%\VignetteEngine{knitr::knitr} %\VignetteIndexEntry{Asymptotic Distribution of the Markowitz Portfolio} %\VignetteKeyword{Finance} %\VignetteKeyword{Sharpe} %\VignettePackage{SharpeR} \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{SharpeR} \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/SharpeRatio") #opts_chunk$set(fig.path="figure/",dev=c("pdf","cairo_ps")) opts_chunk$set(fig.path="figure/SharpeRatio",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(SharpeR) 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} 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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 $\sqrt{\qiform{\pvsig}{\pvmu}} - \fracc{\rfr}{\Rbuj}$. \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 %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 follow 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. \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 analogue. \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 \wrapProj{\pvsig}{\zerJ}\pvmu,\\ c &= \frac{\Rbuj}{\sqrt{\qform{\wrapProj{\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 $\wrapProj{\pvsm}{\zerJt}$ are \begin{equation*} \wrapProj{\pvsm}{\zerJt} = \twobytwo{ 1 + \qform{\wrapProj{\pvsig}{\zerJ}}{\pvmu} }{ -\tr{\pvmu}\wrapProj{\pvsig}{\zerJ}}{ -\wrapProj{\pvsig}{\zerJ}\pvmu}{ \wrapProj{\pvsig}{\zerJ}}. \end{equation*} In particular, elements $2$ through $\nlatf+1$ of $-\fvech{\wrapProj{\pvsm}{\zerJt}}$ 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 $\fvech{\wrapProj{\pvsm}{\zerJt}}$ minus one. \end{lemma}%UNFOLD The asymptotic distribution of $\fvech{\wrapProj{\pvsm}{\zerJt}}$ 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{\wrapProj{\svsm}{\zerJt}} - \fvech{\wrapProj{\pvsm}{\zerJt}}} \rightsquigarrow \normlaw{0,\qoform{\pvvar}{\Mtx{H}}}, \label{eqn:mvclt_zer_isvsm} \end{equation} where \begin{equation*} \Mtx{H} = - \EXD{\wrapParens{\AkronA{\tr{\zerJt}}} \wrapParens{\AkronA{\minv{\wrapParens{\qoform{\pvsm}{\zerJt}}}}} \wrapParens{\AkronA{\zerJt}}}. \end{equation*} \end{theorem}%UNFOLD \begin{proof}%FOLDUP By the multivariate delta method, it suffices to prove that $$ \Mtx{H} = \dbyd{\fvech{\wrapProj{\svsm}{\zerJt}}}{\fvech{\pvsm}}. $$ Via \lemmaref{misc_derivs}, it suffices to prove that $$ \dbyd{\wrapProj{\pvsm}{\zerJt}}{\pvsm} = - \wrapParens{\AkronA{\tr{\zerJt}}} \wrapParens{\AkronA{\minv{\wrapParens{\qoform{\pvsm}{\zerJt}}}}} \wrapParens{\AkronA{\zerJt}}. $$ 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{\wrapProj{\pvsm}{\zerJt}}{\pvsm} &= \dbyd{\wrapProj{\pvsm}{\zerJt}}{\minv{\wrapParens{\qoform{\pvsm}{\zerJt}}}} \dbyd{\minv{\wrapParens{\qoform{\pvsm}{\zerJt}}}}{\qoform{\pvsm}{\zerJt}} \dbyd{\qoform{\pvsm}{\zerJt}}{\pvsm}\\ &= \wrapParens{\AkronA{\tr{\zerJt}}} \dbyd{\minv{\wrapParens{\qoform{\pvsm}{\zerJt}}}}{\qoform{\pvsm}{\zerJt}} \wrapParens{\AkronA{\zerJt}}. \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 $\pportwoptFoo{\Rbuj,\zerJ,}$ 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} - \wrapProj{\pvsig}{\hejG}\pvmu},\\ c &= \frac{\Rbuj}{\sqrt{\qiform{\pvsig}{\pvmu} - \qform{\wrapProj{\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 $$ \pportw = c_3\wrapBracks{\minvAB{\pvsig}{\pvmu} - \wrapProj{\pvsig}{\hejG}\pvmu}. $$ 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 $\psnrsqoptG{\hejG} = \qiform{\pvsig}{\pvmu} - \qform{\wrapProj{\pvsig}{\hejG}}{\pvmu}.$ 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 $$ \Delhej\minv{\pvsm} \defeq \minv{\pvsm} - \wrapProj{\pvsm}{\hejGt}.$$ \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{\wrapProj{\pvsig}{\hejG}}{\pvmu} }{ -\tr{\pvmu}\minv{\pvsig} + \tr{\pvmu}\wrapProj{\pvsig}{\hejG}}{ -\minv{\pvsig}\pvmu + \wrapProj{\pvsig}{\hejG}\pvmu}{ \minv{\pvsig} - \wrapProj{\pvsig}{\hejG}}. \end{equation*} %\twobytwosym{ \qiform{\pvsig}{\pvmu} - \wrapProj{\pvsig}{\hejG} }{ %-\wrapBracks{\tr{\pvmu}\minv{\pvsig} - \tr{\pvmu}\wrapProj{\pvsig}{\hejG}}}{ %\minv{\pvsig} - \wrapProj{\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 $\qiform{\svsig}{\svmu} - \qform{\wrapProj{\svsig}{\hejG}}{\svmu}$, 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}} - \wrapParens{\AkronA{\tr{\hejGt}}} \wrapParens{\AkronA{\minv{\wrapParens{\qoform{\pvsm}{\hejGt}}}}} \wrapParens{\AkronA{\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 %$$ %\dbyd{\wrapProj{\pvsm}{\hejGt}}{\pvsm} = %-\wrapParens{\AkronA{\tr{\hejGt}}} \wrapParens{\AkronA{\minv{\wrapParens{\qoform{\pvsm}{\hejGt}}}}} %\wrapParens{\AkronA{\hejGt}}. %$$ %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{\wrapProj{\pvsm}{\hejGt}}{\pvsm} %&= %\dbyd{\wrapProj{\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 \wrapProj{\pvsig}{\zerJ}\pvmu,\\ %%c &= \frac{\Rbuj}{\sqrt{\qform{\wrapProj{\pvsig}{\zerJ}}{\pvmu}}}. %%\end{split} %%\end{equation*} %When $\rfr > 0$ the solution is unique. %\end{lemma}%UNFOLD %2FIX: continue. %%UNFOLD \subsection{Conditional Heteroskedasticity}%FOLDUP 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 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 `Markowitz coefficient'. 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 Markowitz coefficient, 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: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 Markowitz coefficient (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 %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{SharpeR,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