\documentclass[11pt]{article} \usepackage{../Math545} \usepackage{amsmath,amssymb} \pagestyle{plain} \renewcommand{\labelenumi}{\arabic{enumi}.} \renewcommand{\labelenumii}{\arabic{enumi}.\arabic{enumii}} \renewcommand{\labelenumiii}{\arabic{enumi}.\arabic{enumii}.\arabic{enumiii}} \setlength{\topmargin}{-0.75 in} \setlength{\textheight}{9.5 in} \parindent0in \newenvironment{packed_enum}{ \begin{enumerate} \setlength{\itemsep}{12pt} \setlength{\parskip}{0pt} \setlength{\parsep}{0pt} \setlength{\topsep}{0pt} }{\end{enumerate}} \begin{document} <>= library(knitr) # global chunk options opts_chunk$set(cache=TRUE, autodep=TRUE) options(scipen=999) options(repos=c(CRAN="https://cloud.r-project.org/")) @ \begin{center} \textsclarge{MATH 545: Introduction to Time Series Analysis} \smallskip \textsc{Partial Autocorrelation} \end{center} The partial autocorrelation function, $\pi_X(h)$, for a stationary time series has been defined in two ways. First, we have the first principles definition that defines \begin{equation}\label{eq:pacf1} \pi_X(h) = \Corr[(X_t - \E[X_t|X_{(t+1):(t+h-1)}]),(X_{t+h} - \E[X_{t+h}|X_{(t+1):(t+h-1)}])] \qquad h \geq 2 \end{equation} with $\pi_X(0) = 1$ and $\pi_X(1) = \rho_X(1)$. Secondly, we have from the Levinson-Durbin algorithm that $\pi_X(h)$ is $\phi_{h,h}$, which is the last element of the optimal one-step ahead forecast vector obtained by solving \begin{equation}\label{eq:pacf2} \begin{bmatrix} \gamma_X(0) & \gamma_X(1) & \gamma_X(2) & \cdots & \gamma_X(h-1)\\[6pt] \gamma_X(1) & \gamma_X(0) & \gamma_X(1) & \cdots & \gamma_X(h-2)\\[6pt] \gamma_X(2) & \gamma_X(1) & \gamma_X(0) & \cdots & \gamma_X(h-2)\\[6pt] \vdots & \vdots & \vdots & \ddots & \vdots \\[6pt] \gamma_X(h-1) & \gamma_X(h-2) & \gamma_X(h-3) & \cdots & \gamma_X(0) \end{bmatrix} \begin{bmatrix} \phi_{h,1} \\[6pt] \phi_{h,2} \\[6pt] \phi_{h,3} \\[6pt] \vdots \\[6pt] \phi_{h,h} \end{bmatrix} = \begin{bmatrix} \gamma_X(1)\\[6pt] \gamma_X(2) \\[6pt] \gamma_X(3)\\[6pt] \vdots \\[6pt] \gamma_X(h) \end{bmatrix}. \end{equation} To see that these definitions are equivalent under the assumption that the conditional expectations in \eqref{eq:pacf1} are linear, first denote \begin{equation} \widehat X_{n+1} = \sum_{i=1}^n \phi_{n,i} X_{n-i+1} \equiv \sum_{i=1}^n \widehat a_i X_{i} \label{eq:optfore} \end{equation} say, for zero mean stationary process $\{X_t\}$ as usual. Recall that the minimum mean square error (or OLS) criterion dictates that for $i=1,\ldots,n$, \begin{equation}\label{eq:orthog1} \E\left[ (X_{n+1} - \widehat X_{n+1} ) X_{n-i+1} \right] = 0 \qquad \therefore \qquad \E\left[\widehat X_{n+1} X_{n-i+1} \right] = \E\left[ X_{n+1} X_{n-i+1} \right] \end{equation} which, after stacking the results for $i=1,\ldots,n$, is identical to \eqref{eq:pacf2} with $h=n$) as \[ \E\left[\widehat X_{n+1} X_{n-i+1} \right] = \E\left[\sum_{j=1}^n \phi_{n,j} X_{n-j+1} X_{n-i+1} \right] = \sum_{j=1}^n \phi_{n,j} \gamma_X(i-j). \] Let \begin{itemize} \item $\calH$ denote the space of linear combinations of $X_1,\ldots,X_n$, \smallskip \item $\calS$ denote the space of linear combinations of $X_2,\ldots,X_n$, %\[ %\calS = \left\{X \in \R: X = \sum_{i=2}^n s_i X_i, \ (s_2,\ldots,s_n)^\top \in \R^{n-1} \right\}. %\] and let the optimal linear forecast of $X_{n+1}$ based on $\calS$ be denoted \[ \widehat X_{n+1}^\calS = \sum_{i=2}^n \widehat s_i X_i \] and let the optimal \textbf{backward} forecast of $X_1$ based on $X_2,\ldots,X_n$ be \[ \widehat X_1^\calS = \sum_{i=2}^n \widehat s_i^\ast X_i \] with optimal coefficients determined by the equivalent of \eqref{eq:orthog1}, that is, for $i=2,\ldots,n$ \begin{equation}\label{eq:orthog2} \E\left[ (X_{1} - \widehat X_1^\calS ) X_{i} \right] = 0 . \end{equation} \smallskip \item $\calT$ denote the space of scalar multiples of $X_1 - \widehat X_1^\calS$. \end{itemize} Consider arbitrary $X \in \calH$; we can write \[ X = \sum_{i=1}^n a_i X_i = \sum_{i=2}^n a_i X_i + a_1 X_1 = \sum_{i=2}^n a_i X_i + \left( a_1 X_1 - a_1 \sum_{i=2}^n \widehat s_i^\ast X_i \right) + a_1 \sum_{i=2}^n \widehat s_i^\ast X_i \] so therefore \[ X = \sum_{i=2}^n (a_i - a_1 \widehat s_i^\ast) X_i + a_1 \left(X_1 - \sum_{i=2}^n \widehat s_i^\ast X_i \right) = \sum_{i=2}^n a_i^\dag X_i + a_1 \left(X_1 - \widehat X_1^\calS \right) \] so that $\calH = \calS \bigoplus \calT$. By \eqref{eq:optfore}, as $\widehat X_{n+1} \in \calH$, we have \begin{equation}\label{eq:optdecomp} \widehat X_{n+1} = \sum_{i=1}^n \widehat a_i X_{i} = \sum_{i=2}^n \widehat s_i X_i + c \left(X_1 - \sum_{i=2}^n \widehat s_i^\ast X_i \right) \end{equation} By \eqref{eq:orthog2} which states that $(X_{1} - \widehat X_1^\calS )$ is uncorrelated with each of $X_2,\ldots,X_n$, we have that $\calS$ and $\calT$ are \textbf{orthogonal} subspaces of $\calH$, in the sense that any element of $\calS$ in uncorrelated with any element of $\calT$. \smallskip \begin{itemize} \item By stationarity, the vectors $(X_1,\ldots,X_n)^\top $ and $(X_n,\ldots,X_1)^\top$ have the \textbf{same} autocovariance structure, so optimally forecasting $X_1$ \textbf{backwards} using $X_2,\ldots,X_n$ requires the \textbf{same} coefficients as forecasting $X_n$ \textbf{forwards} using $X_1,\ldots,X_{n-1}$, that is, $\widehat s_i^\ast = \phi_{n-1,i-1}$ for $i=2,\ldots,n$ and \[ \widehat X_1^\calS = \sum_{i=2}^{n} \phi_{n-1,i-1} X_{i}. \] \item Also by stationarity, the vectors $(X_1,\ldots,X_n)^\top$ and $(X_2,\ldots,X_{n+1})^\top$ \textbf{also} have the \textbf{same} autocovariance structure, so comparing with \eqref{eq:optfore}, we have $\widehat s_i = \phi_{n-1,n-i+1}$ for $i=2,\ldots,n$ and \[ \widehat X_{n+1}^\calS = \sum_{i=2}^{n} \phi_{n-1,n-i+1} X_{i}. \] \end{itemize} Then we may write that \begin{align*} \widehat X_{n+1} & = \sum_{i=2}^n \phi_{n-1,n-i+1} X_i + c \left(X_1 - \sum_{i=2}^n \phi_{n-1,i-1} X_i \right) \\[6pt] & = c X_1 + \sum_{i=2}^n (\phi_{n-1,n-i+1} - c \phi_{n-1,i-1}) X_i \\[6pt] & = c X_1 + \sum_{i=1}^{n-1} (\phi_{n-1,n-i} - c \phi_{n-1,i}) X_{i+1} \qquad (i \longrightarrow i-1)\\[6pt] & = c X_1 + \sum_{i=1}^{n-1} (\phi_{n-1,i} - c \phi_{n-1,n-i}) X_{n-i+1} \qquad (i \longrightarrow n-i) \end{align*} Thus by comparing with \eqref{eq:optfore}, we have the Levinson-Durbin recursion as $c = \phi_{n,n}$ and for $i=1,\ldots,n-1$, \[ \phi_{n,i} = \phi_{n-1,i} - \phi_{n,n} \phi_{n-1,n-i} . \] \pagebreak Now from \eqref{eq:optdecomp}, we have \[ \widehat X_{n+1} = \widehat X_{n+1}^\calS + c(X_1 - \widehat X_1^\calS) \] so therefore, multiplying through by $(X_1 - \widehat X_1^\calS)$ and taking expectations, we have \begin{equation}\label{eq:final} \E[\widehat X_{n+1} (X_1 - \widehat X_1^\calS)] = \E[\widehat X_{n+1}^\calS (X_1 - \widehat X_1^\calS)] + c\E[(X_1 - \widehat X_1^\calS)^2]. \end{equation} \begin{itemize} \item For the right hand side, we have \[ \E[\widehat X_{n+1} (X_1 - \widehat X_1^\calS)] = \E[(\widehat X_{n+1} - X_{n+1}) (X_1 - \widehat X_1^\calS)] + \E[X_{n+1} (X_1 - \widehat X_1^\calS)]. \] But by \eqref{eq:orthog1}, $\E[(\widehat X_{n+1} - X_{n+1}) (X_1 - \widehat X_1^\calS)] = 0$, as $X_1 - \widehat X_1^\calS$ is a function of $X_1,\ldots,X_n$. Therefore \[ \E[\widehat X_{n+1} (X_1 - \widehat X_1^\calS)] = \E[X_{n+1} (X_1 - \widehat X_1^\calS)]. \] \item For the left hand side, $\E[\widehat X_{n+1}^\calS (X_1 - \widehat X_1^\calS)] = 0$ by \eqref{eq:orthog2}, as $\widehat X_{n+1}^\calS$ is a function of $X_2,\ldots,X_n$. \end{itemize} Therefore from \eqref{eq:final} we have \begin{align*} c = \frac{\E[X_{n+1} (X_1 - \widehat X_1^\calS)]}{\E[(X_1 - \widehat X_1^\calS)^2]} & = \frac{\E[(X_{n+1}-\widehat X_{n+1}^\calS + \widehat X_{n+1}^\calS) (X_1 - \widehat X_1^\calS)]}{\E[(X_1 - \widehat X_1^\calS)^2]}\\[6pt] & = \frac{\E[(X_{n+1}-\widehat X_{n+1}^\calS) (X_1 - \widehat X_1^\calS)]}{\E[(X_1 - \widehat X_1^\calS)^2]} + \frac{\E[\widehat X_{n+1}^\calS (X_1 - \widehat X_1^\calS)]}{\E[(X_1 - \widehat X_1^\calS)^2]}\\[6pt] & = \frac{\E[(X_{n+1}-\widehat X_{n+1}^\calS ) (X_1 - \widehat X_1^\calS)]}{\E[(X_1 - \widehat X_1^\calS)^2]} + 0\\[6pt] & = \frac{\Cov[(X_{n+1}-\widehat X_{n+1}^\calS ),(X_1 - \widehat X_1^\calS)]}{\Var[(X_1 - \widehat X_1^\calS)]} \end{align*} as in this setting, under the linearity assumption \[ \widehat X_{n+1}^\calS \equiv \E[X_{n+1}|X_{2:n}] \qquad \widehat X_{1}^\calS \equiv \E[X_{1}|X_{2:n}]. \] Also, by stationarity \[ \Var[(X_1 - \widehat X_1^\calS)] = \Var[(X_{n+1} - \widehat X_{n+1}^\calS)] \] so therefore \begin{align*} c & = \frac{\Cov[(X_{n+1}-\E[X_{n+1}|X_{2:n}]),(X_1 - \E[X_{1}|X_{2:n}])]}{\sqrt{\Var[(X_{n+1} - \E[X_{n+1}|X_{2:n}])] \Var[(X_1 - \E[X_{1}|X_{2:n}])]}} \\[6pt] & = \Corr[(X_{n+1} - \E[X_{n+1}|X_{2:n}] ),(X_1 - \E[X_{1}|X_{2:n}])] \end{align*} thereby demonstrating the equivalent definitions of the pacf, as this relationship holds for $n=2,3,\ldots$. <>= @ \end{document}