Type:	Package
Title:	Quantile Autoregressive Distributed Lag Model
Version:	1.0.1
Date:	2026-03-07
Description:	Implements the Quantile Autoregressive Distributed Lag (QARDL) model of Cho, Kim and Shin (2015) <doi:10.1016/j.jeconom.2015.01.003>. Estimates quantile-specific long-run (beta), short-run autoregressive (phi), and impact (gamma) parameters. Features include BIC-based automatic lag selection, Error Correction Model (ECM) parameterization, Wald tests for parameter constancy across quantiles, rolling/recursive QARDL estimation, Monte Carlo simulation, and publication-ready output tables.
License:	GPL-3
URL:	https://github.com/muhammedalkhalaf/qardlr
BugReports:	https://github.com/muhammedalkhalaf/qardlr/issues
Depends:	R (≥ 3.5.0)
Imports:	quantreg (≥ 5.95), stats, MASS
Suggests:	testthat (≥ 3.0.0)
Encoding:	UTF-8
LazyData:	true
RoxygenNote:	7.3.3
Config/testthat/edition:	3
NeedsCompilation:	no
Packaged:	2026-03-09 13:37:40 UTC; acad_
Author:	Muhammad Alkhalaf [aut, cre, cph], Merwan Roudane [ctb] (Original Stata implementation), Jin Seo Cho [ctb] (Original methodology), Tae-Hwan Kim [ctb] (Original methodology), Yongcheol Shin [ctb] (Original methodology)
Maintainer:	Muhammad Alkhalaf <muhammedalkhalaf@gmail.com>
Repository:	CRAN
Date/Publication:	2026-03-13 13:10:02 UTC

qardlr: Quantile Autoregressive Distributed Lag Model

Description

The qardlr package implements the Quantile Autoregressive Distributed Lag (QARDL) model of Cho, Kim and Shin (2015). It provides tools for estimating quantile-specific long-run equilibrium relationships and short-run dynamics.

Main Functions

• qardl: Estimate QARDL model

• qardl_wald: Wald tests for parameter constancy

• qardl_rolling: Rolling window QARDL estimation

• qardl_simulate: Monte Carlo simulation

• qardl_table: Publication-ready tables

• qardl_bic_select: BIC-based lag selection

Key Features

• Quantile regression across multiple tau values

• BIC-based automatic lag selection (p, q)

• Error Correction Model (ECM) parameterization

• Long-run (beta), short-run AR (phi), and impact (gamma) parameters

• Wald tests for parameter constancy across quantiles

• Rolling/recursive QARDL estimation

• Monte Carlo simulation for finite-sample properties

• Publication-ready output tables (text, LaTeX, HTML)

Author(s)

Maintainer: Muhammad Alkhalaf muhammedalkhalaf@gmail.com (ORCID)

Other contributors:

• Merwan Roudane (Original Stata implementation) [contributor]

• Jin Seo Cho (Original methodology) [contributor]

• Tae-Hwan Kim (Original methodology) [contributor]

• Yongcheol Shin (Original methodology) [contributor]

References

Cho, J.S., Kim, T.-H., and Shin, Y. (2015). Quantile cointegration in the autoregressive distributed-lag modeling framework. Journal of Econometrics, 188(1), 281-300. doi:10.1016/j.jeconom.2015.01.003

Build HTML Table

Description

Build HTML Table

Usage

build_html_table(x, include, stars, digits, caption)

Build LaTeX Table

Description

Build LaTeX Table

Usage

build_latex_table(x, include, stars, digits, caption, label)

Build Text Table

Description

Build Text Table

Usage

build_text_table(x, include, stars, digits)

Coefficients Method for QARDL

Description

Extract coefficients from a QARDL model.

Usage

## S3 method for class 'qardl'
coef(object, type = c("all", "beta", "phi", "gamma"), ...)

Arguments

Value

Matrix or list of coefficient matrices.

Compute ECM Parameterization

Description

Computes Error Correction Model parameters.

Usage

compute_ecm(est, y, X, p, q, tau, constant = TRUE)

Arguments

Value

List containing ECM parameters.

Compute Long-Run Parameters

Description

Computes long-run (beta) parameters from QARDL estimates.

Usage

compute_longrun(est, k, tau)

Arguments

Value

List containing beta, beta_se, beta_cov, rho, rho_se.

Data Generating Process for QARDL

Description

Internal function to generate data from a QARDL DGP.

Usage

dgp_qardl(nobs, p, q, k, phi, gamma, sigma_u, sigma_x)

Arguments

Value

List with y and X.

Get Significance Stars

Description

Get Significance Stars

Usage

get_stars(pval)

Get HTML Stars

Description

Get HTML Stars

Usage

get_stars_html(pval)

Get LaTeX Stars

Description

Get LaTeX Stars

Usage

get_stars_latex(pval)

Plot Rolling QARDL Results

Description

Creates time series plots of rolling QARDL parameter estimates.

Usage

## S3 method for class 'qardl_rolling'
plot(x, which = c("beta", "phi", "gamma", "rho"), var = 1, tau_idx = NULL, ...)

Arguments

Value

Invisible NULL.

Predict Method for QARDL

Description

Generate predictions from a fitted QARDL model.

Usage

## S3 method for class 'qardl'
predict(object, newdata = NULL, tau = NULL, ...)

Arguments

Value

Matrix of predicted quantiles.

Print QARDL Results

Description

Print method for QARDL estimation results.

Usage

## S3 method for class 'qardl'
print(x, digits = 4, ...)

Arguments

Value

Invisible x.

Print Monte Carlo Results

Description

Print Monte Carlo Results

Usage

## S3 method for class 'qardl_mc'
print(x, digits = 4, ...)

Arguments

Value

Invisible x.

Print Rolling QARDL Results

Description

Print Rolling QARDL Results

Usage

## S3 method for class 'qardl_rolling'
print(x, ...)

Arguments

Value

Invisible x.

Print QARDL Wald Test Results

Description

Print QARDL Wald Test Results

Usage

## S3 method for class 'qardl_wald'
print(x, digits = 4, ...)

Arguments

Value

Invisible x.

Print BIC Grid

Description

Prints a formatted BIC grid for lag selection.

Usage

print_bic_grid(bic_result, digits = 3)

Arguments

Value

Invisible NULL. Called for side effect of printing.

Examples

data(qardl_sim)
y <- qardl_sim$y
X <- as.matrix(qardl_sim[, c("x1", "x2")])
bic_result <- qardl_bic_select(y, X, pmax = 4, qmax = 4)
print_bic_grid(bic_result)

Print Detailed Parameter Table

Description

Internal function to print detailed parameter table with t-stats and p-values.

Usage

print_detailed_table(est, se, tau, nobs, row_names, row_label, digits = 4)

Print Parameter Table

Description

Internal function to print formatted parameter table.

Usage

print_param_table(est, se, tau, nobs, row_label, digits = 4)

Quantile Autoregressive Distributed Lag Model Estimation

Description

Estimates the Quantile ARDL (QARDL) model of Cho, Kim & Shin (2015). The model estimates quantile-specific long-run equilibrium relationships and short-run dynamics between a dependent variable and covariates.

Usage

qardl(
  formula,
  data,
  tau = c(0.25, 0.5, 0.75),
  p = 0L,
  q = 0L,
  pmax = 7L,
  qmax = 7L,
  ecm = FALSE,
  constant = TRUE
)

Arguments

Details

The QARDL(p,q) model is specified as:

Q_{y_t}(\tau | \mathcal{F}_{t-1}) = c(\tau) + \sum_{i=1}^{p} \phi_i(\tau) y_{t-i} + \sum_{j=0}^{q-1} \gamma'_j(\tau) x_{t-j}

Long-run parameters are computed as:

\beta(\tau) = \frac{\sum_{j=0}^{q-1} \gamma_j(\tau)}{1 - \sum_{i=1}^{p} \phi_i(\tau)}

The speed of adjustment (ECM coefficient) is:

\rho(\tau) = \sum_{i=1}^{p} \phi_i(\tau) - 1

Negative \rho(\tau) indicates convergence to long-run equilibrium.

Value

An object of class "qardl" containing:

beta

Long-run parameters matrix (k x ntau)

beta_se

Standard errors for beta

phi

Short-run AR parameters matrix (p x ntau)

phi_se

Standard errors for phi

gamma

Short-run impact parameters matrix (k x ntau)

gamma_se

Standard errors for gamma

rho

Speed of adjustment parameters (ECM coefficient)

tau

Vector of estimated quantiles

p

AR lag order used

q

Distributed lag order used

nobs

Number of observations

k

Number of covariates

call

The matched call

formula

The model formula

data

The data used

qr_fits

List of quantreg fit objects

bic_grid

BIC grid if lag selection was performed

ecm

Whether ECM parameterization was used

References

Cho, J.S., Kim, T.-H., & Shin, Y. (2015). Quantile cointegration in the autoregressive distributed-lag modeling framework. Journal of Econometrics, 188(1), 281-300. doi:10.1016/j.jeconom.2015.01.003

See Also

qardl_rolling, qardl_simulate, qardl_wald, summary.qardl

Examples

# Load example data
data(qardl_sim)

# Basic QARDL estimation with automatic lag selection
fit <- qardl(y ~ x1 + x2, data = qardl_sim, tau = c(0.25, 0.50, 0.75))
summary(fit)

# QARDL with specified lags
fit2 <- qardl(y ~ x1 + x2, data = qardl_sim, tau = c(0.1, 0.5, 0.9), p = 2, q = 2)
print(fit2)

# QARDL-ECM parameterization
fit_ecm <- qardl(y ~ x1 + x2, data = qardl_sim, tau = c(0.25, 0.50, 0.75), ecm = TRUE)
summary(fit_ecm)

BIC-Based Lag Order Selection for QARDL

Description

Automatically selects optimal lag orders (p, q) for the QARDL model using the Bayesian Information Criterion (BIC) evaluated at the median quantile (tau = 0.5).

Usage

qardl_bic_select(y, X, pmax = 7L, qmax = 7L, constant = TRUE)

Arguments

Details

The BIC is computed using the Schwarz criterion at the median quantile:

BIC(p, q) = \log(\hat{\sigma}^2_{\tau=0.5}) + \frac{k_{pq} \log(n)}{n}

where k_{pq} is the number of parameters (p AR terms + q*k impact terms + constant) and \hat{\sigma}^2 is the estimated residual variance.

Value

A list containing:

p_opt

Optimal AR lag order

q_opt

Optimal distributed lag order

bic_grid

Matrix of BIC values (pmax x qmax)

bic_min

Minimum BIC value

References

Cho, J.S., Kim, T.-H., & Shin, Y. (2015). Quantile cointegration in the autoregressive distributed-lag modeling framework. Journal of Econometrics, 188(1), 281-300. doi:10.1016/j.jeconom.2015.01.003

See Also

qardl

Examples

data(qardl_sim)
y <- qardl_sim$y
X <- as.matrix(qardl_sim[, c("x1", "x2")])
bic_result <- qardl_bic_select(y, X, pmax = 5, qmax = 5)
print(bic_result$bic_grid)

Core QARDL Estimation

Description

Internal function for QARDL parameter estimation.

Usage

qardl_estimate(y, X, p, q, tau, constant = TRUE)

Arguments

Value

List of estimated parameters and their covariances.

Rolling Window QARDL Estimation

Description

Performs rolling or recursive window QARDL estimation to assess parameter stability over time.

Usage

qardl_rolling(
  formula,
  data,
  tau = c(0.25, 0.5, 0.75),
  p = 1L,
  q = 1L,
  window = 0L,
  method = c("rolling", "recursive"),
  constant = TRUE
)

Arguments

Details

Rolling window estimation helps detect structural breaks and assess parameter stability. The function estimates QARDL models on successive windows of data and tracks how parameters evolve over time.

For method = "rolling", a fixed window of size window is used. For method = "recursive", the window expands from window to the full sample.

Value

An object of class "qardl_rolling" containing:

beta

3D array of long-run parameters (k x ntau x nwindows)

phi

3D array of AR parameters (p x ntau x nwindows)

gamma

3D array of impact parameters (k x ntau x nwindows)

rho

Matrix of ECM coefficients (nwindows x ntau)

wald_beta

Matrix of beta constancy Wald statistics

wald_phi

Matrix of phi constancy Wald statistics

wald_gamma

Matrix of gamma constancy Wald statistics

dates

Vector of end dates for each window

window

Window size used

method

Method used ("rolling" or "recursive")

tau

Vector of quantiles

References

Cho, J.S., Kim, T.-H., & Shin, Y. (2015). Quantile cointegration in the autoregressive distributed-lag modeling framework. Journal of Econometrics, 188(1), 281-300. doi:10.1016/j.jeconom.2015.01.003

See Also

qardl, plot.qardl_rolling

Examples

data(qardl_sim)

# Rolling estimation with 50-observation window
roll <- qardl_rolling(y ~ x1 + x2, data = qardl_sim,
                      tau = c(0.25, 0.50, 0.75), p = 2, q = 2, window = 50)
print(roll)

# Recursive estimation
recur <- qardl_rolling(y ~ x1 + x2, data = qardl_sim,
                       tau = c(0.50), p = 2, q = 2,
                       window = 50, method = "recursive")
print(recur)

Simulated QARDL Dataset

Description

A simulated dataset for demonstrating QARDL estimation. The data is generated from a QARDL(2,2) process with two covariates.

Usage

qardl_sim

Format

A data frame with 200 observations and 3 variables:

y

Dependent variable generated from QARDL process

x1

First covariate (I(1) random walk)

x2

Second covariate (I(1) random walk)

Details

The data generating process follows:

y_t = 0.4 y_{t-1} + 0.2 y_{t-2} + 0.5 x_{1t} + 0.3 x_{2t} + u_t

where u_t \sim N(0, 1) and x_{it} are independent random walks.

True parameters:

• \phi_1 = 0.4, \phi_2 = 0.2

• \gamma_1 = 0.5, \gamma_2 = 0.3

• \beta_1 = 0.5/(1-0.6) = 1.25, \beta_2 = 0.3/(1-0.6) = 0.75

References

Cho, J.S., Kim, T.-H., & Shin, Y. (2015). Quantile cointegration in the autoregressive distributed-lag modeling framework. Journal of Econometrics, 188(1), 281-300. doi:10.1016/j.jeconom.2015.01.003

Examples

data(qardl_sim)
head(qardl_sim)
summary(qardl_sim)

# Estimate QARDL model
fit <- qardl(y ~ x1 + x2, data = qardl_sim, tau = c(0.25, 0.50, 0.75), p = 2, q = 2)
summary(fit)

Monte Carlo Simulation for QARDL

Description

Performs Monte Carlo simulation to assess the finite-sample properties of QARDL estimators under specified data generating processes.

Usage

qardl_simulate(
  nobs = 200L,
  reps = 1000L,
  tau = c(0.25, 0.5, 0.75),
  p = 1L,
  q = 1L,
  k = 1L,
  beta_true = NULL,
  phi_true = NULL,
  gamma_true = NULL,
  sigma_u = 1,
  sigma_x = 1,
  seed = NULL,
  parallel = FALSE,
  ncores = NULL
)

Arguments

Details

The data generating process is:

y_t = \sum_{i=1}^{p} \phi_i y_{t-i} + \sum_{j=1}^{k} \gamma_j x_{jt} + u_t

where u_t \sim N(0, \sigma_u^2) and x_{jt} follows a random walk with innovations \sim N(0, \sigma_x^2).

Value

An object of class "qardl_mc" containing:

beta_sim

Array of simulated beta estimates (k x ntau x reps)

phi_sim

Array of simulated phi estimates (p x ntau x reps)

gamma_sim

Array of simulated gamma estimates (k x ntau x reps)

beta_true

True beta values

phi_true

True phi values

gamma_true

True gamma values

bias_beta

Bias in beta estimates

rmse_beta

RMSE of beta estimates

coverage_beta

Empirical coverage of 95% CI for beta

reps

Number of replications

nobs

Sample size

tau

Vector of quantiles

References

Cho, J.S., Kim, T.-H., & Shin, Y. (2015). Quantile cointegration in the autoregressive distributed-lag modeling framework. Journal of Econometrics, 188(1), 281-300. doi:10.1016/j.jeconom.2015.01.003

See Also

qardl, print.qardl_mc

Examples

# Small simulation for illustration
mc <- qardl_simulate(nobs = 100, reps = 50, tau = c(0.25, 0.50, 0.75),
                     p = 1, q = 1, k = 1, seed = 123)
print(mc)

Generate Publication-Ready QARDL Tables

Description

Creates formatted tables suitable for academic publications from QARDL estimation results.

Usage

qardl_table(
  x,
  type = c("text", "latex", "html"),
  include = c("beta", "gamma"),
  stars = TRUE,
  digits = 3,
  caption = NULL,
  label = NULL
)

Arguments

Value

Character string containing the formatted table.

Examples

data(qardl_sim)
fit <- qardl(y ~ x1 + x2, data = qardl_sim, tau = c(0.25, 0.50, 0.75), p = 2, q = 2)
cat(qardl_table(fit, type = "text"))

Wald Tests for QARDL Parameter Constancy

Description

Performs Wald tests for parameter constancy across quantiles in a QARDL model. Tests whether parameters are equal across different quantile levels.

Usage

qardl_wald(
  object,
  type = c("all", "beta", "phi", "gamma", "rho"),
  pairwise = FALSE
)

Arguments

Details

The Wald test statistic is computed as:

W = (R\hat{\theta} - r)' [R \hat{V} R']^{-1} (R\hat{\theta} - r) \sim \chi^2(q)

where R is a restriction matrix testing equality across quantiles, \hat{\theta} is the vector of parameter estimates, and \hat{V} is the estimated covariance matrix.

Value

An object of class "qardl_wald" containing:

tests

Data frame of test results with columns: test, statistic, df, pvalue

pairwise_tests

Data frame of pairwise test results (if pairwise = TRUE)

type

Type of test performed

tau

Vector of quantiles

References

Cho, J.S., Kim, T.-H., & Shin, Y. (2015). Quantile cointegration in the autoregressive distributed-lag modeling framework. Journal of Econometrics, 188(1), 281-300. doi:10.1016/j.jeconom.2015.01.003

See Also

qardl, print.qardl_wald

Examples

data(qardl_sim)
fit <- qardl(y ~ x1 + x2, data = qardl_sim, tau = c(0.25, 0.50, 0.75), p = 2, q = 2)
wald_results <- qardl_wald(fit)
print(wald_results)

# Pairwise tests
wald_pairwise <- qardl_wald(fit, pairwise = TRUE)
print(wald_pairwise)

Summary of QARDL Results

Description

Provides a detailed summary of QARDL estimation results including parameter estimates, standard errors, t-statistics, p-values, and diagnostic tests.

Usage

## S3 method for class 'qardl'
summary(object, wald = TRUE, digits = 4, ...)

Arguments

Value

An object of class "summary.qardl" (invisibly).

Variance-Covariance Method for QARDL

Description

Extract variance-covariance matrices from a QARDL model.

Usage

## S3 method for class 'qardl'
vcov(object, type = c("all", "beta", "phi", "gamma"), ...)

Arguments

Value

Array or list of covariance arrays.

Wald Constancy Test

Description

Internal function to perform Wald test for parameter constancy.

Usage

wald_constancy_test(params, cov_array, nobs, param_name)

Arguments

Value

List with statistic, df, pvalue.

Pairwise Wald Tests

Description

Internal function for pairwise parameter equality tests.

Usage

wald_pairwise_tests(object, type)

Arguments

Value

Data frame of pairwise test results.