Concatenate (With Space)

Description

Concatenate and form strings (with space separation)

Usage

x %s% y

Arguments

Value

A string combining x and y with a space separating them

Examples

`%s%` <- CPAT:::`%s%`
"Hello" %s% "world"

Concatenate (Without Space)

Description

Concatenate and form strings (no space separation)

Usage

x %s0% y

Arguments

Value

A string combining x and y

Examples

`%s0%` <- CPAT:::`%s0%`
"Hello" %s0% "world"

Package Attach Hook Function

Description

Hook triggered when package attached

Usage

.onAttach(lib, pkg)

Arguments

Examples

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

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

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)

Create Package Startup Message

Description

Makes package startup message.

Usage

CPAT_startup_message()

Examples

CPAT:::CPAT_startup_message()

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")

Arguments

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")

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")

Arguments

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")

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 t_T satisfying t_T \to \infty and t_T/T \to 0 as T \to \infty. (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

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")

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

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)

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)

Arguments

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)

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

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

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

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

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

Details

This is the estimator

\hat{\sigma}^2_{T,t} = T^{-1}\left(\sum_{s = 1}^t \left(X_s - \bar{X}_t\right)^2 + \sum_{s = t + 1}^{T}\left(X_s - \tilde{X}_{T - t} \right)^2\right)

where \bar{X}_t = t^{-1}\sum_{s = 1}^t X_s and \tilde{X}_{T - t} = (T - t)^{-1} \sum_{s = t + 1}^{T} X_s. 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)

Rényi-Type Statistic Limiting Distribution Density Function

Description

Function for computing the value of the density function of the limiting distribution of the Rényi-type statistic.

Usage

dZn(x, summands = NULL)

Arguments

Value

Value of the density function at x

Examples

CPAT:::dZn(1)

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

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

Value

List with components w containing the vector of weights and upper, the index of the largest non-zero entry in w

Examples

CPAT:::getLongRunWeights(10, 1)

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

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")

Rènyi-Type Statistic CDF

Description

CDF for the limiting distribution of the Rènyi-type statistic.

Usage

pZn(q, summands = NULL)

Arguments

Value

If Z is the random variable following the limiting distribution, the quantity P(Z \leq q)

Examples

CPAT:::pZn(0.1)

Darling-Erdös Statistic CDF

Description

CDF for the limiting distribution of the Darling-Erdös statistic.

Usage

pdarling_erdos(q)

Arguments

Value

If Z is the random variable with this distribution, the quantity P(Z \leq q)

Examples

CPAT:::pdarling_erdos(0.1)

Hidalgo-Seo Statistic CDF

Description

CDF of the limiting distribution of the Hidalgo-Seo statistic

Usage

phidalgo_seo(q)

Arguments

Value

If Z is the random variable following the limiting distribution, the quantity P(Z \leq q)

Examples

CPAT:::phidalgo_seo(0.1)

Kolmogorov CDF

Description

CDF of the Kolmogorov distribution.

Usage

pkolmogorov(q, summands = ceiling(q * sqrt(72) + 3/2))

Arguments

Value

If Z is the random variable following the Kolmogorov distribution, the quantity P(Z \leq q)

Examples

CPAT:::pkolmogorov(0.1)

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

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)

Darling-Erdös Statistic Limiting Distribution Quantile Function

Description

Quantile function for the limiting distribution of the Darling-Erdös statistic.

Usage

qdarling_erdos(p)

Arguments

Value

The quantile associated with p

Examples

CPAT:::qdarling_erdos(0.5)

Hidalgo-Seo Statistic Limiting Distribution Quantile Function

Description

Quantile function for the limiting distribution of the Hidalgo-Seo statistic

Usage

qhidalgo_seo(p)

Arguments

Value

A The quantile associated with p

Examples

CPAT:::qhidalgo_seo(0.5)

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

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)

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

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)

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

Value

A vector of simulated realizations of the CUSUM statistic

Examples

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

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

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))

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

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))

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

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))

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

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

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))

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

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))

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

Details

The definition of the statistic is

T^{-1/2} \max_{1 \leq t \leq T} \hat{\sigma}_{t,T}^{-1} \left| \sum_{s = 1}^t X_s - \frac{t}{T}\sum_{s = 1}^T \right|

A more general version is

T^{-1/2} \max_{t_T \leq t \leq T - t_T} \hat{\sigma}_{t,T}^{-1} \left(\frac{t}{T} \left(\frac{T - t}{T}\right)\right)^{\tau} \left| \sum_{s = 1}^t X_s - \frac{t}{T}\sum_{s = 1}^T \right|

The parameter kn corresponds to the trimming parameter t_T and the parameter tau corresponds to \tau.

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")

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)

Arguments

Details

The definition of the statistic is

\max_{t_T \leq t \leq T - t_T} \hat{\sigma}_{t,T}^{-1} \left|t^{-1}\sum_{s = 1}^{t}X_s - (T - t)^{-1}\sum_{s = t + 1}^{T} X_s \right|

The parameter kn corresponds to the trimming parameter t_T.

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")

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

Details

If \bar{A}_T(\tau, t_T) is the weighted and trimmed CUSUM statistic with weighting parameter \tau and trimming parameter t_T (see stat_Vn), then the Darling-Erdös statistic is

l(a_T) \bar{A}_T(1/2, 1) - u(b_T)

with l(x) = \sqrt{2 \log x} and u(x) = 2 \log x + \frac{1}{2} \log \log x - \frac{1}{2} \log \pi (\log x is the natural logarithm of x). The parameter a corresponds to a_T and b to b_T; 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")

Compute the Hidalgo-Seo Statistic

Description

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

Usage

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

Arguments

Details

For a data set x_t with n observations, the test statistic is

\max_{1 \leq s \leq n - 1} (\mathcal{LM}(s) - B_n)/A_n

where \hat{u}_t = x_t - \bar{x} (\bar{x} is the sample mean), a_n = (2 \log \log n)^{1/2}, b_n = a_n^2 - \frac{1}{2} \log \log \log n - \log \Gamma (1/2), A_n = b_n / a_n^2, B_n = b_n^2/a_n^2, \hat{\Delta} = \hat{\sigma}^2 = n^{-1} \sum_{t = 1}^{n} \hat{u}_t^2, and \mathcal{LM}(s) = n (n - s)^{-1} s^{-1} \hat{\Delta}^{-1} \left( \sum_{t = 1}^{s} \hat{u}_t\right)^2.

If corr is FALSE, then the residuals are assumed to be uncorrelated. Otherwise, the residuals are assumed to be correlated and \hat{\Delta} = \hat{\gamma}(0) + 2 \sum_{j = 1}^{\lfloor \sqrt{n} \rfloor} (1 - \frac{j}{\sqrt{n}}) \hat{\gamma}(j) with \hat{\gamma}(j) = \frac{1}{n}\sum_{t = 1}^{n - j} \hat{u}_t \hat{u}_{t + 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)

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)