Package 'esreg' reference manual

Title:	Joint Quantile and Expected Shortfall Regression
Description:	Simultaneous modeling of the quantile and the expected shortfall of a response variable given a set of covariates, see Dimitriadis and Bayer (2019) <doi:10.1214/19-EJS1560>.
Authors:	Sebastian Bayer [aut, cre], Timo Dimitriadis [aut]
Maintainer:	Sebastian Bayer <[email protected]>
License:	GPL-3
Version:	0.6.2
Built:	2024-10-30 06:50:17 UTC
Source:	CRAN

Joint Quantile and Expected Shortfall Regression

Description

Estimates a joint linear regression model for the pair (VaR, ES):

$Q_\alpha(Y | Xq) = Xq'\beta_q$

$ES_\alpha(Y | Xe) = Xe'\beta_e$

Usage

esreg(...)

## S3 method for class 'formula'
esreg(
  formula,
  data = parent.frame(),
  alpha,
  g1 = 2L,
  g2 = 1L,
  early_stopping = 10,
  ...
)

## Default S3 method:
esreg(xq, xe, y, alpha, g1 = 2L, g2 = 1L, early_stopping = 10, ...)
esreg(...)

## S3 method for class 'formula'
esreg(
  formula,
  data = parent.frame(),
  alpha,
  g1 = 2L,
  g2 = 1L,
  early_stopping = 10,
  ...
)

## Default S3 method:
esreg(xq, xe, y, alpha, g1 = 2L, g2 = 1L, early_stopping = 10, ...)

Arguments

`...`	Further arguments (does not apply here)
`formula`	Formula: y ~ x1 + x2 ... \| x1 + x2 ... where the first part after the response variable specifies the quantile equation and the second the expected shortfall part. If only one set of regressors is provided it is used for both model specifications.
`data`	data.frame that holds the variables
`alpha`	Probability level
`g1`	1, 2 (see G1_fun, G1_prime_fun), defaults to 1
`g2`	1, 2, 3, 4, 5 (see G2_curly_fun, G2_fun, G2_prime_fun). defaults to 2
`early_stopping`	Stop the iterated local search if there is no improvement in early_stopping steps.
`xq`	Explanatory variables for the quantile regression equation
`xe`	Explanatory variables for the expected shortfall regression equation
`y`	Response vector

Value

An esreg object

References

A Joint Quantile and Expected Shortfall Regression Framework

Examples

# Simulate data (DGP-(2) in the linked paper)
set.seed(0)
x <- rchisq(1000, df=1)
y <- -x + (1 + 0.5 * x) * rnorm(1000)

# True quantile and expected shortfall regression parameters (for alpha=0.025)
alpha=0.025
true_pars <- c(-1.959964, -1.979982, -2.337803, -2.168901)

# Estimate the model using the standard settings
fit <- esreg(y ~ x, alpha=alpha)

# Compare the different variance-covariance estimators
cov1 <- vcov(object=fit, sparsity="iid", sigma_est="ind")
cov2 <- vcov(object=fit, sparsity="nid", sigma_est="scl_N")
cov3 <- vcov(object=fit, sparsity="nid", sigma_est="scl_sp")

print("Comparison of the variance-covariance estimators")
print(cbind(Truth=true_pars,
            Estimate=coef(fit),
            SE_iid_ind=sqrt(diag(cov1)),
            SE_nid_N=sqrt(diag(cov2)),
            SE_nid_sp=sqrt(diag(cov3))))

# Compares estimates using different G2 functions
fit1 <- esreg(y ~ x, alpha=alpha, g2=1)
fit2 <- esreg(y ~ x, alpha=alpha, g2=2)
fit3 <- esreg(y ~ x, alpha=alpha, g2=3)
fit4 <- esreg(y ~ x, alpha=alpha, g2=4)
fit5 <- esreg(y ~ x, alpha=alpha, g2=5)
fits <- sapply(list(fit1, fit2, fit3, fit4, fit5), coef)
colnames(fits) <- sapply(1:5, function(i) esreg:::.G_function_names(1, i)[2])
print("Comparison of the five G2 functions")
print(rbind(Truth=true_pars, t(fits)))

# Usage of different covariates
x <- rchisq(1000, df=1)
noise <- rnorm(1000)
y <- -x + (1 + 0.5 * x) * rnorm(1000)
fit <- esreg(y ~ x | x + noise, alpha=0.025)
print("Using different covariates for VaR and ES")
print(summary(fit))

# Simulate data (DGP-(2) in the linked paper)
set.seed(0)
x <- rchisq(1000, df=1)
y <- -x + (1 + 0.5 * x) * rnorm(1000)

# True quantile and expected shortfall regression parameters (for alpha=0.025)
alpha=0.025
true_pars <- c(-1.959964, -1.979982, -2.337803, -2.168901)

# Estimate the model using the standard settings
fit <- esreg(y ~ x, alpha=alpha)

# Compare the different variance-covariance estimators
cov1 <- vcov(object=fit, sparsity="iid", sigma_est="ind")
cov2 <- vcov(object=fit, sparsity="nid", sigma_est="scl_N")
cov3 <- vcov(object=fit, sparsity="nid", sigma_est="scl_sp")

print("Comparison of the variance-covariance estimators")
print(cbind(Truth=true_pars,
            Estimate=coef(fit),
            SE_iid_ind=sqrt(diag(cov1)),
            SE_nid_N=sqrt(diag(cov2)),
            SE_nid_sp=sqrt(diag(cov3))))

# Compares estimates using different G2 functions
fit1 <- esreg(y ~ x, alpha=alpha, g2=1)
fit2 <- esreg(y ~ x, alpha=alpha, g2=2)
fit3 <- esreg(y ~ x, alpha=alpha, g2=3)
fit4 <- esreg(y ~ x, alpha=alpha, g2=4)
fit5 <- esreg(y ~ x, alpha=alpha, g2=5)
fits <- sapply(list(fit1, fit2, fit3, fit4, fit5), coef)
colnames(fits) <- sapply(1:5, function(i) esreg:::.G_function_names(1, i)[2])
print("Comparison of the five G2 functions")
print(rbind(Truth=true_pars, t(fits)))

# Usage of different covariates
x <- rchisq(1000, df=1)
noise <- rnorm(1000)
y <- -x + (1 + 0.5 * x) * rnorm(1000)
fit <- esreg(y ~ x | x + noise, alpha=0.025)
print("Using different covariates for VaR and ES")
print(summary(fit))

This function matches the estfun function of the sandwich package and returns the estimating functions for the fitted model. It can for instance be used for an OPG estimator of the sigma matrix. For esreg, the dimension of the estimating functions is n x (kq + ke).

Usage

estfun.esreg(x, ...)
estfun.esreg(x, ...)

Arguments

`x`	An esreg object
`...`	Further arguments (does not apply here)

Lambda Matrix

Description

Estimate the lambda matrix.

Usage

lambda_matrix(object, sparsity, bandwidth_estimator, misspec)
lambda_matrix(object, sparsity, bandwidth_estimator, misspec)

Arguments

`object`	An esreg object
`sparsity`	The estimator to be used for the sparsity in $\Lambda$ , see density_quantile_function iid - Piecewise linear interpolation of the distribution nid - Hendricks and Koenker sandwich
`bandwidth_estimator`	The bandwidth estimator to be used for the iid and nid sparsity estimator, see density_quantile_function Bofinger Chamberlain Hall-Sheather
`misspec`	if TRUE, the estimator accounts for potential misspecification in the model

Sigma Matrix

Description

Estimate the sigma matrix.

Usage

sigma_matrix(object, sigma_est, misspec)
sigma_matrix(object, sigma_est, misspec)

Arguments

object

An esreg object

sigma_est

The estimator to be used for $\Sigma$ , see conditional_truncated_variance

ind - Variance over all negative residuals
scl_N - Scaling with the normal distribution
scl_sp - Scaling with the kernel density function

misspec

if TRUE, the estimator accounts for potential misspecification in the model

Covariance Estimation

Description

Estimate the variance-covariance matrix of the joint (VaR, ES) estimator

Usage

## S3 method for class 'esreg'
vcov(object, method = "asymptotic", ...)
## S3 method for class 'esreg'
vcov(object, method = "asymptotic", ...)

Arguments

`object`	An esreg object
`method`	For asymptotic use vcovA, for boot use vcovB
`...`	All possible values which can be passed to vcovA and vcovB

Asymptotic Covariance Estimation

Description

Estimate the variance-covariance matrix of the joint (VaR, ES) estimator by the sandwich formula:

$\lambda^{-1} \Sigma \lambda^{-1}$

Several estimators are available for both matrices and the default options are selected to take into account possible misspecifications in the underlying data.

Usage

vcovA(
  object,
  sigma_est = "scl_sp",
  sparsity = "nid",
  misspec = TRUE,
  bandwidth_estimator = "Hall-Sheather"
)
vcovA(
  object,
  sigma_est = "scl_sp",
  sparsity = "nid",
  misspec = TRUE,
  bandwidth_estimator = "Hall-Sheather"
)

Arguments

`object`	An esreg object
`sigma_est`	The estimator to be used for $\Sigma$ , see conditional_truncated_variance ind - Variance over all negative residuals scl_N - Scaling with the normal distribution scl_sp - Scaling with the kernel density function
`sparsity`	The estimator to be used for the sparsity in $\Lambda$ , see density_quantile_function iid - Piecewise linear interpolation of the distribution nid - Hendricks and Koenker sandwich
`misspec`	if TRUE, the estimator accounts for potential misspecification in the model
`bandwidth_estimator`	The bandwidth estimator to be used for the iid and nid sparsity estimator, see density_quantile_function Bofinger Chamberlain Hall-Sheather

Bootstrap Covariance Estimation

Description

Estimate the variance-covariance matrix of the joint (VaR, ES) estimator using the bootstrap.

Usage

vcovB(object, bootstrap_method = "iid", B = 1000)
vcovB(object, bootstrap_method = "iid", B = 1000)

Arguments

object

An esreg object

bootstrap_method

The bootstrap sampling scheme to be used

iid - The iid bootstrap of Efron (1979)

B

The number of bootstrap iterations

`r`	Vector of returns
`q`	Vector of quantiles
`e`	Vector of expected shortfalls
`alpha`	Probability level
`g1`	1, 2, see G1_fun
`g2`	1, 2, 3, 4, 5, see G2_curly_fun, G2_fun
`return_mean`	If TRUE returns the average tick loss, else the individual values

Package 'esreg'

Help Index

Joint Loss Function

Description

Usage

Arguments

References

Joint Quantile and Expected Shortfall Regression

Description

Usage

Arguments

Value

References

See Also

Examples

Estimating function

Description

Usage

Arguments

Lambda Matrix

Description

Usage

Arguments

Sigma Matrix

Description

Usage

Arguments

Covariance Estimation

Description

Usage

Arguments

Asymptotic Covariance Estimation

Description

Usage

Arguments

Bootstrap Covariance Estimation

Description

Usage

Arguments