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Package ‘CPAT’
October 15, 2018
Title Change Point Analysis Tests
Version 0.1.0
Description Implements several statistical tests for structural change in R.
Depends R (>= 3.2)
Suggests cointReg (>= 0.2), foreach (>= 1.4), doParallel (>= 1.0),
ggplot2 (>= 2.2), dplyr (>= 0.7), tikzDevice (>= 0.12),
testthat (>= 2.0)
Imports stats (>= 3.2), utils (>= 3.2), grDevices (>= 3.2), Rdpack (>=
0.9), methods (>= 3.2), Rcpp (>= 0.12), purrr (>= 0.2)
RdMacros Rdpack
SystemRequirements GNU make
License MIT + file LICENSE
Encoding UTF-8
LazyData true
LinkingTo Rcpp, RcppArmadillo
RoxygenNote 6.1.0
NeedsCompilation yes
Author Curtis Miller [aut, cre]
Maintainer Curtis Miller 

R topics documented:
.onAttach . . . . . . .
Andrews.test . . . . .
andrews_test . . . . . .
andrews_test_reg . . .
banks . . . . . . . . .
CPAT_startup_message
cpt_consistent_var . .
CUSUM.test . . . . . .
DE.test . . . . . . . .
dZn . . . . . . . . . .
ff . . . . . . . . . . . .
getLongRunWeights .
get_lrv_vec . . . . . .

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. 2
. 3
. 3
. 4
. 5
. 5
. 6
. 6
. 7
. 8
. 9
. 10
. 10

2

.onAttach
HR.test . . . .
HS.test . . . . .
pdarling_erdos
phidalgo_seo .
pkolmogorov .
pZn . . . . . .
qdarling_erdos
qhidalgo_seo .
qkolmogorov .
qZn . . . . . .
rchangepoint .
sim_de_stat . .
sim_hs_stat . .
sim_Vn . . . .
sim_Vn_stat . .
sim_Zn . . . .
sim_Zn_stat . .
stat_de . . . . .
stat_hs . . . . .
stat_Vn . . . .
stat_Zn . . . .
%s% . . . . . .
%s0% . . . . .

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Index

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11
12
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15
15
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30
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31

.onAttach

Package Attach Hook Function

Description
Hook triggered when package attached
Usage
.onAttach(lib, pkg)
Arguments
lib

a character string giving the library directory where the package defining the
namespace was found

pkg

a character string giving the name of the package

Examples
CPAT:::.onAttach(.libPaths()[1], "CPAT")

Andrews.test

Andrews.test

3

Andrews’ Test for End-of-Sample Structural Change

Description
Performs Andrews’ test for end-of-sample structural change, as described in (Andrews 2003). This
function works for both univariate and multivariate data depending on the nature of x and whether
formula is specified. This function is thus an interface to andrews_test and andrews_test_reg;
see the documentation of those functions for more details.
Usage
Andrews.test(x, M, formula = NULL)
Arguments
x

Data to test for change in mean (either a vector or data.frame)

M

Numeric index of the location of the first potential change point

formula

The regression formula, which will be passed to lm

Value
A htest-class object containing the results of the test
References
Andrews DWK (2003). “End-of-Sample Instability Tests.” Econometrica, 71(6), 1661–1694. ISSN
00129682, 14680262, https://www.jstor.org/stable/1555535.
Examples
Andrews.test(rnorm(1000), M = 900)
x <- rnorm(1000)
y <- 1 + 2 * x + rnorm(1000)
df <- data.frame(x, y)
Andrews.test(df, y ~ x, M = 900)

andrews_test

Univariate Andrews Test for End-of-Sample Structural Change

Description
This implements Andrews’ test for end-of-sample change, as described by Andrews (2003). This
test was derived for detecting a change in univariate data. See (Andrews 2003) for a description of
the test.
Usage
andrews_test(x, M, pval = TRUE, stat = TRUE)

4

andrews_test_reg

Arguments
x

Vector of the data to test

M

Numeric index of the location of the first potential change point

pval

If TRUE, return a p-value

stat

If TRUE, return a test statistic

Value
If both pval and stat are TRUE, a list containing both; otherwise, a number for one or the other,
depending on which is TRUE
References
Andrews DWK (2003). “End-of-Sample Instability Tests.” Econometrica, 71(6), 1661–1694. ISSN
00129682, 14680262, https://www.jstor.org/stable/1555535.
Examples
CPAT:::andrews_test(rnorm(1000), M = 900)

andrews_test_reg

Multivariate Andrews’ Test for End-of-Sample Structural Change

Description
This implements Andrews’ test for end-of-sample change, as described by Andrews (2003). This
test was derived for detecting a change in multivarate data, aso originally described. See (Andrews
2003) for a description of the test.
Usage
andrews_test_reg(formula, data, M, pval = TRUE, stat = TRUE)
Arguments
formula

The regression formula, which will be passed to lm

data

data.frame containing the data

M

Numeric index of the location of the first potential change point

pval

If TRUE, return a p-value

stat

If TRUE, return a test statistic

Value
If both pval and stat are TRUE, a list containing both; otherwise, a number for one or the other,
depending on which is TRUE
References
Andrews DWK (2003). “End-of-Sample Instability Tests.” Econometrica, 71(6), 1661–1694. ISSN
00129682, 14680262, https://www.jstor.org/stable/1555535.

banks

5

Examples
x <- rnorm(1000)
y <- 1 + 2 * x + rnorm(1000)
df <- data.frame(x, y)
CPAT:::andrews_test_reg(y ~ x, data = df, M = 900)

banks

Bank Portfolio Returns

Description
Data set representing the returns of an industry portfolio representing the banking industry based on
company four-digit SIC codes, obtained from the data library maintained by Kenneth French. Data
ranges from July 1, 1926 to October 31, 2017.
Usage
banks
Format
A data frame with 24099 rows and 1 variable:
Banks The return of a portfolio representing the banking industry
Row names are dates in YYYY-MM-DD format.
Source
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html

CPAT_startup_message

Create Package Startup Message

Description
Makes package startup message.
Usage
CPAT_startup_message()
Examples
CPAT:::CPAT_startup_message()

6

CUSUM.test

cpt_consistent_var

Variance Estimation Consistent Under Change

Description
Estimate the variance (using the sum of squared errors) with an estimator that is consistent when
the mean changes at a known point.
Usage
cpt_consistent_var(x, k)
Arguments
x

A numeric vector for the data set

k

The potential change point at which the data set is split

Details
This is the estimator
2
σ̂T,t

=T

−1

t
X

Xs − X̄t

s=1


2

+

T

X

Xs − X̃T −t

2

!

s=t+1

Pt
PT
where X̄t = t−1 s=1 Xs and X̃T −t = (T − t)−1 s=t+1 Xs . In this implementation, T is
computed automatically as length(x) and k corresponds to t, a potential change point.
Value
The estimated change-consistent variance
Examples
CPAT:::cpt_consistent_var(c(rnorm(500, mean = 0), rnorm(500, mean = 1)), k = 500)

CUSUM.test

CUSUM Test

Description
Performs the (univariate) CUSUM test for change in mean, as described in (Rice et al. ). This
is effectively an interface to stat_Vn; see its documentation for more details. p-values are computed using pkolmogorov, which represents the limiting distribution of the statistic under the null
hypothesis.
Usage
CUSUM.test(x, use_kernel_var = FALSE, stat_plot = FALSE,
kernel = "ba", bandwidth = "and")

DE.test

7

Arguments
x

Data to test for change in mean

use_kernel_var Set to TRUE to use kernel methods for long-run variance estimation (typically
used when the data is believed to be 
correlated); if FALSE, then the long-run vari
2 

2 PT
Pt
2
−1
ance is estimated using σ̂T,t = T
+ s=t+1 Xs − X̃T −t
,
s=1 Xs − X̄t
P
P
t
T
where X̄t = t−1 s=1 Xs and X̃T −t = (T − t)−1 s=t+1 Xs
stat_plot

Whether to create a plot of the values of the statistic at all potential change points

kernel

If character, the identifier of the kernel function as used in cointReg (see getLongRunVar);
if function, the kernel function to be used for long-run variance estimation (default is the Bartlett kernel in cointReg)

bandwidth

If character, the identifier for how to compute the bandwidth as defined in cointReg (see getBandwidth); if function, a function to use for computing the bandwidth; if numeric, the bandwidth value to use (the default is to use Andrews’
method, as used in cointReg)

Value
A htest-class object containing the results of the test
References
Rice G, Miller C, Horváth L (????). “A new class of change point test of Rényi type.” in-press.
Examples
CUSUM.test(rnorm(1000))
CUSUM.test(rnorm(1000), use_kernel_var = TRUE, kernel = "bo",
bandwidth = "nw")

DE.test

Darling-Erdös Test

Description
Performs the (univariate) Darling-Erdös test for change in mean, as described in (Rice et al. ). This
is effectively an interface to stat_de; see its documentation for more details. p-values are computed
using pdarling_erdos, which represents the limiting distribution of the test statistic under the null
hypothesis when a and b are chosen appropriately. (Change those parameters at your own risk!)
Usage
DE.test(x, a = log, b = log, use_kernel_var = FALSE,
stat_plot = FALSE, kernel = "ba", bandwidth = "and")

8

dZn

Arguments
x

Data to test for change in mean

a

The function that will be composed with l(x) = (2 log x)1/2

b

The function that will be composed with u(x) = 2 log x + 21 log log x − 21 log π

use_kernel_var Set to TRUE to use kernel methods for long-run variance estimation (typically
used when the data is believed to be 
correlated); if FALSE, then the long-run vari
2 

2 PT
Pt
2
−1
ance is estimated using σ̂T,t = T
+ s=t+1 Xs − X̃T −t
,
s=1 Xs − X̄t
P
P
t
T
where X̄t = t−1 s=1 Xs and X̃T −t = (T − t)−1 s=t+1 Xs
stat_plot

Whether to create a plot of the values of the statistic at all potential change points

kernel

If character, the identifier of the kernel function as used in cointReg (see getLongRunVar);
if function, the kernel function to be used for long-run variance estimation (default is the Bartlett kernel in cointReg)

bandwidth

If character, the identifier for how to compute the bandwidth as defined in cointReg (see getBandwidth); if function, a function to use for computing the bandwidth; if numeric, the bandwidth value to use (the default is to use Andrews’
method, as used in cointReg)

Value
A htest-class object containing the results of the test
References
Rice G, Miller C, Horváth L (????). “A new class of change point test of Rényi type.” in-press.
Examples
DE.test(rnorm(1000))
DE.test(rnorm(1000), use_kernel_var = TRUE, kernel = "bo", bandwidth = "nw")

dZn

Rényi-Type Statistic Limiting Distribution Density Function

Description
Function for computing the value of the density function of the limiting distribution of the Rényitype statistic.
Usage
dZn(x, summands = NULL)
Arguments
x

Point at which to evaluate the density function (note that this parameter is not
vectorized)

summands

Number of summands to use in summation (the default should be machine accurate)

ff

9

Value
Value of the density function at x
Examples
CPAT:::dZn(1)

ff

Fama-French Five Factors

Description
Data set containing the five factors described by Fama and French (2015), from the data library
maintained by Kenneth French. Data ranges from July 1, 1963 to October 31, 2017.
Usage
ff
Format
A data frame with 13679 rows and 6 variables:
Mkt.RF Market excess returns
RF The risk-free rate of return
SMB The return on a diversified portfolio of small stocks minus return on a diversified portfolio of
big stocks
HML The return of a portfolio of stocks with a high book-to-market (B/M) ratio minus the return
of a portfolio of stocks with a low B/M ratio
RMW The return of a portfolio of stocks with robust profitability minus a portfolio of stocks with
weak profitability
CMA The return of a portfolio of stocks with conservative investment minus the return of a portfolio of stocks with aggressive investment
Row names are dates in YYYYMMDD format.
Source
http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html

10

get_lrv_vec

getLongRunWeights

Weights for Long-Run Variance

Description
Compute some weights for long-run variance. This code comes directly from the source code of
cointReg; see getLongRunWeights.
Usage
getLongRunWeights(n, bandwidth, kernel = "ba")
Arguments
n
bandwidth
kernel

Length of weights’ vector
A number for the bandwidth
The kernel function; see getLongRunVar for possible values

Value
List with components w containing the vector of weights and upper, the index of the largest nonzero entry in w
Examples
CPAT:::getLongRunWeights(10, 1)

get_lrv_vec

Long-Run Variance Estimation With Possible Change Points

Description
Computes the estimates of the long-run variance in a change point context, as described in (Rice et
al. ). By default it uses kernel and bandwidth selection as used in the package cointReg, though
changing the parameters kernel and bandwidth can change this behavior. If cointReg is not installed, the Bartlett internal (defined internally) will be used and the bandwidth will be the square
root of the sample size.
Usage
get_lrv_vec(dat, kernel = "ba", bandwidth = "and")
Arguments
dat
kernel

bandwidth

The data vector
If character, the identifier of the kernel function as used in cointReg (see getLongRunVar);
if function, the kernel function to be used for long-run variance estimation (default is the Bartlett kernel in cointReg)
If character, the identifier for how to compute the bandwidth as defined in cointReg (see getBandwidth); if function, a function to use for computing the bandwidth; if numeric, the bandwidth value to use (the default is to use Andrews’
method, as used in cointReg)

HR.test

11

Value
A vector of estimates of the long-run variance
References
Rice G, Miller C, Horváth L (????). “A new class of change point test of Rényi type.” in-press.
Examples
x <- rnorm(1000)
CPAT:::get_lrv_vec(x)
CPAT:::get_lrv_vec(x, kernel = "pa", bandwidth = "nw")

HR.test

Rényi-Type Test

Description
Performs the (univariate) Rényi-type test for change in mean, as described in (Rice et al. ). This is
effectively an interface to stat_Zn; see its documentation for more details. p-values are computed
using pZn, which represents the limiting distribution of the test statistic under the null hypothesis,
which represents the limiting distribution of the test statistic under the null hypothesis when kn
represents a sequence tT satisfying tT → ∞ and tT /T → 0 as T → ∞. (log and sqrt should be
good choices.)
Usage
HR.test(x, kn = log, use_kernel_var = FALSE, stat_plot = FALSE,
kernel = "ba", bandwidth = "and")
Arguments
x

Data to test for change in mean

kn

A function corresponding to the trimming parameter tT ; by default, the square
root function

use_kernel_var Set to TRUE to use kernel methods for long-run variance estimation (typically
used when the data is believed to be 
correlated); if FALSE, then the long-run vari
2 

2 PT
Pt
2
−1
+ s=t+1 Xs − X̃T −t
,
ance is estimated using σ̂T,t = T
s=1 Xs − X̄t
P
P
t
T
where X̄t = t−1 s=1 Xs and X̃T −t = (T − t)−1 s=t+1 Xs ; if custom_var
is not NULL, this argument is ignored
stat_plot

Whether to create a plot of the values of the statistic at all potential change points

kernel

If character, the identifier of the kernel function as used in cointReg (see getLongRunVar);
if function, the kernel function to be used for long-run variance estimation (default is the Bartlett kernel in cointReg)

bandwidth

If character, the identifier for how to compute the bandwidth as defined in cointReg (see getBandwidth); if function, a function to use for computing the bandwidth; if numeric, the bandwidth value to use (the default is to use Andrews’
method, as used in cointReg)

12

HS.test

Value
A htest-class object containing the results of the test
References
Rice G, Miller C, Horváth L (????). “A new class of change point test of Rényi type.” in-press.
Examples
HR.test(rnorm(1000))
HR.test(rnorm(1000), use_kernel_var = TRUE, kernel = "bo", bandwidth = "nw")

HS.test

Hidalgo-Seo Test

Description
Performs the (univariate) Hidalgo-Seo test for change in mean, as described in (Rice et al. ). This is
effectively an interface to stat_hs; see its documentation for more details. p-values are computed
using phidalgo_seo, which represents the limiting distribution of the test statistic when the null
hypothesis is true.
Usage
HS.test(x, corr = TRUE, stat_plot = FALSE)
Arguments
x

Data to test for change in mean

corr

If TRUE, the long-run variance will be computed under the assumption of correlated residuals; ignored if custom_var is not NULL or use_kernel_var is TRUE

stat_plot

Whether to create a plot of the values of the statistic at all potential change points

Value
A htest-class object containing the results of the test
References
Rice G, Miller C, Horváth L (????). “A new class of change point test of Rényi type.” in-press.
Examples
HS.test(rnorm(1000))
HS.test(rnorm(1000), corr = FALSE)

pdarling_erdos

13

pdarling_erdos

Darling-Erdös Statistic CDF

Description
CDF for the limiting distribution of the Darling-Erdös statistic.
Usage
pdarling_erdos(q)
Arguments
q

Quantile input to CDF

Value
If Z is the random variable with this distribution, the quantity P (Z ≤ q)
Examples
CPAT:::pdarling_erdos(0.1)

phidalgo_seo

Hidalgo-Seo Statistic CDF

Description
CDF of the limiting distribution of the Hidalgo-Seo statistic
Usage
phidalgo_seo(q)
Arguments
q

Quantile input to CDF

Value
If Z is the random variable following the limiting distribution, the quantity P (Z ≤ q)
Examples
CPAT:::phidalgo_seo(0.1)

14

pZn

pkolmogorov

Kolmogorov CDF

Description
CDF of the Kolmogorov distribution.
Usage
pkolmogorov(q, summands = ceiling(q * sqrt(72) + 3/2))
Arguments
q

Quantile input to CDF

summands

Number of summands for infinite sum (the default should have machine accuracy)

Value
If Z is the random variable following the Kolmogorov distribution, the quantity P (Z ≤ q)
Examples
CPAT:::pkolmogorov(0.1)

pZn

Rènyi-Type Statistic CDF

Description
CDF for the limiting distribution of the Rènyi-type statistic.
Usage
pZn(q, summands = NULL)
Arguments
q

Quantile input to CDF

summands

Number of summands for infinite sum; if NULL, automatically determined

Value
If Z is the random variable following the limiting distribution, the quantity P (Z ≤ q)
Examples
CPAT:::pZn(0.1)

qdarling_erdos

15

qdarling_erdos

Darling-Erdös Statistic Limiting Distribution Quantile Function

Description
Quantile function for the limiting distribution of the Darling-Erdös statistic.
Usage
qdarling_erdos(p)
Arguments
p

The probability associated with the desired quantile

Value
The quantile associated with p
Examples
CPAT:::qdarling_erdos(0.5)

qhidalgo_seo

Hidalgo-Seo Statistic Limiting Distribution Quantile Function

Description
Quantile function for the limiting distribution of the Hidalgo-Seo statistic
Usage
qhidalgo_seo(p)
Arguments
p

The probability associated with the desired quantile

Value
A The quantile associated with p
Examples
CPAT:::qhidalgo_seo(0.5)

16

qZn

qkolmogorov

Kolmogorov Distribution Quantile Function

Description
Quantile function for the Kolmogorov distribution.
Usage
qkolmogorov(p, summands = 500, interval = c(0, 100),
tol = .Machine$double.eps, ...)
Arguments
p

Value of the CDF at the quantile

summands
Number of summands for infinite sum
interval, tol, ...
Arguments to be passed to uniroot
Details
This function uses uniroot for finding this quantity, and many of the the accepted parameters are
arguments for that function; see its documentation for more details.
Value
The quantile associated with p
Examples
CPAT:::qkolmogorov(0.5)

qZn

Rènyi-Type Statistic Quantile Function

Description
Quantile function for the limiting distribution of the Rènyi-type statistic.
Usage
qZn(p, summands = 500, interval = c(0, 100),
tol = .Machine$double.eps, ...)
Arguments
p

Value of the CDF at the quantile

summands
Number of summands for infinite sum
interval, tol, ...
Arguments to be passed to uniroot

rchangepoint

17

Details
This function uses uniroot for finding this quantity, and many of the the accepted parameters are
arguments for that function; see its documentation for more details.
Value
The quantile associated with p
Examples
CPAT:::qZn(0.5)

rchangepoint

Simulate Univariate Data With a Single Change Point

Description
This function simulates univariate data with a structural change.
Usage
rchangepoint(n, changepoint = NULL, mean1 = 0, mean2 = 0,
dist = rnorm, meanparam = "mean", ...)
Arguments
n

An integer for the data set’s sample size

changepoint

An integer for where the change point occurs

mean1

The mean prior to the change point

mean2

The mean after the change point

dist

The function with which random data will be generated

meanparam

A string for the parameter in dist representing the mean

...

Other arguments to be passed to dist

Details
This function generates artificial change point data, where up to the specified change point the data
has one mean, and after the point it has a different mean. By default, the function simulates standard
Normal data with no change. If changepoint is NULL, then by default the change point will be at
about the middle of the data.
Value
A vector of the simulated data
Examples
CPAT:::rchangepoint(500)
CPAT:::rchangepoint(500, changepoint = 10, mean2 = 2, sd = 2)
CPAT:::rchangepoint(500, changepoint = 250, dist = rexp, meanparam = "rate",
mean1 = 1, mean2 = 2)

18

sim_de_stat

sim_de_stat

Darling-Erdös Statistic Simulation

Description
Simulates multiple realizations of the Darling-Erdös statistic.
Usage
sim_de_stat(size, a = log, b = log, use_kernel_var = FALSE,
kernel = "ba", bandwidth = "and", n = 500, gen_func = rnorm,
args = NULL, parallel = FALSE)
Arguments
size

Number of realizations to simulate

a

The function that will be composed wit l(x) = (2 log(x))1/2

b

The function that will be composed with u(x) = 2 log(x) +
1
2 log(pi)

1
2

log(log(x)) −

use_kernel_var Set to TRUE to use kernel-based long-run variance estimation (FALSE means this
is not employed)
kernel

If character, the identifier of the kernel function as used in the cointReg (see
documentation for cointReg::getLongRunVar); if function, the kernel function to be used for long-run variance estimation (default is the Bartlett kernel in
cointReg); this parameter has no effect if use_kernel_var is FALSE

bandwidth

If character, the identifier of how to compute the bandwidth as defined in the
cointReg package (see documentation for cointReg::getLongRunVar); if function, a function to use for computing the bandwidth; if numeric, the bandwidth
to use (the default behavior is to use the Andrews (1991) method, as used in
cointReg); this parameter has no effect if use_kernel_var is FALSE

n

The sample size for each realization

gen_func

The function generating the random sample from which the statistic is computed

args

A list of arguments to be passed to gen_func

parallel

Whether to use the foreach and doParallel packages to parallelize simulation
(which needs to be initialized in the global namespace before use)

Details
If use_kernel_var is set to TRUE, long-run variance estimation using kernel-based techniques will
be employed; otherwise, a technique resembling standard variance estimation will be employed.
Any technique employed, though, will account for the potential break points, as described in Rice
et al. (). See the documentation for stat_de for more details.
The parameters kernel and bandwidth control parameters for long-run variance estimation using
kernel methods. These parameters will be passed directly to stat_de.
Value
A vector of simulated realizations of the Darling-Erdös statistic

sim_hs_stat

19

References
Andrews DWK (1991). “Heteroskedasticity and Autocorrelation Consistent Covariance Matrix
Estimation.” Econometrica, 59(3), 817-858.
Rice G, Miller C, Horváth L (????). “A new class of change point test of Rényi type.” in-press.
Examples
CPAT:::sim_de_stat(100)
CPAT:::sim_de_stat(100, use_kernel_var = TRUE,
gen_func = CPAT:::rchangepoint,
args = list(changepoint = 250, mean2 = 1))

sim_hs_stat

Hidalgo-Seo Statistic Simulation

Description
Simulates multiple realizations of the Hidalgo-Seo statistic.
Usage
sim_hs_stat(size, corr = TRUE, gen_func = rnorm, args = NULL,
n = 500, parallel = FALSE, use_kernel_var = FALSE, kernel = "ba",
bandwidth = "and")
Arguments
size

Number of realizations to simulate

corr

Whether long-run variance should be computed under the assumption of correlated residuals

gen_func

The function generating the random sample from which the statistic is computed

args

A list of arguments to be passed to gen_func

n

The sample size for each realization

parallel

Whether to use the foreach and doParallel packages to parallelize simulation
(which needs to be initialized in the global namespace before use)

use_kernel_var Set to TRUE to use kernel-based long-run variance estimation (FALSE means this
is not employed); TODO: NOT CURRENTLY IMPLEMENTED
kernel

If character, the identifier of the kernel function as used in the cointReg (see
documentation for cointReg::getLongRunVar); if function, the kernel function to be used for long-run variance estimation (default is the Bartlett kernel in
cointReg); this parameter has no effect if use_kernel_var is FALSE; TODO:
NOT CURRENTLY IMPLEMENTED

bandwidth

If character, the identifier of how to compute the bandwidth as defined in the
cointReg package (see documentation for cointReg::getLongRunVar); if function, a function to use for computing the bandwidth; if numeric, the bandwidth
to use (the default behavior is to use the Andrews (1991) method, as used in
cointReg); this parameter has no effect if use_kernel_var is FALSE; TODO:
NOT CURRENTLY IMPLEMENTED

20

sim_Vn

Details
If corr is TRUE, then the residuals of the data-generating process are assumed to be correlated and
the test accounts for this in long-run variance estimation; see the documentation for stat_hs for
more details. Otherwise, the sample variance is the estimate for the long-run variance, as described
in Hidalgo and Seo (2013).
Value
A vector of simulated realizations of the Hidalgo-Seo statistic
References
Andrews DWK (1991). “Heteroskedasticity and Autocorrelation Consistent Covariance Matrix
Estimation.” Econometrica, 59(3), 817-858.
Hidalgo J, Seo MH (2013). “Testing for structural stability in the whole sample.” Journal of
Econometrics, 175(2), 84 - 93. ISSN 0304-4076, doi: 10.1016/j.jeconom.2013.02.008, http:
//www.sciencedirect.com/science/article/pii/S0304407613000626.
Examples
CPAT:::sim_hs_stat(100)
CPAT:::sim_hs_stat(100, gen_func = CPAT:::rchangepoint,
args = list(changepoint = 250, mean2 = 1))

sim_Vn

CUSUM Statistic Simulation (Assuming Variance)

Description
Simulates multiple realizations of the CUSUM statistic when the long-run variance of the data is
known.
Usage
sim_Vn(size, n = 500, gen_func = rnorm, sd = 1, args = NULL)
Arguments
size

Number of realizations to simulate

n

The sample size for each realization

gen_func

The function generating the random sample from which the statistic is computed

sd

The square root of the second moment of the data

args

A list of arguments to be passed to gen_func

Value
A vector of simulated realizations of the CUSUM statistic

sim_Vn_stat

21

Examples
CPAT:::sim_Vn(100)
CPAT:::sim_Vn(100, gen_func = CPAT:::rchangepoint,
args = list(changepoint = 250, mean2 = 1))

sim_Vn_stat

CUSUM Statistic Simulation

Description
Simulates multiple realizations of the CUSUM statistic.
Usage
sim_Vn_stat(size, kn = function(n) {
1 }, tau = 0,
use_kernel_var = FALSE, kernel = "ba", bandwidth = "and",
n = 500, gen_func = rnorm, args = NULL, parallel = FALSE)
Arguments
size

Number of realizations to simulate

kn

A function returning a positive integer that is used in the definition of the trimmed
CUSUSM statistic effectively setting the bounds over which the maximum is
taken

tau

The weighting parameter for the weighted CUSUM statistic (defaults to zero for
no weighting)

use_kernel_var Set to TRUE to use kernel-based long-run variance estimation (FALSE means this
is not employed)
kernel

If character, the identifier of the kernel function as used in the cointReg (see
documentation for cointReg::getLongRunVar); if function, the kernel function to be used for long-run variance estimation (default is the Bartlett kernel in
cointReg); this parameter has no effect if use_kernel_var is FALSE

bandwidth

If character, the identifier of how to compute the bandwidth as defined in the
cointReg package (see documentation for cointReg::getLongRunVar); if function, a function to use for computing the bandwidth; if numeric, the bandwidth
to use (the default behavior is to use the method described in (Andrews 1991),
as used in cointReg); this parameter has no effect if use_kernel_var is FALSE

n

The sample size for each realization

gen_func

The function generating the random sample from which the statistic is computed

args

A list of arguments to be passed to gen_func

parallel

Whether to use the foreach and doParallel packages to parallelize simulation
(which needs to be initialized in the global namespace before use)

22

sim_Zn

Details
This differs from sim_Vn() in that the long-run variance is estimated with this function, while
sim_Vn() assumes the long-run variance is known. Estimation can be done in a variety of ways. If
use_kernel_var is set to TRUE, long-run variance estimation using kernel-based techniques will be
employed; otherwise, a technique resembling standard variance estimation will be employed. Any
technique employed, though, will account for the potential break points, as described in Rice et al.
(). See the documentation for stat_Vn for more details.
The parameters kernel and bandwidth control parameters for long-run variance estimation using
kernel methods. These parameters will be passed directly to stat_Vn.
Versions of the CUSUM statistic, such as the weighted or trimmed statistics, can be simulated with
the function by passing values to kn and tau; again, see the documentation for stat_Vn.
Value
A vector of simulated realizations of the CUSUM statistic
References
Andrews DWK (1991). “Heteroskedasticity and Autocorrelation Consistent Covariance Matrix
Estimation.” Econometrica, 59(3), 817-858.
Rice G, Miller C, Horváth L (????). “A new class of change point test of Rényi type.” in-press.
Examples
CPAT:::sim_Vn_stat(100)
CPAT:::sim_Vn_stat(100, kn = function(n) {floor(0.1 * n)}, tau = 1/3,
use_kernel_var = TRUE, gen_func = CPAT:::rchangepoint,
args = list(changepoint = 250, mean2 = 1))

sim_Zn

Rènyi-Type Statistic Simulation (Assuming Variance)

Description
Simulates multiple realizations of the Rènyi-type statistic when the long-run variance of the data is
known.
Usage
sim_Zn(size, kn, n = 500, gen_func = rnorm, args = NULL, sd = 1)
Arguments
size

Number of realizations to simulate

kn

A function returning a positive integer that is used in the definition of the Rènyitype statistic effectively setting the bounds over which the maximum is taken

n

The sample size for each realization

gen_func

The function generating the random sample from which the statistic is computed

args

A list of arguments to be passed to gen_func

sd

The square root of the second moment of the data

sim_Zn_stat

23

Value
A vector of simulated realizations of the Rènyi-type statistic
Examples
CPAT:::sim_Zn(100, kn = function(n) {floor(log(n))})
CPAT:::sim_Zn(100, kn = function(n) {floor(log(n))},
gen_func = CPAT:::rchangepoint, args = list(changepoint = 250,
mean2 = 1))

sim_Zn_stat

Rènyi-Type Statistic Simulation

Description
Simulates multiple realizations of the Rènyi-type statistic.
Usage
sim_Zn_stat(size, kn = function(n) {
floor(sqrt(n)) },
use_kernel_var = FALSE, kernel = "ba", bandwidth = "and",
n = 500, gen_func = rnorm, args = NULL, parallel = FALSE)
Arguments
size

Number of realizations to simulate

kn

A function returning a positive integer that is used in the definition of the Rènyitype statistic effectively setting the bounds over which the maximum is taken

use_kernel_var Set to TRUE to use kernel-based long-run variance estimation (FALSE means this
is not employed)
kernel

If character, the identifier of the kernel function as used in the cointReg (see
documentation for cointReg::getLongRunVar); if function, the kernel function to be used for long-run variance estimation (default is the Bartlett kernel in
cointReg); this parameter has no effect if use_kernel_var is FALSE

bandwidth

If character, the identifier of how to compute the bandwidth as defined in the
cointReg package (see documentation for cointReg::getLongRunVar); if function, a function to use for computing the bandwidth; if numeric, the bandwidth
to use (the default behavior is to use the Andrews (1991) method, as used in
cointReg); this parameter has no effect if use_kernel_var is FALSE

n

The sample size for each realization

gen_func

The function generating the random sample from which the statistic is computed

args

A list of arguments to be passed to gen_func

parallel

Whether to use the foreach and doParallel packages to parallelize simulation
(which needs to be initialized in the global namespace before use)

24

stat_de

Details
This differs from sim_Zn() in that the long-run variance is estimated with this function, while
sim_Zn() assumes the long-run variance is known. Estimation can be done in a variety of ways. If
use_kernel_var is set to TRUE, long-run variance estimation using kernel-based techniques will be
employed; otherwise, a technique resembling standard variance estimation will be employed. Any
technique employed, though, will account for the potential break points, as described in Rice et al.
(). See the documentation for stat_Zn for more details.
The parameters kernel and bandwidth control parameters for long-run variance estimation using
kernel methods. These parameters will be passed directly to stat_Zn.
Value
A vector of simulated realizations of the Rènyi-type statistic
References
Andrews DWK (1991). “Heteroskedasticity and Autocorrelation Consistent Covariance Matrix
Estimation.” Econometrica, 59(3), 817-858.
Rice G, Miller C, Horváth L (????). “A new class of change point test of Rényi type.” in-press.
Examples
CPAT:::sim_Zn_stat(100)
CPAT:::sim_Zn_stat(100, kn = function(n) {floor(log(n))},
use_kernel_var = TRUE, gen_func = CPAT:::rchangepoint,
args = list(changepoint = 250, mean2 = 1))

stat_de

Compute the Darling-Erdös Statistic

Description
This function computes the Darling-Erdös statistic.
Usage
stat_de(dat, a = log, b = log, estimate = FALSE,
use_kernel_var = FALSE, custom_var = NULL, kernel = "ba",
bandwidth = "and", get_all_vals = FALSE)
Arguments
dat
a
b
estimate
use_kernel_var

The data vector
The function that will be composed with l(x) = (2 log x)1/2
The function that will be composed with u(x) = 2 log x + 21 log log x − 21 log π
Set to TRUE to return the estimated location of the change point
Set to TRUE to use kernel methods for long-run variance estimation (typically
used when the data is believed to be 
correlated); if FALSE, then the long-run vari
2 

2 PT
Pt
2
−1
,
ance is estimated using σ̂T,t = T
+ s=t+1 Xs − X̃T −t
s=1 Xs − X̄t
P
P
t
T
where X̄t = t−1 s=1 Xs and X̃T −t = (T − t)−1 s=t+1 Xs

stat_hs

25

custom_var

Can be a vector the same length as dat consisting of variance-like numbers
at each potential change point (so each entry of the vector would be the "best
estimate" of the long-run variance if that location were where the change point
occured) or a function taking two parameters x and k that can be used to generate
this vector, with x representing the data vector and k the position of a potential
change point; if NULL, this argument is ignored

kernel

If character, the identifier of the kernel function as used in cointReg (see getLongRunVar);
if function, the kernel function to be used for long-run variance estimation (default is the Bartlett kernel in cointReg)

bandwidth

If character, the identifier for how to compute the bandwidth as defined in cointReg (see getBandwidth); if function, a function to use for computing the bandwidth; if numeric, the bandwidth value to use (the default is to use Andrews’
method, as used in cointReg)

get_all_vals

If TRUE, return all values for the statistic at every tested point in the data set

Details
If ĀT (τ, tT ) is the weighted and trimmed CUSUM statistic with weighting parameter τ and trimming parameter tT (see stat_Vn), then the Darling-Erdös statistic is
l(aT )ĀT (1/2, 1) − u(bT )
√
with l(x) = 2 log x and u(x) = 2 log x + 12 log log x − 12 log π (log x is the natural logarithm of
x). The parameter a corresponds to aT and b to bT ; these are both log by default.
See (Rice et al. ) to learn more.
Value
If both estimate and get_all_vals are FALSE, the value of the test statistic; otherwise, a list that
contains the test statistic and the other values requested (if both are TRUE, the test statistic is in the
first position and the estimated changg point in the second)
References
Rice G, Miller C, Horváth L (????). “A new class of change point test of Rényi type.” in-press.
Examples
CPAT:::stat_de(rnorm(1000))
CPAT:::stat_de(rnorm(1000), use_kernel_var = TRUE, bandwidth = "nw", kernel = "bo")

stat_hs

Compute the Hidalgo-Seo Statistic

Description
This function computes the Hidalgo-Seo statistic for a change in mean model.

26

stat_hs

Usage
stat_hs(dat, estimate = FALSE, corr = TRUE, get_all_vals = FALSE,
custom_var = NULL, use_kernel_var = FALSE, kernel = "ba",
bandwidth = "and")
Arguments
dat
estimate
corr

The data vector
Set to TRUE to return the estimated location of the change point
If TRUE, the long-run variance will be computed under the assumption of correlated residuals; ignored if custom_var is not NULL or use_kernel_var is TRUE
get_all_vals
If TRUE, return all values for the statistic at every tested point in the data set
custom_var
Can be a vector the same length as dat consisting of variance-like numbers
at each potential change point (so each entry of the vector would be the "best
estimate" of the long-run variance if that location were where the change point
occured) or a function taking two parameters x and k that can be used to generate
this vector, with x representing the data vector and k the position of a potential
change point; if NULL, this argument is ignored
use_kernel_var Set to TRUE to use kernel methods for long-run variance estimation (typically
used when the data is believed to be 
correlated); if FALSE, then the long-run vari
2 

2 PT
Pt
−1
2
X
−
X̄
+
X
−
X̃
,
ance is estimated using σ̂T,t = T
s
t
s
T −t
s=1
s=t+1
P
P
t
T
where X̄t = t−1 s=1 Xs and X̃T −t = (T − t)−1 s=t+1 Xs ; if custom_var
is not NULL, this argument is ignored
kernel
If character, the identifier of the kernel function as used in cointReg (see getLongRunVar);
if function, the kernel function to be used for long-run variance estimation (default is the Bartlett kernel in cointReg)
bandwidth
If character, the identifier for how to compute the bandwidth as defined in cointReg (see getBandwidth); if function, a function to use for computing the bandwidth; if numeric, the bandwidth value to use (the default is to use Andrews’
method, as used in cointReg)
Details
For a data set xt with n observations, the test statistic is
max (LM(s) − Bn )/An

1≤s≤n−1

where ût = xt − x̄ (x̄ is the sample mean), an = (2 log log n)1/2 , bn = a2n − 12 log log log n −
ˆ = σ̂ 2 = n−1 Pn û2t , and LM(s) = n(n −
log Γ(1/2), An = bn /a2n , Bn = b2n /a2n , ∆
t=1
ˆ −1 (Ps ût )2 .
s)−1 s−1 ∆
t=1
If corr is FALSE, then the residuals are assumed
to be uncorrelated. Otherwise, the residuals are
Pb√nc
Pn−j
ˆ
assumed to be correlated and ∆ = γ̂(0) + 2 j=1 (1 − √jn )γ̂(j) with γ̂(j) = n1 t=1 ût ût+j .
This statistic was presented in (Hidalgo and Seo 2013).
Value
If both estimate and get_all_vals are FALSE, the value of the test statistic; otherwise, a list that
contains the test statistic and the other values requested (if both are TRUE, the test statistic is in the
first position and the estimated change point in the second)

stat_Vn

27

References
Hidalgo J, Seo MH (2013). “Testing for structural stability in the whole sample.” Journal of
Econometrics, 175(2), 84 - 93. ISSN 0304-4076, doi: 10.1016/j.jeconom.2013.02.008, http:
//www.sciencedirect.com/science/article/pii/S0304407613000626.
Examples
CPAT:::stat_hs(rnorm(1000))
CPAT:::stat_hs(rnorm(1000), corr = FALSE)

stat_Vn

Compute the CUSUM Statistic

Description
This function computes the CUSUM statistic (and can compute weighted/trimmed variants, depending on the values of kn and tau).
Usage
stat_Vn(dat, kn = function(n) {
1 }, tau = 0, estimate = FALSE,
use_kernel_var = FALSE, custom_var = NULL, kernel = "ba",
bandwidth = "and", get_all_vals = FALSE)
Arguments
dat
kn

The data vector
A function corresponding to the trimming parameter tT in the trimmed CUSUM
variant; by default, is a function returning 1 (for no trimming)
tau
The weighting parameter τ for the weighted CUSUM statistic; by default, is 0
(for no weighting)
estimate
Set to TRUE to return the estimated location of the change point
use_kernel_var Set to TRUE to use kernel methods for long-run variance estimation (typically
used when the data is believed to be 
correlated); if FALSE, then the long-run vari
2 

2 PT
Pt
−1
2
+ s=t+1 Xs − X̃T −t
,
ance is estimated using σ̂T,t = T
s=1 Xs − X̄t
P
P
t
T
where X̄t = t−1 s=1 Xs and X̃T −t = (T − t)−1 s=t+1 Xs
custom_var
Can be a vector the same length as dat consisting of variance-like numbers
at each potential change point (so each entry of the vector would be the "best
estimate" of the long-run variance if that location were where the change point
occured) or a function taking two parameters x and k that can be used to generate
this vector, with x representing the data vector and k the position of a potential
change point; if NULL, this argument is ignored
kernel
If character, the identifier of the kernel function as used in cointReg (see getLongRunVar);
if function, the kernel function to be used for long-run variance estimation (default is the Bartlett kernel in cointReg)
bandwidth
If character, the identifier for how to compute the bandwidth as defined in cointReg (see getBandwidth); if function, a function to use for computing the bandwidth; if numeric, the bandwidth value to use (the default is to use Andrews’
method, as used in cointReg)
get_all_vals
If TRUE, return all values for the statistic at every tested point in the data set

28

stat_Zn

Details
The definition of the statistic is

−1
T −1/2 max σ̂t,T
1≤t≤T

t
X

Xs −

s=1

T
t X
T s=1

A more general version is

T

−1/2

max

tT ≤t≤T −tT

−1
σ̂t,T



t
T



T −t
T

τ X
t
s=1

Xs −

T
t X
T s=1

The parameter kn corresponds to the trimming parameter tT and the parameter tau corresponds to
τ.
See (Rice et al. ) for more details.
Value
If both estimate and get_all_vals are FALSE, the value of the test statistic; otherwise, a list that
contains the test statistic and the other values requested (if both are TRUE, the test statistic is in the
first position and the estimated change point in the second)
References
Rice G, Miller C, Horváth L (????). “A new class of change point test of Rényi type.” in-press.
Examples
CPAT:::stat_Vn(rnorm(1000))
CPAT:::stat_Vn(rnorm(1000), kn = function(n) {0.1 * n}, tau = 1/2)
CPAT:::stat_Vn(rnorm(1000), use_kernel_var = TRUE, bandwidth = "nw", kernel = "bo")

stat_Zn

Compute the Rényi-Type Statistic

Description
This function computes the Rényi-type statistic.
Usage
stat_Zn(dat, kn = function(n) {
floor(sqrt(n)) }, estimate = FALSE,
use_kernel_var = FALSE, custom_var = NULL, kernel = "ba",
bandwidth = "and", get_all_vals = FALSE)

stat_Zn

29

Arguments
dat

The data vector

kn

A function corresponding to the trimming parameter tT ; by default, the square
root function

estimate

Set to TRUE to return the estimated location of the change point

use_kernel_var Set to TRUE to use kernel methods for long-run variance estimation (typically
used when the data is believed to be 
correlated); if FALSE, then the long-run vari
2 

2 PT
Pt
2
−1
ance is estimated using σ̂T,t = T
+ s=t+1 Xs − X̃T −t
,
s=1 Xs − X̄t
P
P
T
t
where X̄t = t−1 s=1 Xs and X̃T −t = (T − t)−1 s=t+1 Xs ; if custom_var
is not NULL, this argument is ignored
custom_var

Can be a vector the same length as dat consisting of variance-like numbers
at each potential change point (so each entry of the vector would be the "best
estimate" of the long-run variance if that location were where the change point
occured) or a function taking two parameters x and k that can be used to generate
this vector, with x representing the data vector and k the position of a potential
change point; if NULL, this argument is ignored

kernel

If character, the identifier of the kernel function as used in cointReg (see getLongRunVar);
if function, the kernel function to be used for long-run variance estimation (default is the Bartlett kernel in cointReg)

bandwidth

If character, the identifier for how to compute the bandwidth as defined in cointReg (see getBandwidth); if function, a function to use for computing the bandwidth; if numeric, the bandwidth value to use (the default is to use Andrews’
method, as used in cointReg)

get_all_vals

If TRUE, return all values for the statistic at every tested point in the data set

Details
The definition of the statistic is

max

tT ≤t≤T −tT

−1 −1
t
σ̂t,T

t
X

Xs − (T − t)−1

s=1

T
X

Xs

s=t+1

The parameter kn corresponds to the trimming parameter tT .
Value
If both estimate and get_all_vals are FALSE, the value of the test statistic; otherwise, a list that
contains the test statistic and the other values requested (if both are TRUE, the test statistic is in the
first position and the estimated change point in the second)
Examples
CPAT:::stat_Zn(rnorm(1000))
CPAT:::stat_Zn(rnorm(1000), kn = function(n) {floor(log(n))})
CPAT:::stat_Zn(rnorm(1000), use_kernel_var = TRUE, bandwidth = "nw",
kernel = "bo")

30

%s0%

%s%

Concatenate (With Space)

Description
Concatenate and form strings (with space separation)
Usage
x %s% y
Arguments
x

One object

y

Another object

Value
A string combining x and y with a space separating them
Examples
`%s%` <- CPAT:::`%s%`
"Hello" %s% "world"

%s0%

Concatenate (Without Space)

Description
Concatenate and form strings (no space separation)
Usage
x %s0% y
Arguments
x

One object

y

Another object

Value
A string combining x and y
Examples
`%s0%` <- CPAT:::`%s0%`
"Hello" %s0% "world"

Index

Andrews.test, 3
andrews_test, 3, 3
andrews_test_reg, 3, 4

sim_hs_stat, 19
sim_Vn, 20
sim_Vn_stat, 21
sim_Zn, 22
sim_Zn_stat, 23
sqrt, 11
stat_de, 7, 18, 24
stat_hs, 12, 20, 25
stat_Vn, 6, 22, 25, 27
stat_Zn, 11, 24, 28

banks, 5

uniroot, 16, 17

∗Topic datasets
banks, 5
ff, 9
.onAttach, 2
%s0%, 30
%s%, 30

CPAT_startup_message, 5
cpt_consistent_var, 6
CUSUM.test, 6
DE.test, 7
dZn, 8
ff, 9
get_lrv_vec, 10
getBandwidth, 7, 8, 10, 11, 25–27, 29
getLongRunVar, 7, 8, 10, 11, 25–27, 29
getLongRunWeights, 10, 10
HR.test, 11
HS.test, 12
lm, 3, 4
log, 11
pdarling_erdos, 7, 13
phidalgo_seo, 12, 13
pkolmogorov, 6, 14
pZn, 11, 14
qdarling_erdos, 15
qhidalgo_seo, 15
qkolmogorov, 16
qZn, 16
rchangepoint, 17
sim_de_stat, 18
31



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