strucchange/0000755000175400001440000000000012534551563012764 5ustar zeileisusersstrucchange/inst/0000755000175400001440000000000012534542351013734 5ustar zeileisusersstrucchange/inst/CITATION0000644000175400001440000000506612220001265015062 0ustar zeileisuserscitHeader("There are several publications about strucchange which describe different aspects of the software. The main reference is the JSS paper.") citEntry(entry="Article", title = "strucchange: An R Package for Testing for Structural Change in Linear Regression Models", author = personList(as.person("Achim Zeileis"), as.person("Friedrich Leisch"), as.person("Kurt Hornik"), as.person("Christian Kleiber")), journal = "Journal of Statistical Software", year = "2002", volume = "7", number = "2", pages = "1--38", url = "http://www.jstatsoft.org/v07/i02/", textVersion = paste("Achim Zeileis, Friedrich Leisch, Kurt Hornik and Christian Kleiber (2002).", "strucchange: An R Package for Testing for Structural Change in Linear Regression Models.", "Journal of Statistical Software, 7(2), 1-38.", "URL http://www.jstatsoft.org/v07/i02/"), header = "To cite strucchange in publications use:" ) citEntry(entry="Article", title = "Testing and Dating of Structural Changes in Practice", author = personList(as.person("Achim Zeileis"), as.person("Christian Kleiber"), as.person("Walter Kr\\\"amer"), as.person("Kurt Hornik")), journal = "Computational Statistics \\& Data Analysis", year = "2003", volume = "44", pages = "109--123", textVersion = paste("Achim Zeileis, Christian Kleiber, Walter Kraemer and Kurt Hornik (2003).", "Testing and Dating of Structural Changes in Practice.", "Computational Statistics & Data Analysis, 44, 109-123."), header = "If methods related to breakpoints() are used also cite:" ) citEntry(entry = "Article", title = "Implementing a Class of Structural Change Tests: An Econometric Computing Approach", author = "Achim Zeileis", journal = "Computational Statistics \\& Data Analysis", year = "2006", volume = "50", pages = "2987--3008", textVersion = paste("Achim Zeileis (2006).", "Implementing a Class of Structural Change Tests: An Econometric Computing Approach.", "Computational Statistics & Data Analysis, 50, 2987-3008."), header = "If methods related to gefp() and efpFunctional() are used also cite:" ) strucchange/inst/doc/0000755000175400001440000000000012534542351014501 5ustar zeileisusersstrucchange/inst/doc/strucchange-intro.pdf0000644000175400001440000062206012534542353020643 0ustar zeileisusers%PDF-1.5 % 1 0 obj << /Type /ObjStm /Length 4896 /Filter /FlateDecode /N 92 /First 774 >> stream x\[s6~?UaJ Vl'/$nF3 KÛʒ 4dLMT"Lt"JLbLlN\<QY¥0($pp\&"xIP2Ӊ9^D(65v9牴,QY''Ji4ρ|5d ]Mh瀺E'yb, $O,%R$.CAJEr\1HK MrC>*pƁƹD°xHe;FΝ`@"-Fgq#W$ +@4 <р,ր,rՀ,M7V'PK 5 +p>`•0 Y2H <'b803p-*5Md1נ04b,  [31=f7ut 2Mh-sV+"&[8M@4$rb; $s1n+ae],H0,9HسzY-'(ė/? 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Since \cite{Z-papers:Zeileis+Leisch+Hornik:2002} various extensions were added to the package, in particular related to breakpoint estimation \citep[also know as ``dating'', discussed in][]{Z-papers:Zeileis+Kleiber+Kraemer:2003} and to structural change tests in other parametric models \citep{Z-papers:Zeileis:2006}. A more unifying view of the underlying theory is presented in \cite{Z-papers:Zeileis:2005} and \cite{Z-papers:Zeileis+Shah+Patnaik:2010}. Here, we focus on the linear regression model and introduce a unified approach for implementing tests from the fluctuation test and $F$~test framework for this model, illustrating how this approach has been realized in {\tt strucchange}. Enhancing the standard significance test approach the package contains methods to fit, plot and test empirical fluctuation processes (like CUSUM, MOSUM and estimates-based processes) and to compute, plot and test sequences of $F$ statistics with the sup$F$, ave$F$ and exp$F$ test. Thus, it makes powerful tools available to display information about structural changes in regression relationships and to assess their significance. Furthermore, it is described how incoming data can be monitored. \end{abstract} \noindent {\bf Keywords:} structural change, CUSUM, MOSUM, recursive estimates, moving estimates, monitoring, \textsf{R}, \textsf{S}.\\ \section{Introduction} \label{sec:intro} The problem of detecting structural changes in linear regression relationships has been an important topic in statistical and econometric research. The most important classes of tests on structural change are the tests from the generalized fluctuation test framework \citep{Z:Kuan+Hornik:1995} on the one hand and tests based on $F$ statistics \citep{Z:Hansen:1992,Z:Andrews:1993,Z:Andrews+Ploberger:1994} on the other. The first class includes in particular the CUSUM and MOSUM tests and the fluctuation test, while the Chow and the sup$F$ test belong to the latter. A topic that gained more interest rather recently is to monitor structural change, i.e., to start after a history phase (without structural changes) to analyze new observations and to be able to detect a structural change as soon after its occurrence as possible.\\ This paper concerns ideas and methods for implementing generalized fluctuation tests as well as $F$ tests in a comprehensive and flexible way, that reflects the common features of the testing procedures. It also offers facilities to display the results in various ways.\\ This paper is organized as follows: In Section~\ref{sec:model} the standard linear regression model upon which all tests are based will be described and the testing problem will be specified. Section~\ref{sec:data} introduces a data set which is also available in the package and which is used for the examples in this paper. The following sections \ref{sec:fluctests}, \ref{sec:Ftests} and \ref{sec:monitor} will then explain the tests, how they are implemented in {\tt strucchange} and give examples for each. Section \ref{sec:fluctests} is concerned with computing empirical fluctuation processes, with plotting them and the corresponding boundaries and finally with testing for structural change based on these processes. Analogously, Section~\ref{sec:Ftests} introduces the $F$ statistics and their plotting and testing methods before Section~\ref{sec:monitor} extends the tools from Section~\ref{sec:fluctests} for the monitoring case. \section{The model} \label{sec:model} Consider the standard linear regression model \begin{equation} \label{model1} y_i = x_i^\top \beta_i + u_i \qquad (i = 1, \dots, n), \end{equation} where at time $i$, $y_i$ is the observation of the dependent variable, $x_i = (1, x_{i2}, \dots, x_{ik})^\top$ is a $k \times 1$ vector of observations of the independent variables, with the first component equal to unity, $u_i$ are iid(0, $\sigma^2$), and $\beta_i$ is the $k \times 1$ vector of regression coefficients. Tests on structural change are concerned with testing the null hypothesis of ``no structural change'' \begin{equation} \label{null-hypothesis} H_0: \quad \beta_i = \beta_0 \qquad (i = 1, \dots, n) \end{equation} against the alternative that the coefficient vector varies over time, with certain tests being more or less suitable (i.e., having good or poor power) for certain patterns of deviation from the null hypothesis.\\ It is assumed that the regressors are nonstochastic with $||x_i|| = O(1)$ and that \begin{equation} \label{A5} \frac{1}{n} \sum_{i=1}^n x_i x_i^\top \quad \longrightarrow \quad Q\end{equation} for some finite regular matrix $Q$. These are strict regularity conditions excluding trends in the data which are assumed for simplicity. For some tests these assumptions can be extended to dynamic models without changing the main properties of the tests; but as these details are not part of the focus of this work they are omitted here.\\ In what follows $\hat \beta^{(i, j)}$ is the ordinary least squares (OLS) estimate of the regression coefficients based on the observations $i+1, \dots, i+j$, and $\hat \beta^{(i)} = \hat \beta^{(0,i)}$ is the OLS estimate based on all observations up to $i$. Hence $\hat \beta^{(n)}$ is the common OLS estimate in the linear regression model. Similarly $X^{(i)}$ is the regressor matrix based on all observations up to $i$. The OLS residuals are denoted as $\hat u_i = y_i - x_i^\top \hat \beta^{(n)}$ with the variance estimate $\hat{\sigma}^2 = \frac{1}{n-k} \sum_{i=1}^n \hat u_i^2$. Another type of residuals that are often used in tests on structural change are the recursive residuals \begin{equation} \label{rr} \tilde u_i \; = \; \frac{y_i - x_i^\top \hat \beta^{(i-1)}}{\sqrt{1+x_i^\top \left(X^{(i-1)^\top} X^{(i-1)} \right)^{-1}x_i}} \qquad (i = k+1, \dots, n),\end{equation} which have zero mean and variance $\sigma^2$ under the null hypothesis. The corresponding variance estimate is $\tilde{\sigma}^2 = \frac{1}{n-k} \sum_{i=k+1}^n (\tilde u_i - \bar{ \tilde u})^2$. \section{The data} \label{sec:data} The data used for examples throughout this paper are macroeconomic time series from the USA. The data set contains the aggregated monthly personal income and personal consumption expenditures (in billion US dollars) between January 1959 and February 2001, which are seasonally adjusted at annual rates. It was originally taken from \url{http://www.economagic.com/}, a web site for economic times series. Both time series are depicted in Figure \ref{fig:USIncExp}. \setkeys{Gin}{width=0.6\textwidth} \begin{figure}[htbp] \begin{center} <>= library("strucchange") data("USIncExp") plot(USIncExp, plot.type = "single", col = 1:2, ylab = "billion US$") legend(1960, max(USIncExp), c("income", "expenditures"), lty = c(1,1), col = 1:2, bty = "n") @ \caption{\label{fig:USIncExp} Personal income and personal consumption expenditures in the US} \end{center} \end{figure} The data is available in the {\tt strucchange} package: it can be loaded and a suitable subset chosen by <>= library("strucchange") data("USIncExp") USIncExp2 <- window(USIncExp, start = c(1985,12)) @ We use a simple error correction model (ECM) for the consumption function similar to \cite{sc:Hansen:1992a}: \begin{eqnarray} \label{ecm.model} \Delta c_t & = & \beta_1 + \beta_2 \; e_{t-1} + \beta_3 \; \Delta i_t + u_t, \\ \label{coint.model} e_t & = & c_t - \alpha_1 - \alpha_2 \; i_t, \end{eqnarray} where $c_t$ is the consumption expenditure and $i_t$ the income. We estimate the cointegration equation (\ref{coint.model}) by OLS and use the residuals $\hat e_t$ as regressors in equation (\ref{ecm.model}), in which we will test for structural change. Thus, the dependent variable is the increase in expenditure and the regressors are the cointegration residuals and the increments of income (and a constant). To compute the cointegration residuals and set up the model equation we need the following steps in \textsf{R}: <>= coint.res <- residuals(lm(expenditure ~ income, data = USIncExp2)) coint.res <- lag(ts(coint.res, start = c(1985,12), freq = 12), k = -1) USIncExp2 <- cbind(USIncExp2, diff(USIncExp2), coint.res) USIncExp2 <- window(USIncExp2, start = c(1986,1), end = c(2001,2)) colnames(USIncExp2) <- c("income", "expenditure", "diff.income", "diff.expenditure", "coint.res") ecm.model <- diff.expenditure ~ coint.res + diff.income @ Figure~\ref{fig:ts} shows the transformed time series necessary for estimation of equation (\ref{ecm.model}). \begin{figure}[htbp] \begin{center} <>= plot(USIncExp2[,3:5], main = "") @ \caption{\label{fig:ts} Time series used -- first differences and cointegration residuals} \end{center} \end{figure} In the following sections we will apply the methods introduced to test for structural change in this model. \section{Generalized fluctuation tests} \label{sec:fluctests} The generalized fluctuation tests fit a model to the given data and derive an empirical process, that captures the fluctuation either in residuals or in estimates. For these empirical processes the limiting processes are known, so that boundaries can be computed, whose crossing probability under the null hypothesis is $\alpha$. If the empirical process path crosses these boundaries, the fluctuation is improbably large and hence the null hypothesis should be rejected (at significance level $\alpha$). \subsection{Empirical fluctuation processes: function \texttt{efp}} Given a formula that describes a linear regression model to be tested the function {\tt efp} creates an object of class {\tt "efp"} which contains a fitted empirical fluctuation process of a specified type. The types available will be described in detail in this section.\\ {\bf CUSUM processes}: The first type of processes that can be computed are CUSUM processes, which contain cumulative sums of standardized residuals. \cite{Z:Brown+Durbin+Evans:1975} suggested to consider cumulative sums of recursive residuals: \begin{equation}\label{Rec-CUSUM} W_n(t) \; = \; \frac{1}{\tilde \sigma \sqrt{\eta}}\sum_{i=k+1}^{k +\lfloor t\eta \rfloor} \tilde u_i \qquad (0 \le t \le 1),\end{equation} where $\eta = n-k$ is the number of recursive residuals and $\lfloor t\eta \rfloor$ is the integer part of $t\eta$.\\ Under the null hypothesis the limiting process for the empirical fluctuation process $W_n(t)$ is the Standard Brownian Motion (or Wiener Process) $W(t)$. More precisely the following functional central limit theorem (FCLT) holds: \begin{equation}\label{lim(Rec-CUSUM)} W_n \Longrightarrow W, \end{equation} as $n \rightarrow \infty$, where $\Rightarrow$ denotes weak convergence of the associated probability measures.\\ Under the alternative, if there is just a single structural change point $t_0$, the recursive residuals will only have zero mean up to $t_0$. Hence the path of the process should be close to 0 up to $t_0$ and leave its mean afterwards. \cite{Z:Kraemer+Ploberger+Alt:1988} show that the main properties of the CUSUM quantity remain the same even under weaker assumptions, in particular in dynamic models. Therefore {\tt efp} has the logical argument {\tt dynamic}; if set to {\tt TRUE} the lagged observations $y_{t-1}$ will be included as regressors.\\ \cite{Z:Ploberger+Kraemer:1992} suggested to base a structural change test on cumulative sums of the common OLS residuals. Thus, the OLS-CUSUM type empirical fluctuation process is defined by: \begin{equation} \label{OLS-CUSUM} W_n^0(t) \quad = \quad \frac{1}{\hat \sigma \sqrt{n}} \sum_{i=1}^{\lfloor nt \rfloor} \hat u_i \qquad (0 \le t \le 1). \end{equation} The limiting process for $W_n^0(t)$ is the standard Brownian bridge $W^0(t) = W(t) - t W(1)$. It starts in 0 at $t = 0$ and it also returns to 0 for $t = 1$. Under a single structural shift alternative the path should have a peak around $t_0$.\\ These processes are available in the function {\tt efp} by specifying the argument {\tt type} to be either {\tt "Rec-CUSUM"} or {\tt "OLS-CUSUM"}, respectively.\\ {\bf MOSUM processes}: Another possibility to detect a structural change is to analyze moving sums of residuals (instead of using cumulative sums of the same residuals). The resulting empirical fluctuation process does then not contain the sum of all residuals up to a certain time~$t$ but the sum of a fixed number of residuals in a data window whose size is determined by the bandwidth parameter $h \in (0,1)$ and which is moved over the whole sample period. Hence the Recursive MOSUM process is defined by \begin{eqnarray} \label{Rec-MOSUM} M_n(t | h) & = & \frac{1}{\tilde \sigma \sqrt{\eta}} \sum_{i = k+\lfloor N_\eta t \rfloor +1}^{k+\lfloor N_\eta t \rfloor +\lfloor \eta h \rfloor} \tilde u_i \qquad (0 \le t \le 1-h) \\ \label{Rec-MOSUM2} & = & W_n \left( \frac{ \lfloor N_\eta t \rfloor + \lfloor \eta h\rfloor}{\eta} \right) - W_n \left( \frac{ \lfloor N_\eta t \rfloor}{\eta} \right), \end{eqnarray} where $N_\eta = (\eta - \lfloor \eta h \rfloor)/(1-h)$. Similarly the OLS-based MOSUM process is defined by \begin{eqnarray} \label{OLS-MOSUM} M_n^0 (t | h) & = & \frac{1}{\hat \sigma \sqrt{n}} \left( \sum_{i = \lfloor N_n t \rfloor +1}^{\lfloor N_n t \rfloor +\lfloor nh \rfloor} \hat u_i \right) \qquad (0 \le t \le 1 - h) \\ \label{OLS-MOSUM2} & = & W_n^0 \left( \frac{ \lfloor N_n t \rfloor + \lfloor n h\rfloor}{n} \right) - W_n^0 \left( \frac{ \lfloor N_n t \rfloor}{n} \right), \end{eqnarray} where $N_n = (n - \lfloor n h \rfloor)/(1-h)$. As the representations (\ref{Rec-MOSUM2}) and (\ref{OLS-MOSUM2}) suggest, the limiting process for the empirical MOSUM processes are the increments of a Brownian motion or a Brownian bridge respectively. This is shown in detail in \cite{Z:Chu+Hornik+Kuan:1995}.\\ If again a single structural shift is assumed at $t_0$, then both MOSUM paths should also have a strong shift around $t_0$.\\ The MOSUM processes will be computed if {\tt type} is set to {\tt "Rec-MOSUM"} or {\tt "OLS-MOSUM"}, respectively.\\ {\bf Estimates-based processes}: Instead of defining fluctuation processes on the basis of residuals they can be equally well based on estimates of the unknown regression coefficients. With the same ideas as for the residual-based CUSUM- and MOSUM-type processes the $k \times 1$-vector $\beta$ is either estimated recursively with a growing number of observations or with a moving data window of constant bandwidth $h$ and then compared to the estimates based on the whole sample. The former idea leads to the fluctuation process in the spirit of \cite{Z:Ploberger+Kraemer+Kontrus:1989} which is defined by \begin{equation} \label{fluctuation} Y_n \left(t \right) = \frac{\sqrt{i}}{\hat \sigma \sqrt{n}} \left({X^{(i)}}^\top X^{(i)} \right)^{\frac{1}{2}} \left( \hat \beta^{(i)} - \hat \beta^{(n)} \right), \end{equation} where $i = \lfloor k + t(n-k) \rfloor$ with $t \in [0,1]$. And the latter gives the moving estimates (ME) process introduced by \cite{Z:Chu+Hornik+Kuan:1995a}: \begin{equation} \label{ME} Z_n \left( \left. t \right| h \right) = \frac{\sqrt{\lfloor nh \rfloor}}{\hat \sigma \sqrt{n}} \left({X^{(\lfloor nt \rfloor, \lfloor nh \rfloor)}}^\top X^{(\lfloor nt \rfloor, \lfloor nh \rfloor)} \right)^{\frac{1}{2}} \left( \hat \beta^{(\lfloor nt \rfloor,\lfloor nh \rfloor)} - \hat \beta^{(n)} \right), \end{equation} where $0 \le t \le 1-h$. Both are $k$-dimensional empirical processes. Thus, the limiting processes are a $k$-dimensional Brownian Bridge or the increments thereof respectively. Instead of rescaling the processes for each $i$ they can also be standardized by $\left( {X^{(n)}}^\top X^{(n)} \right)^{\frac{1}{2}}$. This has the advantage that it has to be calculated only once, but \cite{Z:Kuan+Chen:1994} showed that if there are dependencies between the regressors the rescaling improves the empirical size of the resulting test. Heuristically the rescaled empirical fluctuation process ``looks'' more like its theoretic counterpart.\\ Under a single shift alternative the recursive estimates processes should have a peak and the moving estimates process should again have a shift close to the shift point $t_0$.\\ For {\tt type="fluctuation"} the function {\tt efp} returns the recursive estimates process, whereas for {\tt "ME"} the moving estimates process is returned.\\ All six processes may be fitted using the function {\tt efp}. For our example we want to fit an OLS-based CUSUM process, and a moving estimates (ME) process with bandwidth $h = 0.2$. The commands are simply <>= ocus <- efp(ecm.model, type="OLS-CUSUM", data=USIncExp2) me <- efp(ecm.model, type="ME", data=USIncExp2, h=0.2) @ These return objects of class {\tt "efp"} which contain mainly the empirical fluctuation processes and a few further attributes like the process type. The process itself is of class {\tt "ts"} (the basic time series class in \textsf{R}), which either preserves the time properties of the dependent variable if this is a time series (like in our example), or which is standardized to the interval $[0,1]$ (or a subinterval). For the MOSUM and ME processes the centered interval $[h/2, 1-h/2]$ is chosen rather than $[0,1-h]$ as in (\ref{Rec-MOSUM}) and (\ref{OLS-MOSUM}).\\ Any other process type introduced in this section can be fitted by setting the {\tt type} argument. The fitted process can then be printed, plotted or tested with the corresponding test on structural change. For the latter appropriate boundaries are needed; the concept of boundaries for fluctuation processes is explained in the next section. \subsection{Boundaries and plotting} The idea that is common to all generalized fluctuation tests is that the null hypothesis of ``no structural change'' should be rejected when the fluctuation of the empirical process $\mathit{efp}(t)$ gets improbably large compared to the fluctuation of the limiting process. For the one-dimensional residual-based processes this comparison is performed by some appropriate boundary $b(t)$, that the limiting process just crosses with a given probability $\alpha$. Thus, if $\mathit{efp}(t)$ crosses either $b(t)$ or $-b(t)$ for any $t$ then it has to be concluded that the fluctuation is improbably large and the null hypothesis can be rejected at confidence level $\alpha$. The procedure for the $k$-dimensional estimates-based processes is similar, but instead of a boundary for the process itself a boundary for $||\mathit{efp}_i(t)||$ is used, where $||\cdot||$ is an appropriate functional which is applied component-wise. We have implemented the functionals `max' and `range'. The null hypothesis is rejected if $||\mathit{efp}_i(t)||$ gets larger than a constant $\lambda$, which depends on the confidence level $\alpha$, for any $i = 1, \dots, k$.\\ The boundaries for the MOSUM processes are also constants, i.e., of form $b(t) = \lambda$, which seems natural as the limiting processes are stationary. The situation for the CUSUM processes is different though. Both limiting processes, the Brownian motion and the Brownian bridge, respectively, are not stationary. It would seem natural to use boundaries that are proportional to the standard deviation function of the corresponding theoretic process, i.e., \begin{eqnarray} \label{bound:Rec-CUS1} b(t) &=& \lambda \cdot \sqrt{t}\\ \label{bound:OLS-CUS1} b(t) &=& \lambda \cdot \sqrt{t(1-t)} \end{eqnarray} for the Recursive CUSUM and the OLS-based CUSUM path respectively, where $\lambda$ determines the confidence level. But the boundaries that are commonly used are linear, because a closed form solution for the crossing probability is known. So the standard boundaries for the two proccess are of type \begin{eqnarray} \label{bound:Rec-CUS} b(t) &=& \lambda \cdot (1+2t)\\ \label{bound:OLS-CUS} b(t) &=& \lambda. \end{eqnarray} They were chosen because they are tangential to the boundaries (\ref{bound:Rec-CUS1}) and (\ref{bound:OLS-CUS1}) respectively in $t = 0.5$. However, \cite{Zo:Zeileis:2000a} examined the properties of the alternative boundaries (\ref{bound:Rec-CUS1}) and (\ref{bound:OLS-CUS1}) and showed that the resulting OLS-based CUSUM test has better power for structural changes early and late in the sample period.\\ Given a fitted empirical fluctuation process the boundaries can be computed very easily using the function {\tt boundary}, which returns a time series object with the same time properties as the given fluctuation process: <>= bound.ocus <- boundary(ocus, alpha=0.05) @ It is also rather convenient to plot the process with its boundaries for some confidence level $\alpha$ (by default 0.05) to see whether the path exceeds the boundaries or not: This is demonstrated in Figure~\ref{fig:ocus}. \begin{figure}[hbtp] \begin{center} <>= plot(ocus) @ \caption{\label{fig:ocus} OLS-based CUSUM process} \end{center} \end{figure} It can be seen that the OLS-based CUSUM process exceeds its boundary; hence there is evidence for a structural change. Furthermore the process seems to indicate two changes: one in the first half of the 1990s and another one at the end of 1998.\\ It is also possible to suppress the boundaries and add them afterwards, e.g. in another color <>= plot(ocus, boundary = FALSE) lines(bound.ocus, col = 4) lines(-bound.ocus, col = 4) @ For estimates-based processes it is only sensible to use time series plots if the functional `max' is used because it is equivalent to rejecting the null hypothesis when $\max_{i=1, \dots, k} ||\mathit{efp}(t)||$ gets large or when $\max_t \max_{i=1, \dots, k} \mathit{efp}_i(t)$ gets large. This again is equivalent to any one of the (one-dimensinal) processes $\mathit{efp}_i(t)$ for $i = 1, \dots, k$ exceeding the boundary. The $k$-dimensional process can also be plotted by specifying the parameter {\tt functional} (which defaults to {\tt "max"}) as {\tt NULL}: \begin{figure}[h] \begin{center} <>= plot(me, functional = NULL) @ \caption{\label{fig:me-null} 3-dimensional moving estimates process} \end{center} \end{figure} The output from \textsf{R} can be seen in Figure~\ref{fig:me-null}, where the three parts of the plot show the processes that correspond to the estimate of the regression coefficients of the intercept, the cointegration residuals and the increments of income, respectively. All three paths show two shifts: the first shift starts at the beginning of the sample period and ends in about 1991 and the second shift occurs at the very end of the sample period. The shift that causes the significance seems to be the strong first shift in the process for the intercept and the cointegration residuals, because these cross their boundaries. Thus, the ME test leads to similar results as the OLS-based CUSUM test, but provides a little more information about the nature of the structural change. \subsection{Significance testing with empirical fluctuation processes} Although calculating and plotting the empiricial fluctuation process with its boundaries provides and visualizes most of the information, it might still be necessary or desirable to carry out a traditional significance test. This can be done easily with the function {\tt sctest} (\underline{s}tructural \underline{c}hange \underline{test}) which returns an object of class {\tt "htest"} (\textsf{R}'s standard class for statistical test results) containing in particular the test statistic and the corresponding $p$ value. The test statistics reflect what was described by the crossing of boundaries in the previous section. Hence the test statistic is $S_r$ from (\ref{statr}) for the residual-based processes and $S_e$ from (\ref{state}) for the estimates-based processes: \begin{eqnarray} \label{statr} S_r & = & \max_t \frac{\mathit{efp}(t)}{f(t)},\\ \label{state} S_e & = & \max ||\mathit{efp}(t)||, \end{eqnarray} where $f(t)$ depends on the shape of the boundary, i.e., $b(t) = \lambda \cdot f(t)$. For most boundaries is $f(t) \equiv 1$, but the linear boundary for the Recursive CUSUM test for example has shape $f(t) = 1 + 2t$.\\ It is either possible to supply {\tt sctest} with a fitted empirical fluctuation process or with a formula describing the model that should be tested. Thus, the commands <>= sctest(ocus) @ and <>= sctest(ecm.model, type="OLS-CUSUM", data=USIncExp2) @ lead to equivalent results. {\tt sctest} is a generic function which has methods not only for fluctuation tests, but all structural change tests (on historic data) introduced in this paper including the $F$ tests described in the next section. \section{$F$ tests} \label{sec:Ftests} A rather different approach to investigate whether the null hypothesis of ``no structural change'' holds, is to use $F$ test statistics. An important difference is that the alternative is specified: whereas the generalized fluctuation tests are suitable for various patterns of structural changes, the $F$ tests are designed to test against a single shift alternative. Thus, the alternative can be formulated on the basis of the model (\ref{model1}) \begin{equation} \label{H1} \beta_i = \left\{ \begin{array}{l} \beta_A \quad (1 \le i \le i_0) \\ \beta_B \quad (i_0 < i \le n) \end{array} \right., \end{equation} where $i_0$ is some change point in the interval $(k, n-k)$. \cite{Z:Chow:1960} was the first to suggest such a test on structural change for the case where the (potential) change point $i_0$ is known. He proposed to fit two separate regressions for the two subsamples defined by $i_0$ and to reject whenever \begin{equation} \label{Fstat} F_{i_0} \; = \; \frac{{\hat u}^\top {\hat u} - {\hat e}^\top {\hat e}}{{\hat e}^\top {\hat e}/(n-2k)}. \end{equation} is too large, where $\hat e = (\hat u_A, \hat u_B)^\top$ are the residuals from the full model, where the coefficients in the subsamples are estimated separately, and $\hat u$ are the residuals from the restricted model, where the parameters are just fitted once for all observations. The test statistic $F_{i_0}$ has an asymptotic $\chi^2$ distribution with $k$ degrees of freedom and (under the assumption of normality) $F_{i_0}/k$ has an exact $F$ distribution with $k$ and $n-2k$ degrees of freedom. The major drawback of this ``Chow test'' is that the change point has to be known in advance, but there are tests based upon $F$ statistics (Chow statistics), that do not require a specification of a particular change point and which will be introduced in the following sections. \subsection{$F$ statistics: function \texttt{Fstats}} A natural idea to extend the ideas from the Chow test is to calculate the $F$ statistics for all potential change points or for all potential change points in an interval $[\ui, \oi]$ and to reject if any of those statistics get too large. Therefore the first step is to compute the $F$ statistics $F_i$ for $k < \ui \le i \le \oi < n-k$, which can be easily done using the function {\tt Fstats}. Again the model to be tested is specified by a formula interface and the parameters $\ui$ and $\oi$ are respresented by {\tt from} and {\tt to}, respectively. Alternatively to indices of observations these two parameters can also be specified by fractions of the sample; the default is to take {\tt from = 0.15} and implicitly {\tt to = 0.85}. To compute the $F$ test statistics for all potential change points between January 1990 and June 1999 the appropriate command would be: <>= fs <- Fstats(ecm.model, from = c(1990, 1), to = c(1999,6), data = USIncExp2) @ This returns an object of class {\tt "Fstats"} which mainly contains a time series of $F$ statistics. Analogously to the empiricial fluctuation processes these objects can be printed, plotted and tested. \subsection{Boundaries and plotting} The computation of boundaries and plotting of $F$ statistics is rather similar to that of empirical fluctuation processes introduced in the previous section. Under the null hypthesis of no structural change boundaries can be computed such that the asymptotic probability that the supremum (or the mean) of the statistics $F_i$ (for $\ui \le i \le \oi$) exceeds this boundary is $\alpha$. So the following command \begin{figure}[htbp] \begin{center} <>= plot(fs) @ \caption{\label{fig:Fstats} $F$ statistics} \end{center} \end{figure} plots the process of $F$ statistics with its boundary; the output can be seen in Figure~\ref{fig:Fstats}. As the $F$ statistics cross their boundary, there is evidence for a structural change (at the level $\alpha = 0.05$). The process has a clear peak in 1998, which mirrors the results from the analysis by empirical fluctuation processes and tests, respectively, that also indicated a break in the late 1990s.\\ It is also possible to plot the $p$ values instead of the $F$ statistics themselves by <>= plot(fs, pval=TRUE) @ which leads to equivalent results. Furthermore it is also possible to set up the boundaries for the average instead of the supremum by: <>= plot(fs, aveF=TRUE) @ In this case another dashed line for the observed mean of the $F$ statistics will be drawn. \subsection{Significance testing with $F$ statistics} As already indicated in the previous section, there is more than one possibility to aggregate the series of $F$ statistics into a test statistic. \cite{Z:Andrews:1993} and \cite{Z:Andrews+Ploberger:1994} respectively suggested three different test statistics and examined their asymptotic distribution: \begin{eqnarray} \label{supF} \mbox{\textnormal{sup}}F & = & \sup_{\ui\le i \le \oi} F_i, \\ \label{aveF} \mbox{\textnormal{ave}}F & = & \frac{1}{\oi - \ui+ 1} \sum_{i = \ui}^{\oi} F_i, \\ \label{expF} \mbox{\textnormal{exp}}F & = & \log \left( \frac{1}{\oi - \ui+ 1} \sum_{i = \ui}^{\oi} \exp ( 0.5 \cdot F_i) \right). \end{eqnarray} The sup$F$ statistic in (\ref{supF}) and the ave$F$ statistic from (\ref{aveF}) respectively reflect the testing procedures that have been described above. Either the null hypothesis is rejected when the maximal or the mean $F$ statistic gets too large. A third possibility is to reject when the exp$F$ statistic from (\ref{expF}) gets too large. The ave$F$ and exp$F$ test have certain optimality properties \citep{Z:Andrews+Ploberger:1994}. The tests can be carried out in the same way as the fluctuation tests: either by supplying the fitted {\tt Fstats} object or by a formula that describes the model to be tested. Hence the commands <>= sctest(fs, type="expF") @ and <>= sctest(ecm.model, type = "expF", from = 49, to = 162, data = USIncExp2) @ lead to equivalent output. The $p$ values are computed based on \cite{Z:Hansen:1997}.\footnote{The authors thank Bruce Hansen, who wrote the original code for computing $p$ values for $F$ statistics in \textsf{GAUSS}, for putting his code at disposal for porting to \textsf{R}.} \section{Monitoring with the generalized fluctuation test} \label{sec:monitor} In the previous sections we were concerned with the retrospective detection of structural changes in \emph{given} data sets. Over the last years several structural change tests have been extended to monitoring of linear regression models where new data arrive over time \citep{Z:Chu+Stinchcombe+White:1996,Z:Leisch+Hornik+Kuan:2000}. Such forward looking tests are closely related to sequential tests. When new observations arrive, estimates are computed sequentially from all available data (historical sample plus newly arrived data) and compared to the estimate based only on the historical sample. As in the retrospective case, the hypothesis of no structural change is rejected if the difference between these two estimates gets too large. The standard linear regression model~(\ref{model1}) is generalized to \begin{equation} y_i = x_i^\top \beta_i + u_i \qquad (i = 1, \dots, n, n+1, \ldots), \end{equation} i.e., we expect new observations to arrive after time $n$ (when the monitoring begins). The sample $\{(x_1,y_1),\ldots,(x_n,y_n)\}$ will be called the \emph{historic sample}, the corresponding time period $1,\ldots,n$ the \emph{history period}. Currently monitoring has only been developed for recursive \citep{Z:Chu+Stinchcombe+White:1996} and moving \citep{Z:Leisch+Hornik+Kuan:2000} estimates tests. The respective limiting processes are---as in the retrospective case---the Brownian Bridge and increments of the Brownian Bridge. The empirical processes are rescaled to map the history period to the interval [0,1] of the Brownian Bridge. For recursive estimates there exists a closed form solution for boundary functions, such that the limiting Brownian Bridge stays within the boundaries on the interval $(1,\infty)$ with probability $1-\alpha$. Note that the monitoring period consisting of all data arriving after the history period corresponds to the Brownian Bridge after time 1. For moving estimates, only the growth rate of the boundaries can be derived analytically and critical values have to be simulated. Consider that we want to monitor our ECM during the 1990s for structural change, using years 1986--1989 as the history period. First we cut the historic sample from the complete data set and create an object of class \texttt{"mefp"}: <>= USIncExp3 <- window(USIncExp2, start = c(1986, 1), end = c(1989,12)) me.mefp <- mefp(ecm.model, type = "ME", data = USIncExp3, alpha = 0.05) @ Because monitoring is a sequential test procedure, the significance level has to be specified \emph{in advance}, i.e., when the object of class \texttt{"mefp"} is created. The \texttt{"mefp"} object can now be monitored repeatedly for structural changes. Let us assume we get new observations for the year 1990. Calling function \texttt{monitor} on \texttt{me.mefp} automatically updates our monitoring object for the new observations and runs a sequential test for structural change on each new observation (no structural break is detected in 1990): <>= USIncExp3 <- window(USIncExp2, start = c(1986, 1), end = c(1990,12)) me.mefp <- monitor(me.mefp) @ Then new data for the years 1991--2001 arrive and we repeat the monitoring: <>= USIncExp3 <- window(USIncExp2, start = c(1986, 1)) me.mefp <- monitor(me.mefp) me.mefp @ The software informs us that a structural break has been detected at observation \#72, which corresponds to December 1991. Boundary and plotting methods for \texttt{"mefp"} objects work (almost) exactly as their \texttt{"efp"} counterparts, only the significance level \texttt{alpha} cannot be specified, because it is specified when the \texttt{"mefp"} object is created. The output of \texttt{plot(me.mefp)} can be seen in Figure~\ref{fig:monitor}. \begin{figure}[htbp] \begin{center} <>= plot(me.mefp) @ \caption{\label{fig:monitor} Monitoring structural change with bandwidth $h=1$} \end{center} \end{figure} Instead of creating an {\tt "mefp"} object using the formula interface like above, it could also be done re-using an existing \texttt{"efp"} object, e.g.: <>= USIncExp3 <- window(USIncExp2, start = c(1986, 1), end = c(1989,12)) me.efp <- efp(ecm.model, type = "ME", data = USIncExp3, h = 0.5) me.mefp <- mefp(me.efp, alpha=0.05) @ If now again the new observations up to February 2001 arrive, we can monitor the data <>= USIncExp3 <- window(USIncExp2, start = c(1986, 1)) me.mefp <- monitor(me.mefp) @ and discover the structural change even two observations earlier as we used the bandwidth {\tt h=0.5} instead of {\tt h=1}. Due to this we have not one history estimate that is being compared with the new moving estimates, but we have a history process, which can be seen on the left in Figure~\ref{fig:monitor2}. This plot can simply be generated by \texttt{plot(me.mefp)}. \begin{figure}[htbp] \begin{center} <>= plot(me.mefp) @ \caption{\label{fig:monitor2} Monitoring structural change with bandwidth $h=0.5$} \end{center} \end{figure} The results of the monitoring emphasize the results of the historic tests: the moving estimates process has two strong shifts, the first around 1992 and the second around 1998. \section{Conclusions} \label{sec:conclusions} In this paper, we have described the {\tt strucchange} package that implements methods for testing for structural change in linear regression relationships. It provides a unified framework for displaying information about structural changes flexibly and for assessing their significance according to various tests.\\ Containing tests from the generalized fluctuation test framework as well as tests based on $F$ statistics (Chow test ststistics) the package extends standard significance testing procedures: There are methods for fitting empirical fluctuation processes (CUSUM, MOSUM and estimates-based processes), computing an appropriate boundary, plotting these results and finally carrying out a formal significance test. Analogously a sequence of $F$ statistics with the corresponding boundary can be computed, plotted and tested. Finally the methods for estimates-based fluctuation processes have extensions to monitor incoming data. In addition to these methods for the linear regression model, the \texttt{strucchange} package contains infrastructure for testing, monitoring, and dating structural changes in other parametric models, e.g., estimated by maximum likelihood. Details about the underlying theory can be found in \cite{Z-papers:Zeileis:2005}, \cite{Z-papers:Zeileis+Hornik:2007}, and \cite{Z-papers:Zeileis+Shah+Patnaik:2010}. The corresponding functions in \texttt{strucchange} are presented in \cite{Z-papers:Zeileis+Kleiber+Kraemer:2003} and \cite{Z-papers:Zeileis:2006}. \section*{Acknowledgments} The research of Achim Zeileis, Friedrich Leisch and Kurt Hornik was supported by the Austrian Science Foundation (FWF) under grant SFB\#010 (`Adaptive Information Systems and Modeling in Economics and Management Science').\\ The work of Christian Kleiber was supported by the Deutsche Forschungsgemeinschaft, Sonderforschungsbereich 475. \bibliography{strucchange} \bibliographystyle{abbrvnat} \newpage \begin{appendix} \section{Implementation details for $p$ values} An important and useful tool concerning significance tests are $p$ values, especially for application in a software package. Their implementation is therefore crucial and in this section we will give more detail about the implementation in the {\tt strucchange} package.\\ For the CUSUM tests with linear boundaries there are rather good approximations to the asymptotic $p$ value functions given in \cite{Zo:Zeileis:2000a}. For the recursive estimates fluctuation test there is a series expansion, which is evaluated for the first hundred terms. For all other tests from the generalized fluctuation test framework the $p$ values are computed by linear interpolation from tabulated critical values. For the Recursive CUSUM test with alternative boundaries $p$ values from the interval $[0.001, 1]$ and $[0.001, 0.999]$ for the OLS-based version respectively are approximated from tables given in \cite{Zo:Zeileis:2000}. The critical values for the Recursive MOSUM test for levels in $[0.01, 0.2]$ are taken from \cite{Z:Chu+Hornik+Kuan:1995}, while the critical values for the levels in $[0.01, 0.1]$ for the OLS-based MOSUM and the ME test are given in \cite{Z:Chu+Hornik+Kuan:1995a}; the parameter $h$ is in both cases interpolated for values in $[0.05, 0.5]$.\\ The $p$ values for the sup$F$, ave$F$ and exp$F$ test are approximated based on \cite{Z:Hansen:1997}, who also wrote the original code in \textsf{GAUSS}, which we merely ported to \textsf{R}. The computation uses tabulated simulated regression coefficients. \end{appendix} \end{document} strucchange/inst/doc/strucchange-intro.R0000644000175400001440000001255712534542351020275 0ustar zeileisusers### R code from vignette source 'strucchange-intro.Rnw' ################################################### ### code chunk number 1: data ################################################### library("strucchange") data("USIncExp") plot(USIncExp, plot.type = "single", col = 1:2, ylab = "billion US$") legend(1960, max(USIncExp), c("income", "expenditures"), lty = c(1,1), col = 1:2, bty = "n") ################################################### ### code chunk number 2: subset ################################################### library("strucchange") data("USIncExp") USIncExp2 <- window(USIncExp, start = c(1985,12)) ################################################### ### code chunk number 3: ecm-setup ################################################### coint.res <- residuals(lm(expenditure ~ income, data = USIncExp2)) coint.res <- lag(ts(coint.res, start = c(1985,12), freq = 12), k = -1) USIncExp2 <- cbind(USIncExp2, diff(USIncExp2), coint.res) USIncExp2 <- window(USIncExp2, start = c(1986,1), end = c(2001,2)) colnames(USIncExp2) <- c("income", "expenditure", "diff.income", "diff.expenditure", "coint.res") ecm.model <- diff.expenditure ~ coint.res + diff.income ################################################### ### code chunk number 4: ts-used ################################################### plot(USIncExp2[,3:5], main = "") ################################################### ### code chunk number 5: efp ################################################### ocus <- efp(ecm.model, type="OLS-CUSUM", data=USIncExp2) me <- efp(ecm.model, type="ME", data=USIncExp2, h=0.2) ################################################### ### code chunk number 6: efp-boundary ################################################### bound.ocus <- boundary(ocus, alpha=0.05) ################################################### ### code chunk number 7: OLS-CUSUM ################################################### plot(ocus) ################################################### ### code chunk number 8: efp-boundary2 (eval = FALSE) ################################################### ## plot(ocus, boundary = FALSE) ## lines(bound.ocus, col = 4) ## lines(-bound.ocus, col = 4) ################################################### ### code chunk number 9: ME-null ################################################### plot(me, functional = NULL) ################################################### ### code chunk number 10: efp-sctest (eval = FALSE) ################################################### ## sctest(ocus) ################################################### ### code chunk number 11: efp-sctest2 ################################################### sctest(ecm.model, type="OLS-CUSUM", data=USIncExp2) ################################################### ### code chunk number 12: Fstats ################################################### fs <- Fstats(ecm.model, from = c(1990, 1), to = c(1999,6), data = USIncExp2) ################################################### ### code chunk number 13: Fstats-plot ################################################### plot(fs) ################################################### ### code chunk number 14: pval-plot (eval = FALSE) ################################################### ## plot(fs, pval=TRUE) ################################################### ### code chunk number 15: aveF-plot (eval = FALSE) ################################################### ## plot(fs, aveF=TRUE) ################################################### ### code chunk number 16: Fstats-sctest (eval = FALSE) ################################################### ## sctest(fs, type="expF") ################################################### ### code chunk number 17: Fstats-sctest2 ################################################### sctest(ecm.model, type = "expF", from = 49, to = 162, data = USIncExp2) ################################################### ### code chunk number 18: mefp ################################################### USIncExp3 <- window(USIncExp2, start = c(1986, 1), end = c(1989,12)) me.mefp <- mefp(ecm.model, type = "ME", data = USIncExp3, alpha = 0.05) ################################################### ### code chunk number 19: monitor1 ################################################### USIncExp3 <- window(USIncExp2, start = c(1986, 1), end = c(1990,12)) me.mefp <- monitor(me.mefp) ################################################### ### code chunk number 20: monitor2 ################################################### USIncExp3 <- window(USIncExp2, start = c(1986, 1)) me.mefp <- monitor(me.mefp) me.mefp ################################################### ### code chunk number 21: monitor-plot ################################################### plot(me.mefp) ################################################### ### code chunk number 22: mefp2 ################################################### USIncExp3 <- window(USIncExp2, start = c(1986, 1), end = c(1989,12)) me.efp <- efp(ecm.model, type = "ME", data = USIncExp3, h = 0.5) me.mefp <- mefp(me.efp, alpha=0.05) ################################################### ### code chunk number 23: monitor3 ################################################### USIncExp3 <- window(USIncExp2, start = c(1986, 1)) me.mefp <- monitor(me.mefp) ################################################### ### code chunk number 24: monitor-plot2 ################################################### plot(me.mefp) strucchange/tests/0000755000175400001440000000000012220001265014103 5ustar zeileisusersstrucchange/tests/Examples/0000755000175400001440000000000012534534151015676 5ustar zeileisusersstrucchange/tests/Examples/strucchange-Ex.Rout.save0000644000175400001440000021061712534542314022375 0ustar zeileisusers R version 3.2.0 (2015-04-16) -- "Full of Ingredients" Copyright (C) 2015 The R Foundation for Statistical Computing Platform: x86_64-pc-linux-gnu (64-bit) R is free software and comes with ABSOLUTELY NO WARRANTY. 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Type 'q()' to quit R. > pkgname <- "strucchange" > source(file.path(R.home("share"), "R", "examples-header.R")) > options(warn = 1) > library('strucchange') Loading required package: zoo Attaching package: 'zoo' The following objects are masked from 'package:base': as.Date, as.Date.numeric Loading required package: sandwich > > base::assign(".oldSearch", base::search(), pos = 'CheckExEnv') > cleanEx() > nameEx("BostonHomicide") > ### * BostonHomicide > > flush(stderr()); flush(stdout()) > > ### Name: BostonHomicide > ### Title: Youth Homicides in Boston > ### Aliases: BostonHomicide > ### Keywords: datasets > > ### ** Examples > > data("BostonHomicide") > attach(BostonHomicide) > > ## data from Table 1 > tapply(homicides, year, mean) 1992 1993 1994 1995 1996 1997 1998 3.083333 4.000000 3.166667 3.833333 2.083333 1.250000 0.800000 > populationBM[0:6*12 + 7] [1] 12977 12455 12272 12222 11895 12038 NA > tapply(ahomicides25, year, mean) 1992 1993 1994 1995 1996 1997 1998 3.250000 4.166667 3.916667 4.166667 2.666667 2.333333 1.400000 > tapply(ahomicides35, year, mean) 1992 1993 1994 1995 1996 1997 1998 0.8333333 1.0833333 1.3333333 1.1666667 1.0833333 0.7500000 0.4000000 > population[0:6*12 + 7] [1] 228465 227218 226611 231367 230744 228696 NA > unemploy[0:6*12 + 7] [1] 20.2 18.8 15.9 14.7 13.8 12.6 NA > > ## model A > ## via OLS > fmA <- lm(homicides ~ populationBM + season) > anova(fmA) Analysis of Variance Table Response: homicides Df Sum Sq Mean Sq F value Pr(>F) populationBM 1 14.364 14.3642 3.7961 0.05576 . season 11 47.254 4.2959 1.1353 0.34985 Residuals 64 242.174 3.7840 --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 > ## as GLM > fmA1 <- glm(homicides ~ populationBM + season, family = poisson) > anova(fmA1, test = "Chisq") Analysis of Deviance Table Model: poisson, link: log Response: homicides Terms added sequentially (first to last) Df Deviance Resid. Df Resid. Dev Pr(>Chi) NULL 76 115.649 populationBM 1 4.9916 75 110.657 0.02547 * season 11 18.2135 64 92.444 0.07676 . --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 > > ## model B & C > fmB <- lm(homicides ~ populationBM + season + ahomicides25) > fmC <- lm(homicides ~ populationBM + season + ahomicides25 + unemploy) > > detach(BostonHomicide) > > > > cleanEx() > nameEx("DJIA") > ### * DJIA > > flush(stderr()); flush(stdout()) > > ### Name: DJIA > ### Title: Dow Jones Industrial Average > ### Aliases: DJIA > ### Keywords: datasets > > ### ** Examples > > data("DJIA") > ## look at log-difference returns > djia <- diff(log(DJIA)) > plot(djia) > > ## convenience functions > ## set up a normal regression model which > ## explicitely also models the variance > normlm <- function(formula, data = list()) { + rval <- lm(formula, data = data) + class(rval) <- c("normlm", "lm") + return(rval) + } > estfun.normlm <- function(obj) { + res <- residuals(obj) + ef <- NextMethod(obj) + sigma2 <- mean(res^2) + rval <- cbind(ef, res^2 - sigma2) + colnames(rval) <- c(colnames(ef), "(Variance)") + return(rval) + } > > ## normal model (with constant mean and variance) for log returns > m1 <- gefp(djia ~ 1, fit = normlm, vcov = meatHAC, sandwich = FALSE) > plot(m1, aggregate = FALSE) > ## suggests a clear break in the variance (but not the mean) > > ## dating > bp <- breakpoints(I(djia^2) ~ 1) > plot(bp) > ## -> clearly one break > bp Optimal 2-segment partition: Call: breakpoints.formula(formula = I(djia^2) ~ 1) Breakpoints at observation number: 89 Corresponding to breakdates: 0.552795 > time(djia)[bp$breakpoints] [1] "1973-03-16" > > ## visualization > plot(djia) > abline(v = time(djia)[bp$breakpoints], lty = 2) > lines(time(djia)[confint(bp)$confint[c(1,3)]], rep(min(djia), 2), col = 2, type = "b", pch = 3) > > > > cleanEx() > nameEx("Fstats") > ### * Fstats > > flush(stderr()); flush(stdout()) > > ### Name: Fstats > ### Title: F Statistics > ### Aliases: Fstats print.Fstats > ### Keywords: regression > > ### ** Examples > > ## Nile data with one breakpoint: the annual flows drop in 1898 > ## because the first Ashwan dam was built > data("Nile") > plot(Nile) > > ## test the null hypothesis that the annual flow remains constant > ## over the years > fs.nile <- Fstats(Nile ~ 1) > plot(fs.nile) > sctest(fs.nile) supF test data: fs.nile sup.F = 75.93, p-value = 2.22e-16 > ## visualize the breakpoint implied by the argmax of the F statistics > plot(Nile) > lines(breakpoints(fs.nile)) > > ## UK Seatbelt data: a SARIMA(1,0,0)(1,0,0)_12 model > ## (fitted by OLS) is used and reveals (at least) two > ## breakpoints - one in 1973 associated with the oil crisis and > ## one in 1983 due to the introduction of compulsory > ## wearing of seatbelts in the UK. > data("UKDriverDeaths") > seatbelt <- log10(UKDriverDeaths) > seatbelt <- cbind(seatbelt, lag(seatbelt, k = -1), lag(seatbelt, k = -12)) > colnames(seatbelt) <- c("y", "ylag1", "ylag12") > seatbelt <- window(seatbelt, start = c(1970, 1), end = c(1984,12)) > plot(seatbelt[,"y"], ylab = expression(log[10](casualties))) > > ## compute F statistics for potential breakpoints between > ## 1971(6) (corresponds to from = 0.1) and 1983(6) (corresponds to > ## to = 0.9 = 1 - from, the default) > ## compute F statistics > fs <- Fstats(y ~ ylag1 + ylag12, data = seatbelt, from = 0.1) > ## this gives the same result > fs <- Fstats(y ~ ylag1 + ylag12, data = seatbelt, from = c(1971, 6), + to = c(1983, 6)) > ## plot the F statistics > plot(fs, alpha = 0.01) > ## plot F statistics with aveF boundary > plot(fs, aveF = TRUE) > ## perform the expF test > sctest(fs, type = "expF") expF test data: fs exp.F = 6.4247, p-value = 0.008093 > > > > cleanEx() > nameEx("GermanM1") > ### * GermanM1 > > flush(stderr()); flush(stdout()) > > ### Encoding: UTF-8 > > ### Name: GermanM1 > ### Title: German M1 Money Demand > ### Aliases: GermanM1 historyM1 monitorM1 > ### Keywords: datasets > > ### ** Examples > > data("GermanM1") > ## Lütkepohl et al. (1999) use the following model > LTW.model <- dm ~ dy2 + dR + dR1 + dp + m1 + y1 + R1 + season > ## Zeileis et al. (2005) use > M1.model <- dm ~ dy2 + dR + dR1 + dp + ecm.res + season > > > ## historical tests > ols <- efp(LTW.model, data = GermanM1, type = "OLS-CUSUM") > plot(ols) > re <- efp(LTW.model, data = GermanM1, type = "fluctuation") > plot(re) > fs <- Fstats(LTW.model, data = GermanM1, from = 0.1) > plot(fs) > > ## monitoring > M1 <- historyM1 > ols.efp <- efp(M1.model, type = "OLS-CUSUM", data = M1) > newborder <- function(k) 1.5778*k/118 > ols.mefp <- mefp(ols.efp, period = 2) > ols.mefp2 <- mefp(ols.efp, border = newborder) > M1 <- GermanM1 > ols.mon <- monitor(ols.mefp) Break detected at observation # 128 > ols.mon2 <- monitor(ols.mefp2) Break detected at observation # 135 > plot(ols.mon) > lines(boundary(ols.mon2), col = 2) > > ## dating > bp <- breakpoints(LTW.model, data = GermanM1) > summary(bp) Optimal (m+1)-segment partition: Call: breakpoints.formula(formula = LTW.model, data = GermanM1) Breakpoints at observation number: m = 1 119 m = 2 42 119 m = 3 48 71 119 m = 4 27 48 71 119 m = 5 27 48 71 98 119 Corresponding to breakdates: m = 1 1990(3) m = 2 1971(2) 1990(3) m = 3 1972(4) 1978(3) 1990(3) m = 4 1967(3) 1972(4) 1978(3) 1990(3) m = 5 1967(3) 1972(4) 1978(3) 1985(2) 1990(3) Fit: m 0 1 2 3 4 5 RSS 3.683e-02 1.916e-02 1.522e-02 1.301e-02 1.053e-02 9.198e-03 BIC -6.974e+02 -7.296e+02 -7.025e+02 -6.653e+02 -6.356e+02 -5.952e+02 > plot(bp) > > plot(fs) > lines(confint(bp)) > > > > cleanEx() > nameEx("Grossarl") > ### * Grossarl > > flush(stderr()); flush(stdout()) > > ### Name: Grossarl > ### Title: Marriages, Births and Deaths in Grossarl > ### Aliases: Grossarl > ### Keywords: datasets > > ### ** Examples > > data("Grossarl") > > ## time series of births, deaths, marriages > ########################################### > > with(Grossarl, plot(cbind(deaths, illegitimate + legitimate, marriages), + plot.type = "single", col = grey(c(0.7, 0, 0)), lty = c(1, 1, 3), + lwd = 1.5, ylab = "annual Grossarl series")) > legend("topright", c("deaths", "births", "marriages"), col = grey(c(0.7, 0, 0)), + lty = c(1, 1, 3), bty = "n") > > ## illegitimate births > ###################### > ## lm + MOSUM > plot(Grossarl$fraction) > fm.min <- lm(fraction ~ politics, data = Grossarl) > fm.ext <- lm(fraction ~ politics + morals + nuptiality + marriages, + data = Grossarl) > lines(ts(fitted(fm.min), start = 1700), col = 2) > lines(ts(fitted(fm.ext), start = 1700), col = 4) > mos.min <- efp(fraction ~ politics, data = Grossarl, type = "OLS-MOSUM") > mos.ext <- efp(fraction ~ politics + morals + nuptiality + marriages, + data = Grossarl, type = "OLS-MOSUM") > plot(mos.min) > lines(mos.ext, lty = 2) > > ## dating > bp <- breakpoints(fraction ~ 1, data = Grossarl, h = 0.1) > summary(bp) Optimal (m+1)-segment partition: Call: breakpoints.formula(formula = fraction ~ 1, h = 0.1, data = Grossarl) Breakpoints at observation number: m = 1 127 m = 2 55 122 m = 3 55 124 180 m = 4 55 122 157 179 m = 5 54 86 122 157 179 m = 6 35 55 86 122 157 179 m = 7 35 55 80 101 122 157 179 m = 8 35 55 79 99 119 139 159 179 Corresponding to breakdates: m = 1 1826 m = 2 1754 1821 m = 3 1754 1823 1879 m = 4 1754 1821 1856 1878 m = 5 1753 1785 1821 1856 1878 m = 6 1734 1754 1785 1821 1856 1878 m = 7 1734 1754 1779 1800 1821 1856 1878 m = 8 1734 1754 1778 1798 1818 1838 1858 1878 Fit: m 0 1 2 3 4 5 6 RSS 1.1088 0.8756 0.6854 0.6587 0.6279 0.6019 0.5917 BIC -460.8402 -497.4625 -535.8459 -533.1857 -532.1789 -530.0501 -522.8510 m 7 8 RSS 0.5934 0.6084 BIC -511.7017 -496.0924 > ## RSS, BIC, AIC > plot(bp) > plot(0:8, AIC(bp), type = "b") > > ## probably use 5 or 6 breakpoints and compare with > ## coding of the factors as used by us > ## > ## politics 1803 1816 1850 > ## morals 1736 1753 1771 1803 > ## nuptiality 1803 1810 1816 1883 > ## > ## m = 5 1753 1785 1821 1856 1878 > ## m = 6 1734 1754 1785 1821 1856 1878 > ## 6 2 5 1 4 3 > > ## fitted models > coef(bp, breaks = 6) (Intercept) 1700 - 1734 0.16933985 1735 - 1754 0.14078070 1755 - 1785 0.09890276 1786 - 1821 0.05955620 1822 - 1856 0.17441529 1857 - 1878 0.22425604 1879 - 1899 0.15414723 > plot(Grossarl$fraction) > lines(fitted(bp, breaks = 6), col = 2) > lines(ts(fitted(fm.ext), start = 1700), col = 4) > > > ## marriages > ############ > ## lm + MOSUM > plot(Grossarl$marriages) > fm.min <- lm(marriages ~ politics, data = Grossarl) > fm.ext <- lm(marriages ~ politics + morals + nuptiality, data = Grossarl) > lines(ts(fitted(fm.min), start = 1700), col = 2) > lines(ts(fitted(fm.ext), start = 1700), col = 4) > mos.min <- efp(marriages ~ politics, data = Grossarl, type = "OLS-MOSUM") > mos.ext <- efp(marriages ~ politics + morals + nuptiality, data = Grossarl, + type = "OLS-MOSUM") > plot(mos.min) > lines(mos.ext, lty = 2) > > ## dating > bp <- breakpoints(marriages ~ 1, data = Grossarl, h = 0.1) > summary(bp) Optimal (m+1)-segment partition: Call: breakpoints.formula(formula = marriages ~ 1, h = 0.1, data = Grossarl) Breakpoints at observation number: m = 1 114 m = 2 39 114 m = 3 39 114 176 m = 4 39 95 115 176 m = 5 39 62 95 115 176 m = 6 39 62 95 115 136 176 m = 7 39 62 95 115 136 156 176 m = 8 21 41 62 95 115 136 156 176 Corresponding to breakdates: m = 1 1813 m = 2 1738 1813 m = 3 1738 1813 1875 m = 4 1738 1794 1814 1875 m = 5 1738 1761 1794 1814 1875 m = 6 1738 1761 1794 1814 1835 1875 m = 7 1738 1761 1794 1814 1835 1855 1875 m = 8 1720 1740 1761 1794 1814 1835 1855 1875 Fit: m 0 1 2 3 4 5 6 7 8 RSS 3832 3059 2863 2723 2671 2634 2626 2626 2645 BIC 1169 1134 1132 1132 1139 1147 1157 1167 1179 > ## RSS, BIC, AIC > plot(bp) > plot(0:8, AIC(bp), type = "b") > > ## probably use 3 or 4 breakpoints and compare with > ## coding of the factors as used by us > ## > ## politics 1803 1816 1850 > ## morals 1736 1753 1771 1803 > ## nuptiality 1803 1810 1816 1883 > ## > ## m = 3 1738 1813 1875 > ## m = 4 1738 1794 1814 1875 > ## 2 4 1 3 > > ## fitted models > coef(bp, breaks = 4) (Intercept) 1700 - 1738 13.487179 1739 - 1794 10.160714 1795 - 1814 12.150000 1815 - 1875 6.885246 1876 - 1899 9.750000 > plot(Grossarl$marriages) > lines(fitted(bp, breaks = 4), col = 2) > lines(ts(fitted(fm.ext), start = 1700), col = 4) > > > > cleanEx() > nameEx("PhillipsCurve") > ### * PhillipsCurve > > flush(stderr()); flush(stdout()) > > ### Name: PhillipsCurve > ### Title: UK Phillips Curve Equation Data > ### Aliases: PhillipsCurve > ### Keywords: datasets > > ### ** Examples > > ## load and plot data > data("PhillipsCurve") > uk <- window(PhillipsCurve, start = 1948) > plot(uk[, "dp"]) > > ## AR(1) inflation model > ## estimate breakpoints > bp.inf <- breakpoints(dp ~ dp1, data = uk, h = 8) > plot(bp.inf) > summary(bp.inf) Optimal (m+1)-segment partition: Call: breakpoints.formula(formula = dp ~ dp1, h = 8, data = uk) Breakpoints at observation number: m = 1 20 m = 2 20 28 m = 3 9 20 28 Corresponding to breakdates: m = 1 1967 m = 2 1967 1975 m = 3 1956 1967 1975 Fit: m 0 1 2 3 RSS 0.03068 0.02672 0.01838 0.01786 BIC -162.34174 -156.80265 -160.70385 -150.78479 > > ## fit segmented model with three breaks > fac.inf <- breakfactor(bp.inf, breaks = 2, label = "seg") > fm.inf <- lm(dp ~ 0 + fac.inf/dp1, data = uk) > summary(fm.inf) Call: lm(formula = dp ~ 0 + fac.inf/dp1, data = uk) Residuals: Min 1Q Median 3Q Max -0.046987 -0.014861 -0.003593 0.006286 0.058081 Coefficients: Estimate Std. Error t value Pr(>|t|) fac.infseg1 0.024501 0.011176 2.192 0.0353 * fac.infseg2 -0.000775 0.017853 -0.043 0.9656 fac.infseg3 0.017603 0.015007 1.173 0.2489 fac.infseg1:dp1 0.274012 0.269892 1.015 0.3171 fac.infseg2:dp1 1.343369 0.224521 5.983 9.05e-07 *** fac.infseg3:dp1 0.683410 0.130106 5.253 8.07e-06 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 0.02325 on 34 degrees of freedom Multiple R-squared: 0.9237, Adjusted R-squared: 0.9103 F-statistic: 68.64 on 6 and 34 DF, p-value: < 2.2e-16 > > ## Results from Table 2 in Bai & Perron (2003): > ## coefficient estimates > coef(bp.inf, breaks = 2) (Intercept) dp1 1948 - 1967 0.0245010729 0.2740125 1968 - 1975 -0.0007750299 1.3433686 1976 - 1987 0.0176032179 0.6834098 > ## corresponding standard errors > sqrt(sapply(vcov(bp.inf, breaks = 2), diag)) 1948 - 1967 1968 - 1975 1976 - 1987 (Intercept) 0.008268814 0.01985539 0.01571339 dp1 0.199691273 0.24969992 0.13622996 > ## breakpoints and confidence intervals > confint(bp.inf, breaks = 2) Confidence intervals for breakpoints of optimal 3-segment partition: Call: confint.breakpointsfull(object = bp.inf, breaks = 2) Breakpoints at observation number: 2.5 % breakpoints 97.5 % 1 18 20 25 2 26 28 34 Corresponding to breakdates: 2.5 % breakpoints 97.5 % 1 1965 1967 1972 2 1973 1975 1981 > > ## Phillips curve equation > ## estimate breakpoints > bp.pc <- breakpoints(dw ~ dp1 + du + u1, data = uk, h = 5, breaks = 5) > ## look at RSS and BIC > plot(bp.pc) > summary(bp.pc) Optimal (m+1)-segment partition: Call: breakpoints.formula(formula = dw ~ dp1 + du + u1, h = 5, breaks = 5, data = uk) Breakpoints at observation number: m = 1 26 m = 2 20 28 m = 3 9 25 30 m = 4 11 16 25 30 m = 5 11 16 22 27 32 Corresponding to breakdates: m = 1 1973 m = 2 1967 1975 m = 3 1956 1972 1977 m = 4 1958 1963 1972 1977 m = 5 1958 1963 1969 1974 1979 Fit: m 0 1 2 3 4 5 RSS 3.409e-02 1.690e-02 1.062e-02 7.835e-03 5.183e-03 3.388e-03 BIC -1.508e+02 -1.604e+02 -1.605e+02 -1.542e+02 -1.523e+02 -1.509e+02 > > ## fit segmented model with three breaks > fac.pc <- breakfactor(bp.pc, breaks = 2, label = "seg") > fm.pc <- lm(dw ~ 0 + fac.pc/dp1 + du + u1, data = uk) > summary(fm.pc) Call: lm(formula = dw ~ 0 + fac.pc/dp1 + du + u1, data = uk) Residuals: Min 1Q Median 3Q Max -0.041392 -0.011516 0.000089 0.010036 0.044539 Coefficients: Estimate Std. Error t value Pr(>|t|) fac.pcseg1 0.06574 0.01169 5.623 3.24e-06 *** fac.pcseg2 0.06231 0.01883 3.310 0.00232 ** fac.pcseg3 0.18093 0.05388 3.358 0.00204 ** du -0.14408 0.58218 -0.247 0.80611 u1 -0.87516 0.37274 -2.348 0.02523 * fac.pcseg1:dp1 0.09373 0.24053 0.390 0.69936 fac.pcseg2:dp1 1.23143 0.20498 6.008 1.06e-06 *** fac.pcseg3:dp1 0.01618 0.25667 0.063 0.95013 --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 0.02021 on 32 degrees of freedom Multiple R-squared: 0.9655, Adjusted R-squared: 0.9569 F-statistic: 112 on 8 and 32 DF, p-value: < 2.2e-16 > > ## Results from Table 3 in Bai & Perron (2003): > ## coefficient estimates > coef(fm.pc) fac.pcseg1 fac.pcseg2 fac.pcseg3 du u1 0.06574278 0.06231337 0.18092502 -0.14408073 -0.87515585 fac.pcseg1:dp1 fac.pcseg2:dp1 fac.pcseg3:dp1 0.09372759 1.23143008 0.01617826 > ## corresponding standard errors > sqrt(diag(vcov(fm.pc))) fac.pcseg1 fac.pcseg2 fac.pcseg3 du u1 0.01169149 0.01882668 0.05388166 0.58217571 0.37273955 fac.pcseg1:dp1 fac.pcseg2:dp1 fac.pcseg3:dp1 0.24052539 0.20497973 0.25666903 > ## breakpoints and confidence intervals > confint(bp.pc, breaks = 2, het.err = FALSE) Confidence intervals for breakpoints of optimal 3-segment partition: Call: confint.breakpointsfull(object = bp.pc, breaks = 2, het.err = FALSE) Breakpoints at observation number: 2.5 % breakpoints 97.5 % 1 19 20 21 2 27 28 29 Corresponding to breakdates: 2.5 % breakpoints 97.5 % 1 1966 1967 1968 2 1974 1975 1976 > > > > cleanEx() > nameEx("RealInt") > ### * RealInt > > flush(stderr()); flush(stdout()) > > ### Name: RealInt > ### Title: US Ex-post Real Interest Rate > ### Aliases: RealInt > ### Keywords: datasets > > ### ** Examples > > ## load and plot data > data("RealInt") > plot(RealInt) > > ## estimate breakpoints > bp.ri <- breakpoints(RealInt ~ 1, h = 15) > plot(bp.ri) > summary(bp.ri) Optimal (m+1)-segment partition: Call: breakpoints.formula(formula = RealInt ~ 1, h = 15) Breakpoints at observation number: m = 1 79 m = 2 47 79 m = 3 24 47 79 m = 4 24 47 64 79 m = 5 16 31 47 64 79 Corresponding to breakdates: m = 1 1980(3) m = 2 1972(3) 1980(3) m = 3 1966(4) 1972(3) 1980(3) m = 4 1966(4) 1972(3) 1976(4) 1980(3) m = 5 1964(4) 1968(3) 1972(3) 1976(4) 1980(3) Fit: m 0 1 2 3 4 5 RSS 1214.9 645.0 456.0 445.2 444.9 449.6 BIC 555.7 499.8 473.3 480.1 489.3 499.7 > > ## fit segmented model with three breaks > fac.ri <- breakfactor(bp.ri, breaks = 3, label = "seg") > fm.ri <- lm(RealInt ~ 0 + fac.ri) > summary(fm.ri) Call: lm(formula = RealInt ~ 0 + fac.ri) Residuals: Min 1Q Median 3Q Max -4.5157 -1.3674 -0.0578 1.3248 6.0990 Coefficients: Estimate Std. Error t value Pr(>|t|) fac.riseg1 1.8236 0.4329 4.213 5.57e-05 *** fac.riseg2 0.8661 0.4422 1.959 0.053 . fac.riseg3 -1.7961 0.3749 -4.791 5.83e-06 *** fac.riseg4 5.6429 0.4329 13.036 < 2e-16 *** --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 2.121 on 99 degrees of freedom Multiple R-squared: 0.6842, Adjusted R-squared: 0.6714 F-statistic: 53.62 on 4 and 99 DF, p-value: < 2.2e-16 > > ## setup kernel HAC estimator > vcov.ri <- function(x, ...) kernHAC(x, kernel = "Quadratic Spectral", + prewhite = 1, approx = "AR(1)", ...) > > ## Results from Table 1 in Bai & Perron (2003): > ## coefficient estimates > coef(bp.ri, breaks = 3) (Intercept) 1961(1) - 1966(4) 1.8236167 1967(1) - 1972(3) 0.8660848 1972(4) - 1980(3) -1.7961384 1980(4) - 1986(3) 5.6428896 > ## corresponding standard errors > sapply(vcov(bp.ri, breaks = 3, vcov = vcov.ri), sqrt) 1961(1) - 1966(4) 1967(1) - 1972(3) 1972(4) - 1980(3) 1980(4) - 1986(3) 0.1857577 0.1499849 0.5026749 0.5887460 > ## breakpoints and confidence intervals > confint(bp.ri, breaks = 3, vcov = vcov.ri) Confidence intervals for breakpoints of optimal 4-segment partition: Call: confint.breakpointsfull(object = bp.ri, breaks = 3, vcov. = vcov.ri) Breakpoints at observation number: 2.5 % breakpoints 97.5 % 1 18 24 35 2 33 47 48 3 77 79 81 Corresponding to breakdates: Warning: Overlapping confidence intervals 2.5 % breakpoints 97.5 % 1 1965(2) 1966(4) 1969(3) 2 1969(1) 1972(3) 1972(4) 3 1980(1) 1980(3) 1981(1) > > ## Visualization > plot(RealInt) > lines(as.vector(time(RealInt)), fitted(fm.ri), col = 4) > lines(confint(bp.ri, breaks = 3, vcov = vcov.ri)) Warning: Overlapping confidence intervals > > > > cleanEx() > nameEx("SP2001") > ### * SP2001 > > flush(stderr()); flush(stdout()) > > ### Name: SP2001 > ### Title: S&P 500 Stock Prices > ### Aliases: SP2001 > ### Keywords: datasets > > ### ** Examples > > ## load and transform data > ## (DAL: Delta Air Lines, LU: Lucent Technologies) > data("SP2001") > stock.prices <- SP2001[, c("DAL", "LU")] > stock.returns <- diff(log(stock.prices)) > > ## price and return series > plot(stock.prices, ylab = c("Delta Air Lines", "Lucent Technologies"), main = "") > plot(stock.returns, ylab = c("Delta Air Lines", "Lucent Technologies"), main = "") > > ## monitoring of DAL series > myborder <- function(k) 1.939*k/28 > x <- as.vector(stock.returns[, "DAL"][1:28]) > dal.cusum <- mefp(x ~ 1, type = "OLS-CUSUM", border = myborder) > dal.mosum <- mefp(x ~ 1, type = "OLS-MOSUM", h = 0.5, period = 4) > x <- as.vector(stock.returns[, "DAL"]) > dal.cusum <- monitor(dal.cusum) Break detected at observation # 29 > dal.mosum <- monitor(dal.mosum) Break detected at observation # 29 > > ## monitoring of LU series > x <- as.vector(stock.returns[, "LU"][1:28]) > lu.cusum <- mefp(x ~ 1, type = "OLS-CUSUM", border = myborder) > lu.mosum <- mefp(x ~ 1, type = "OLS-MOSUM", h = 0.5, period = 4) > x <- as.vector(stock.returns[, "LU"]) > lu.cusum <- monitor(lu.cusum) > lu.mosum <- monitor(lu.mosum) > > ## pretty plotting > ## (needs some work because lm() does not keep "zoo" attributes) > cus.bound <- zoo(c(rep(NA, 27), myborder(28:102)), index(stock.returns)) > mos.bound <- as.vector(boundary(dal.mosum)) > mos.bound <- zoo(c(rep(NA, 27), mos.bound[1], mos.bound), index(stock.returns)) > > ## Lucent Technologies: CUSUM test > plot(zoo(c(lu.cusum$efpprocess, lu.cusum$process), index(stock.prices)), + ylim = c(-1, 1) * coredata(cus.bound)[102], xlab = "Time", ylab = "empirical fluctuation process") > abline(0, 0) > abline(v = as.Date("2001-09-10"), lty = 2) > lines(cus.bound, col = 2) > lines(-cus.bound, col = 2) > > ## Lucent Technologies: MOSUM test > plot(zoo(c(lu.mosum$efpprocess, lu.mosum$process), index(stock.prices)[-(1:14)]), + ylim = c(-1, 1) * coredata(mos.bound)[102], xlab = "Time", ylab = "empirical fluctuation process") > abline(0, 0) > abline(v = as.Date("2001-09-10"), lty = 2) > lines(mos.bound, col = 2) > lines(-mos.bound, col = 2) > > ## Delta Air Lines: CUSUM test > plot(zoo(c(dal.cusum$efpprocess, dal.cusum$process), index(stock.prices)), + ylim = c(-1, 1) * coredata(cus.bound)[102], xlab = "Time", ylab = "empirical fluctuation process") > abline(0, 0) > abline(v = as.Date("2001-09-10"), lty = 2) > lines(cus.bound, col = 2) > lines(-cus.bound, col = 2) > > ## Delta Air Lines: MOSUM test > plot(zoo(c(dal.mosum$efpprocess, dal.mosum$process), index(stock.prices)[-(1:14)]), + ylim = range(dal.mosum$process), xlab = "Time", ylab = "empirical fluctuation process") > abline(0, 0) > abline(v = as.Date("2001-09-10"), lty = 2) > lines(mos.bound, col = 2) > lines(-mos.bound, col = 2) > > > > cleanEx() > nameEx("USIncExp") > ### * USIncExp > > flush(stderr()); flush(stdout()) > > ### Name: USIncExp > ### Title: Income and Expenditures in the US > ### Aliases: USIncExp > ### Keywords: datasets > > ### ** Examples > > ## These example are presented in the vignette distributed with this > ## package, the code was generated by Stangle("strucchange-intro.Rnw") > > ################################################### > ### chunk number 1: data > ################################################### > library("strucchange") > data("USIncExp") > plot(USIncExp, plot.type = "single", col = 1:2, ylab = "billion US$") > legend(1960, max(USIncExp), c("income", "expenditures"), + lty = c(1,1), col = 1:2, bty = "n") > > > ################################################### > ### chunk number 2: subset > ################################################### > library("strucchange") > data("USIncExp") > USIncExp2 <- window(USIncExp, start = c(1985,12)) > > > ################################################### > ### chunk number 3: ecm-setup > ################################################### > coint.res <- residuals(lm(expenditure ~ income, data = USIncExp2)) > coint.res <- lag(ts(coint.res, start = c(1985,12), freq = 12), k = -1) > USIncExp2 <- cbind(USIncExp2, diff(USIncExp2), coint.res) > USIncExp2 <- window(USIncExp2, start = c(1986,1), end = c(2001,2)) > colnames(USIncExp2) <- c("income", "expenditure", "diff.income", + "diff.expenditure", "coint.res") > ecm.model <- diff.expenditure ~ coint.res + diff.income > > > ################################################### > ### chunk number 4: ts-used > ################################################### > plot(USIncExp2[,3:5], main = "") > > > ################################################### > ### chunk number 5: efp > ################################################### > ocus <- efp(ecm.model, type="OLS-CUSUM", data=USIncExp2) > me <- efp(ecm.model, type="ME", data=USIncExp2, h=0.2) > > > ################################################### > ### chunk number 6: efp-boundary > ################################################### > bound.ocus <- boundary(ocus, alpha=0.05) > > > ################################################### > ### chunk number 7: OLS-CUSUM > ################################################### > plot(ocus) > > > ################################################### > ### chunk number 8: efp-boundary2 > ################################################### > plot(ocus, boundary = FALSE) > lines(bound.ocus, col = 4) > lines(-bound.ocus, col = 4) > > > ################################################### > ### chunk number 9: ME-null > ################################################### > plot(me, functional = NULL) > > > ################################################### > ### chunk number 10: efp-sctest > ################################################### > sctest(ocus) OLS-based CUSUM test data: ocus S0 = 1.5511, p-value = 0.01626 > > > ################################################### > ### chunk number 11: efp-sctest2 > ################################################### > sctest(ecm.model, type="OLS-CUSUM", data=USIncExp2) OLS-based CUSUM test data: ecm.model S0 = 1.5511, p-value = 0.01626 > > > ################################################### > ### chunk number 12: Fstats > ################################################### > fs <- Fstats(ecm.model, from = c(1990, 1), to = c(1999,6), data = USIncExp2) > > > ################################################### > ### chunk number 13: Fstats-plot > ################################################### > plot(fs) > > > ################################################### > ### chunk number 14: pval-plot > ################################################### > plot(fs, pval=TRUE) > > > ################################################### > ### chunk number 15: aveF-plot > ################################################### > plot(fs, aveF=TRUE) > > > ################################################### > ### chunk number 16: Fstats-sctest > ################################################### > sctest(fs, type="expF") expF test data: fs exp.F = 8.9955, p-value = 0.001311 > > > ################################################### > ### chunk number 17: Fstats-sctest2 > ################################################### > sctest(ecm.model, type = "expF", from = 49, to = 162, data = USIncExp2) expF test data: ecm.model exp.F = 8.9955, p-value = 0.001311 > > > ################################################### > ### chunk number 18: mefp > ################################################### > USIncExp3 <- window(USIncExp2, start = c(1986, 1), end = c(1989,12)) > me.mefp <- mefp(ecm.model, type = "ME", data = USIncExp3, alpha = 0.05) > > > ################################################### > ### chunk number 19: monitor1 > ################################################### > USIncExp3 <- window(USIncExp2, start = c(1986, 1), end = c(1990,12)) > me.mefp <- monitor(me.mefp) > > > ################################################### > ### chunk number 20: monitor2 > ################################################### > USIncExp3 <- window(USIncExp2, start = c(1986, 1)) > me.mefp <- monitor(me.mefp) Break detected at observation # 72 > me.mefp Monitoring with ME test (moving estimates test) Initial call: mefp.formula(formula = ecm.model, type = "ME", data = USIncExp3, alpha = 0.05) Last call: monitor(obj = me.mefp) Significance level : 0.05 Critical value : 3.109524 History size : 48 Last point evaluated : 182 Structural break at : 72 Parameter estimate on history : (Intercept) coint.res diff.income 18.9299679 -0.3893141 0.3156597 Last parameter estimate : (Intercept) coint.res diff.income 27.94869106 0.00983451 0.13314662 > > > ################################################### > ### chunk number 21: monitor-plot > ################################################### > plot(me.mefp) > > > ################################################### > ### chunk number 22: mefp2 > ################################################### > USIncExp3 <- window(USIncExp2, start = c(1986, 1), end = c(1989,12)) > me.efp <- efp(ecm.model, type = "ME", data = USIncExp3, h = 0.5) > me.mefp <- mefp(me.efp, alpha=0.05) > > > ################################################### > ### chunk number 23: monitor3 > ################################################### > USIncExp3 <- window(USIncExp2, start = c(1986, 1)) > me.mefp <- monitor(me.mefp) Break detected at observation # 70 > > > ################################################### > ### chunk number 24: monitor-plot2 > ################################################### > plot(me.mefp) > > > > > cleanEx() > nameEx("boundary.Fstats") > ### * boundary.Fstats > > flush(stderr()); flush(stdout()) > > ### Name: boundary.Fstats > ### Title: Boundary for F Statistics > ### Aliases: boundary.Fstats > ### Keywords: regression > > ### ** Examples > > ## Load dataset "nhtemp" with average yearly temperatures in New Haven > data("nhtemp") > ## plot the data > plot(nhtemp) > > ## test the model null hypothesis that the average temperature remains > ## constant over the years for potential break points between 1941 > ## (corresponds to from = 0.5) and 1962 (corresponds to to = 0.85) > ## compute F statistics > fs <- Fstats(nhtemp ~ 1, from = 0.5, to = 0.85) > ## plot the p values without boundary > plot(fs, pval = TRUE, alpha = 0.01) > ## add the boundary in another colour > lines(boundary(fs, pval = TRUE, alpha = 0.01), col = 2) > > > > cleanEx() > nameEx("boundary.efp") > ### * boundary.efp > > flush(stderr()); flush(stdout()) > > ### Name: boundary.efp > ### Title: Boundary for Empirical Fluctuation Processes > ### Aliases: boundary.efp > ### Keywords: regression > > ### ** Examples > > ## Load dataset "nhtemp" with average yearly temperatures in New Haven > data("nhtemp") > ## plot the data > plot(nhtemp) > > ## test the model null hypothesis that the average temperature remains constant > ## over the years > ## compute OLS-CUSUM fluctuation process > temp.cus <- efp(nhtemp ~ 1, type = "OLS-CUSUM") > ## plot the process without boundaries > plot(temp.cus, alpha = 0.01, boundary = FALSE) > ## add the boundaries in another colour > bound <- boundary(temp.cus, alpha = 0.01) > lines(bound, col=4) > lines(-bound, col=4) > > > > cleanEx() > nameEx("boundary.mefp") > ### * boundary.mefp > > flush(stderr()); flush(stdout()) > > ### Name: boundary.mefp > ### Title: Boundary Function for Monitoring of Structural Changes > ### Aliases: boundary.mefp > ### Keywords: regression > > ### ** Examples > > df1 <- data.frame(y=rnorm(300)) > df1[150:300,"y"] <- df1[150:300,"y"]+1 > me1 <- mefp(y~1, data=df1[1:50,,drop=FALSE], type="ME", h=1, + alpha=0.05) > me2 <- monitor(me1, data=df1) Break detected at observation # 183 > > plot(me2, boundary=FALSE) > lines(boundary(me2), col="green", lty="44") > > > > cleanEx() > nameEx("breakdates") > ### * breakdates > > flush(stderr()); flush(stdout()) > > ### Name: breakdates > ### Title: Breakdates Corresponding to Breakpoints > ### Aliases: breakdates breakdates.breakpoints > ### breakdates.confint.breakpoints > ### Keywords: regression > > ### ** Examples > > ## Nile data with one breakpoint: the annual flows drop in 1898 > ## because the first Ashwan dam was built > data("Nile") > plot(Nile) > > bp.nile <- breakpoints(Nile ~ 1) > summary(bp.nile) Optimal (m+1)-segment partition: Call: breakpoints.formula(formula = Nile ~ 1) Breakpoints at observation number: m = 1 28 m = 2 28 83 m = 3 28 68 83 m = 4 28 45 68 83 m = 5 15 30 45 68 83 Corresponding to breakdates: m = 1 1898 m = 2 1898 1953 m = 3 1898 1938 1953 m = 4 1898 1915 1938 1953 m = 5 1885 1900 1915 1938 1953 Fit: m 0 1 2 3 4 5 RSS 2835157 1597457 1552924 1538097 1507888 1659994 BIC 1318 1270 1276 1285 1292 1311 > plot(bp.nile) > > ## compute breakdates corresponding to the > ## breakpoints of minimum BIC segmentation > breakdates(bp.nile) [1] 1898 > > ## confidence intervals > ci.nile <- confint(bp.nile) > breakdates(ci.nile) 2.5 % breakpoints 97.5 % 1 1895 1898 1902 > ci.nile Confidence intervals for breakpoints of optimal 2-segment partition: Call: confint.breakpointsfull(object = bp.nile) Breakpoints at observation number: 2.5 % breakpoints 97.5 % 1 25 28 32 Corresponding to breakdates: 2.5 % breakpoints 97.5 % 1 1895 1898 1902 > > plot(Nile) > lines(ci.nile) > > > > cleanEx() > nameEx("breakfactor") > ### * breakfactor > > flush(stderr()); flush(stdout()) > > ### Name: breakfactor > ### Title: Factor Coding of Segmentations > ### Aliases: breakfactor > ### Keywords: regression > > ### ** Examples > > ## Nile data with one breakpoint: the annual flows drop in 1898 > ## because the first Ashwan dam was built > data("Nile") > plot(Nile) > > ## compute breakpoints > bp.nile <- breakpoints(Nile ~ 1) > > ## fit and visualize segmented and unsegmented model > fm0 <- lm(Nile ~ 1) > fm1 <- lm(Nile ~ breakfactor(bp.nile, breaks = 1)) > > lines(fitted(fm0), col = 3) > lines(fitted(fm1), col = 4) > lines(bp.nile, breaks = 1) > > > > cleanEx() > nameEx("breakpoints") > ### * breakpoints > > flush(stderr()); flush(stdout()) > > ### Name: breakpoints > ### Title: Dating Breaks > ### Aliases: breakpoints breakpoints.formula breakpoints.breakpointsfull > ### breakpoints.Fstats summary.breakpoints summary.breakpointsfull > ### plot.breakpointsfull plot.summary.breakpointsfull print.breakpoints > ### print.summary.breakpointsfull lines.breakpoints coef.breakpointsfull > ### vcov.breakpointsfull fitted.breakpointsfull residuals.breakpointsfull > ### df.residual.breakpointsfull > ### Keywords: regression > > ### ** Examples > > ## Nile data with one breakpoint: the annual flows drop in 1898 > ## because the first Ashwan dam was built > data("Nile") > plot(Nile) > > ## F statistics indicate one breakpoint > fs.nile <- Fstats(Nile ~ 1) > plot(fs.nile) > breakpoints(fs.nile) Optimal 2-segment partition: Call: breakpoints.Fstats(obj = fs.nile) Breakpoints at observation number: 28 Corresponding to breakdates: 1898 > lines(breakpoints(fs.nile)) > > ## or > bp.nile <- breakpoints(Nile ~ 1) > summary(bp.nile) Optimal (m+1)-segment partition: Call: breakpoints.formula(formula = Nile ~ 1) Breakpoints at observation number: m = 1 28 m = 2 28 83 m = 3 28 68 83 m = 4 28 45 68 83 m = 5 15 30 45 68 83 Corresponding to breakdates: m = 1 1898 m = 2 1898 1953 m = 3 1898 1938 1953 m = 4 1898 1915 1938 1953 m = 5 1885 1900 1915 1938 1953 Fit: m 0 1 2 3 4 5 RSS 2835157 1597457 1552924 1538097 1507888 1659994 BIC 1318 1270 1276 1285 1292 1311 > > ## the BIC also chooses one breakpoint > plot(bp.nile) > breakpoints(bp.nile) Optimal 2-segment partition: Call: breakpoints.breakpointsfull(obj = bp.nile) Breakpoints at observation number: 28 Corresponding to breakdates: 1898 > > ## fit null hypothesis model and model with 1 breakpoint > fm0 <- lm(Nile ~ 1) > fm1 <- lm(Nile ~ breakfactor(bp.nile, breaks = 1)) > plot(Nile) > lines(ts(fitted(fm0), start = 1871), col = 3) > lines(ts(fitted(fm1), start = 1871), col = 4) > lines(bp.nile) > > ## confidence interval > ci.nile <- confint(bp.nile) > ci.nile Confidence intervals for breakpoints of optimal 2-segment partition: Call: confint.breakpointsfull(object = bp.nile) Breakpoints at observation number: 2.5 % breakpoints 97.5 % 1 25 28 32 Corresponding to breakdates: 2.5 % breakpoints 97.5 % 1 1895 1898 1902 > lines(ci.nile) > > > ## UK Seatbelt data: a SARIMA(1,0,0)(1,0,0)_12 model > ## (fitted by OLS) is used and reveals (at least) two > ## breakpoints - one in 1973 associated with the oil crisis and > ## one in 1983 due to the introduction of compulsory > ## wearing of seatbelts in the UK. > data("UKDriverDeaths") > seatbelt <- log10(UKDriverDeaths) > seatbelt <- cbind(seatbelt, lag(seatbelt, k = -1), lag(seatbelt, k = -12)) > colnames(seatbelt) <- c("y", "ylag1", "ylag12") > seatbelt <- window(seatbelt, start = c(1970, 1), end = c(1984,12)) > plot(seatbelt[,"y"], ylab = expression(log[10](casualties))) > > ## testing > re.seat <- efp(y ~ ylag1 + ylag12, data = seatbelt, type = "RE") > plot(re.seat) > > ## dating > bp.seat <- breakpoints(y ~ ylag1 + ylag12, data = seatbelt, h = 0.1) > summary(bp.seat) Optimal (m+1)-segment partition: Call: breakpoints.formula(formula = y ~ ylag1 + ylag12, h = 0.1, data = seatbelt) Breakpoints at observation number: m = 1 46 m = 2 46 157 m = 3 46 70 157 m = 4 46 70 108 157 m = 5 46 70 120 141 160 m = 6 46 70 89 108 141 160 m = 7 46 70 89 107 125 144 162 m = 8 18 46 70 89 107 125 144 162 Corresponding to breakdates: m = 1 1973(10) m = 2 1973(10) 1983(1) m = 3 1973(10) 1975(10) 1983(1) m = 4 1973(10) 1975(10) 1978(12) 1983(1) m = 5 1973(10) 1975(10) 1979(12) 1981(9) 1983(4) m = 6 1973(10) 1975(10) 1977(5) 1978(12) 1981(9) 1983(4) m = 7 1973(10) 1975(10) 1977(5) 1978(11) 1980(5) 1981(12) 1983(6) m = 8 1971(6) 1973(10) 1975(10) 1977(5) 1978(11) 1980(5) 1981(12) 1983(6) Fit: m 0 1 2 3 4 5 6 RSS 0.3297 0.2967 0.2676 0.2438 0.2395 0.2317 0.2258 BIC -602.8611 -601.0539 -598.9042 -594.8774 -577.2905 -562.4880 -546.3632 m 7 8 RSS 0.2244 0.2231 BIC -526.7295 -506.9886 > lines(bp.seat, breaks = 2) > > ## minimum BIC partition > plot(bp.seat) > breakpoints(bp.seat) Optimal 1-segment partition: Call: breakpoints.breakpointsfull(obj = bp.seat) Breakpoints at observation number: NA Corresponding to breakdates: NA > ## the BIC would choose 0 breakpoints although the RE and supF test > ## clearly reject the hypothesis of structural stability. Bai & > ## Perron (2003) report that the BIC has problems in dynamic regressions. > ## due to the shape of the RE process of the F statistics choose two > ## breakpoints and fit corresponding models > bp.seat2 <- breakpoints(bp.seat, breaks = 2) > fm0 <- lm(y ~ ylag1 + ylag12, data = seatbelt) > fm1 <- lm(y ~ breakfactor(bp.seat2)/(ylag1 + ylag12) - 1, data = seatbelt) > > ## plot > plot(seatbelt[,"y"], ylab = expression(log[10](casualties))) > time.seat <- as.vector(time(seatbelt)) > lines(time.seat, fitted(fm0), col = 3) > lines(time.seat, fitted(fm1), col = 4) > lines(bp.seat2) > > ## confidence intervals > ci.seat2 <- confint(bp.seat, breaks = 2) > ci.seat2 Confidence intervals for breakpoints of optimal 3-segment partition: Call: confint.breakpointsfull(object = bp.seat, breaks = 2) Breakpoints at observation number: 2.5 % breakpoints 97.5 % 1 33 46 56 2 144 157 171 Corresponding to breakdates: 2.5 % breakpoints 97.5 % 1 1972(9) 1973(10) 1974(8) 2 1981(12) 1983(1) 1984(3) > lines(ci.seat2) > > > > cleanEx() > nameEx("catL2BB") > ### * catL2BB > > flush(stderr()); flush(stdout()) > > ### Name: catL2BB > ### Title: Generators for efpFunctionals along Categorical Variables > ### Aliases: catL2BB ordL2BB ordwmax > ### Keywords: regression > > ### ** Examples > > ## artificial data > set.seed(1) > d <- data.frame( + x = runif(200, -1, 1), + z = factor(rep(1:4, each = 50)), + err = rnorm(200) + ) > d$y <- rep(c(0.5, -0.5), c(150, 50)) * d$x + d$err > > ## empirical fluctuation process > scus <- gefp(y ~ x, data = d, fit = lm, order.by = ~ z) > > ## chi-squared-type test (unordered LM-type test) > LMuo <- catL2BB(scus) > plot(scus, functional = LMuo) > sctest(scus, functional = LMuo) M-fluctuation test data: scus f(efp) = 12.375, p-value = 0.05411 > > ## ordinal maxLM test (with few replications only to save time) > maxLMo <- ordL2BB(scus, nrep = 10000) > plot(scus, functional = maxLMo) > sctest(scus, functional = maxLMo) M-fluctuation test data: scus f(efp) = 9.0937, p-value = 0.03173 > > ## ordinal weighted double maximum test > WDM <- ordwmax(scus) > plot(scus, functional = WDM) > sctest(scus, functional = WDM) M-fluctuation test data: scus f(efp) = 3.001, p-value = 0.015 > > > > cleanEx() > nameEx("confint.breakpointsfull") > ### * confint.breakpointsfull > > flush(stderr()); flush(stdout()) > > ### Name: confint.breakpointsfull > ### Title: Confidence Intervals for Breakpoints > ### Aliases: confint.breakpointsfull lines.confint.breakpoints > ### print.confint.breakpoints > ### Keywords: regression > > ### ** Examples > > ## Nile data with one breakpoint: the annual flows drop in 1898 > ## because the first Ashwan dam was built > data("Nile") > plot(Nile) > > ## dating breaks > bp.nile <- breakpoints(Nile ~ 1) > ci.nile <- confint(bp.nile, breaks = 1) > lines(ci.nile) > > > > cleanEx() > nameEx("durab") > ### * durab > > flush(stderr()); flush(stdout()) > > ### Name: durab > ### Title: US Labor Productivity > ### Aliases: durab > ### Keywords: datasets > > ### ** Examples > > data("durab") > ## use AR(1) model as in Hansen (2001) and Zeileis et al. (2005) > durab.model <- y ~ lag > > ## historical tests > ## OLS-based CUSUM process > ols <- efp(durab.model, data = durab, type = "OLS-CUSUM") > plot(ols) > ## F statistics > fs <- Fstats(durab.model, data = durab, from = 0.1) > plot(fs) > > ## F statistics based on heteroskadisticy-consistent covariance matrix > fsHC <- Fstats(durab.model, data = durab, from = 0.1, + vcov = function(x, ...) vcovHC(x, type = "HC", ...)) > plot(fsHC) > > ## monitoring > Durab <- window(durab, start=1964, end = c(1979, 12)) > ols.efp <- efp(durab.model, type = "OLS-CUSUM", data = Durab) > newborder <- function(k) 1.723 * k/192 > ols.mefp <- mefp(ols.efp, period=2) > ols.mefp2 <- mefp(ols.efp, border=newborder) > Durab <- window(durab, start=1964) > ols.mon <- monitor(ols.mefp) Break detected at observation # 437 > ols.mon2 <- monitor(ols.mefp2) Break detected at observation # 416 > plot(ols.mon) > lines(boundary(ols.mon2), col = 2) > ## Note: critical value for linear boundary taken from Table III > ## in Zeileis et al. 2005: (1.568 + 1.896)/2 = 1.732 is a linear > ## interpolation between the values for T = 2 and T = 3 at > ## alpha = 0.05. A typo switched 1.732 to 1.723. > > ## dating > bp <- breakpoints(durab.model, data = durab) > summary(bp) Optimal (m+1)-segment partition: Call: breakpoints.formula(formula = durab.model, data = durab) Breakpoints at observation number: m = 1 418 m = 2 221 530 m = 3 114 225 530 m = 4 114 221 418 531 m = 5 114 221 319 418 531 Corresponding to breakdates: m = 1 1981(12) m = 2 1965(7) 1991(4) m = 3 1956(8) 1965(11) 1991(4) m = 4 1956(8) 1965(7) 1981(12) 1991(5) m = 5 1956(8) 1965(7) 1973(9) 1981(12) 1991(5) Fit: m 0 1 2 3 4 5 RSS 5.586e-02 5.431e-02 5.325e-02 5.220e-02 5.171e-02 5.157e-02 BIC -4.221e+03 -4.220e+03 -4.213e+03 -4.207e+03 -4.194e+03 -4.176e+03 > plot(summary(bp)) > > plot(ols) > lines(breakpoints(bp, breaks = 1), col = 3) > lines(breakpoints(bp, breaks = 2), col = 4) > plot(fs) > lines(breakpoints(bp, breaks = 1), col = 3) > lines(breakpoints(bp, breaks = 2), col = 4) > > > > cleanEx() > nameEx("efp") > ### * efp > > flush(stderr()); flush(stdout()) > > ### Name: efp > ### Title: Empirical Fluctuation Processes > ### Aliases: efp print.efp > ### Keywords: regression > > ### ** Examples > > ## Nile data with one breakpoint: the annual flows drop in 1898 > ## because the first Ashwan dam was built > data("Nile") > plot(Nile) > > ## test the null hypothesis that the annual flow remains constant > ## over the years > ## compute OLS-based CUSUM process and plot > ## with standard and alternative boundaries > ocus.nile <- efp(Nile ~ 1, type = "OLS-CUSUM") > plot(ocus.nile) > plot(ocus.nile, alpha = 0.01, alt.boundary = TRUE) > ## calculate corresponding test statistic > sctest(ocus.nile) OLS-based CUSUM test data: ocus.nile S0 = 2.9518, p-value = 5.409e-08 > > ## UK Seatbelt data: a SARIMA(1,0,0)(1,0,0)_12 model > ## (fitted by OLS) is used and reveals (at least) two > ## breakpoints - one in 1973 associated with the oil crisis and > ## one in 1983 due to the introduction of compulsory > ## wearing of seatbelts in the UK. > data("UKDriverDeaths") > seatbelt <- log10(UKDriverDeaths) > seatbelt <- cbind(seatbelt, lag(seatbelt, k = -1), lag(seatbelt, k = -12)) > colnames(seatbelt) <- c("y", "ylag1", "ylag12") > seatbelt <- window(seatbelt, start = c(1970, 1), end = c(1984,12)) > plot(seatbelt[,"y"], ylab = expression(log[10](casualties))) > > ## use RE process > re.seat <- efp(y ~ ylag1 + ylag12, data = seatbelt, type = "RE") > plot(re.seat) > plot(re.seat, functional = NULL) > sctest(re.seat) RE test (recursive estimates test) data: re.seat RE = 1.6311, p-value = 0.02904 > > > > cleanEx() > nameEx("efpFunctional") > ### * efpFunctional > > flush(stderr()); flush(stdout()) > > ### Name: efpFunctional > ### Title: Functionals for Fluctuation Processes > ### Aliases: efpFunctional simulateBMDist maxBM maxBB maxBMI maxBBI maxL2BB > ### meanL2BB rangeBM rangeBB rangeBMI rangeBBI > ### Keywords: regression > > ### ** Examples > > > data("BostonHomicide") > gcus <- gefp(homicides ~ 1, family = poisson, vcov = kernHAC, + data = BostonHomicide) > plot(gcus, functional = meanL2BB) > gcus Generalized Empirical M-Fluctuation Process Call: gefp(homicides ~ 1, family = poisson, vcov = kernHAC, data = BostonHomicide) Fitted model: Call: fit(formula = ..1, family = ..2, data = data) Coefficients: (Intercept) 1.017 Degrees of Freedom: 76 Total (i.e. Null); 76 Residual Null Deviance: 115.6 Residual Deviance: 115.6 AIC: 316.5 > sctest(gcus, functional = meanL2BB) M-fluctuation test data: gcus f(efp) = 0.93375, p-value = 0.005 > > y <- rnorm(1000) > x1 <- runif(1000) > x2 <- runif(1000) > > ## supWald statistic computed by Fstats() > fs <- Fstats(y ~ x1 + x2, from = 0.1) > plot(fs) > sctest(fs) supF test data: fs sup.F = 12.252, p-value = 0.1161 > > ## compare with supLM statistic > scus <- gefp(y ~ x1 + x2, fit = lm) > plot(scus, functional = supLM(0.1)) > sctest(scus, functional = supLM(0.1)) M-fluctuation test data: scus f(efp) = 12.258, p-value = 0.1158 > > ## seatbelt data > data("UKDriverDeaths") > seatbelt <- log10(UKDriverDeaths) > seatbelt <- cbind(seatbelt, lag(seatbelt, k = -1), lag(seatbelt, k = -12)) > colnames(seatbelt) <- c("y", "ylag1", "ylag12") > seatbelt <- window(seatbelt, start = c(1970, 1), end = c(1984,12)) > > scus.seat <- gefp(y ~ ylag1 + ylag12, data = seatbelt) > > ## double maximum test > plot(scus.seat) > ## range test > plot(scus.seat, functional = rangeBB) > ## Cramer-von Mises statistic (Nyblom-Hansen test) > plot(scus.seat, functional = meanL2BB) > ## supLM test > plot(scus.seat, functional = supLM(0.1)) > > > > cleanEx() > nameEx("gefp") > ### * gefp > > flush(stderr()); flush(stdout()) > > ### Name: gefp > ### Title: Generalized Empirical M-Fluctuation Processes > ### Aliases: gefp print.gefp sctest.gefp plot.gefp time.gefp print.gefp > ### Keywords: regression > > ### ** Examples > > data("BostonHomicide") > gcus <- gefp(homicides ~ 1, family = poisson, vcov = kernHAC, + data = BostonHomicide) > plot(gcus, aggregate = FALSE) > gcus Generalized Empirical M-Fluctuation Process Call: gefp(homicides ~ 1, family = poisson, vcov = kernHAC, data = BostonHomicide) Fitted model: Call: fit(formula = ..1, family = ..2, data = data) Coefficients: (Intercept) 1.017 Degrees of Freedom: 76 Total (i.e. Null); 76 Residual Null Deviance: 115.6 Residual Deviance: 115.6 AIC: 316.5 > sctest(gcus) M-fluctuation test data: gcus f(efp) = 1.669, p-value = 0.007613 > > > > cleanEx() > nameEx("logLik.breakpoints") > ### * logLik.breakpoints > > flush(stderr()); flush(stdout()) > > ### Name: logLik.breakpoints > ### Title: Log Likelihood and Information Criteria for Breakpoints > ### Aliases: logLik.breakpoints logLik.breakpointsfull AIC.breakpointsfull > ### Keywords: regression > > ### ** Examples > > ## Nile data with one breakpoint: the annual flows drop in 1898 > ## because the first Ashwan dam was built > data("Nile") > plot(Nile) > > bp.nile <- breakpoints(Nile ~ 1) > summary(bp.nile) Optimal (m+1)-segment partition: Call: breakpoints.formula(formula = Nile ~ 1) Breakpoints at observation number: m = 1 28 m = 2 28 83 m = 3 28 68 83 m = 4 28 45 68 83 m = 5 15 30 45 68 83 Corresponding to breakdates: m = 1 1898 m = 2 1898 1953 m = 3 1898 1938 1953 m = 4 1898 1915 1938 1953 m = 5 1885 1900 1915 1938 1953 Fit: m 0 1 2 3 4 5 RSS 2835157 1597457 1552924 1538097 1507888 1659994 BIC 1318 1270 1276 1285 1292 1311 > plot(bp.nile) > > ## BIC of partitions with0 to 5 breakpoints > plot(0:5, AIC(bp.nile, k = log(bp.nile$nobs)), type = "b") > ## AIC > plot(0:5, AIC(bp.nile), type = "b") > > ## BIC, AIC, log likelihood of a single partition > bp.nile1 <- breakpoints(bp.nile, breaks = 1) > AIC(bp.nile1, k = log(bp.nile1$nobs)) [1] 1270.084 > AIC(bp.nile1) [1] 1259.663 > logLik(bp.nile1) 'log Lik.' -625.8315 (df=4) > > > > cleanEx() > nameEx("mefp") > ### * mefp > > flush(stderr()); flush(stdout()) > > ### Name: mefp > ### Title: Monitoring of Empirical Fluctuation Processes > ### Aliases: mefp mefp.formula mefp.efp print.mefp monitor > ### Keywords: regression > > ### ** Examples > > df1 <- data.frame(y=rnorm(300)) > df1[150:300,"y"] <- df1[150:300,"y"]+1 > > ## use the first 50 observations as history period > e1 <- efp(y~1, data=df1[1:50,,drop=FALSE], type="ME", h=1) > me1 <- mefp(e1, alpha=0.05) > > ## the same in one function call > me1 <- mefp(y~1, data=df1[1:50,,drop=FALSE], type="ME", h=1, + alpha=0.05) > > ## monitor the 50 next observations > me2 <- monitor(me1, data=df1[1:100,,drop=FALSE]) > plot(me2) > > # and now monitor on all data > me3 <- monitor(me2, data=df1) Break detected at observation # 183 > plot(me3) > > > ## Load dataset "USIncExp" with income and expenditure in the US > ## and choose a suitable subset for the history period > data("USIncExp") > USIncExp3 <- window(USIncExp, start=c(1969,1), end=c(1971,12)) > ## initialize the monitoring with the formula interface > me.mefp <- mefp(expenditure~income, type="ME", rescale=TRUE, + data=USIncExp3, alpha=0.05) > > ## monitor the new observations for the year 1972 > USIncExp3 <- window(USIncExp, start=c(1969,1), end=c(1972,12)) > me.mefp <- monitor(me.mefp) > > ## monitor the new data for the years 1973-1976 > USIncExp3 <- window(USIncExp, start=c(1969,1), end=c(1976,12)) > me.mefp <- monitor(me.mefp) Break detected at observation # 58 > plot(me.mefp, functional = NULL) > > > > cleanEx() > nameEx("plot.Fstats") > ### * plot.Fstats > > flush(stderr()); flush(stdout()) > > ### Name: plot.Fstats > ### Title: Plot F Statistics > ### Aliases: plot.Fstats lines.Fstats > ### Keywords: hplot > > ### ** Examples > > ## Load dataset "nhtemp" with average yearly temperatures in New Haven > data("nhtemp") > ## plot the data > plot(nhtemp) > > ## test the model null hypothesis that the average temperature remains > ## constant over the years for potential break points between 1941 > ## (corresponds to from = 0.5) and 1962 (corresponds to to = 0.85) > ## compute F statistics > fs <- Fstats(nhtemp ~ 1, from = 0.5, to = 0.85) > ## plot the F statistics > plot(fs, alpha = 0.01) > ## and the corresponding p values > plot(fs, pval = TRUE, alpha = 0.01) > ## perform the aveF test > sctest(fs, type = "aveF") aveF test data: fs ave.F = 10.81, p-value = 2.059e-06 > > > > cleanEx() > nameEx("plot.efp") > ### * plot.efp > > flush(stderr()); flush(stdout()) > > ### Name: plot.efp > ### Title: Plot Empirical Fluctuation Process > ### Aliases: plot.efp lines.efp > ### Keywords: hplot > > ### ** Examples > > ## Load dataset "nhtemp" with average yearly temperatures in New Haven > data("nhtemp") > ## plot the data > plot(nhtemp) > > ## test the model null hypothesis that the average temperature remains > ## constant over the years > ## compute Rec-CUSUM fluctuation process > temp.cus <- efp(nhtemp ~ 1) > ## plot the process > plot(temp.cus, alpha = 0.01) > ## and calculate the test statistic > sctest(temp.cus) Recursive CUSUM test data: temp.cus S = 1.2724, p-value = 0.002902 > > ## compute (recursive estimates) fluctuation process > ## with an additional linear trend regressor > lin.trend <- 1:60 > temp.me <- efp(nhtemp ~ lin.trend, type = "fluctuation") > ## plot the bivariate process > plot(temp.me, functional = NULL) > ## and perform the corresponding test > sctest(temp.me) RE test (recursive estimates test) data: temp.me RE = 1.4938, p-value = 0.04558 > > > > cleanEx() > nameEx("plot.mefp") > ### * plot.mefp > > flush(stderr()); flush(stdout()) > > ### Name: plot.mefp > ### Title: Plot Methods for mefp Objects > ### Aliases: plot.mefp lines.mefp > ### Keywords: hplot > > ### ** Examples > > df1 <- data.frame(y=rnorm(300)) > df1[150:300,"y"] <- df1[150:300,"y"]+1 > me1 <- mefp(y~1, data=df1[1:50,,drop=FALSE], type="ME", h=1, + alpha=0.05) > me2 <- monitor(me1, data=df1) Break detected at observation # 183 > > plot(me2) > > > > cleanEx() > nameEx("recresid") > ### * recresid > > flush(stderr()); flush(stdout()) > > ### Name: recresid > ### Title: Recursive Residuals > ### Aliases: recresid recresid.default recresid.formula recresid.lm > ### Keywords: regression > > ### ** Examples > > x <- rnorm(100) + rep(c(0, 2), each = 50) > rr <- recresid(x ~ 1) > plot(cumsum(rr), type = "l") > > plot(efp(x ~ 1, type = "Rec-CUSUM")) > > > > cleanEx() > nameEx("root.matrix") > ### * root.matrix > > flush(stderr()); flush(stdout()) > > ### Name: root.matrix > ### Title: Root of a Matrix > ### Aliases: root.matrix > ### Keywords: algebra > > ### ** Examples > > X <- matrix(c(1,2,2,8), ncol=2) > test <- root.matrix(X) > ## control results > X [,1] [,2] [1,] 1 2 [2,] 2 8 > test %*% test [,1] [,2] [1,] 1 2 [2,] 2 8 > > > > cleanEx() > nameEx("scPublications") > ### * scPublications > > flush(stderr()); flush(stdout()) > > ### Name: scPublications > ### Title: Structural Change Publications > ### Aliases: scPublications > ### Keywords: datasets > > ### ** Examples > > ## construct time series: > ## number of sc publications in econometrics/statistics > data("scPublications") > > ## select years from 1987 and > ## `most important' journals > pub <- scPublications > pub <- subset(pub, year > 1986) > tab1 <- table(pub$journal) > nam1 <- names(tab1)[as.vector(tab1) > 9] ## at least 10 papers > tab2 <- sapply(levels(pub$journal), function(x) min(subset(pub, journal == x)$year)) > nam2 <- names(tab2)[as.vector(tab2) < 1991] ## started at least in 1990 > nam <- nam1[nam1 %in% nam2] > pub <- subset(pub, as.character(journal) %in% nam) > pub$journal <- factor(pub$journal) > pub_data <- pub > > ## generate time series > pub <- with(pub, tapply(type, year, table)) > pub <- zoo(t(sapply(pub, cbind)), 1987:2006) > colnames(pub) <- levels(pub_data$type) > > ## visualize > plot(pub, ylim = c(0, 35)) > > > > cleanEx() > nameEx("sctest.Fstats") > ### * sctest.Fstats > > flush(stderr()); flush(stdout()) > > ### Name: sctest.Fstats > ### Title: supF-, aveF- and expF-Test > ### Aliases: sctest.Fstats > ### Keywords: htest > > ### ** Examples > > ## Load dataset "nhtemp" with average yearly temperatures in New Haven > data(nhtemp) > ## plot the data > plot(nhtemp) > > ## test the model null hypothesis that the average temperature remains > ## constant over the years for potential break points between 1941 > ## (corresponds to from = 0.5) and 1962 (corresponds to to = 0.85) > ## compute F statistics > fs <- Fstats(nhtemp ~ 1, from = 0.5, to = 0.85) > ## plot the F statistics > plot(fs, alpha = 0.01) > ## and the corresponding p values > plot(fs, pval = TRUE, alpha = 0.01) > ## perform the aveF test > sctest(fs, type = "aveF") aveF test data: fs ave.F = 10.81, p-value = 2.059e-06 > > > > cleanEx() > nameEx("sctest.default") > ### * sctest.default > > flush(stderr()); flush(stdout()) > > ### Name: sctest.default > ### Title: Structural Change Tests in Parametric Models > ### Aliases: sctest.default > ### Keywords: htest > > ### ** Examples > > ## Zeileis and Hornik (2007), Section 5.3, Figure 6 > data("Grossarl") > m <- glm(cbind(illegitimate, legitimate) ~ 1, family = binomial, data = Grossarl, + subset = time(fraction) <= 1800) > sctest(m, order.by = 1700:1800, functional = "CvM") M-fluctuation test data: m f(efp) = 3.5363, p-value = 0.005 > > > > cleanEx() > nameEx("sctest.efp") > ### * sctest.efp > > flush(stderr()); flush(stdout()) > > ### Name: sctest.efp > ### Title: Generalized Fluctuation Tests > ### Aliases: sctest.efp > ### Keywords: htest > > ### ** Examples > > ## Load dataset "nhtemp" with average yearly temperatures in New Haven > data("nhtemp") > ## plot the data > plot(nhtemp) > > ## test the model null hypothesis that the average temperature remains > ## constant over the years compute OLS-CUSUM fluctuation process > temp.cus <- efp(nhtemp ~ 1, type = "OLS-CUSUM") > ## plot the process with alternative boundaries > plot(temp.cus, alpha = 0.01, alt.boundary = TRUE) > ## and calculate the test statistic > sctest(temp.cus) OLS-based CUSUM test data: temp.cus S0 = 2.0728, p-value = 0.0003709 > > ## compute moving estimates fluctuation process > temp.me <- efp(nhtemp ~ 1, type = "ME", h = 0.2) > ## plot the process with functional = "max" > plot(temp.me) > ## and perform the corresponding test > sctest(temp.me) ME test (moving estimates test) data: temp.me ME = 1.5627, p-value = 0.01 > > > > cleanEx() > nameEx("sctest.formula") > ### * sctest.formula > > flush(stderr()); flush(stdout()) > > ### Name: sctest.formula > ### Title: Structural Change Tests in Linear Regression Models > ### Aliases: sctest.formula > ### Keywords: htest > > ### ** Examples > > ## Example 7.4 from Greene (1993), "Econometric Analysis" > ## Chow test on Longley data > data("longley") > sctest(Employed ~ Year + GNP.deflator + GNP + Armed.Forces, data = longley, + type = "Chow", point = 7) Chow test data: Employed ~ Year + GNP.deflator + GNP + Armed.Forces F = 3.9268, p-value = 0.06307 > > ## which is equivalent to segmenting the regression via > fac <- factor(c(rep(1, 7), rep(2, 9))) > fm0 <- lm(Employed ~ Year + GNP.deflator + GNP + Armed.Forces, data = longley) > fm1 <- lm(Employed ~ fac/(Year + GNP.deflator + GNP + Armed.Forces), data = longley) > anova(fm0, fm1) Analysis of Variance Table Model 1: Employed ~ Year + GNP.deflator + GNP + Armed.Forces Model 2: Employed ~ fac/(Year + GNP.deflator + GNP + Armed.Forces) Res.Df RSS Df Sum of Sq F Pr(>F) 1 11 4.8987 2 6 1.1466 5 3.7521 3.9268 0.06307 . --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 > > ## estimates from Table 7.5 in Greene (1993) > summary(fm0) Call: lm(formula = Employed ~ Year + GNP.deflator + GNP + Armed.Forces, data = longley) Residuals: Min 1Q Median 3Q Max -0.9058 -0.3427 -0.1076 0.2168 1.4377 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 1.169e+03 8.359e+02 1.399 0.18949 Year -5.765e-01 4.335e-01 -1.330 0.21049 GNP.deflator -1.977e-02 1.389e-01 -0.142 0.88940 GNP 6.439e-02 1.995e-02 3.227 0.00805 ** Armed.Forces -1.015e-04 3.086e-03 -0.033 0.97436 --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 0.6673 on 11 degrees of freedom Multiple R-squared: 0.9735, Adjusted R-squared: 0.9639 F-statistic: 101.1 on 4 and 11 DF, p-value: 1.346e-08 > summary(fm1) Call: lm(formula = Employed ~ fac/(Year + GNP.deflator + GNP + Armed.Forces), data = longley) Residuals: Min 1Q Median 3Q Max -0.47717 -0.18950 0.02089 0.14836 0.56493 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 1.678e+03 9.390e+02 1.787 0.12413 fac2 2.098e+03 1.786e+03 1.174 0.28473 fac1:Year -8.352e-01 4.847e-01 -1.723 0.13563 fac2:Year -1.914e+00 7.913e-01 -2.419 0.05194 . fac1:GNP.deflator -1.633e-01 1.762e-01 -0.927 0.38974 fac2:GNP.deflator -4.247e-02 2.238e-01 -0.190 0.85576 fac1:GNP 9.481e-02 3.815e-02 2.485 0.04747 * fac2:GNP 1.123e-01 2.269e-02 4.951 0.00258 ** fac1:Armed.Forces -2.467e-03 6.965e-03 -0.354 0.73532 fac2:Armed.Forces -2.579e-02 1.259e-02 -2.049 0.08635 . --- Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 Residual standard error: 0.4372 on 6 degrees of freedom Multiple R-squared: 0.9938, Adjusted R-squared: 0.9845 F-statistic: 106.9 on 9 and 6 DF, p-value: 6.28e-06 > > > > cleanEx() > nameEx("solveCrossprod") > ### * solveCrossprod > > flush(stderr()); flush(stdout()) > > ### Name: solveCrossprod > ### Title: Inversion of X'X > ### Aliases: solveCrossprod > ### Keywords: algebra > > ### ** Examples > > X <- cbind(1, rnorm(100)) > solveCrossprod(X) [,1] [,2] [1,] 0.010148448 -0.001363317 [2,] -0.001363317 0.012520432 > solve(crossprod(X)) [,1] [,2] [1,] 0.010148448 -0.001363317 [2,] -0.001363317 0.012520432 > > > > cleanEx() > nameEx("supLM") > ### * supLM > > flush(stderr()); flush(stdout()) > > ### Name: supLM > ### Title: Generators for efpFunctionals along Continuous Variables > ### Aliases: supLM maxMOSUM > ### Keywords: regression > > ### ** Examples > > ## seatbelt data > data("UKDriverDeaths") > seatbelt <- log10(UKDriverDeaths) > seatbelt <- cbind(seatbelt, lag(seatbelt, k = -1), lag(seatbelt, k = -12)) > colnames(seatbelt) <- c("y", "ylag1", "ylag12") > seatbelt <- window(seatbelt, start = c(1970, 1), end = c(1984,12)) > > ## empirical fluctuation process > scus.seat <- gefp(y ~ ylag1 + ylag12, data = seatbelt) > > ## supLM test > plot(scus.seat, functional = supLM(0.1)) > ## MOSUM test > plot(scus.seat, functional = maxMOSUM(0.25)) > ## double maximum test > plot(scus.seat) > ## range test > plot(scus.seat, functional = rangeBB) > ## Cramer-von Mises statistic (Nyblom-Hansen test) > plot(scus.seat, functional = meanL2BB) > > > > ### *