| Type: | Package |
| Title: | Modified Iterative Cumulative Sum of Squares Algorithm |
| Version: | 0.2.0 |
| Maintainer: | Andreu Sansó <andreu.sanso@uib.eu> |
| Description: | Companion package of Carrion-i-Silvestre & Sansó (2023): "Generalized Extreme Value Approximation to the CUMSUMQ Test for Constant Unconditional Variance in Heavy-Tailed Time Series". It implements the Modified Iterative Cumulative Sum of Squares Algorithm, which is an extension of the Iterative Cumulative Sum of Squares (ICSS) Algorithm of Inclan and Tiao (1994), and it checks for changes in the unconditional variance of a time series controlling for the tail index of the underlying distribution. The fourth order moment is estimated non-parametrically to avoid the size problems when the innovations are non-Gaussian (see, Sansó et al., 2004). Critical values and p-values are generated using a Generalized Extreme Value distribution approach. References Carrion-i-Silvestre J.J & Sansó A (2023) https://www.ub.edu/irea/working_papers/2023/202309.pdf. Inclan C & Tiao G.C (1994) <doi:10.1080/01621459.1994.10476824>, Sansó A & Aragó V & Carrion-i-Silvestre J.L (2004) https://dspace.uib.es/xmlui/bitstream/handle/11201/152078/524035.pdf. |
| License: | GPL-2 |
| Depends: | R (≥ 3.5.0), dplyr |
| Imports: | methods |
| Encoding: | UTF-8 |
| LazyData: | true |
| RoxygenNote: | 7.2.3 |
| Suggests: | testthat (≥ 3.0.0) |
| Config/testthat/edition: | 3 |
| NeedsCompilation: | no |
| Packaged: | 2024-08-22 11:00:17 UTC; HP |
| Author: | Josep Lluís Carrion-i-Silvestre [aut], Andreu Sansó [aut, cre] |
| Repository: | CRAN |
| Date/Publication: | 2024-08-22 18:50:02 UTC |
Hill's estimator of the tail index
Description
Computes the estimator of the tail index proposed by Hill (1975).
Usage
alpha_hill(x, k)
Arguments
x |
A numeric vector. |
k |
Fraction of the upper tail to be used to estimate of the tail index. |
Value
-
alpha: Estimated tail index. -
sd.alpha: Standard error. -
s: Number of observations used in the estimation.
References
B. Hill (1975): A Simple General Approach to Inference About the Tail of a Distribution. The Annals of Mathematical Statistics 3, 1163-1174.
See Also
Examples
alpha_hill(rnorm(500),k=0.1)
Nicolau & Rodrigues estimator of the tail index
Description
Computes the estimator of the tail index proposed by Nicolau & Rodrigues (2019).
Usage
alpha_nr(y, k)
Arguments
y |
A numeric vector. |
k |
Fraction of the upper tail to be used to estimate of the tail index. |
Value
-
alpha: Estimated tail index. -
sd.alpha: Standard error.
References
J. Nicolau and P.M.M. Rodrigues (2019): A new regression-based tail index estimator. The Review of Economics and Statistics 101, 667-680.
See Also
Examples
alpha_nr(rnorm(500),k=0.1)
Critical Values
Description
Computes critical values for the kappa_test statistic
Usage
cv.kappa(t, alpha, sig.lev)
Arguments
t |
Sample size. |
alpha |
Value of the tail index between 2 and 4. |
sig.lev |
Significance level. |
Value
Critical value.
References
J.L. Carrion-i-Silvestre & A. Sansó (2023): Generalized Extreme Value Approximation to the CUMSUMQ Test for Constant Unconditional Variance in Heavy-Tailed Time Series.
See Also
Data used in the examples
Description
Log returns of the exchange rate South African Rand versus United States Dollar.
Usage
data(logReturnsRandDollar)Value
Time series with 7705 observations.
Author(s)
J.L. Carrion-i-Silvestre and A. Sansó
Source
Paulo Rodrigues
References
J.L. Carrion-i-Silvestre & A. Sansó (2023): Generalized Extreme Value Approximation to the CUMSUMQ Test for Constant Unconditional Variance in Heavy-Tailed Time Series.
Examples
data(logReturnsRandDollar)
names(data)
estimate.alpha
Description
Computes the estimator of the tail index (alpha) using Hill (1975) or Nicolau & Rodrigues (2019) estimators and tests both the hypothesis that alpha is bigger or equal to 4, and that alpha is lower or equal to 2.
Usage
estimate.alpha(x, sig.lev = 0.05, tail.est = "Hill", k = 0.1)
Arguments
x |
A numeric vector. |
sig.lev |
Significance level. The default value is 0.05. |
tail.est |
Estimator of the tail index. The default value is "Hill", which uses Hill's (1975) estimator. "NR" uses Nicolau & Rodrigues (2019) estimator. |
k |
Fraction of the upper tail to be used to estimate of the tail index. The default value is 0.1. |
Details
Used internally by micss.
Value
-
alpha: Value of the tail index (alpha) to be used in micss. -
alpha.fit: Estimated tail index. -
t4: Test of the null hypothesis alpha>=4 against alpha<4. -
t2: Test of the null hypothesis alpha<=2 against alpha>2.
References
B. Hill (1975): A Simple General Approach to Inference About the Tail of a Distribution. The Annals of Mathematical Statistics 3, 1163-1174.
J. Nicolau and P.M.M. Rodrigues (2019): A new regression-based tail index estimator. The Review of Economics and Statistics 101, 667-680.
Iterative Cumulative Sum of Squares Algorithm
Description
Implements the ICSS algorithm of Inclan and Tiao (1994) using the CUMSUMQ test detailed in Carrion-i-Silvestre & Sansó (2023)
Usage
icss(e,sig.lev=0.05,kmax=NULL,alpha=NULL)
Arguments
e |
A numeric vector. Stationary variable on which the constancy of unconditional variance is tested. |
sig.lev |
Significance level. The default value is 0.05 |
kmax |
Maximum lag to be used for the estimation of the long-run fourth order moment. If not reported, an automatic procedure computes it depending on the sample size. |
alpha |
Tail index. If not reported, it is set at 4, which was the implicit assumption of Inclan & Tiao (1994). Values between 2 and 4 are allowed because this function is used by micss. |
Value
nb |
Number of breaks found by the algorithm. |
tb |
Vector with the time breaks. |
Author(s)
J.L. Carrion-i-Silvestre and A. Sanso.
References
J.L. Carrion-i-Silvestre & A. Sansó (2023): Generalized Extreme Value Approximation to the CUMSUMQ Test for Constant Unconditional Variance in Heavy-Tailed Time Series.
C. Inclan & G.C. Tiao (1994): Use of Cumulative Sums of Squares for Retrospective Detection of Changes of Variance. Journal of the American Statistical Association 89, 913-923.
See Also
Examples
set.seed(2)
e <- c(stats::rnorm(200),3*stats::rnorm(200))
o <- icss(e)
print.icss(o)
CUMSUMQ test to test for changes in the unconditional variance
Description
Computes the CUMSUMQ test to test for changes in the unconditional variance and reports the p-value adapted to the tail index and sample size
Usage
kappa_test(e,sig.lev=0.05,alpha=NULL,kmax=NULL)
Arguments
e |
A numeric vector. Stationary variable on which the constancy of unconditional variance is tested. |
sig.lev |
Significance level. The default value is 0.05. |
alpha |
Tail index. Must be a number between 2 and 4. The default value is 4. |
kmax |
Maximum lag to be used for the estimation of the long-run fourth order moment. If not reported, an automatic procedure computes it depending on the sample size. |
Details
It is only computed if the sample size is greater than 25 observations.
Value
kappa |
CUMSUMQ test. |
tb |
Possible time of the break (with maximum value of the statistic). |
cv |
critical value at the specified significance level. |
p.val |
p-value. |
Author(s)
J.L. Carrion-i-Silvestre and A. Sanso.
References
J.L. Carrion-i-Silvestre & A. Sansó (2023): Generalized Extreme Value Approximation to the CUMSUMQ Test for Constant Unconditional Variance in Heavy-Tailed Time Series.
See Also
Examples
data(logReturnsRandDollar)
e <- whitening(data$rand.dollar)$e # whitening
kappa_test(e)
Data used in the examples
Description
Log returns of the exchange rate South African Rand versus United States Dollar.
Usage
data(logReturnsRandDollar)Value
Time series with 7705 observations.
Author(s)
J.L. Carrion-i-Silvestre and A. Sansó.
Source
Paulo Rodrigues
References
J.L. Carrion-i-Silvestre & A. Sansó (2023): Generalized Extreme Value Approximation to the CUMSUMQ Test for Constant Unconditional Variance in Heavy-Tailed Time Series.
Examples
data(logReturnsRandDollar)
names(data)
# The following example replicates some of the results of Table 6 in
# Carrion-i-Silvestres & Sanso (2023)
data(logReturnsRandDollar)
e <- whitening(data$rand.dollar)$e # pre-whitening
m <- micss(e)
print.micss(m)
lrv.spc.bartlett
Description
Estimation of the long-run variance using the Barlett window.
Usage
lrv.spc.bartlett(x, kmax = NULL)
Arguments
x |
Stationary variable. A numeric vector. |
kmax |
Maximum lag to be used for the long-run estimation of the variance. |
Details
Estimates the log-run fourth order moment when x are the squares of a variable.
Value
Estimation of the long-run variance.
References
D. Sul, P.C.B. Phillips & C.Y. Choi (2005): Prewhitening Bias in HAC Estimation, Oxford Bulletin of Economics and Statistics 67, 517-546.
D.W.K. Andrews & J.C. Monahan (1992): An Improved Heteroskedasticity and Autocorrelation Consistent Covariance Matrix Estimator. Econometrica 60, 953-966.
Examples
lrv.spc.bartlett(rnorm(100))
Modiffied Iterative Cumulative Sum of Squares Algorithm
Description
Implements the MICSS algorithm of Carrion-i-Silvestre & Sansó (2023).
Usage
micss(e,sig.lev=0.05,kmax=NULL,alpha=NULL,tail.est="NR",k=0.1)
Arguments
e |
A numeric vector. Stationary variable on which the constancy of unconditional variance is tested. |
sig.lev |
Significance level. The default value is 0.05. |
kmax |
Maximum lag to be used for the estimation of the long-run fourth order moment. If not reported, an automatic procedure computes it depending on the sample size. |
alpha |
Tail index. If not reported, it is estimated automatically. |
tail.est |
Estimator of the tail index. The default value is "NR", which uses Nicolau & Rodrigues (2019) estimator. "Hill" uses the Hill's (1975) estimator. |
k |
Fraction of the upper tail to be used to estimate of the tail index. The default value is 0.1. |
Details
The tail index is estimated using the absolute values.
Value
icss |
An object with the output of the icss algorithm. |
alpha |
An object with the output of the estimate.alpha. |
Author(s)
J.L. Carrion-i-Silvestre and A. Sansó.
References
J.L. Carrion-i-Silvestre & A. Sansó (2023): Generalized Extreme Value Approximation to the CUMSUMQ Test for Constant Unconditional Variance in Heavy-Tailed Time Series.
B. Hill (1975): A Simple General Approach to Inference About the Tail of a Distribution. The Annals of Mathematical Statistics 3, 1163-1174.
J. Nicolau & P.M.M. Rodrigues (2019): A new regression-based tail index estimator. The Review of Economics and Statistics 101, 667-680.
See Also
icss
estimate.alpha
print.micss
plot.icss
Examples
set.seed(2)
e <- c(stats::rnorm(200),3*stats::rnorm(200))
o <- micss(e)
print.micss(o)
# The following example replicates some of the results of Table 6 in
# Carrion-i-Silvestres & Sanso (2023)
data(logReturnsRandDollar)
e <- whitening(data$rand.dollar)$e # pre-whitening
m <- micss(e)
print.micss(m)
P-values
Description
Computes p-values for the kappa_test statistic
Usage
p.val.kappa(x, t, alpha)
Arguments
x |
Value of the statistic. |
t |
Sample size. |
alpha |
Value of the tail index between 2 and 4. |
Value
p-value.
References
J.L. Carrion-i-Silvestre & A. Sansó (2023): Generalized Extreme Value Approximation to the CUMSUMQ Test for Constant Unconditional Variance in Heavy-Tailed Time Series.
See Also
plot.icss
Description
Plots the output of the ICSS algorithm.
Usage
## S3 method for class 'icss'
plot(x, type = "std", title = NULL, ...)
Arguments
x |
|
type |
Type of graphic. 3 possibilities: "std", which is the default, plots the absolute value of the variable and the standard deviation; "var" plots the squares of the variable and the variance; "res.std" plots the standardized residuals. |
title |
Title of the graphic. |
... |
Further arguments passed to or from other methods. |
Value
No return value. It generates a plot the output of micss or icss
Examples
set.seed(2)
e <- c(stats::rnorm(200),3*stats::rnorm(200))
o <- micss(e)
plot.icss(o,title="Example of the MICSS algorithm")
print.icss
Description
Prints the output of icss.
Usage
## S3 method for class 'icss'
print(x, ...)
Arguments
x |
An object with the output of the icss algorithm. |
... |
Further arguments passed to or from other methods. |
Details
Used internally by icss.
Value
No return value. It prints the output of icss
Examples
set.seed(2)
e <- c(stats::rnorm(200),3*stats::rnorm(200))
o <- icss(e)
print.icss(o)
print.micss
Description
Prints the output of micss.
Usage
## S3 method for class 'micss'
print(x, ...)
Arguments
x |
An object with the output of the micss algorithm. |
... |
Further arguments passed to or from other methods. |
Value
No return value. It prints the output of micss
Examples
set.seed(2)
e <- c(stats::rnorm(200),3*stats::rnorm(200))
o <- micss(e)
print.micss(o)
sr_loc
Description
Computes the location parameter for the GEV approximation to the distribution of kappa_test statistic
Usage
sr_loc(t, alpha, lmax = NULL)
Arguments
t |
Sample size |
alpha |
Value of the tail index between 2 and 4 |
lmax |
Maximum lag to be used for the long-run estimation of the fourth order moment of the innovations. If not specified it is generated automatically depending on the sample size. |
Details
used internally by cv.kappa and p.val.kappa
Value
Location parameter
References
J.L. Carrion-i-Silvestre & A. Sansó (2023): Generalized Extreme Value Approximation to the CUMSUMQ Test for Constant Unconditional Variance in Heavy-Tailed Time Series.
sr_scale
Description
Computes the scale parameter for the GEV approximation to the distribution of kappa_test statistic
Usage
sr_scale(t, alpha, lmax = NULL)
Arguments
t |
Sample size |
alpha |
Value of the tail index between 2 and 4 |
lmax |
Maximum lag to be used for the long-run estimation of the fourth order moment of the innovations. If not specified it is generated automatically depending on the sample size. |
Details
Used internally by cv.kappa and p.val.kappa
Value
Scale parameter
References
J.L. Carrion-i-Silvestre & A. Sansó (2023): Generalized Extreme Value Approximation to the CUMSUMQ Test for Constant Unconditional Variance in Heavy-Tailed Time Series.
sr_shape
Description
Computes the scale parameter for the GEV approximation to the distribution of kappa_test statistic
Usage
sr_shape(t, alpha, lmax = NULL)
Arguments
t |
Sample size |
alpha |
Value of the tail index between 2 and 4 |
lmax |
Maximum lag to be used for the long-run estimation of the fourth order moment of the innovations. If not specified it is generated automatically depending on the sample size. |
Details
Used internally by cv.kappa and p.val.kappa
Value
Shape parameter
References
J.L. Carrion-i-Silvestre & A. Sansó (2023): Generalized Extreme Value Approximation to the CUMSUMQ Test for Constant Unconditional Variance in Heavy-Tailed Time Series.
step2a
Description
Computes step 2a of the icss Algorithm.
Usage
step2a(e, sig.lev, kmax, alpha)
Arguments
e |
Stationary variable on which the constancy of unconditional variance is tested. |
sig.lev |
Significance level. |
kmax |
Maximum lag to be used for the long-run estimation of the fourth order moment of the innovations. |
alpha |
Tail index. |
Details
Used internally by icss.
Value
-
tb: Time of the break.
References
C. Inclan & G.C. Tiao (1994): Use of Cumulative Sums of Squares for Retrospective Detection of Changes of Variance. Journal of the American Statistical Association 89, 913-923.
step2b
Description
Computes step 2b of the icss Algorithm.
Usage
step2b(e, sig.lev, kmax, alpha)
Arguments
e |
Stationary variable on which the constancy of unconditional variance is tested. |
sig.lev |
Significance level. |
kmax |
Maximum lag to be used for the long-run estimation of the fourth order moment of the innovations. |
alpha |
Tail index. |
Details
Used internally by icss.
Value
-
tb: Time of the break.
References
C. Inclan & G.C. Tiao (1994): Use of Cumulative Sums of Squares for Retrospective Detection of Changes of Variance. Journal of the American Statistical Association 89, 913-923.
step2c
Description
Computes step 2c of the icss Algorithm.
Usage
step2c(e, bb, sig.lev, kmax, alpha)
Arguments
e |
Stationary variable on which the constancy of unconditional variance is tested. |
bb |
The output of function steps1.to.2b. |
sig.lev |
Significance level. |
kmax |
Maximum lag to be used for the long-run estimation of the fourth order moment of the innovations. |
alpha |
Tail index. |
Details
Used internally by icss.
Value
-
nb: Number of breaks. -
tb: Time of the breaks.
References
C. Inclan & G.C. Tiao (1994): Use of Cumulative Sums of Squares for Retrospective Detection of Changes of Variance. Journal of the American Statistical Association 89, 913-923.
step3
Description
Computes step 3 of the icss Algorithm.
Usage
step3(e, bb, sig.lev, kmax, alpha)
Arguments
e |
Stationary variable on which the constancy of unconditional variance is tested. |
bb |
The output of step2c, A list with the number of break and the time of the breaks. |
sig.lev |
Significance level. |
kmax |
Maximum lag to be used for the long-run estimation of the fourth order moment of the innovations. |
alpha |
Tail index. |
Details
Used internally by icss.
Value
-
nb: Number of breaks. -
tb: Time of the breaks.
References
C. Inclan & G.C. Tiao (1994): Use of Cumulative Sums of Squares for Retrospective Detection of Changes of Variance. Journal of the American Statistical Association 89, 913-923.
steps1.to.2b
Description
Computes steps 1 to 2b of the icss Algorithm.
Usage
steps1.to.2b(e, sig.lev, kmax, alpha)
Arguments
e |
Stationary variable on which the constancy of unconditional variance is tested. |
sig.lev |
Significance level. |
kmax |
Maximum lag to be used for the long-run estimation of the fourth order moment of the innovations. |
alpha |
Tail index. |
Details
Used internally by icss.
Value
-
nb: Number of breaks. -
tb: Time of the breaks. -
kappa.max: Maximum value of the kappa_test if there is no break. -
p.val: p-value.
References
C. Inclan & G.C. Tiao (1994): Use of Cumulative Sums of Squares for Retrospective Detection of Changes of Variance. Journal of the American Statistical Association 89, 913-923.
taula.icss
Description
When there are breaks in the unconditional variance, it creates a matrix with the different periods and estimated variances.
Usage
taula.icss(x)
Arguments
x |
An object with the output of icss. |
Details
Used internally by print.icss. If there are no break returns NA value.
Value
Matrix with the different periods and estimated variances.
taula.micss
Description
When there are breaks in the unconditional variance, it creates a matrix with the different periods, estimated variances, values of kappa_test and p.val.kappa.
Usage
taula.micss(x, alpha)
Arguments
x |
An object with the output of icss. |
alpha |
Value of the tail index between 2 and 4 |
Details
Used internally by print.micss.
Value
Matrix with the different periods, estimated variances, values of kappa_test and p.val.kappa.
var.est.icss
Description
Computes the variance of each period according to the breaks found with icss or micss.
Usage
var.est.icss(x)
Arguments
x |
Value
A vector with the variances.
Whitening
Description
Eliminates the autocorrelation of a variable using an AR model.
Usage
whitening(y, kmax = NULL)
Arguments
y |
A numeric vector. Variable to be whiten. |
kmax |
Maximum lag to be used for the long-run estimation of the variance. If not specified uses [12*(t/100)^(1/4)]. |
Details
Selects the model using the Bayes Information Criterium.
Value
-
e: Whiten variable. -
rho: Vector of autoregressive parameters. -
lag: number of lags used.
Examples
whitening(rnorm(100))