Package 'qardlr' reference manual

Title:	Quantile Autoregressive Distributed Lag Model
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.
Authors:	Muhammad Alkhalaf [aut, cre, cph] (ORCID: <https://orcid.org/0009-0002-2677-9246>), Jin Seo Cho [ctb] (Original methodology), Tae-Hwan Kim [ctb] (Original methodology), Yongcheol Shin [ctb] (Original methodology)
Maintainer:	Muhammad Alkhalaf <[email protected]>
License:	GPL-3
Version:	1.0.1
Built:	2026-05-28 07:28:55 UTC
Source:	https://github.com/muhammedalkhalaf/qardlr

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

Arguments

object

An object of class "qardl".

type

Character. Which coefficients to extract: "beta", "phi", "gamma", or "all". Default is "all".

...

Additional arguments (unused).

Value

Matrix or list of coefficient matrices.

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

Arguments

x

Object of class "qardl_rolling".

which

Character. Which parameter to plot: "beta", "phi", "gamma", or "rho". Default is "beta".

var

Integer or character. Which variable to plot (for beta/gamma). Default is 1.

tau_idx

Integer. Which quantile index to plot. Default is all.

...

Additional arguments passed to plot.

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

Arguments

object

An object of class "qardl".

newdata

Optional data frame for prediction. If NULL, uses the original data.

tau

Quantile(s) for prediction. Default uses all fitted quantiles.

...

Additional arguments (unused).

Value

Matrix of predicted quantiles.

Print BIC Grid

Description

Prints a formatted BIC grid for lag selection.

Usage

print_bic_grid(bic_result, digits = 3)
print_bic_grid(bic_result, digits = 3)

Arguments

bic_result

Result from qardl_bic_select.

digits

Number of decimal places. Default is 3.

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)

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 QARDL Results

Description

Print method for QARDL estimation results.

Usage

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

Arguments

x

An object of class "qardl".

digits

Number of decimal places. Default is 4.

...

Additional arguments (unused).

Value

Invisible x.

Print Monte Carlo Results

Description

Print Monte Carlo Results

Usage

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

Arguments

x

Object of class "qardl_mc".

digits

Number of decimal places. Default is 4.

...

Additional arguments (unused).

Value

Invisible x.

Print Rolling QARDL Results

Description

Print Rolling QARDL Results

Usage

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

Arguments

x

Object of class "qardl_rolling".

...

Additional arguments (unused).

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

Arguments

x

Object of class "qardl_wald".

digits

Number of decimal places. Default is 4.

...

Additional arguments (unused).

Value

Invisible x.

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

Arguments

formula

A formula of the form y ~ x1 + x2 + ... where y is the dependent variable and x1, x2, ... are covariates.

data

A data frame containing the variables in the formula.

tau

Numeric vector of quantiles to estimate. Must be in (0, 1). Default is c(0.25, 0.50, 0.75).

p

Integer. AR lag order for the dependent variable. If 0, automatically selected via BIC. Default is 0.

q

Integer. Distributed lag order for covariates. If 0, automatically selected via BIC. Default is 0.

pmax

Integer. Maximum AR lag order for BIC selection. Default is 7.

qmax

Integer. Maximum DL lag order for BIC selection. Default is 7.

ecm

Logical. If TRUE, estimate Error Correction Model parameterization. Default is FALSE.

constant

Logical. If TRUE, include an intercept. Default is TRUE.

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

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)

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

Arguments

y

Numeric vector of dependent variable.

X

Matrix of covariates.

pmax

Integer. Maximum AR lag order to consider. Default is 7.

qmax

Integer. Maximum distributed lag order to consider. Default is 7.

constant

Logical. Include intercept. Default is TRUE.

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

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)

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)

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
)
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

formula

A formula of the form y ~ x1 + x2 + ....

data

A data frame containing the variables.

tau

Numeric vector of quantiles. Default is c(0.25, 0.50, 0.75).

p

Integer. AR lag order. Default is 1.

q

Integer. Distributed lag order. Default is 1.

window

Integer. Rolling window size. If 0, uses 10% of sample size.

method

Character. Either "rolling" (fixed window) or "recursive" (expanding window). Default is "rolling".

constant

Logical. Include intercept. Default is TRUE.

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

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)

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
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

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)

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
)
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

nobs

Integer. Sample size for each simulation. Default is 200.

reps

Integer. Number of Monte Carlo replications. Default is 1000.

tau

Numeric vector of quantiles. Default is c(0.25, 0.50, 0.75).

p

Integer. AR lag order. Default is 1.

q

Integer. Distributed lag order. Default is 1.

k

Integer. Number of covariates. Default is 1.

beta_true

Numeric vector. True long-run parameters (length k). Default is rep(1, k).

phi_true

Numeric vector. True AR parameters (length p). Default is rep(0.5, p).

gamma_true

Numeric vector. True impact parameters (length k). Default is rep(0.3, k).

sigma_u

Numeric. Standard deviation of the error term. Default is 1.

sigma_x

Numeric. Standard deviation of covariate innovations. Default is 1.

seed

Integer. Random seed for reproducibility. Default is NULL.

parallel

Logical. Use parallel processing. Default is FALSE.

ncores

Integer. Number of cores for parallel processing. Default is parallel::detectCores - 1.

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

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)

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

Arguments

x

An object of class "qardl".

type

Character. Type of table: "latex", "html", or "text". Default is "text".

include

Character vector. Which parameters to include: "beta", "phi", "gamma", "rho". Default is c("beta", "gamma").

stars

Logical. Include significance stars. Default is TRUE.

digits

Integer. Number of decimal places. Default is 3.

caption

Character. Table caption. Default is NULL.

label

Character. LaTeX label. Default is NULL.

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"))

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

Arguments

object

An object of class "qardl".

type

Character string specifying which parameters to test: "all" (default), "beta" (long-run), "phi" (AR), "gamma" (short-run impact), or "rho" (ECM speed of adjustment).

pairwise

Logical. If TRUE, perform pairwise tests between adjacent quantiles. Default is FALSE.

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

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)

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

Arguments

object

An object of class "qardl".

wald

Logical. Include Wald tests for parameter constancy. Default is TRUE.

digits

Number of decimal places. Default is 4.

...

Additional arguments (unused).

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

Arguments

object

An object of class "qardl".

type

Character. Which covariance to extract: "beta", "phi", "gamma", or "all". Default is "all".

...

Additional arguments (unused).

Value

Array or list of covariance arrays.

Package 'qardlr'

Help Index

Coefficients Method for QARDL

Description

Usage

Arguments

Value

Plot Rolling QARDL Results

Description

Usage

Arguments

Value

Predict Method for QARDL

Description

Usage

Arguments

Value

Print BIC Grid

Description

Usage

Arguments

Value

Examples

Print QARDL Results

Description

Usage

Arguments

Value

Print Monte Carlo Results

Description

Usage

Arguments

Value

Print Rolling QARDL Results

Description

Usage

Arguments

Value

Print QARDL Wald Test Results

Description

Usage

Arguments

Value

Quantile Autoregressive Distributed Lag Model Estimation

Description

Usage

Arguments

Details

Value

References

See Also

Examples

BIC-Based Lag Order Selection for QARDL

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Rolling Window QARDL Estimation

Description

Usage

Arguments

Details

Value

References

See Also

Examples

Simulated QARDL Dataset

Description

Usage

Format

Details

References

Examples

Monte Carlo Simulation for QARDL

Description

Usage