1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305

\name{breakpoints}
\alias{breakpoints}
\alias{breakpoints.formula}
\alias{breakpoints.breakpointsfull}
\alias{breakpoints.Fstats}
\alias{summary.breakpoints}
\alias{summary.breakpointsfull}
\alias{plot.breakpointsfull}
\alias{plot.summary.breakpointsfull}
\alias{print.breakpoints}
\alias{print.summary.breakpointsfull}
\alias{lines.breakpoints}
\alias{coef.breakpointsfull}
\alias{vcov.breakpointsfull}
\alias{fitted.breakpointsfull}
\alias{residuals.breakpointsfull}
\alias{df.residual.breakpointsfull}
\encoding{latin1}
\title{Dating Breaks}
\description{
Computation of breakpoints in regression relationships. Given a number
of breaks the function computes the optimal breakpoints.
}
\usage{
\method{breakpoints}{formula}(formula, h = 0.15, breaks = NULL,
data = list(), hpc = c("none", "foreach"), \dots)
\method{breakpoints}{breakpointsfull}(obj, breaks = NULL, \dots)
\method{summary}{breakpointsfull}(object, breaks = NULL, sort = TRUE,
format.times = NULL, \dots)
\method{lines}{breakpoints}(x, breaks = NULL, lty = 2, \dots)
\method{coef}{breakpointsfull}(object, breaks = NULL, names = NULL, \dots)
\method{fitted}{breakpointsfull}(object, breaks = NULL, \dots)
\method{residuals}{breakpointsfull}(object, breaks = NULL, \dots)
\method{vcov}{breakpointsfull}(object, breaks = NULL, names = NULL,
het.reg = TRUE, het.err = TRUE, vcov. = NULL, sandwich = TRUE, \dots)
}
\arguments{
\item{formula}{a symbolic description for the model in which breakpoints
will be estimated.}
\item{h}{minimal segment size either given as fraction relative to the
sample size or as an integer giving the minimal number of observations
in each segment.}
\item{breaks}{integer specifying the maximal number of breaks to be calculated.
By default the maximal number allowed by \code{h} is used.}
\item{data}{an optional data frame containing the variables in the model. By
default the variables are taken from the environment which \code{breakpoints} is
called from.}
\item{hpc}{a character specifying the high performance computing support.
Default is \code{"none"}, can be set to \code{"foreach"}.}
\item{\dots}{currently not used.}
\item{obj, object}{an object of class \code{"breakpointsfull"}.}
\item{sort}{logical. If set to \code{TRUE} \code{summary} tries to match
the breakpoints from partitions with different numbers of breaks.}
\item{format.times}{logical. If set to \code{TRUE} a vector of
strings with the formatted breakdates is printed. See \code{\link{breakdates}}
for more information.}
\item{x}{an object of class \code{"breakpoints"}.}
\item{lty}{line type.}
\item{names}{a character vector giving the names of the segments. If of length
1 it is taken to be a generic prefix, e.g. \code{"segment"}.}
\item{het.reg}{logical. Should heterogeneous regressors be assumed? If set
to \code{FALSE} the distribution of the regressors is assumed to be
homogeneous over the segments.}
\item{het.err}{logical. Should heterogeneous errors be assumed? If set
to \code{FALSE} the distribution of the errors is assumed to be
homogeneous over the segments.}
\item{vcov.}{a function to extract the covariance matrix
for the coefficients of a fitted model of class \code{"lm"}.}
\item{sandwich}{logical. Is the function \code{vcov.} the sandwich
estimator or only the middle part?}
}
\details{
All procedures in this package are concerned with testing or assessing
deviations from stability in the classical linear regression model
\deqn{y_i = x_i^\top \beta + u_i}{y_i = x_i' b + u_i}
In many applications it is reasonable to assume
that there are \eqn{m} breakpoints, where the coefficients shift from
one stable regression relationship to a different one. Thus,
there are \eqn{m+1} segments in which the regression coefficients are
constant, and the model can be rewritten as
\deqn{y_i = x_i^\top \beta_j + u_i
\qquad (i = i_{j1} + 1, \dots, i_j, \quad j = 1, \dots, m+1)}{y_i =
x_i' b_j + u_i (i = i_{j1} + 1, \dots, i_j, j = 1, \dots, m+1)}
where \eqn{j} denotes the segment index. In practice the breakpoints \eqn{i_j}
are rarely given exogenously, but have to be estimated.
\code{breakpoints} estimates these breakpoints by minimizing the residual sum of
squares (RSS) of the equation above.
The foundation for estimating breaks in time series regression models
was given by Bai (1994) and was extended to multiple breaks by Bai (1997ab)
and Bai & Perron (1998). \code{breakpoints} implements the algorithm
described in Bai & Perron (2003) for simultaneous estimation of
multiple breakpoints. The distribution function used for the confidence
intervals for the breakpoints is given in Bai (1997b). The ideas behind
this implementation are described in Zeileis et al. (2003).
The algorithm for computing the optimal breakpoints given the number
of breaks is based on a dynamic programming approach. The underlying
idea is that of the Bellman principle. The main computational effort
is to compute a triangular RSS matrix, which gives the residual
sum of squares for a segment starting at observation \eqn{i} and
ending at \eqn{i'} with \eqn{i} < \eqn{i'}.
Given a \code{formula} as the first argument, \code{breakpoints} computes
an object of class \code{"breakpointsfull"} which inherits from \code{"breakpoints"}.
This contains in particular the triangular RSS
matrix and functions to extract an optimal segmentation. A \code{summary}
of this object will give the breakpoints (and associated) breakdates
for all segmentations up to the maximal number of breaks together
with the associated RSS and BIC. These will be plotted if \code{plot}
is applied and thus visualize the minimum BIC estimator of the number
of breakpoints. From an object of class \code{"breakpointsfull"} an
arbitrary number of \code{breaks} (admissible by the minimum segment
size \code{h}) can be extracted by another application of
\code{breakpoints}, returning an object of class \code{"breakpoints"}.
This contains only the breakpoints for the specified number of breaks
and some model properties (number of observations, regressors, time
series properties and the associated RSS) but not the triangular RSS
matrix and related extractor functions. The set of breakpoints which
is associated by default with a \code{"breakpointsfull"} object is
the minimum BIC partition.
Breakpoints are the number of observations that are the last in one
segment, it is also possible to compute the corresponding \code{breakdates}
which are the breakpoints on the underlying time scale. The breakdates
can be formatted which enhances readability in particular for quarterly
or monthly time series. For example the breakdate \code{2002.75} of a monthly
time series will be formatted to \code{"2002(10)"}. See \code{\link{breakdates}}
for more details.
From a \code{"breakpointsfull"} object confidence intervals for the breakpoints
can be computed using the method of \code{\link{confint}}.
The breakdates corresponding to the breakpoints can again be computed
by \code{\link{breakdates}}. The breakpoints and their confidence
intervals can be visualized by \code{lines}. Convenience functions are
provided for extracting the coefficients and covariance matrix, fitted
values and residuals of segmented models.
The log likelihood as well as some information criteria can be computed
using the methods for the \code{\link{logLik}} and \code{\link{AIC}}. As
for linear models the log likelihood is computed on a normal model and
the degrees of freedom are the number of regression coefficients multiplied
by the number of segments plus the number of estimated breakpoints plus
1 for the error variance. More details can be found on the help page of
the method \code{\link{logLik.breakpoints}}.
As the maximum of a sequence of F statistics is equivalent to the minimum
OLS estimator of the breakpoint in a 2segment partition it can be
extracted by \code{breakpoints} from an object of class \code{"Fstats"}
as computed by \code{\link{Fstats}}. However, this cannot be used to extract
a larger number of breakpoints.
For illustration see the commented examples below and Zeileis et al. (2003).
Optional support for high performance computing is available, currently using
\code{\link[foreach]{foreach}} for the dynamic programming algorithm.
If \code{hpc = "foreach"} is to be used, a parallel backend should be registered
before. See \code{\link[foreach]{foreach}} for more information.
}
\section{value}{
An object of class \code{"breakpoints"} is a list with the following
elements:
\describe{
\item{breakpoints}{the breakpoints of the optimal partition with the
number of breaks specified (set to \code{NA} if the optimal 1segment
solution is reported),}
\item{RSS}{the associated RSS,}
\item{nobs}{the number of observations,}
\item{nreg}{the number of regressors,}
\item{call}{the function call,}
\item{datatsp}{the time series properties \code{tsp} of the data,
if any, \code{c(1/nobs, 1, nobs)} otherwise.}
}
If applied to a \code{formula} as first argument, \code{breakpoints} returns an object of class
\code{"breakpointsfull"} (which inherits from \code{"breakpoints"}), that
contains some additional (or slightly different) elements such as:
\describe{
\item{breakpoints}{the breakpoints of the minimum BIC partition,}
\item{RSS}{a function which takes two arguments \code{i,j} and computes
the residual sum of squares for a segment starting at observation \code{i} and
ending at \code{j} by looking up the corresponding element in the triangular
RSS matrix \code{RSS.triang},}
\item{RSS.triang}{a list encoding the triangular RSS matrix.}
}
}
\references{
Bai J. (1994), Least Squares Estimation of a Shift in Linear Processes,
\emph{Journal of Time Series Analysis}, \bold{15}, 453472.
Bai J. (1997a), Estimating Multiple Breaks One at a Time,
\emph{Econometric Theory}, \bold{13}, 315352.
Bai J. (1997b), Estimation of a Change Point in Multiple Regression Models,
\emph{Review of Economics and Statistics}, \bold{79}, 551563.
Bai J., Perron P. (1998), Estimating and Testing Linear Models With Multiple Structural
Changes, \emph{Econometrica}, \bold{66}, 4778.
Bai J., Perron P. (2003), Computation and Analysis of Multiple Structural Change
Models, \emph{Journal of Applied Econometrics}, \bold{18}, 122.
Zeileis A., Kleiber C., Krmer W., Hornik K. (2003), Testing and Dating of
Structural Changes in Practice, \emph{Computational Statistics and Data Analysis},
\bold{44}, 109123. doi:10.1016/S01679473(03)000306.
Zeileis A., Shah A., Patnaik I. (2010), Testing, Monitoring, and Dating Structural
Changes in Exchange Rate Regimes, \emph{Computational Statistics and Data Analysis},
\bold{54}(6), 16961706. doi:10.1016/j.csda.2009.12.005.
}
\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)
lines(breakpoints(fs.nile))
## or
bp.nile < breakpoints(Nile ~ 1)
summary(bp.nile)
## the BIC also chooses one breakpoint
plot(bp.nile)
breakpoints(bp.nile)
## 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
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)
lines(bp.seat, breaks = 2)
## minimum BIC partition
plot(bp.seat)
breakpoints(bp.seat)
## 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
lines(ci.seat2)
}
\concept{breakpoint estimation}
\concept{changepoint estimation}
\concept{segmented regression}
\keyword{regression}
