| Type: | Package |
| Title: | Joint Quantile and Expected Shortfall Regression |
| Version: | 0.6.2 |
| Date: | 2023-05-12 |
| 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>. |
| License: | GPL-3 |
| Encoding: | UTF-8 |
| Imports: | quantreg, Rcpp, stats, Formula |
| LinkingTo: | Rcpp, RcppArmadillo |
| RoxygenNote: | 7.2.3 |
| NeedsCompilation: | yes |
| Packaged: | 2023-05-13 08:05:33 UTC; sebastian |
| Author: | Sebastian Bayer [aut, cre], Timo Dimitriadis [aut] |
| Maintainer: | Sebastian Bayer <sebastian.bayer@uni-konstanz.de> |
| Repository: | CRAN |
| Date/Publication: | 2023-05-13 08:30:02 UTC |
Specification Function
Description
G1
Usage
G1_fun(z, type)
Arguments
z |
Data |
type |
Choice of the G1 function:
|
Specification Function
Description
G1_prime
Usage
G1_prime_fun(z, type)
Arguments
z |
Data |
type |
Choice of the G1_prime function:
|
Specification Function
Description
G1_prime_prime
Usage
G1_prime_prime_fun(z, type)
Arguments
z |
Data |
type |
Choice of the G1_prime_prime function:
|
Specification Function
Description
G2_curly
Usage
G2_curly_fun(z, type)
Arguments
z |
Data |
type |
Choice of the G2_curly function:
|
Specification Function
Description
G2
Usage
G2_fun(z, type)
Arguments
z |
Data |
type |
Choice of the G2 function:
|
Specification Function
Description
G2_prime
Usage
G2_prime_fun(z, type)
Arguments
z |
Data |
type |
Choice of the G2_prime function:
|
Specification Function
Description
G2_prime_prime
Usage
G2_prime_prime(z, type)
Arguments
z |
Data |
type |
Choice of the G2_prime_prime function:
|
Vectorized call to the G1 / G2 functions
Description
Vectorized call to the G1 / G2 functions
Usage
G_vec(z, g, type)
Arguments
z |
Vector |
g |
String, either G1, G1_prime, G2_curly, G2 or G2_curly |
type |
Numeric, for G1: 1-2; G2: 1-5 (see G1_fun, G1_prime_fun, G2_curly_fun, G2_fun, G2_prime_fun) |
Cumulative Density Function at Quantile
Description
Returns the cumulative density function evaluated at quantile predictions. For a correctly specified model this should yield a value close to the quantile level.
Usage
cdf_at_quantile(y, x, q)
Arguments
y |
Vector of dependent data |
x |
Matrix of covariates including the intercept |
q |
Vector of quantile predictions |
Conditional Mean and Sigma
Description
Estimate the conditional mean and sigma of the dependent data conditional on covariates x
Usage
conditional_mean_sigma(y, x)
Arguments
y |
Vector of dependent data |
x |
Matrix of covariates including the intercept |
Conditional truncated variance
Description
Estimate the variance of y given x and given y <= 0. If approach is:
ind - Variance of all y where y <= 0.
scl_N or scl_sp - Assumes a location-scale model: y = x'b + (x'g)e. First, it estimates b and g using PMLE. Then, it computes the conditional truncated variance by integrating the truncated density of y.
Usage
conditional_truncated_variance(y, x, approach)
Arguments
y |
Vector of dependent data |
x |
Matrix of covariates including the intercept |
approach |
ind, scl_N or scl_sp |
Density Quantile Function
Description
Estimate the density quantile function
Usage
density_quantile_function(y, x, u, alpha, sparsity, bandwidth_estimator)
Arguments
y |
Vector of dependent data |
x |
Matrix of covariates |
u |
Quantile residuals |
alpha |
Probability level |
sparsity |
iid or ind |
bandwidth_estimator |
Bofinger, Chamberlain or Hall-Sheather |
References
For the iid and nid method, see Koenker (1994), and Hendricks and Koenker (1992). For the bandwidth types, see Bofinger (1975), Chamberlain (1994), and Hall and Sheather(1988).
Joint Loss Function
Description
Computes the joint (VaR, ES) loss
Usage
esr_loss(r, q, e, alpha, g1 = 2L, g2 = 1L, return_mean = TRUE)
Arguments
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 |
References
Fissler and Ziegel (2016)
Joint (VaR, ES) loss for a linear predictor
Description
Returns the loss for the parameter vector b
Usage
esr_rho_lp(b, y, xq, xe, alpha, g1 = 2L, g2 = 1L)
Arguments
b |
Parameter vector |
y |
Vector of dependent data |
xq |
Matrix of covariates for the quantile part |
xe |
Matrix of covariates for the expected shortfall part |
alpha |
Probability level |
g1 |
1, 2 (see G1_fun) |
g2 |
1, 2, 3, 4, 5 (see G2_curly_fun, G2_fun) |
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, ...)
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
See Also
vcov.esreg for covariance estimation
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))
Estimating function
Description
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, ...)
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)
Arguments
object |
An esreg object |
sparsity |
The estimator to be used for the sparsity in
|
bandwidth_estimator |
The bandwidth estimator to be used for the iid and nid sparsity estimator, see density_quantile_function
|
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)
Arguments
object |
An esreg object |
sigma_est |
The estimator to be used for
|
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", ...)
Arguments
object |
An esreg object |
method |
|
... |
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"
)
Arguments
object |
An esreg object |
sigma_est |
The estimator to be used for
|
sparsity |
The estimator to be used for the sparsity in
|
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
|
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)
Arguments
object |
An esreg object |
bootstrap_method |
The bootstrap sampling scheme to be used
|
B |
The number of bootstrap iterations |