\documentclass[notitlepage]{article} \usepackage{../../Math545} \usepackage{listings} \usepackage{numprint} \usepackage{enumerate} \usepackage{bbm} \usepackage{amsfonts,amsmath} \setlength{\parindent}{0in} \lstloadlanguages{R} \definecolor{keywordcolor}{rgb}{0,0.6,0.6} \definecolor{delimcolor}{rgb}{0.461,0.039,0.102} \definecolor{Rcommentcolor}{rgb}{0.101,0.043,0.432} \lstdefinestyle{Rsettings}{ basicstyle=\ttfamily, breaklines=true, showstringspaces=false, keywords={if, else, function, theFunction, tmp}, % Write as many keywords otherkeywords={}, commentstyle=\itshape\color{Rcommentcolor}, keywordstyle=\color{keywordcolor}, moredelim=[s][\color{delimcolor}]{"}{"}, } \lstset{basicstyle=\ttfamily, numbers=none, literate={~} {$\sim$}{2}} \def\ctitle{MATH 545: Introduction to Time Series Analysis} \def\ttitle{Winter 2025} \def\stitle{Notation and Simulation for Time Series Models} \begin{document} \maketitle We can define time series models using simple constructions. Consider first the sequence of random variables $\{Z_t\}_{t \geq 1}$, where $Z_t \sim Normal(0,1)$ with the random variables (mutually) independent. Using these variables, we may define the simple time series $\{X_t\}_{t \geq 1}$. For example, suppose \[ X_t = Z_t - 5 Z_{t-1} + 4 Z_{t-2}. \] We define the backshift operator, $\B$, essentially, as the operator that shifts time back by one unit; in particular, we have \[ \B Z_t = Z_{t-1}. \] This operator exhibits standard arithmetic and algebraic properties: for example \begin{itemize} \item $\B^2 Z_t = \B (\B Z_t) = \B (Z_{t-1}) = Z_{t-2}$ \smallskip \item $(c_1+c_2 \B) Z_t = c_1 Z_t + c_2 Z_{t-1}$ \smallskip \item $(1-\B)^2 Z_t = (1-2\B + \B^2)Z_t = Z_t - 2 Z_{t-1} + Z_{t-2}$ \end{itemize} and so on. Therefore we may write \[ X_t = Z_t - 5 Z_{t-1} + 4 Z_{t-2} = \left(1 -5 \B + 4 \B^2 \right) Z_t. \] For simulation, we generate $Z_{-1}$ and $Z_{-2}$ from the same $Normal(0,1)$ distribution. <>= set.seed(2101) #Set the random number generator seed value n<-200 #Set the sample size Z<-rnorm(n) #Generate Z[1],...Z[n] Z.1<-rnorm(1) #Generate Z[-1] Z.2<-rnorm(1) #Generate Z[-2] Z.extra<-c(Z.2,Z.1,Z) #Concatenate X<-Z.extra[3:(n+2)]-5*Z.extra[2:(n+1)]+4*Z.extra[1:n] #Define the time series X @ An alternate way to do this is to use the \texttt{filter} function: <>= X.f<-filter(Z.extra,c(1,-5,4))[2:(n+1)] sum(X.f-X) #X and X.f are identical @ We can also create a special time series object in \texttt{R} using the \texttt{ts} function: for example, if we wanted the observations to represent quarterly data, starting from 2010, 3rd quarter, we would write <>= X.t<-ts(X, frequency = 4, start = c(2010, 3)) @ With 200 observations, this amounts to 40 years worth of data. \medskip The plotting outcomes are slightly different: <>= par(mar=c(4,4,1,0)) #Set up the plotting margins plot(1:n,X,xlab='t',type='l') #Construct a line plot points(1:n,X,pch=19,cex=0.5) #Add the points on top of the line plot(X.t,xlab='t',type='l') #Construct a line plot points(X.t,pch=19,cex=0.5) #Add the points on top of the line @ Note that the time axis is now plotted in years. \pagebreak We can use elementary functions in \texttt{R} to summarize the simulated data: <>= mean(X) #Mean var(X) #Variance @ In time series, we are interested in covariance and correlation at different \textit{lags}, $h=0,1,2,\ldots$; recall that for a stationary process \[ \gamma_X(h) = \Cov[X_t,X_{t+h}] \qquad h=0,\pm 1, \pm 2,\ldots \] with $\gamma_X(-h) = \gamma_X(h)$, and \[ \rho_X(h) = \Corr[X_t,X_{t+h}] = \frac{\gamma_X(h)}{\gamma_X(0)} \qquad h=0,\pm 1, \pm 2,\ldots \] define the \textit{autocovariance function} (ACVF) and \textit{autocorrelation function} (ACF). We can compute both using the \texttt{acf} function: <>= acf(X,type='covariance',plot=F) #Covariance acf(X,type='correlation',plot=F) #Correlation @ Notice that the variance calculation using \texttt{acf} at lag $h=0$ gives a different answer to that obtained using \texttt{var}: <>= (v0<-var(X)) #Variance using var (v1<-acf(X,type='covariance',plot=F)$acf[1]) #Variance using acf @ This is because the \texttt{acf} uses the formula \begin{equation} \label{eq:gammahat} \widehat \gamma_X(h) = \frac{1}{n-h} \sum_{t=1}^{n-h} (x_{t} - \overline{x})(x_{t+h} - \overline{x}) \end{equation} that is, dividing by $n$ not $n-1$ for the variance calculation (with $h=0$). We may plot the ACVF or ACF using the \texttt{'plot'} argument to \texttt{acf} <>= par(mar=c(4,4,1,0)) #Set up the plotting margins acf(X,type='correlation',plot=T,ylim=range(-1,1)) #Autocorrelation plot @ The dashed horizontal lines either side of zero give an indication of statistical significance; if the ACF estimate $\widehat \rho_X(h)$ exceeds the horizontal dashed line, then they correlation at that lag may be considered significantly different from zero. \medskip \textbf{Note:} Recall that a random walk time series process is formed by setting \[ X_t = \sum\limits_{i=1}^t Z_i; \] with $\{Z_t\} \sim \WN(0,\sigma_Z^2)$. Recall that for such a process \[ \rho_X(t,h) = \Corr[X_t,X_{t+h}] = \frac{t}{\sqrt{t(t+h)}}. \] This process is non-stationary and because of this, the estimates derived using equation \eqref{eq:gammahat} -- which is derived assuming stationarity -- are inappropriate. Because of this, the random walk has a characteristic sample ACF pattern with very slow decay. <>= n<-2000 Z<-rnorm(n) X.rw<-cumsum(Z) #Form the cumulative sum of the Zs par(mar=c(4,4,1,0)) #Set up the plotting margins plot(1:n,X.rw,xlab='t',type='l',ylim=range(-50,50)) #Construct a line plot points(1:n,X.rw,pch=19,cex=0.5) #Add the points on top of the line acf(X.rw,type='correlation',plot=T,ylim=range(-1,1),lag.max=100) #Autocorrelation plot @ An ACF plot such as this is strongly indicative of non-stationarity. \end{document}