{equation*}
via \lemmaref{deriv_vech_matrix_inverse}. Perform a change of variables.
%Noting that $\dbyd{\fvec{\Mtx{A}}}{\fvech{\Mtx{A}}} = \Dupp,$ we have
Via \lemmaref{misc_derivs},
\begin{equation*}
\drbydr[2]{\log\FOOpdf{}{\svsm}{\pvsm}}{\wrapParens{\fvech{\minv{\pvsm}}}}
= - \half[\ssiz] \qform{\wrapParens{\AkronA{\pvsm}}}{\Dupp}.
\end{equation*}
Using \lemmaref{deriv_vech_matrix_inverse}, perform another change of
variables to find
\begin{equation*}
\drbydr[2]{\log\FOOpdf{}{\svsm}{\pvsm}}{\wrapParens{\fvech{\pvsm}}}
= - \half[\ssiz]
\qform{ \qform{\wrapParens{\AkronA{\pvsm}}}{\Dupp}
}{\wrapBracks{\EXD{\wrapParens{\minv{\pvsm}\kron\minv{\pvsm}}}}}.
\end{equation*}
Finally, perform the change of variables to get the Hessian with
respect to $\Unun\fvech{\pvsm}$.
Since the Fisher Information is negative the expected value of this
Hessian, the result follows. \cite{pawitanIAL}
\end{proof}%UNFOLD
%\begin{corollary}%FOLDUP
%By change of variables, and again using \lemmaref{deriv_vech_matrix_inverse},
%the Fisher Information with respect to \pvsm is
%\begin{equation}
%\FishIf[\ssiz]{\pvsm} = \half[\ssiz]
%\qform{\wrapParens{\qoElim{\wrapParens{\AkronA{\pvsm}}}}}{\wrapBracks{\qoElim{\wrapParens{\AkronA{\minv{\pvsm}}}}}}.
%\end{equation}
%\end{corollary}%UNFOLD
Thus the analogue of \theoremref{inv_distribution} for Gaussian returns
is given by the following theorem.
\begin{theorem}%FOLDUP
\label{theorem:theta_asym_var_gaussian}
Let \svsm be the unbiased sample estimate of
\pvsm, based on \ssiz \iid samples of \vreti, assumed multivariate
Gaussian.
Then, asymptotically in \ssiz,
\begin{equation}
\sqrt{\ssiz}\wrapParens{\fvech{\svsm} - \fvech{\pvsm}}
\rightsquigarrow
\normlaw{0,\pvvar},
\label{eqn:mvclt_isvsm_gaussian}
\end{equation}
where the first row and column of \pvvar are all zero, and the lower
right block part is
\begin{equation*}
2\minv{\wrapBracks{
\qoform{ \qform{ \qform{\wrapParens{\AkronA{\pvsm}}}{\Dupp}
}{\wrapBracks{\EXD{\wrapParens{\minv{\pvsm}\kron\minv{\pvsm}}}}} }{\Unun}}}.
\end{equation*}
\end{theorem}%UNFOLD
\begin{proof}%FOLDUP
Under `the appropriate regularity conditions,'
\cite{wasserman2004all,pawitanIAL}
\begin{equation}
\label{eqn:converge_theta}
\wrapParens{\Unun\fvech{{\svsm}} - \Unun\fvech{{\pvsm}}}
\rightsquigarrow
\normlaw{0,\minv{\wrapBracks{\FishIf[\ssiz]{\Unun\fvech{{\pvsm}}}}}},
\end{equation}
and the result follows from
\lemmaref{theta_fish_inf}, and the fact that the first elements of both
\vech{\svsm} and \vech{\pvsm} are a deterministic $1$.
\end{proof}%UNFOLD
The `plug-in' estimator of the covariance substitutes in \svsm for
\pvsm in the right hand side of \eqnref{mvclt_isvsm_gaussian}.
The following conjecture is true in the $\nlatf=1$ case. Use of
the Sherman-Morrison-Woodbury formula might aid in a proof.
\begin{conjecture}%FOLDUP
\label{conjecture:theta_asym_var_gaussian}
For the Gaussian case, asymptotically in \ssiz,
\begin{equation}
\sqrt{\ssiz}\wrapParens{\fvech{\minv{\svsm}} - \fvech{\minv{\pvsm}}}
\rightsquigarrow
\normlaw{0,2 \minv{\wrapBracks{\qform{\wrapParens{\AkronA{\pvsm}}}{\Dupp}}} -
2\ogram{\basev[1]}}.
\end{equation}
\end{conjecture}%UNFOLD
A check of \theoremref{theta_asym_var_gaussian} and an illustration of
\conjectureref{theta_asym_var_gaussian} are given in the appendix.
\subsection{Likelihood ratio test on \txtMP}%FOLDUP
\providecommand{\lrtA}[1][i]{\mathSUB{\Mtx{A}}{#1}}
\providecommand{\lrta}[1][i]{\mathSUB{a}{#1}}
%This section is based on Dempster's ``Covariance Selection''. \cite{dempster1972}
Consider the null hypothesis
\begin{equation}
H_0: \trace{\lrtA[i]\minv{\pvsm}} = \lrta[i],\,i=1,\ldots,m.
\label{eqn:lrt_null_back}
\end{equation}
The constraints have to be sensible.
For example, they cannot
violate the positive definiteness of
\minv{\pvsm}, symmetry, \etc
Without loss of generality, we can assume
that the \lrtA[i] are symmetric, since \pvsm is symmetric, and for
symmetric \Mtx{G} and square \Mtx{H},
$\trace{\Mtx{G}\Mtx{H}} = \trace{\Mtx{G}\half\wrapParens{\Mtx{H} +
\tr{\Mtx{H}}}}$, and so we could replace any non-symmetric \lrtA[i] with
$\half\wrapParens{\lrtA[i] + \tr{\lrtA[i]}}$.
Employing the Lagrange multiplier technique, the maximum likelihood
estimator under the null hypothesis, call it \pvsm[0], solves the
following equation
\begin{equation}
\begin{split}
0 &= \drbydr{\log\FOOpdf{}{\svsm}{\pvsm}}{\minv{\pvsm}}
- \sum_i \lambda_i
\drbydr{\trace{\lrtA[i]\minv{\pvsm}}}{\minv{\pvsm}},\notag \\
&= - \pvsm[0] + \svsm - \sum_i \lambda_i \lrtA[i],\notag.
\end{split}
\end{equation}
Thus the MLE under the null is
\begin{equation}
\pvsm[0] = \svsm - \sum_i \lambda_i \lrtA[i].
\label{eqn:lrt_mle_soln}
\end{equation}
The maximum likelihood estimator under the constraints has to be
found numerically by solving for the $\lambda_i$, subject
to the constraints in \eqnref{lrt_null_back}.
This framework slightly generalizes Dempster's
``Covariance Selection,'' \cite{dempster1972} which
reduces to the case where each \lrta[i] is zero, and
each \lrtA[i] is a matrix of all zeros except two (symmetric) ones
somewhere in the lower right $\nlatf \times \nlatf$ sub matrix. In
all other respects, however, the solution here follows Dempster.
\providecommand{\vitrlam}[2]{\vectUL{\lambda}{\wrapNeParens{#1}}{#2}}
\providecommand{\sitrlam}[2]{\mathUL{\lambda}{\wrapNeParens{#1}}{#2}}
\providecommand{\vitrerr}[2]{\vectUL{\epsilon}{\wrapNeParens{#1}}{#2}}
\providecommand{\sitrerr}[2]{\mathUL{\epsilon}{\wrapNeParens{#1}}{#2}}
An iterative technique for finding the MLE based on a Newton step
would proceed as follow. \cite{nocedal2006numerical}
Let \vitrlam{0}{} be some initial estimate of the
vector of $\lambda_i$. (A good initial estimate can likely be had
by abusing the asymptotic normality
result from \subsecref{dist_markoport}.)
The residual of the \kth{k} estimate, \vitrlam{k}{} is
\begin{equation}
\vitrerr{k}{i} \defeq
\trace{\lrtA[i]\minv{\wrapBracks{\svsm - \sum_j \sitrlam{k}{j} \lrtA[j]}}} - \lrta[i].
\end{equation}
The Jacobian of this residual with respect to the \kth{l} element of \vitrlam{k}
is
\begin{equation}
\begin{split}
\drbydr{\vitrerr{k}{i}}{\sitrlam{k}{l}} &=
\trace{\lrtA[i]\minv{\wrapBracks{\svsm - \sum_j \sitrlam{k}{j} \lrtA[j]}}
\lrtA[l] \minv{\wrapBracks{\svsm - \sum_j \sitrlam{k}{j} \lrtA[j]}}},\\
&= \tr{\fvec{\lrtA[i]}} \wrapParens{\AkronA{\minv{\wrapBracks{\svsm - \sum_j
\sitrlam{k}{j}\lrtA[j]}}}} \fvec{\lrtA[l]}.
\end{split}
\end{equation}
Newton's method is then the iterative scheme
\begin{equation}
\vitrlam{k+1}{} \leftarrow \vitrlam{k}{} -
\minv{\wrapParens{\drbydr{\vitrerr{k}{}}{\vitrlam{k}{}}}} \vitrerr{k}.
\end{equation}
%Note there is no reason one must use a Newton scheme to maximize the
%likelihood. High quality optimization routines (\eg BFGS) which require
%only the gradient to be
%There is no reason to restrict oneself to a Newton solver. High quality
%optimization routines which require ...
%\cite{nocedal2006numerical}
When (if?) the iterative scheme converges on the optimum, plugging in
\vitrlam{k}{} into \eqnref{lrt_mle_soln} gives the MLE under the null.
The likelihood ratio test statistic is
\begin{equation}
\begin{split}
-2\log\Lambda &\defeq
-2\log\wrapParens{\frac{\FOOlik{}{\svsm}{\pvsm[0]}}{\FOOlik{}{\svsm}{\pvsm[\mbox{unrestricted }\txtMLE]}}},\\
&= \ssiz\wrapParens{\logdet{\pvsm[0]\minv{\svsm}} +
\trace{\wrapBracks{\minv{\pvsm[0]} - \minv{\svsm}}\svsm}},\\
&= \ssiz\wrapParens{\logdet{\pvsm[0]\minv{\svsm}} +
\trace{\minv{\pvsm[0]}\svsm} - \wrapBracks{\nlatf + 1}},
\end{split}
\label{eqn:wilks_lambda_def}
\end{equation}
using the fact that \svsm is the unrestricted MLE, per
\corollaryref{theta_mle}.
By Wilks' Theorem, under the null
hypothesis, $-2\log\Lambda$ is, asymptotically in \ssiz, distributed as
a chi-square with $m$ degrees of freedom. \cite{wilkstheorem1938}
%UNFOLD
%UNFOLD
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Extensions}%FOLDUP
For large samples, Wald statistics of the elements of the Markowitz
portfolio computed using the procedure outlined above tend to be
very similar to the t-statistics produced by the procedure of
Britten-Jones. \cite{BrittenJones1999}
However, the technique proposed here admits a number of interesting
extensions.
The script for each of these extensions is the same:
define, then solve, some portfolio optimization problem;
show that the solution can be defined in terms of some transformation
of $\minv{\pvsm}$, giving an implicit recipe for constructing the
sample portfolio based on the same transformation of $\minv{\svsm}$;
find the asymptotic distribution of the sample portfolio in terms of \pvvar.
%which can be estimated empirically, or via assuming Gaussian returns.
To simplify notation, we need the following definitions and a lemma.
\begin{definition}[Risk Projection]%FOLDUP
\label{definition:risk_projection}
Define the covariance-projection operator as
\begin{equation}
\proj{\pvsig}{\Mtx{A}} \defeq \wrapProj{\pvsig}{\Mtx{A}},
\label{eqn:def_proj}
\end{equation}
for conformable matrix \Mtx{A}. The derivative of this operator
will be shown to be the following operator:
\begin{equation}
\broj{\pvsig}{\Mtx{A}} \defeq
\wrapParens{\AkronA{\tr{\Mtx{A}}}}
\wrapParens{\AkronA{\minv{\wrapParens{\qoform{\pvsig}{\Mtx{A}}}}}}
\wrapParens{\AkronA{\Mtx{A}}}.
\label{eqn:def_broj}
\end{equation}
\end{definition}%UNFOLD
\begin{lemma}[Derivative of covariance-projection]%FOLDUP
\label{lemma:der_cov_proj}
For comformable \pvsig and \Mtx{A},
\begin{equation}
\dbyd{\proj{\pvsig}{\Mtx{A}}}{\pvsig} =
\broj{\pvsig}{\Mtx{A}}.
\label{eqn:del_proj_is_broj}
\end{equation}
\end{lemma}%UNFOLD
\begin{proof}%FOLDUP
A well-known fact regarding matrix manipulation \cite{magnus1999matrix} is
$$
\fvec{\Mtx{A}\Mtx{B}\Mtx{C}} = \wrapParens{\Mtx{A}\kron\tr{\Mtx{C}}}
\fvec{\Mtx{B}},\quad\mbox{therefore,}\quad
\dbyd{\Mtx{A}\Mtx{B}\Mtx{C}}{\Mtx{B}} = \Mtx{A}\kron\tr{\Mtx{C}}.
$$
Using this twice after the chain rule, we have:
\begin{equation*}
\begin{split}
\dbyd{\proj{\pvsig}{\Mtx{A}}}{\pvsig}
&=
\dbyd{\proj{\pvsig}{\Mtx{A}}}{\minv{\wrapParens{\qoform{\pvsig}{\Mtx{A}}}}}
\dbyd{\minv{\wrapParens{\qoform{\pvsig}{\Mtx{A}}}}}{\qoform{\pvsig}{\Mtx{A}}}
\dbyd{\qoform{\pvsig}{\Mtx{A}}}{\pvsig},\\
&=
\wrapParens{\AkronA{\tr{\Mtx{A}}}}
\dbyd{\minv{\wrapParens{\qoform{\pvsig}{\Mtx{A}}}}}{\qoform{\pvsig}{\Mtx{A}}}
\wrapParens{\AkronA{\Mtx{A}}}.
\end{split}
\end{equation*}
\lemmaref{deriv_vech_matrix_inverse} gives the middle term, completing the
proof.
\end{proof}%UNFOLD
\subsection{Subspace constraint} %FOLDUP
\label{subsec:subspace_constraint}
Consider the \emph{constrained} portfolio optimization problem
\begin{equation}
\max_{\substack{\pportw : \zerJc \pportw = \vzero,\\
\qform{\pvsig}{\pportw} \le \Rbuj^2}}
\frac{\trAB{\pportw}{\pvmu} - \rfr}{\sqrt{\qform{\pvsig}{\pportw}}},
\label{eqn:portopt_zer}
\end{equation}
where $\zerJc$ is a $\wrapParens{\nlatf - \nlatfzer} \times \nlatf$
matrix of rank $\nlatf - \nlatfzer$,
\rfr is the disastrous rate, and $\Rbuj > 0$ is the
risk budget. Let the rows of \zerJ span the null space of the rows of
\zerJc; that is, $\zerJc \tr{\zerJ} = \mzero$, and $\ogram{\zerJ} = \eye$.
We can interpret the orthogonality constraint $\zerJc \pportw = \vzero$ as
stating that \pportw must be a linear combination of the columns of
\tr{\zerJ}, thus $\pportw = \trAB{\zerJ}{\pportx}$. The columns of
\tr{\zerJ} may be considered `baskets' of assets to which our investments
are restricted.
We can rewrite the portfolio optimization problem in terms of solving
for \pportx, but then find the asymptotic distribution of the resultant
\pportw. Note that the expected return and covariance of the portfolio
\pportx are, respectively, \trAB{\pportx}{\zerJ\pvmu} and
\qform{\qoform{\pvsig}{\zerJ}}{\pportx}. Thus we can plug in
$\zerJ\pvmu$ and \qoform{\pvsig}{\zerJ} into
\lemmaref{sr_optimal_portfolio} to get the following analogous lemma.
\begin{lemma}[subspace constrained \txtSR optimal portfolio]%FOLDUP
\label{lemma:subsp_cons_sr_optimal_portfolio}
Assuming the rows of \zerJ span the null space of the rows of \zerJc,
$\zerJ\pvmu \ne \vzero$, and \pvsig is invertible,
the portfolio optimization problem
\begin{equation}
\max_{\substack{\pportw : \zerJc \pportw = \vzero,\\
\qform{\pvsig}{\pportw} \le \Rbuj^2}}
\frac{\trAB{\pportw}{\pvmu} - \rfr}{\sqrt{\qform{\pvsig}{\pportw}}},
\label{eqn:portopt_zer_I}
\end{equation}
for $\rfr \ge 0, \Rbuj > 0$ is solved by
\begin{equation*}
\begin{split}
\pportwoptFoo{\Rbuj,\zerJ,}
&\defeq c \proj{\pvsig}{\zerJ}\pvmu,\\
c &= \frac{\Rbuj}{\sqrt{\qform{\proj{\pvsig}{\zerJ}}{\pvmu}}}.
\end{split}
\end{equation*}
When $\rfr > 0$ the solution is unique.
\end{lemma}%UNFOLD
We can easily find the asymptotic distribution of
\sportwoptFoo{\Rbuj,\zerJ,}, the sample analogue of the optimal portfolio
in \lemmaref{subsp_cons_sr_optimal_portfolio}. First define the
subspace second moment.
%\begin{definition}[Subspace Second Moment]%FOLDUP
%\label{definition:subspace_second_moment}
%Let \zerJt be the $\wrapParens{1+\nlatfzer}\times\wrapParens{\nlatf+1}$
%matrix,
%$$
%\zerJt \defeq \twobytwossym{1}{0}{\zerJ}.
%$$
%Define the `subspace second moment' as
%$$
%\pvsm[\zerJ] \defeq \qoform{\pvsm}{\zerJt}.
%$$
%\end{definition}%UNFOLD
\begin{definition}%FOLDUP
\label{definition:subspace_second_moment}
Let \zerJt be the $\wrapParens{1+\nlatfzer}\times\wrapParens{\nlatf+1}$
matrix,
$$
\zerJt \defeq \twobytwossym{1}{0}{\zerJ}.
$$
%Define the `subspace second moment' as
%$$
%\pvsm[\zerJ] \defeq \qoform{\pvsm}{\zerJt}.
%$$
\end{definition}%UNFOLD
Simple algebra proves the following lemma.
\begin{lemma}%FOLDUP
The elements of $\proj{\pvsm}{\zerJt}$
are
\begin{equation*}
\proj{\pvsm}{\zerJt}
=
\twobytwo{ 1 + \qform{\proj{\pvsig}{\zerJ}}{\pvmu} }{
-\tr{\pvmu}\proj{\pvsig}{\zerJ}}{
-\proj{\pvsig}{\zerJ}\pvmu}{
\proj{\pvsig}{\zerJ}}.
\end{equation*}
In particular, elements $2$ through $\nlatf+1$ of
$-\fvech{\proj{\pvsm}{\zerJt}}$ are the portfolio
$\sportwoptFoo{\Rbuj,\zerJ,}$
defined in \lemmaref{subsp_cons_sr_optimal_portfolio}, up to the scaling
constant $c$ which is the ratio of \Rbuj to the square root of the first
element of $\fvech{\proj{\pvsm}{\zerJt}}$ minus one.
\end{lemma}%UNFOLD
The asymptotic distribution of $\fvech{\proj{\pvsm}{\zerJt}}$
is given by the following theorem, which is the analogue
of \theoremref{inv_distribution}.
\begin{theorem}%FOLDUP
\label{theorem:subzer_inv_distribution}
Let \svsm be the unbiased sample estimate of
\pvsm, based on \ssiz \iid samples of \vreti.
Let \zerJt
be defined as in \definitionref{subspace_second_moment}.
Let \pvvar be the variance of $\fvech{\ogram{\avreti}}$.
Then, asymptotically in \ssiz,
\begin{equation}
\sqrt{\ssiz}\wrapParens{\fvech{\proj{\svsm}{\zerJt}} -
\fvech{\proj{\pvsm}{\zerJt}}}
\rightsquigarrow
\normlaw{0,\qoform{\pvvar}{\Mtx{H}}},
\label{eqn:mvclt_zer_isvsm}
\end{equation}
where
\begin{equation*}
\Mtx{H} = - \EXD{\broj{\pvsm}{\zerJt}}.
\end{equation*}
\end{theorem}%UNFOLD
\begin{proof}%FOLDUP
By the multivariate delta method, it suffices to prove that
$$
\Mtx{H} = \dbyd{\fvech{\proj{\svsm}{\zerJt}}}{\fvech{\pvsm}}.
$$
This follows from \lemmaref{der_cov_proj} and \lemmaref{misc_derivs}.
\end{proof}
An analogue of \corollaryref{portwoptR_dist} gives the asymptotic
distribution of $\pportwoptFoo{\Rbuj,\zerJ,}$
defined in \lemmaref{subsp_cons_sr_optimal_portfolio}.
%UNFOLD
%UNFOLD
\subsection{Hedging constraint} %FOLDUP
\label{subsec:hedging_constraint}
% 2FIX: nstrat nstrathej fuck. as in, does this make sense in SharpeRatio.rnw?
% these show up as k and p, and what are you doing?
Consider, now, the constrained portfolio optimization problem,
\begin{equation}
\max_{\substack{\pportw : \hejG\pvsig \pportw = \vzero,\\
\qform{\pvsig}{\pportw} \le \Rbuj^2}}
\frac{\trAB{\pportw}{\pvmu} - \rfr}{\sqrt{\qform{\pvsig}{\pportw}}},
\label{eqn:portopt_hej}
\end{equation}
where $\hejG$ is now a $\nlatfhej \times \nlatf$ matrix of
rank \nlatfhej.
We can interpret the \hejG constraint as stating that the covariance of
the returns of a feasible portfolio with the returns of a portfolio
whose weights are in a given row of \hejG shall equal zero.
In the garden variety application of this problem, \hejG consists of
\nlatfhej rows of the identity matrix;
in this case, feasible portfolios are `hedged' with respect
to the \nlatfhej assets selected by \hejG
(although they may hold some position in the hedged assets).
\begin{lemma}[constrained \txtSR optimal portfolio]%FOLDUP
\label{lemma:cons_sr_optimal_portfolio}
Assuming $\pvmu \ne \vzero$, and \pvsig is invertible,
the portfolio optimization problem
\begin{equation}
\max_{\substack{\pportw : \hejG\pvsig \pportw = \vzero,\\
\qform{\pvsig}{\pportw} \le \Rbuj^2}}
\frac{\trAB{\pportw}{\pvmu} - \rfr}{\sqrt{\qform{\pvsig}{\pportw}}},
\label{eqn:cons_port_prob}
\end{equation}
for $\rfr \ge 0, \Rbuj > 0$ is solved by
% (2FIX nomenclature)
\begin{equation*}
\begin{split}
\pportwoptFoo{\Rbuj,\hejG,} &\defeq c \wrapParens{\minv{\pvsig}{\pvmu} -
\proj{\pvsig}{\hejG}\pvmu},\\
c &= \frac{\Rbuj}{\sqrt{\qiform{\pvsig}{\pvmu} -
\qform{\proj{\pvsig}{\hejG}}{\pvmu}}}.
\end{split}
\end{equation*}
When $\rfr > 0$ the solution is unique.
\end{lemma}%UNFOLD
\begin{proof}%FOLDUP
By the Lagrange multiplier technique, the optimal portfolio
solves the following equations:
\begin{equation*}
\begin{split}
0 &= c_1 \pvmu - c_2 \pvsig \pportw - \gamma_1 \pvsig \pportw -
\pvsig\trAB{\hejG}{\vect{\gamma_2}},\\
\qform{\pvsig}{\pportw} &\le \Rbuj^2,\\
\hejG\pvsig \pportw &= \vzero,
\end{split}
\end{equation*}
where $\gamma_i$ are Lagrange multipliers, and $c_1, c_2$ are
scalar constants.
Solving the first equation gives
$$
\pportw = c_3\wrapBracks{\minvAB{\pvsig}{\pvmu} -
\trAB{\hejG}{\vect{\gamma_2}}}.
$$
Reconciling this with the hedging equation we have
$$
\vzero
= \hejG\pvsig \pportw
= c_3 \hejG\pvsig \wrapBracks{\minvAB{\pvsig}{\pvmu} -
\trAB{\hejG}{\vect{\gamma_2}}},
$$
and therefore
$\vect{\gamma_2} = \minvAB{\wrapParens{\qoform{\pvsig}{\hejG}}}{\hejG}\pvmu.$
Thus
$$
\pportw = c_3\wrapBracks{\minvAB{\pvsig}{\pvmu} -
\proj{\pvsig}{\hejG}\pvmu}.
$$
Plugging this into the objective reduces the problem to the
univariate optimization
\begin{equation*}
\max_{c_3 :\, c_3^2 \le \fracc{\Rbuj^2}{\psnrsqoptG{\hejG}}}
\sign{c_3} \psnroptG{\hejG} - \frac{\rfr}{\abs{c_3}\psnroptG{\hejG}},
\end{equation*}
where $\psnrsqoptG{\hejG} = \qiform{\pvsig}{\pvmu} -
\qform{\proj{\pvsig}{\hejG}}{\pvmu}.$
The optimum
occurs for $c = \fracc{\Rbuj}{\psnroptG{\hejG}}$, moreover the optimum
is unique when $\rfr > 0$.
\end{proof}%UNFOLD
The optimal hedged
portfolio in \lemmaref{cons_sr_optimal_portfolio} is, up to scaling,
the difference of the unconstrained optimal portfolio
from \lemmaref{sr_optimal_portfolio} and the subspace constrained
portfolio in \lemmaref{subsp_cons_sr_optimal_portfolio}. This `delta'
analogy continues for the rest of this section.
\begin{definition}[Delta Inverse Second Moment]%FOLDUP
\label{definition:delta_inv_second_moment}
Let \hejGt be the $\wrapParens{1+\nlatfhej}\times\wrapParens{\nlatf+1}$
matrix,
$$
\hejGt \defeq \twobytwossym{1}{0}{\hejG}.
$$
Define the `delta inverse second moment' as
$$
\Delhej\minv{\pvsm} \defeq \minv{\pvsm} -
\proj{\pvsm}{\hejGt}.$$
\end{definition}%UNFOLD
Simple algebra proves the following lemma.
\begin{lemma}%FOLDUP
The elements of $\Delhej\minv{\pvsm}$
are
\begin{equation*}
\Delhej\minv{\pvsm} =
\twobytwo{ \qiform{\pvsig}{\pvmu} - \qform{\proj{\pvsig}{\hejG}}{\pvmu} }{
-\tr{\pvmu}\minv{\pvsig} + \tr{\pvmu}\proj{\pvsig}{\hejG}}{
-\minv{\pvsig}\pvmu + \proj{\pvsig}{\hejG}\pvmu}{
\minv{\pvsig} - \proj{\pvsig}{\hejG}}.
\end{equation*}
%\twobytwosym{ \qiform{\pvsig}{\pvmu} - \proj{\pvsig}{\hejG} }{
%-\wrapBracks{\tr{\pvmu}\minv{\pvsig} - \tr{\pvmu}\proj{\pvsig}{\hejG}}}{
%\minv{\pvsig} - \proj{\pvsig}{\hejG}}.
In particular, elements $2$ through $\nlatf+1$ of
$-\fvech{\Delhej\minv{\pvsm}}$ are the portfolio $\pportwoptFoo{\Rbuj,\hejG,}$
defined in \lemmaref{cons_sr_optimal_portfolio}, up to the scaling
constant $c$ which is the ratio of \Rbuj to the square root of the first
element of \fvech{\Delhej\minv{\pvsm}}.
\end{lemma}%UNFOLD
The statistic
$\qiform{\svsig}{\svmu} - \qform{\proj{\svsig}{\hejG}}{\svmu}$, for the
case where \hejG is some rows of the $\nlatf \times \nlatf$ identity matrix,
was first proposed by Rao, and its distribution under Gaussian returns was
later found by Giri. \cite{rao1952,giri1964likelihood} This test statistic
may be used for tests of portfolio \emph{spanning} for the case where a
risk-free instrument is traded. \cite{HKspan1987,KanZhou2012}
The asymptotic distribution of $\Delhej\minv{\svsm}$ is given by the
following theorem, which is the analogue of \theoremref{inv_distribution}.
\begin{theorem}%FOLDUP
Let \svsm be the unbiased sample estimate of
\pvsm, based on \ssiz \iid samples of \vreti.
Let $\Delhej\minv{\pvsm}$
be defined as in \definitionref{delta_inv_second_moment}, and similarly
define $\Delhej\minv{\svsm}$. Let \pvvar be the variance of
$\fvech{\ogram{\avreti}}$. Then, asymptotically in \ssiz,
\begin{equation}
\sqrt{\ssiz}\wrapParens{\fvech{\Delhej\minv{\svsm}} - \fvech{\Delhej\minv{\pvsm}}}
\rightsquigarrow
\normlaw{0,\qoform{\pvvar}{\Mtx{H}}},
\label{eqn:mvclt_hej_isvsm}
\end{equation}
where
\begin{equation*}
\Mtx{H} = - \EXD{\wrapBracks{\AkronA{\minv{\pvsm}} - \broj{\pvsm}{\hejGt}}}.
\end{equation*}
\label{theorem:delhej_inv_distribution}
\end{theorem}%UNFOLD
%\begin{proof}%FOLDUP
%By the multivariate delta method, it suffices to prove that
%$$
%\Mtx{H} = \dbyd{\fvech{\Delhej\minv{\pvsm}}}{\fvech{\pvsm}}.
%$$
%Via \lemmaref{misc_derivs},
%%Via \lemmaref{deriv_vech_matrix_inverse},
%%and the proof of \theoremref{inv_distribution},
%it suffices to prove that
%$$
%\dbyd{\proj{\pvsm}{\hejGt}}{\pvsm} =
%-\wrapParens{\AkronA{\tr{\hejGt}}} \wrapParens{\AkronA{\minv{\wrapParens{\qoform{\pvsm}{\hejGt}}}}}
%\wrapParens{\AkronA{\hejGt}}.
%$$
%A well-known fact regarding matrix manipulation \cite{magnus1999matrix} is
%$$
%\fvec{\Mtx{A}\Mtx{B}\Mtx{C}} = \wrapParens{\Mtx{A}\kron\tr{\Mtx{C}}}
%\fvec{\Mtx{B}},\quad\mbox{therefore,}\quad
%\dbyd{\Mtx{A}\Mtx{B}\Mtx{C}}{\Mtx{B}} = \Mtx{A}\kron\tr{\Mtx{C}}.
%$$
%Using this, and the chain rule, we have:
%\begin{equation*}
%\begin{split}
%\dbyd{\proj{\pvsm}{\hejGt}}{\pvsm}
%&=
%\dbyd{\proj{\pvsm}{\hejGt}}{\minv{\wrapParens{\qoform{\pvsm}{\hejGt}}}}
%\dbyd{\minv{\wrapParens{\qoform{\pvsm}{\hejGt}}}}{\qoform{\pvsm}{\hejGt}}
%\dbyd{\qoform{\pvsm}{\hejGt}}{\pvsm}\\
%&=
%\wrapParens{\AkronA{\tr{\hejGt}}}
%\dbyd{\minv{\wrapParens{\qoform{\pvsm}{\hejGt}}}}{\qoform{\pvsm}{\hejGt}}
%\wrapParens{\AkronA{\hejGt}}.
%\end{split}
%\end{equation*}
%\lemmaref{deriv_vech_matrix_inverse} gives the middle term, completing the
%proof.
%\end{proof}
%An analogue of \corollaryref{portwoptR_dist} gives the asymptotic
%distribution of $\pportwoptFoo{\Rbuj,\hejG,}$
%defined in \lemmaref{cons_sr_optimal_portfolio}.
%%UNFOLD
\begin{proof}%FOLDUP
Minor modification of proof of \theoremref{subzer_inv_distribution}.
\end{proof}%UNFOLD
\begin{caution}%FOLDUP
In the hedged portfolio optimization problem considered here,
the optimal portfolio will, in general, hold money in the
row space of \hejG.
For example, in
the garden variety application, where one is hedging out exposure to
`the market' by including a broad market ETF, and taking \hejG to be
the corresponding row of the identity matrix, the final portfolio may
hold some position in that broad market ETF. This is fine for an ETF,
but one may wish to hedge out exposure to an untradeable returns
stream--the returns of an index, say.
Combining the hedging constraint of this section with the
subspace constraint of \subsecref{subspace_constraint} is
simple in the case where the rows of \hejG are spanned
by the rows of \zerJ. The more general case, however, is rather
more complicated.
\end{caution}%UNFOLD
%UNFOLD
%\subsection{Subspace and hedging constraints}%FOLDUP
%\label{subsec:subspace_and_hedge_constraint}
%In the hedged portfolio optimization problem considered in
%\subsecref{hedging_constraint}, the optimal portfolio
%will, in general, hold money in the row space of \hejG. For example, in
%the garden variety application, where one is hedging out exposure to
%`the market' by including a broad market ETF, and taking \hejG to be
%the corresponding row of the identity matrix, the final portfolio may
%hold some position in that broad market ETF. This is fine for an ETF,
%but one may wish to hedge out exposure to an untradeable returns
%stream--the returns of an index, say.
%To deal with this problem, we can combine the constraints of
%\subsecref{hedging_constraint} and \subsecref{subspace_constraint}.
%So let \zerJ, \zerJc, \hejG be as above.
%\begin{lemma}[subspace constrained \txtSR optimal portfolio]%FOLDUP
%\label{lemma:snh_cons_sr_optimal_portfolio}
%Assuming the rows of \zerJ span the null space of the rows of \zerJc,
%$\zerJ\pvmu \ne \vzero$, and \pvsig is invertible,
%the portfolio optimization problem
%\begin{equation}
%\max_{\substack{\pportw : \zerJc \pportw = \vzero,\\
%\hejG\pvsig \pportw = \vzero,\\
%\qform{\pvsig}{\pportw} \le \Rbuj^2}}
%\frac{\trAB{\pportw}{\pvmu} - \rfr}{\sqrt{\qform{\pvsig}{\pportw}}},
%\label{eqn:portopt_zer_III}
%\end{equation}
%for $\rfr \ge 0, \Rbuj > 0$ is solved by
%%\begin{equation*}
%%\begin{split}
%%\pportwoptFoo{\Rbuj,\zerJ,}
%%&\defeq c \proj{\pvsig}{\zerJ}\pvmu,\\
%%c &= \frac{\Rbuj}{\sqrt{\qform{\proj{\pvsig}{\zerJ}}{\pvmu}}}.
%%\end{split}
%%\end{equation*}
%When $\rfr > 0$ the solution is unique.
%\end{lemma}%UNFOLD
%2FIX: continue.
%%UNFOLD
\subsection{Conditional heteroskedasticity}%FOLDUP
\label{subsec:cond_het}
The methods described above ignore `volatility clustering', and assume
homoskedasticity. \cite{stylized_facts,nelson1991,ARCH1987}
To deal with this,
consider a strictly positive scalar random variable, \fvola[i],
observable at the time the investment decision is
required to capture \vreti[i+1]. For reasons to be obvious later, it
is more convenient to think of \fvola[i] as a `quietude' indicator.
Two simple competing models for conditional heteroskedasticity are
\begin{align}
\label{eqn:cond_model_I}
\mbox{(constant):}\quad \Econd{\vreti[i+1]}{\fvola[i]} &=
\fvola[i]^{-1}\pvmu & \Varcond{\vreti[i+1]}{\fvola[i]} &=
\fvola[i]^{-2} \pvsig,\\
\label{eqn:cond_model_II}
\mbox{(floating):}\quad \Econd{\vreti[i+1]}{\fvola[i]} &=
\pvmu & \Varcond{\vreti[i+1]}{\fvola[i]} &=
\fvola[i]^{-2} \pvsig.
\end{align}
Under the model in \eqnref{cond_model_I}, the maximal \txtSR
is $\sqrt{\qiform{\pvsig}{\pvmu}}$, independent of \fvola[i];
under \eqnref{cond_model_II}, it is
is $\fvola[i]\sqrt{\qiform{\pvsig}{\pvmu}}$.
The model names reflect whether or not
the maximal \txtSR varies conditional on \fvola[i].
The optimal portfolio under both models is the same, as stated
in the following lemma, the proof of which follows by simply using
\lemmaref{sr_optimal_portfolio}.
\begin{lemma}[Conditional \txtSR optimal portfolio]%FOLDUP
\label{lemma:cond_sr_optimal_portfolio_I}
Under either the model in \eqnref{cond_model_I} or \eqnref{cond_model_II},
conditional on observing \fvola[i], the portfolio optimization problem
\begin{equation}
\argmax_{\pportw :\, \Varcond{\trAB{\pportw}{\vreti[i+1]}}{\fvola[i]} \le \Rbuj^2}
\frac{\Econd{\trAB{\pportw}{\vreti[i+1]}}{\fvola[i]} -
\rfr}{\sqrt{\Varcond{\trAB{\pportw}{\vreti[i+1]}}{\fvola[i]}}},
\label{eqn:cond_sr_optimal_portfolio_problem_I}
\end{equation}
for $\rfr \ge 0, \Rbuj > 0$ is solved by
\begin{equation}
\pportwopt = \frac{\fvola[i] \Rbuj}{\sqrt{\qiform{\pvsig}{\pvmu}}}
\minvAB{\pvsig}{\pvmu}.
\end{equation}
Moreover, this is the unique solution whenever $\rfr > 0$.
\end{lemma}%UNFOLD
To perform inference on the portfolio \pportwopt from
\lemmaref{cond_sr_optimal_portfolio_I}, under the `constant'
model of \eqnref{cond_model_I}, apply the unconditional techniques
to the sample second moment of $\fvola[i]\avreti[i+1]$.
For
the `floating' model of \eqnref{cond_model_II}, however, some
adjustment to the technique is required.
Define $\aavreti[i+1] \defeq \fvola[i]\avreti[i+1]$; that is,
$\aavreti[i+1] = \asvec{\fvola[i],\fvola[i]\tr{\vreti[i+1]}}$.
Consider the second moment of \aavreti:
\begin{equation}
\pvsm[{\fvola}] \defeq \E{\ogram{\aavreti}} =
\twobytwo{\volavar}{\volavar\tr{\pvmu}}{\volavar\pvmu}{\pvsig +
\qoform{\volavar}{\pvmu}},\quad\mbox{where}\quad
\volavar \defeq \E{\fvola^2}.
%\label{eqn:pvsm_def}
\end{equation}
The inverse of \pvsm[{\fvola}] is
\begin{equation}
\minv{\pvsm[{\fvola}]}
= \twobytwo{\volaivar + \qiform{\pvsig}{\pvmu}}{-\tr{\pvmu}\minv{\pvsig}}{-\minv{\pvsig}\pvmu}{\minv{\pvsig}}
%= \twobytwo{1 + \psnrsqopt}{-\tr{\pportwopt}}{-\pportwopt}{\minv{\pvsig}},
\label{eqn:new_trick_inversion}
\end{equation}
Once again, the optimal portfolio (up to scaling and sign), appears in
\fvech{\minv{\pvsm[{\fvola}]}}.
Similarly, define the sample analogue:
\begin{equation}
\svsm[{\fvola}] \defeq \oneby{\ssiz}\sum_i \ogram{\aavreti[i+1]}.
\end{equation}
We can find the asymptotic distribution of \fvech{\svsm[{\fvola}]}
using the same techniques as in the unconditional case, as in the
following analogue of \theoremref{inv_distribution}:
\begin{theorem}%FOLDUP
\label{theorem:cond_inv_distribution}
Let $\svsm[{\fvola}] \defeq \oneby{\ssiz}\sum_i \ogram{\aavreti[i+1]}$,
based on \ssiz \iid samples of \asvec{\fvola,\tr{\vreti}}.
Let \pvvar be the variance of $\fvech{\ogram{\aavreti}}$.
Then, asymptotically in \ssiz,
%%\begin{equation}
%\begin{multline}
%\sqrt{\ssiz}\wrapParens{\fvech{\minv{\svsm[{\fvola}]}} -
%\fvech{\minv{\pvsm[{\fvola}]}}}
%\rightsquigarrow \\
%\normlaw{0,\qoform{\pvvar}{\wrapBracks{\EXD{\wrapParens{\AkronA{\minv{\pvsm[{\fvola}]}}}}}}}.
%\label{eqn:cond_mvclt_isvsm}
%\end{multline}
%%\end{equation}
\begin{equation}
\label{eqn:cond_mvclt_isvsm}
\sqrt{\ssiz}\wrapParens{\fvech{\minv{\svsm[{\fvola}]}} -
\fvech{\minv{\pvsm[{\fvola}]}}}
\rightsquigarrow
\normlaw{0,\qoform{\pvvar}{\Mtx{H}}},
\end{equation}
where
\begin{equation}
\Mtx{H} = -\EXD{\wrapParens{\AkronA{\minv{\pvsm[{\fvola}]}}}}.
\end{equation}
Furthermore, we may replace \pvvar in this equation with an asymptotically
consistent estimator, \svvar.
\end{theorem}%UNFOLD
The only real difference from the unconditional case is that we cannot
automatically assume that the first row and column of \pvvar is zero
(unless \fvola is actually constant, which misses the point). Moreover,
the shortcut for estimating \pvvar under Gaussian returns is not
valid without some patching, an exercise left for the reader.
Dependence or independence of maximal \txtSR from
volatility is an assumption which, ideally, one could test with
data. A mixed model containing both characteristics can be written
as follows:
\begin{align}
\label{eqn:cond_model_III}
\mbox{(mixed):}\quad \Econd{\vreti[i+1]}{\fvola[i]} &=
\fvola[i]^{-1}\pvmu[0] + \pvmu[1] & \Varcond{\vreti[i+1]}{\fvola[i]} &=
\fvola[i]^{-2} \pvsig.
\end{align}
One could then test whether elements of $\pvmu[0]$ or of $\pvmu[1]$ are
zero. Analyzing this model is somewhat complicated without moving to a
more general framework, as in the sequel.
%UNFOLD
\subsection{Conditional expectation and heteroskedasticity}%FOLDUP
\label{subsec:cond_ret_het}
Suppose you observe random variables $\fvola[i] > 0$, and
\nfac-vector $\vfact[i]$ at some time prior to when the investment
decision is required to capture \vreti[i+1]. It need not be the case
that \fvola[{}] and \vfact[{}] are independent. The general model
is now
\begin{align}
\label{eqn:cond_model_IV}
\mbox{(bi-conditional):}\quad
\Econd{\vreti[i+1]}{\fvola[i],\vfact[i]} &=
\pRegco \vfact[i] &
\Varcond{\vreti[i+1]}{\fvola[i],\vfact[i]} &=
\fvola[i]^{-2} \pvsig,
\end{align}
where \pRegco is some $\nlatf \times \nfac$ matrix. Without the
\fvola[i] term, these are the `predictive regression' equations
commonly used in Tactical Asset
Allocation. \cite{connor1997,herold2004TAA,brandt2009portfolio}
By letting
$\vfact[i] = \asvec{\fvola[i]^{-1},1}$ we recover
the mixed model in \eqnref{cond_model_III}; the bi-conditional model
is considerably more general, however. The conditionally-optimal portfolio
is given by the following lemma. Once again, the proof proceeds simply
by plugging in the conditional expected return and volatility into
\lemmaref{sr_optimal_portfolio}.
\begin{lemma}[Conditional \txtSR optimal portfolio]%FOLDUP
\label{lemma:cond_sr_optimal_portfolio_II}
Under the model in \eqnref{cond_model_IV},
conditional on observing \fvola[i] and \vfact[i],
the portfolio optimization problem
\begin{equation}
\argmax_{\pportw :\, \Varcond{\trAB{\pportw}{\vreti[i+1]}}{\fvola[i],\vfact[i]} \le \Rbuj^2}
\frac{\Econd{\trAB{\pportw}{\vreti[i+1]}}{\fvola[i],\vfact[i]} -
\rfr}{\sqrt{\Varcond{\trAB{\pportw}{\vreti[i+1]}}{\fvola[i],\vfact[i]}}},
\label{eqn:cond_sr_optimal_portfolio_problem_I}
\end{equation}
for $\rfr \ge 0, \Rbuj > 0$ is solved by
\begin{equation}
\pportwopt = \frac{\fvola[i]
\Rbuj}{\sqrt{\qform{\qiform{\pvsig}{\pRegco}}{\vfact[i]}}}
\minvAB{\pvsig}{\pRegco\vfact[i]}.
\end{equation}
Moreover, this is the unique solution whenever $\rfr > 0$.
\end{lemma}%UNFOLD
\begin{caution}
It is emphatically \emph{not} the case that
investing in the portfolio \pportwopt from
\lemmaref{cond_sr_optimal_portfolio_II} at every time step
is long-term \txtSR optimal. One may possibly achieve a higher long-term
\txtSR by down-levering at times when the conditional \txtSR is low.
The optimal long term investment strategy falls under the
rubric of `multiperiod portfolio choice', and is an area of active
research. \cite{mulvey2003advantages,fabozzi2007robust,brandt2009portfolio}
%research. \cite{mulvey2003advantages,fabozzi2007robust,bertsimas2008robust}
\end{caution}
The matrix \minvAB{\pvsig}{\pRegco} is the generalization of the
Markowitz portfolio: it is the multiplier for a model under which
the optimal portfolio is linear in the features \vfact[i] (up to
scaling to satisfy the risk budget). We can think of this matrix
as the `\txtMC'. If an entire column of
\minvAB{\pvsig}{\pRegco} is zero, it suggests that the
corresponding element of \vfact[{}] can be ignored in investment
decisions; if an entire row of \minvAB{\pvsig}{\pRegco} is zero,
it suggests the corresponding instrument delivers no return or
hedging benefit.
Tests on \minvAB{\pvsig}{\pRegco} should be contrasted with
the so-called Multivariate General Linear Hypothesis (MGLH),
which tests the matrix equation
$\Mtx{A}\pRegco\Mtx{C} = \Mtx{T}$, for conformable
$\Mtx{A},\Mtx{C},\Mtx{T}$. \cite{Rencher2002,Muller1984143}
To perform inference on the \txtMC, we can proceed
exactly as above. Let
\begin{equation}
\aavreti[i+1] \defeq \asvec{\fvola[i]\tr{\vfact[i]},\fvola[i]\tr{\vreti[i+1]}}.
\end{equation}
Consider the second moment of \aavreti:
\begin{equation}
\pvsm[{\sfact[]}] \defeq \E{\ogram{\aavreti}} =
\twobytwo{\pfacsig}{\pfacsig\tr{\pRegco}}{\pRegco\pfacsig}{\pvsig +
\qoform{\pfacsig}{\pRegco}},\quad\mbox{where}\quad
\pfacsig \defeq \E{\fvola[{}]^2\ogram{\vfact[{}]}}.
\label{eqn:cond_pvsm_def}
%\label{eqn:pvsm_def}
\end{equation}
The inverse of \pvsm[{\sfact[]}] is
\begin{equation}
\minv{\pvsm[{\sfact[]}]}
= \twobytwo{\minv{\pfacsig} +
\qiform{\pvsig}{\pRegco}}{-\tr{\pRegco}\minv{\pvsig}}{-\minv{\pvsig}\pRegco}{\minv{\pvsig}}
\label{eqn:new_new_trick_inversion}
\end{equation}
Once again, the \txtMC (up to scaling and sign), appears in
\fvech{\minv{\pvsm[{\sfact[]}]}}.
The following theorem is an analogue of, and shares a proof with,
\theoremref{inv_distribution}.
\begin{theorem}%FOLDUP
\label{theorem:cond_inv_distribution_II}
Let $\svsm[{\sfact[]}] \defeq \oneby{\ssiz}\sum_i \ogram{\aavreti[i+1]}$,
based on \ssiz \iid samples of \asvec{\fvola,\tr{\vfact[{}]},\tr{\vreti}},
where
$$
\aavreti[i+1] \defeq \asvec{\fvola[i]\tr{\vfact[i]},\fvola[i]\tr{\vreti[i+1]}}.
$$
Let \pvvar be the variance of $\fvech{\ogram{\aavreti}}$.
Then, asymptotically in \ssiz,
%%\begin{equation}
%\begin{multline}
%\sqrt{\ssiz}\wrapParens{\fvech{\minv{\svsm[{\sfact[]}]}} -
%\fvech{\minv{\pvsm[{\sfact[]}]}}}
%\rightsquigarrow \\
%\normlaw{0,\qoform{\pvvar}{\wrapBracks{\EXD{\wrapParens{\AkronA{\minv{\pvsm[{\sfact[]}]}}}}}}}.
%\label{eqn:cond_mvclt_isvsm_II}
%\end{multline}
%%\end{equation}
\begin{equation}
\sqrt{\ssiz}\wrapParens{\fvech{\minv{\svsm[{\sfact[]}]}} -
\fvech{\minv{\pvsm[{\sfact[]}]}}}
\rightsquigarrow
\normlaw{0,\qoform{\pvvar}{\Mtx{H}}},
\label{eqn:cond_mvclt_isvsm_II}
\end{equation}
where
\begin{equation}
\Mtx{H} = -\EXD{\wrapParens{\AkronA{\minv{\pvsm[{\sfact[]}]}}}}.
\end{equation}
Furthermore, we may replace \pvvar in this equation with an asymptotically
consistent estimator, \svvar.
\end{theorem}%UNFOLD
%UNFOLD
%\subsection{Conditional expectation and heteroskedasticity with hedging constraint}%FOLDUP
%A little work allows us to combine the conditional model of
%\subsecref{cond_ret_het} with the hedging constraint of
%\subsecref{hedging_constraint}. Suppose returns follow the model
%of \eqnref{cond_model_IV}. To prove the following lemma, simply
%plug in $\pRegco\vfact[i]$ for \pvmu, and
%$\fvola[i]^{-2} \pvsig$ for \pvsig into \lemmaref{cons_sr_optimal_portfolio}.
%\begin{lemma}[Hedged Conditional \txtSR optimal portfolio]%FOLDUP
%\label{lemma:hej_cond_sr_optimal_portfolio}
%Let $\hejG$ be a given $\nlatfhej \times \nlatf$ matrix of
%rank \nlatfhej. Under the model in \eqnref{cond_model_IV},
%conditional on observing \fvola[i] and \vfact[i],
%the portfolio optimization problem
%\begin{equation}
%\argmax_{\substack{\pportw :\,\hejG\pvsig \pportw = \vzero,\\
%\Varcond{\trAB{\pportw}{\vreti[i+1]}}{\fvola[i],\vfact[i]} \le \Rbuj^2}}
%\frac{\Econd{\trAB{\pportw}{\vreti[i+1]}}{\fvola[i],\vfact[i]} -
%\rfr}{\sqrt{\Varcond{\trAB{\pportw}{\vreti[i+1]}}{\fvola[i],\vfact[i]}}},
%\label{eqn:hej_cond_sr_optimal_portfolio_problem_I}
%\end{equation}
%for $\rfr \ge 0, \Rbuj > 0$ is solved by
%\begin{equation*}
%\begin{split}
%\pportwoptFoo{\Rbuj,\hejG,} &\defeq c \wrapParens{\minv{\pvsig}{\pRegco} -
%\proj{\pvsig}{\hejG}\pRegco}\vfact[i],\\
%c &= \frac{\fvola[i]\Rbuj}{\sqrt{\qiform{\pvsig}{\wrapParens{\pRegco\vfact[i]}} -
%\qform{\proj{\pvsig}{\hejG}}{\wrapParens{\pRegco\vfact[i]}}}}.
%\end{split}
%\end{equation*}
%Moreover, this is the unique solution whenever $\rfr > 0$.
%\end{lemma}%UNFOLD
%The same cautions regarding multiperiod portfolio choice apply to the
%above lemma. Results analogous to those of \subsecref{hedging_constraint}
%follow, but with one minor modification to the analogue of
%\definitionref{delta_inv_second_moment}.
%\begin{lemma}%FOLDUP
%Let \hejGt now be the
%$\wrapParens{\nfac+\nlatfhej}\times\wrapParens{\nfac+\nlatf}$ matrix,
%$$
%\hejGt \defeq \twobytwossym{\eye_{\nfac}}{0}{\hejG},
%$$
%where the upper right corner is the $\nfac\times\nfac$ identity matrix.
%Define the `delta inverse second moment' as
%\begin{equation}
%\label{eqn:cond_del_hej_def}
%\Delhej\minv{\pvsm[{\sfact[]}]} \defeq \minv{\pvsm[{\sfact[]}]} -
%\proj{\pvsm[{\sfact[]}]}{\hejGt},
%\end{equation}
%where $\pvsm[{\sfact[]}]$ is defined as in \eqnref{cond_pvsm_def}.
%The elements of $\Delhej\minv{\pvsm[{\sfact[]}]}$
%are
%\begin{equation*}
%\Delhej\minv{\pvsm} =
%\twobytwo{ \qiform{\pvsig}{\pRegco} - \qform{\proj{\pvsig}{\hejG}}{\pRegco} }{
%-\tr{\pRegco}\minv{\pvsig} + \tr{\pRegco}\proj{\pvsig}{\hejG}}{
%-\minv{\pvsig}\pRegco + \proj{\pvsig}{\hejG}\pRegco}{
%\minv{\pvsig} - \proj{\pvsig}{\hejG}}.
%\end{equation*}
%In particular, the \txtMC from
%\lemmaref{hej_cond_sr_optimal_portfolio} appears in the lower left
%corner of
%$-\fvech{\Delhej\minv{\pvsm[{\sfact[]}]}}$, and the
%denominator of the constant $c$ from
%\lemmaref{hej_cond_sr_optimal_portfolio} depends on a quadratic form
%of \vfact[i] with the upper right left corner of
%$\fvech{\Delhej\minv{\pvsm[{\sfact[]}]}}$.
%\end{lemma}%UNFOLD
%\begin{theorem}%FOLDUP
%\label{theorem:hej_cond_inv_distribution_II}
%Let $\svsm[{\sfact[]}] \defeq \oneby{\ssiz}\sum_i \ogram{\aavreti[i+1]}$,
%based on \ssiz \iid samples of \asvec{\fvola,\tr{\vfact[{}]},\tr{\vreti}},
%where
%$$
%\aavreti[i+1] \defeq \asvec{\fvola[i]\tr{\vfact[i]},\fvola[i]\tr{\vreti[i+1]}}.
%$$
%Let \pvvar be the variance of $\fvech{\ogram{\aavreti}}$.
%Define
%$\Delhej\minv{\pvsm[{\sfact[]}]}$ as in
%\eqnref{cond_del_hej_def} for given
%$\wrapParens{\nfac+\nlatfhej}\times\wrapParens{\nlatf+\nfac}$ matrix \hejGt.
%Then, asymptotically in \ssiz,
%\begin{equation}
%\sqrt{\ssiz}\wrapParens{\fvech{\Delhej\minv{\svsm[{\sfact[]}]}} -
%\fvech{\Delhej\minv{\pvsm[{\sfact[]}]}}}
%\rightsquigarrow
%\normlaw{0,\qoform{\pvvar}{\Mtx{H}}},
%\label{eqn:hej_cond_mvclt_isvsm_II}
%\end{equation}
%where
%\begin{equation*}
%\Mtx{H} = - \EXD{\wrapBracks{\AkronA{\minv{\pvsm[{\sfact[]}]}} -
%\wrapParens{\AkronA{\tr{\hejGt}}}
%\wrapParens{\AkronA{\minv{\wrapParens{\qoform{\pvsm[{\sfact[]}]}{\hejGt}}}}}
%\wrapParens{\AkronA{\hejGt}}}}.
%\end{equation*}
%Furthermore, we may replace \pvvar in this equation with an asymptotically
%consistent estimator, \svvar.
%\end{theorem}%UNFOLD
%%UNFOLD
\subsection{Conditional expectation and heteroskedasticity with subspace and hedging constraint}%FOLDUP
A little work allows us to combine the conditional model of
\subsecref{cond_ret_het} with the subspace constraint of
\subsecref{subspace_constraint} and the
hedging constraint of \subsecref{hedging_constraint}.
This extension is trivial only in the case
where the rows of \hejG are spanned by the rows of \zerJ. So, for
the remainder of this section, we will assume this is the case.
The problem considered here is the most general case solved in this
note; the previous sections are all specializations of it in one way
or another.
\begin{lemma}[Hedged Conditional \txtSR optimal portfolio]%FOLDUP
\label{lemma:zerhej_cond_sr_optimal_portfolio}
Let $\zerJ$ be a given $\nlatfzer \times \nlatf$ matrix, the rows of which
span the rows of $\hejG$, a given $\nlatfhej \times \nlatf$ matrix of
rank \nlatfhej. Under the model in \eqnref{cond_model_IV},
conditional on observing \fvola[i] and \vfact[i],
the portfolio optimization problem
\begin{equation}
\argmax_{\substack{\pportw :\,\zerJc\pportw = \vzero,\\
\hejG\pvsig \pportw = \vzero,\\
\Varcond{\trAB{\pportw}{\vreti[i+1]}}{\fvola[i],\vfact[i]} \le \Rbuj^2}}
\frac{\Econd{\trAB{\pportw}{\vreti[i+1]}}{\fvola[i],\vfact[i]} -
\rfr}{\sqrt{\Varcond{\trAB{\pportw}{\vreti[i+1]}}{\fvola[i],\vfact[i]}}},
\label{eqn:zerhej_cond_sr_optimal_portfolio_problem_I}
\end{equation}
for $\rfr \ge 0, \Rbuj > 0$ is solved by
\begin{equation*}
\begin{split}
\pportwoptFoo{\Rbuj,\zerJ,\hejG,} &\defeq c \wrapParens{\proj{\pvsig}{\zerJ}\pRegco -
\proj{\pvsig}{\hejG}\pRegco}\vfact[i],\\
c &= \frac{\fvola[i]\Rbuj}{\sqrt{\qform{\proj{\pvsig}{\zerJ}}{\wrapParens{\pRegco\vfact[i]}} -
\qform{\proj{\pvsig}{\hejG}}{\wrapParens{\pRegco\vfact[i]}}}}.
\end{split}
\end{equation*}
Moreover, this is the unique solution whenever $\rfr > 0$.
\end{lemma}%UNFOLD
The same cautions regarding multiperiod portfolio choice apply to the
above lemma. The asymptotic distribution results that follow are minor
modifications of those from previous sections.
The `delta inverse second moment' now explicitly becomes the
difference of two projections:
\begin{definition}[Delta Inverse Second Moment]%FOLDUP
\label{definition:cond_delta_inv_second_moment}
Given \zerJ and \hejG, define
\begin{equation}
\label{eqn:zerJ_twid_def}
\zerJt \defeq \twobytwossym{\eye_{\nfac}}{0}{\zerJ},\,\mbox{and}\,
\hejGt \defeq \twobytwossym{\eye_{\nfac}}{0}{\hejG},
\end{equation}
where $\eye_{\nfac}$ is the $\nfac\times\nfac$ identity matrix.
Define the `delta inverse second moment' as
\begin{equation}
\label{eqn:cond_del_hej_def}
\Delzh\minv{\pvsm[{\sfact[]}]} \defeq
\proj{\pvsm[{\sfact[]}]}{\zerJt} -
\proj{\pvsm[{\sfact[]}]}{\hejGt},
\end{equation}
where $\pvsm[{\sfact[]}]$ is defined in \eqnref{cond_pvsm_def}.
\end{definition}%UNFOLD
Once again, the delta inverse second moment contains the \txtMC,
as in the following lemma.
\begin{lemma}%FOLDUP
Under %the definitions in
\definitionref{cond_delta_inv_second_moment},
%Then $\Delzh\minv{\pvsm[{\sfact[]}]}$ equals
%\begin{multline}
%\Delzh\minv{\pvsm} =\\
\begin{equation*}
\Delzh\minv{\pvsm[{\sfact[]}]} =
\twobytwo{
\qform{\proj{\pvsig}{\zerJ}}{\pRegco} - \qform{\proj{\pvsig}{\hejG}}{\pRegco} }{
-\tr{\pRegco}\proj{\pvsig}{\zerJ} + \tr{\pRegco}\proj{\pvsig}{\hejG}}{
-\proj{\pvsig}{\zerJ}\pRegco + \proj{\pvsig}{\hejG}\pRegco}{
\proj{\pvsig}{\zerJ} - \proj{\pvsig}{\hejG}}.
\end{equation*}
%\end{multline}
\end{lemma}%UNFOLD
%\begin{note}
In particular, the \txtMC from
\lemmaref{zerhej_cond_sr_optimal_portfolio} appears in the lower left
corner of
$-{\Delzh\minv{\pvsm[{\sfact[]}]}}$, and the
denominator of the constant $c$ from
\lemmaref{zerhej_cond_sr_optimal_portfolio} depends on a quadratic form
of \vfact[i] with the upper left corner of
${\Delzh\minv{\pvsm[{\sfact[]}]}}$.
%\end{note}
\begin{theorem}%FOLDUP
\label{theorem:zerhej_cond_inv_distribution_II}
Let $\svsm[{\sfact[]}] \defeq \oneby{\ssiz}\sum_i \ogram{\aavreti[i+1]}$,
based on \ssiz \iid samples of \asvec{\fvola,\tr{\vfact[{}]},\tr{\vreti}},
where
$$
\aavreti[i+1] \defeq \asvec{\fvola[i]\tr{\vfact[i]},\fvola[i]\tr{\vreti[i+1]}}.
$$
Let \pvvar be the variance of $\fvech{\ogram{\aavreti}}$.
Define
$\Delzh\minv{\pvsm[{\sfact[]}]}$ as in
\eqnref{cond_del_hej_def} for the given \zerJt and \hejGt.
%$\wrapParens{\nfac+\nlatfzer}\times\wrapParens{\nlatf+\nfac}$ matrix \zerJt
%and $\wrapParens{\nfac+\nlatfhej}\times\wrapParens{\nlatf+\nfac}$ matrix \hejGt.
Then, asymptotically in \ssiz,
\begin{equation}
\sqrt{\ssiz}\wrapParens{\fvech{\Delzh\minv{\svsm[{\sfact[]}]}} -
\fvech{\Delzh\minv{\pvsm[{\sfact[]}]}}}
\rightsquigarrow
\normlaw{0,\qoform{\pvvar}{\Mtx{H}}},
\label{eqn:zerhej_cond_mvclt_isvsm_II}
\end{equation}
where
\begin{equation*}
\Mtx{H} = - \EXD{\wrapParens{\broj{\pvsm[{\sfact[]}]}{\zerJt} -
\broj{\pvsm[{\sfact[]}]}{\hejGt}}}.
\end{equation*}
Furthermore, we may replace \pvvar in this equation with an asymptotically
consistent estimator, \svvar.
\end{theorem}%UNFOLD
%UNFOLD
%UNFOLD
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Examples}%FOLDUP
%Empirically, the marginal Wald test for zero weighting in the
%\txtMP based on this approximation are nearly
%identical to the \tstat-statistics produced by the procedure
%of Britten-Jones, as shown below. \cite{BrittenJones1999}
%<<'load_data',echo=TRUE>>=
%fname <- system.file('extdata','gmacro_data.rda',package='SharpeR')
%if (fname == "") {
%fname <- 'gmacro_data.rda'
%}
%# poofs ff5.xts ff10.xts fff.xts dp.xts shill.xts
%load(fname)
%@
%<<'me_vs_bjones',echo=TRUE>>=
%nday <- 1024
%nstk <- 5
%# under the null: all returns are zero mean;
%set.seed(as.integer(charToRaw("7fbb2a84-aa4c-4977-8301-539e48355a35")))
%rets <- matrix(rnorm(nday * nstk),nrow=nday)
%# t-stat via Britten-Jones procedure
%bjones.ts <- function(rets) {
%ones.vec <- matrix(1,nrow=dim(rets)[1],ncol=1)
%bjones.mod <- lm(ones.vec ~ rets - 1)
%bjones.sum <- summary(bjones.mod)
%retval <- bjones.sum$coefficients[,3]
%}
%# wald stat via inverse second moment trick
%ism.ws <- function(rets,...) {
%# flipping the sign on returns is idiomatic,
%asymv <- ism_vcov(- as.matrix(rets),...)
%asymv.mu <- asymv$mu[1:asymv$p]
%asymv.Sg <- asymv$Ohat[1:asymv$p,1:asymv$p]
%retval <- asymv.mu / sqrt(diag(asymv.Sg))
%}
%bjones.tstat <- bjones.ts(rets)
%ism.wald <- ism.ws(rets)
%# compare them:
%print(bjones.tstat)
%print(ism.wald)
%# repeat under the alternative;
%set.seed(as.integer(charToRaw("a5f17b28-436b-4d01-a883-85b3e5b7c218")))
%zero.rets <- t(matrix(rnorm(nday * nstk),nrow=nday))
%mu.vals <- (1/sqrt(253)) * seq(-1,1,length.out=nstk)
%rets <- t(zero.rets + mu.vals)
%bjones.tstat <- bjones.ts(rets)
%ism.wald <- ism.ws(rets)
%# compare them:
%print(bjones.tstat)
%print(ism.wald)
%@
%%<<'test_FF3',echo=TRUE>>=
%%ff3 <- read.csv('http://www.quandl.com/api/v1/datasets/KFRENCH/FACTORS_M.csv?&trim_start=1926-07-31&trim_end=2013-10-31&sort_order=desc', colClasses=c('Month'='Date'))
%%require(sandwich)
%%rfr <- ff3[,'RF']
%%hml <- ff3[,'HML'] - rfr
%%smb <- ff3[,'SMB'] - rfr
%%ff.ret <- cbind(mkt=ff3[,'Mkt.RF'],hml,smb)
%%my.ws <- ism.ws(ff.ret)
%%my.ws2 <- ism.ws(ff.ret,vcov.func=vcovHAC)
%%my.bt <- bjones.ts(ff.ret)
%%# conditionally?
%%blahm <- cbind(-ff.ret[(1:(dim(ff.ret)[1]-1)),],ff.ret[(2:dim(ff.ret)[1]),])
%%# futz from here ...
%%XX <- ism_vcov(blahm)
%%@
%% 2FIX: add examples here ...
%%UNFOLD
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% bibliography%FOLDUP
\nocite{markowitz1952portfolio,markowitz1999early,markowitz2012foundations}
%\bibliographystyle{jss}
%\bibliographystyle{siam}
%\bibliographystyle{ieeetr}
\bibliographystyle{plainnat}
%\bibliographystyle{acm}
\bibliography{MarkowitzR,rauto}
%\bibliography{AsymptoticMarkowitz}
%UNFOLD
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\appendix%FOLDUP
\section{Confirming the scalar Gaussian case}
\begin{example}%FOLDUP
To sanity check
\theoremref{theta_asym_var_gaussian},
consider the $\nlatf = 1$ Gaussian case. In this case,
$$
\fvech{\pvsm} = \asvec{1,\pmu,\psig^2 + \pmu^2},\quad\mbox{and}\quad
\fvech{\minv{\pvsm}} =
\asvec{1 + \frac{\pmu^2}{\psig^2},- \frac{\pmu}{\psig^2},\oneby{\psig^2}}.
$$
Let $\smu, \ssig^2$ be the unbiased sample estimates. By well
known results \cite{spiegel2007schaum}, $\smu$ and $\ssig^2$ are independent,
and have asymptotic variances of $\psig^2/\ssiz$ and $2\psig^4/\ssiz$
respectively. By the delta method, the asymptotic variance of
$\Unun \fvech{\svsm}$ and $\fvech{\minv{\svsm}}$ can be computed as
\begin{equation}
\begin{split}
\VAR{\Unun \fvech{\svsm}} &\rightsquigarrow
\oneby{\ssiz}\qform{\twobytwossym{\psig^2}{0}{2\psig^4}}{\twobytwo{1}{2\pmu}{0}{1}},\\
&= \oneby{\ssiz}\twobytwossym{\psig^2}{2\pmu\psig^2}{4\pmu^2\psig^2 +
2\psig^4}.
\label{eqn:gauss_confirm_theta}
\end{split}
\end{equation}
\begin{equation}
\begin{split}
\VAR{\fvech{\minv{\svsm}}} &\rightsquigarrow
\oneby{\ssiz}\qform{\twobytwossym{\psig^2}{0}{2\psig^4}}{\twobythree{\frac{2\pmu}{\psig^2}}{-\frac{1}{\psig^2}}{0}{-\frac{\pmu^2}{\psig^4}}{\frac{\pmu}{\psig^4}}{-\frac{1}{\psig^4}}},\\
&=
\oneby{\ssiz}\gram{\twobythree{2\psnr}{-\frac{1}{\psig}}{0}{-\sqrt{2}\psnr^2}{\sqrt{2}\frac{\psnr}{\psig}}{-\frac{\sqrt{2}}{\psig^2}}},\\
&= \oneby{\ssiz}\threebythreessym{2\psnr^2\wrapParens{2 + \psnr^2}}{-\frac{2\psnr}{\psig}\wrapParens{1+\psnr^2}}{2\frac{\psnr^2}{\psig^2}}{\frac{1 +
2\psnr^2}{\psig^2}}{-\frac{2\psnr}{\psig^3}}{\frac{2}{\psig^4}}.
\label{eqn:gauss_confirm_itheta}
\end{split}
\end{equation}
Now it remains to compute $\VAR{\Unun \fvech{\svsm}}$ via
\theoremref{theta_asym_var_gaussian}, and then
\VAR{\fvech{\minv{\svsm}}} via
\theoremref{inv_distribution}, and confirm they match the values above.
This is a rather tedious computation best left to a computer. Below is
an excerpt of an iPython notebook using Sympy \cite{PER-GRA:2007,sympy}
which performs this computation.
This notebook is available online. \cite{SEP2013example}
% SYMPY from here out%FOLDUP
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}1}]:} \PY{c}{\PYZsh{} confirm the asymptotic distribution of Theta}
\PY{c}{\PYZsh{} for scalar Gaussian case.}
\PY{k+kn}{from} \PY{n+nn}{\PYZus{}\PYZus{}future\PYZus{}\PYZus{}} \PY{k+kn}{import} \PY{n}{division}
\PY{k+kn}{from} \PY{n+nn}{sympy} \PY{k+kn}{import} \PY{o}{*}
\PY{k+kn}{from} \PY{n+nn}{sympy.physics.quantum} \PY{k+kn}{import} \PY{n}{TensorProduct}
\PY{n}{init\PYZus{}printing}\PY{p}{(}\PY{n}{use\PYZus{}unicode}\PY{o}{=}\PY{n+nb+bp}{False}\PY{p}{,} \PY{n}{wrap\PYZus{}line}\PY{o}{=}\PY{n+nb+bp}{False}\PY{p}{,} \PYZbs{}
\PY{n}{no\PYZus{}global}\PY{o}{=}\PY{n+nb+bp}{True}\PY{p}{)}
\PY{n}{mu} \PY{o}{=} \PY{n}{symbols}\PY{p}{(}\PY{l+s}{\PYZsq{}}\PY{l+s}{\PYZbs{}}\PY{l+s}{mu}\PY{l+s}{\PYZsq{}}\PY{p}{)}
\PY{n}{sg} \PY{o}{=} \PY{n}{symbols}\PY{p}{(}\PY{l+s}{\PYZsq{}}\PY{l+s}{\PYZbs{}}\PY{l+s}{sigma}\PY{l+s}{\PYZsq{}}\PY{p}{)}
\PY{c}{\PYZsh{} the elimination, duplication and U\PYZus{}\PYZob{}\PYZhy{}1\PYZcb{} matrices:}
\PY{n}{Elim} \PY{o}{=} \PY{n}{Matrix}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{4}\PY{p}{,}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}
\PY{n}{Dupp} \PY{o}{=} \PY{n}{Matrix}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{,}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}
\PY{n}{Unun} \PY{o}{=} \PY{n}{Matrix}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{,}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}
\PY{k}{def} \PY{n+nf}{Qform}\PY{p}{(}\PY{n}{A}\PY{p}{,}\PY{n}{x}\PY{p}{)}\PY{p}{:}
\PY{l+s+sd}{\PYZdq{}\PYZdq{}\PYZdq{}compute the quadratic form x\PYZsq{}Ax\PYZdq{}\PYZdq{}\PYZdq{}}
\PY{k}{return} \PY{n}{x}\PY{o}{.}\PY{n}{transpose}\PY{p}{(}\PY{p}{)} \PY{o}{*} \PY{n}{A} \PY{o}{*} \PY{n}{x}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}2}]:} \PY{n}{Theta} \PY{o}{=} \PY{n}{Matrix}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{,}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,}\PY{n}{mu}\PY{p}{,}\PY{n}{mu}\PY{p}{,}\PY{n}{mu}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{2} \PY{o}{+} \PY{n}{sg}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{2}\PY{p}{]}\PY{p}{)}
\PY{n}{Theta}
\end{Verbatim}
\texttt{\color{outcolor}Out[{\color{outcolor}2}]:}
\begin{equation*}
\left[\begin{matrix}1 & \mu\\\mu & \mu^{2} + \sigma^{2}\end{matrix}\right]
\end{equation*}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}3}]:} \PY{c}{\PYZsh{} compute tensor products and }
\PY{c}{\PYZsh{} the derivative d vech(Theta\PYZca{}\PYZhy{}1) / d vech(Theta)}
\PY{c}{\PYZsh{} see also \theoremref{inv_distribution}}
\PY{n}{Theta\PYZus{}Theta} \PY{o}{=} \PY{n}{TensorProduct}\PY{p}{(}\PY{n}{Theta}\PY{p}{,}\PY{n}{Theta}\PY{p}{)}
\PY{n}{iTheta\PYZus{}iTheta} \PY{o}{=} \PY{n}{TensorProduct}\PY{p}{(}\PY{n}{Theta}\PY{o}{.}\PY{n}{inv}\PY{p}{(}\PY{p}{)}\PY{p}{,}\PY{n}{Theta}\PY{o}{.}\PY{n}{inv}\PY{p}{(}\PY{p}{)}\PY{p}{)}
\PY{n}{theta\PYZus{}i\PYZus{}deriv} \PY{o}{=} \PY{n}{Elim} \PY{o}{*} \PY{p}{(}\PY{n}{iTheta\PYZus{}iTheta}\PY{p}{)} \PY{o}{*} \PY{n}{Dupp}
\end{Verbatim}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}4}]:} \PY{c}{\PYZsh{} towards \theoremref{theta_asym_var_gaussian}}
\PY{n}{DTTD} \PY{o}{=} \PY{n}{Qform}\PY{p}{(}\PY{n}{Theta\PYZus{}Theta}\PY{p}{,}\PY{n}{Dupp}\PY{p}{)}
\PY{n}{D\PYZus{}DTTD\PYZus{}D} \PY{o}{=} \PY{n}{Qform}\PY{p}{(}\PY{n}{DTTD}\PY{p}{,}\PY{n}{theta\PYZus{}i\PYZus{}deriv}\PY{p}{)}
\PY{n}{iOmega} \PY{o}{=} \PY{n}{Qform}\PY{p}{(}\PY{n}{D\PYZus{}DTTD\PYZus{}D}\PY{p}{,}\PY{n}{Unun}\PY{o}{.}\PY{n}{transpose}\PY{p}{(}\PY{p}{)}\PY{p}{)}
\PY{n}{Omega} \PY{o}{=} \PY{l+m+mi}{2} \PY{o}{*} \PY{n}{iOmega}\PY{o}{.}\PY{n}{inv}\PY{p}{(}\PY{p}{)}
\PY{n}{simplify}\PY{p}{(}\PY{n}{Omega}\PY{p}{)}
\end{Verbatim}
\texttt{\color{outcolor}Out[{\color{outcolor}4}]:}
\begin{equation*}
\left[\begin{matrix}\sigma^{2} & 2 \mu \sigma^{2}\\2 \mu \sigma^{2} & 2 \sigma^{2} \left(2 \mu^{2} + \sigma^{2}\right)\end{matrix}\right]
\end{equation*}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}5}]:} \PY{c}{\PYZsh{} this matches the computation in \eqnref{gauss_confirm_theta}}
\PY{c}{\PYZsh{} on to the inverse:}
\PY{c}{\PYZsh{} actually use \theoremref{inv_distribution}}
\PY{n}{theta\PYZus{}i\PYZus{}deriv\PYZus{}t} \PY{o}{=} \PY{n}{theta\PYZus{}i\PYZus{}deriv}\PY{o}{.}\PY{n}{transpose}\PY{p}{(}\PY{p}{)}
\PY{n}{theta\PYZus{}inv\PYZus{}var} \PY{o}{=} \PY{n}{Qform}\PY{p}{(}\PY{n}{Qform}\PY{p}{(}\PY{n}{Omega}\PY{p}{,}\PY{n}{Unun}\PY{p}{)}\PY{p}{,}\PY{n}{theta\PYZus{}i\PYZus{}deriv\PYZus{}t}\PY{p}{)}
\PY{n}{simplify}\PY{p}{(}\PY{n}{theta\PYZus{}inv\PYZus{}var}\PY{p}{)}
\end{Verbatim}
\texttt{\color{outcolor}Out[{\color{outcolor}5}]:}
\begin{equation*}
\left[\begin{matrix}\frac{2 \mu^{2}}{\sigma^{4}} \left(\mu^{2} + 2 \sigma^{2}\right) & - \frac{2 \mu}{\sigma^{4}} \left(\mu^{2} + \sigma^{2}\right) & \frac{2 \mu^{2}}{\sigma^{4}}\\- \frac{2 \mu}{\sigma^{4}} \left(\mu^{2} + \sigma^{2}\right) & \frac{1}{\sigma^{4}} \left(2 \mu^{2} + \sigma^{2}\right) & - \frac{2 \mu}{\sigma^{4}}\\\frac{2 \mu^{2}}{\sigma^{4}} & - \frac{2 \mu}{\sigma^{4}} & \frac{2}{\sigma^{4}}\end{matrix}\right]
\end{equation*}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}6}]:} \PY{c}{\PYZsh{} this matches the computation in \eqnref{gauss_confirm_itheta}}
\PY{c}{\PYZsh{} now check \conjectureref{theta_asym_var_gaussian}}
\PY{n}{conjec} \PY{o}{=} \PY{n}{Qform}\PY{p}{(}\PY{n}{Theta\PYZus{}Theta}\PY{p}{,}\PY{n}{Dupp}\PY{p}{)}
\PY{n}{e1} \PY{o}{=} \PY{n}{Matrix}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{)}
\PY{n}{convar} \PY{o}{=} \PY{l+m+mi}{2} \PY{o}{*} \PY{p}{(}\PY{n}{conjec}\PY{o}{.}\PY{n}{inv}\PY{p}{(}\PY{p}{)} \PY{o}{\PYZhy{}} \PY{n}{e1} \PY{o}{*} \PY{n}{e1}\PY{o}{.}\PY{n}{transpose}\PY{p}{(}\PY{p}{)}\PY{p}{)}
\PY{n}{simplify}\PY{p}{(}\PY{n}{convar}\PY{p}{)}
\end{Verbatim}
\texttt{\color{outcolor}Out[{\color{outcolor}6}]:}
\begin{equation*}
\left[\begin{matrix}\frac{2 \mu^{2}}{\sigma^{4}} \left(\mu^{2} + 2 \sigma^{2}\right) & - \frac{2 \mu}{\sigma^{4}} \left(\mu^{2} + \sigma^{2}\right) & \frac{2 \mu^{2}}{\sigma^{4}}\\- \frac{2 \mu}{\sigma^{4}} \left(\mu^{2} + \sigma^{2}\right) & \frac{1}{\sigma^{4}} \left(2 \mu^{2} + \sigma^{2}\right) & - \frac{2 \mu}{\sigma^{4}}\\\frac{2 \mu^{2}}{\sigma^{4}} & - \frac{2 \mu}{\sigma^{4}} & \frac{2}{\sigma^{4}}\end{matrix}\right]
\end{equation*}
\begin{Verbatim}[commandchars=\\\{\}]
{\color{incolor}In [{\color{incolor}7}]:} \PY{c}{\PYZsh{} are they the same?}
\PY{n}{simplify}\PY{p}{(}\PY{n}{theta\PYZus{}inv\PYZus{}var} \PY{o}{\PYZhy{}} \PY{n}{convar}\PY{p}{)}
\end{Verbatim}
\texttt{\color{outcolor}Out[{\color{outcolor}7}]:}
\begin{equation*}
\left[\begin{matrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{matrix}\right]
\end{equation*}
%UNFOLD
%% OLD%FOLDUP
%\begin{Verbatim}[commandchars=\\\{\}]
%{\color{incolor}In [{\color{incolor}1}]:} \PY{k+kn}{from} \PY{n+nn}{\PYZus{}\PYZus{}future\PYZus{}\PYZus{}} \PY{k+kn}{import} \PY{n}{division}
%\PY{k+kn}{from} \PY{n+nn}{sympy} \PY{k+kn}{import} \PY{o}{*}
%\PY{k+kn}{from} \PY{n+nn}{sympy.physics.quantum} \PY{k+kn}{import} \PY{n}{TensorProduct}
%\PY{n}{init\PYZus{}printing}\PY{p}{(}\PY{n}{use\PYZus{}unicode}\PY{o}{=}\PY{n+nb+bp}{False}\PY{p}{,} \PY{n}{wrap\PYZus{}line}\PY{o}{=}\PY{n+nb+bp}{False}\PY{p}{,} \PY{n}{no\PYZus{}global}\PY{o}{=}\PY{n+nb+bp}{True}\PY{p}{)}
%\PY{n}{mu} \PY{o}{=} \PY{n}{symbols}\PY{p}{(}\PY{l+s}{\PYZsq{}}\PY{l+s}{\PYZbs{}}\PY{l+s}{mu}\PY{l+s}{\PYZsq{}}\PY{p}{)}
%\PY{n}{sg} \PY{o}{=} \PY{n}{symbols}\PY{p}{(}\PY{l+s}{\PYZsq{}}\PY{l+s}{\PYZbs{}}\PY{l+s}{sigma}\PY{l+s}{\PYZsq{}}\PY{p}{)}
%\PY{c}{\PYZsh{} the elimination, duplication and U\PYZus{}\PYZob{}\PYZhy{}1\PYZcb{} matrices:}
%\PY{n}{Elim} \PY{o}{=} \PY{n}{Matrix}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{4}\PY{p}{,}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}
%\PY{n}{Dupp} \PY{o}{=} \PY{n}{Matrix}\PY{p}{(}\PY{l+m+mi}{4}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{,}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}
%\PY{n}{Unun} \PY{o}{=} \PY{n}{Matrix}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{,}\PY{p}{[}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,} \PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{1}\PY{p}{]}\PY{p}{)}
%\PY{k}{def} \PY{n+nf}{quad\PYZus{}form}\PY{p}{(}\PY{n}{A}\PY{p}{,}\PY{n}{x}\PY{p}{)}\PY{p}{:}
%\PY{l+s+sd}{\PYZdq{}\PYZdq{}\PYZdq{}compute the quadratic form x\PYZsq{}Ax\PYZdq{}\PYZdq{}\PYZdq{}}
%\PY{k}{return} \PY{n}{x}\PY{o}{.}\PY{n}{transpose}\PY{p}{(}\PY{p}{)} \PY{o}{*} \PY{n}{A} \PY{o}{*} \PY{n}{x}
%\end{Verbatim}
%\begin{Verbatim}[commandchars=\\\{\}]
%{\color{incolor}In [{\color{incolor}2}]:} \PY{n}{Theta} \PY{o}{=} \PY{n}{Matrix}\PY{p}{(}\PY{l+m+mi}{2}\PY{p}{,}\PY{l+m+mi}{2}\PY{p}{,}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,}\PY{n}{mu}\PY{p}{,}\PY{n}{mu}\PY{p}{,}\PY{n}{mu}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{2} \PY{o}{+} \PY{n}{sg}\PY{o}{*}\PY{o}{*}\PY{l+m+mi}{2}\PY{p}{]}\PY{p}{)}
%\PY{n}{Theta}
%\end{Verbatim}
%\texttt{\color{outcolor}Out[{\color{outcolor}2}]:}
%\begin{equation*}
%\left[\begin{matrix}1 & \mu\\\mu & \mu^{2} + \sigma^{2}\end{matrix}\right]
%\end{equation*}
%\begin{Verbatim}[commandchars=\\\{\}]
%{\color{incolor}In [{\color{incolor}3}]:} \PY{c}{\PYZsh{} compute tensor products and }
%\PY{c}{\PYZsh{} the derivative d vech(Theta\PYZca{}\PYZhy{}1) / d vech(Theta)}
%\PY{n}{Theta\PYZus{}Theta} \PY{o}{=} \PY{n}{TensorProduct}\PY{p}{(}\PY{n}{Theta}\PY{p}{,}\PY{n}{Theta}\PY{p}{)}
%\PY{n}{iTheta\PYZus{}iTheta} \PY{o}{=} \PY{n}{TensorProduct}\PY{p}{(}\PY{n}{Theta}\PY{o}{.}\PY{n}{inv}\PY{p}{(}\PY{p}{)}\PY{p}{,}\PY{n}{Theta}\PY{o}{.}\PY{n}{inv}\PY{p}{(}\PY{p}{)}\PY{p}{)}
%\PY{n}{theta\PYZus{}i\PYZus{}deriv} \PY{o}{=} \PY{n}{Elim} \PY{o}{*} \PY{p}{(}\PY{n}{iTheta\PYZus{}iTheta}\PY{p}{)} \PY{o}{*} \PY{n}{Dupp}
%\end{Verbatim}
%\begin{Verbatim}[commandchars=\\\{\}]
%{\color{incolor}In [{\color{incolor}4}]:} \PY{c}{\PYZsh{} towards thm \ref{theorem:theta_asym_var_gaussian}}
%\PY{n}{DTTD} \PY{o}{=} \PY{n}{quad\PYZus{}form}\PY{p}{(}\PY{n}{Theta\PYZus{}Theta}\PY{p}{,}\PY{n}{Dupp}\PY{p}{)}
%\PY{n}{D\PYZus{}DTTD\PYZus{}D} \PY{o}{=} \PY{n}{quad\PYZus{}form}\PY{p}{(}\PY{n}{DTTD}\PY{p}{,}\PY{n}{theta\PYZus{}i\PYZus{}deriv}\PY{p}{)}
%\PY{n}{iOmega} \PY{o}{=} \PY{n}{quad\PYZus{}form}\PY{p}{(}\PY{n}{D\PYZus{}DTTD\PYZus{}D}\PY{p}{,}\PY{n}{Unun}\PY{o}{.}\PY{n}{transpose}\PY{p}{(}\PY{p}{)}\PY{p}{)}
%\PY{n}{Omega} \PY{o}{=} \PY{l+m+mi}{2} \PY{o}{*} \PY{n}{iOmega}\PY{o}{.}\PY{n}{inv}\PY{p}{(}\PY{p}{)}
%\PY{n}{simplify}\PY{p}{(}\PY{n}{Omega}\PY{p}{)}
%\end{Verbatim}
%\texttt{\color{outcolor}Out[{\color{outcolor}4}]:}
%\begin{equation*}
%\left[\begin{matrix}\sigma^{2} & 2 \mu \sigma^{2}\\2 \mu \sigma^{2} & 2 \sigma^{2} \left(2 \mu^{2} + \sigma^{2}\right)\end{matrix}\right]
%\end{equation*}
%This matches the computation in \eqnref{gauss_confirm_theta}. On to the
%inverse:
%\begin{Verbatim}[commandchars=\\\{\}]
%{\color{incolor}In [{\color{incolor}5}]:} \PY{c}{\PYZsh{} now use theorem \ref{theorem:inv_distribution}}
%\PY{n}{theta\PYZus{}inv\PYZus{}var} \PY{o}{=} \PY{n}{quad\PYZus{}form}\PY{p}{(}\PY{n}{quad\PYZus{}form}\PY{p}{(}\PY{n}{Omega}\PY{p}{,}\PY{n}{Unun}\PY{p}{)}\PY{p}{,}\PY{n}{theta\PYZus{}i\PYZus{}deriv}\PY{o}{.}\PY{n}{transpose}\PY{p}{(}\PY{p}{)}\PY{p}{)}
%\PY{n}{simplify}\PY{p}{(}\PY{n}{theta\PYZus{}inv\PYZus{}var}\PY{p}{)}
%\end{Verbatim}
%\texttt{\color{outcolor}Out[{\color{outcolor}5}]:}
%\begin{equation*}
%\left[\begin{matrix}\frac{2 \mu^{2}}{\sigma^{4}} \left(\mu^{2} + 2 \sigma^{2}\right) & - \frac{2 \mu}{\sigma^{4}} \left(\mu^{2} + \sigma^{2}\right) & \frac{2 \mu^{2}}{\sigma^{4}}\\- \frac{2 \mu}{\sigma^{4}} \left(\mu^{2} + \sigma^{2}\right) & \frac{1}{\sigma^{4}} \left(2 \mu^{2} + \sigma^{2}\right) & - \frac{2 \mu}{\sigma^{4}}\\\frac{2 \mu^{2}}{\sigma^{4}} & - \frac{2 \mu}{\sigma^{4}} & \frac{2}{\sigma^{4}}\end{matrix}\right]
%\end{equation*}
%This matches the computation in \eqnref{gauss_confirm_itheta}. Check
%the conjecture:
%\begin{Verbatim}[commandchars=\\\{\}]
%{\color{incolor}In [{\color{incolor}6}]:} \PY{c}{\PYZsh{} now conjecture \ref{conjecture:theta_asym_var_gaussian}}
%\PY{n}{conjec} \PY{o}{=} \PY{n}{quad\PYZus{}form}\PY{p}{(}\PY{n}{Theta\PYZus{}Theta}\PY{p}{,}\PY{n}{Dupp}\PY{p}{)}
%\PY{n}{convar} \PY{o}{=} \PY{l+m+mi}{2} \PY{o}{*} \PY{p}{(}\PY{n}{conjec}\PY{o}{.}\PY{n}{inv}\PY{p}{(}\PY{p}{)} \PY{o}{\PYZhy{}} \PY{n}{Matrix}\PY{p}{(}\PY{l+m+mi}{3}\PY{p}{,}\PY{l+m+mi}{3}\PY{p}{,}\PY{p}{[}\PY{l+m+mi}{1}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{,}\PY{l+m+mi}{0}\PY{p}{]}\PY{p}{)}\PY{p}{)}
%\PY{n}{simplify}\PY{p}{(}\PY{n}{convar}\PY{p}{)}
%\end{Verbatim}
%\texttt{\color{outcolor}Out[{\color{outcolor}6}]:}
%\begin{equation*}
%\left[\begin{matrix}\frac{2 \mu^{2}}{\sigma^{4}} \left(\mu^{2} + 2 \sigma^{2}\right) & - \frac{2 \mu}{\sigma^{4}} \left(\mu^{2} + \sigma^{2}\right) & \frac{2 \mu^{2}}{\sigma^{4}}\\- \frac{2 \mu}{\sigma^{4}} \left(\mu^{2} + \sigma^{2}\right) & \frac{1}{\sigma^{4}} \left(2 \mu^{2} + \sigma^{2}\right) & - \frac{2 \mu}{\sigma^{4}}\\\frac{2 \mu^{2}}{\sigma^{4}} & - \frac{2 \mu}{\sigma^{4}} & \frac{2}{\sigma^{4}}\end{matrix}\right]
%\end{equation*}
%\begin{Verbatim}[commandchars=\\\{\}]
%{\color{incolor}In [{\color{incolor}7}]:} \PY{c}{\PYZsh{} are they the same?}
%\PY{n}{simplify}\PY{p}{(}\PY{n}{theta\PYZus{}inv\PYZus{}var} \PY{o}{\PYZhy{}} \PY{n}{convar}\PY{p}{)}
%\end{Verbatim}
%\texttt{\color{outcolor}Out[{\color{outcolor}7}]:}
%\begin{equation*}
%\left[\begin{matrix}0 & 0 & 0\\0 & 0 & 0\\0 & 0 & 0\end{matrix}\right]
%\end{equation*}
%%UNFOLD
\end{example}%UNFOLD
%UNFOLD
\end{document}
%for vim modeline: (do not edit)
% vim:fdm=marker:fmr=FOLDUP,UNFOLD:cms=%%s:syn=rnoweb:ft=rnoweb:nu