Package 'xtpqardl'

Coefficients Method for xtpqardl Objects

Description

Extract estimated coefficients from a Panel Quantile ARDL model.

Usage

## S3 method for class 'xtpqardl'
coef(object, type = c("beta", "rho", "all"), ...)

Arguments

object	An object of class "xtpqardl".
type	Character string specifying which coefficients to extract: "beta" for long-run coefficients (default), "rho" for ECT speed of adjustment, or "all" for both.
...	Additional arguments (currently unused).

Value

A named numeric vector or list of coefficients.

Examples

data(pqardl_sample)
fit <- xtpqardl(
  formula = d_y ~ d_x1 + d_x2,
  data = pqardl_sample,
  id = "country",
  time = "year",
  lr = c("L_y", "x1", "x2"),
  tau = c(0.25, 0.50, 0.75)
)
coef(fit)
coef(fit, type = "rho")

---

Compute Impulse Response Function

Description

Computes the impulse response function (IRF) for Panel Quantile ARDL models, showing the response to a one-unit shock via the error correction mechanism.

Usage

compute_irf(object, horizon = 20)

Arguments

object	An object of class "xtpqardl".
horizon	Integer specifying the number of periods for the IRF. Default is 20.

Details

The IRF for the error correction model is computed as:

$IRF_t(\tau) = (1 + \rho(\tau))^t$

which shows the decay of a unit shock over time through the error correction mechanism. Values approach zero as $t \to \infty$ when $-1 < \rho(\tau) < 0$.

Value

A matrix with rows representing time periods and columns representing quantiles. Each entry shows the response at that period for that quantile.

Examples

data(pqardl_sample)
fit <- xtpqardl(
  formula = d_y ~ d_x1 + d_x2,
  data = pqardl_sample,
  id = "country",
  time = "year",
  lr = c("L_y", "x1", "x2"),
  tau = c(0.25, 0.50, 0.75)
)
irf <- compute_irf(fit, horizon = 15)
print(irf)

---

Simulated Panel Data for PQARDL Estimation

Description

A simulated panel dataset for demonstrating Panel Quantile ARDL estimation. Contains 10 countries observed over 30 years with variables suitable for error correction modeling.

Usage

pqardl_sample

Format

A data frame with 300 rows and 9 variables:

country

Factor indicating the panel unit (10 countries)

year

Integer year variable (1990-2019)

y

Dependent variable in levels (e.g., GDP per capita)

x1

First explanatory variable in levels (e.g., investment)

x2

Second explanatory variable in levels (e.g., trade openness)

L_y

Lagged dependent variable (y at t-1)

d_y

First difference of y

d_x1

First difference of x1

d_x2

First difference of x2

Details

The data are simulated from a panel error correction model with heterogeneous adjustment speeds across countries. The true long-run relationship is:

$y_{it} = \beta_1 x_{1,it} + \beta_2 x_{2,it} + \mu_i + \varepsilon_{it}$

with error correction dynamics:

$\Delta y_{it} = \rho_i (y_{i,t-1} - \beta_1 x_{1,i,t-1} - \beta_2 x_{2,i,t-1}) + \gamma_1 \Delta x_{1,it} + \gamma_2 \Delta x_{2,it} + u_{it}$

where $\rho_i \sim U(-0.6, -0.2)$ varies by panel.

Source

Simulated data for package demonstration.

Examples

data(pqardl_sample)
head(pqardl_sample)

# Check panel structure
table(pqardl_sample$country)

---

Print Method for irf.xtpqardl Objects

Description

Print Method for irf.xtpqardl Objects

Usage

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

Arguments

x	An object of class "irf.xtpqardl".
...	Additional arguments (currently unused).

Value

Invisibly returns the input object.

---

Print Method for summary.xtpqardl Objects

Description

Print Method for summary.xtpqardl Objects

Usage

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

Arguments

x	An object of class "summary.xtpqardl".
digits	Number of significant digits to display. Default is 4.
...	Additional arguments (currently unused).

Value

Invisibly returns the input object.

---

Print Method for wald_test.xtpqardl Objects

Description

Print Method for wald_test.xtpqardl Objects

Usage

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

Arguments

x	An object of class "wald_test.xtpqardl".
...	Additional arguments (currently unused).

Value

Invisibly returns the input object.

---

Print Method for xtpqardl Objects

Description

Print Method for xtpqardl Objects

Usage

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

Arguments

x	An object of class "xtpqardl".
...	Additional arguments (currently unused).

Value

Invisibly returns the input object.

---

Summary Method for xtpqardl Objects

Description

Produces a detailed summary of Panel Quantile ARDL estimation results, including long-run coefficients, ECT speed of adjustment, half-life of adjustment, and short-run parameters by quantile.

Usage

## S3 method for class 'xtpqardl'
summary(object, ...)

Arguments

object	An object of class "xtpqardl".
...	Additional arguments (currently unused).

Value

An object of class "summary.xtpqardl" containing formatted tables of results.

Examples

data(pqardl_sample)
fit <- xtpqardl(
  formula = d_y ~ d_x1 + d_x2,
  data = pqardl_sample,
  id = "country",
  time = "year",
  lr = c("L_y", "x1", "x2"),
  tau = c(0.25, 0.50, 0.75)
)
summary(fit)

---

Variance-Covariance Matrix for xtpqardl Objects

Description

Extract the variance-covariance matrix of the estimated parameters.

Usage

## S3 method for class 'xtpqardl'
vcov(object, type = c("beta", "rho"), ...)

Arguments

object	An object of class "xtpqardl".
type	Character string specifying which covariance matrix to extract: "beta" for long-run coefficients (default), or "rho" for ECT coefficients.
...	Additional arguments (currently unused).

Value

A variance-covariance matrix.

---

Wald Test for Parameter Equality Across Quantiles

Description

Performs Wald tests for the null hypothesis that parameters are equal across different quantiles. Tests both long-run coefficients (beta) and ECT speed of adjustment (rho).

Usage

wald_test(object, joint = TRUE)

## S3 method for class 'xtpqardl'
wald_test(object, joint = TRUE)

Arguments

object	An object of class "xtpqardl".
joint	Logical. If TRUE (default), performs joint test for all coefficients. If FALSE, performs individual tests for each variable.

Details

The Wald test statistic is computed as:

$W = (R\theta)'[R V R']^{-1}(R\theta)$

where $\theta$ is the vector of coefficients, $V$ is the variance-covariance matrix, and $R$ is a restriction matrix testing equality of coefficients across quantiles.

Under the null hypothesis of equal coefficients, $W$ follows a chi-squared distribution with degrees of freedom equal to the number of restrictions.

Value

An object of class "wald_test.xtpqardl" containing:

beta_test

Wald test result for long-run coefficients

rho_test

Wald test result for ECT coefficients

individual_beta

Individual tests for each long-run variable (if joint = FALSE)

tau

Quantiles tested

References

Koenker R, Bassett G (1982). "Tests of Linear Hypotheses and L1 Estimation." Econometrica, 50(6), 1577-1583. doi:10.2307/1913398

Examples

data(pqardl_sample)
fit <- xtpqardl(
  formula = d_y ~ d_x1 + d_x2,
  data = pqardl_sample,
  id = "country",
  time = "year",
  lr = c("L_y", "x1", "x2"),
  tau = c(0.25, 0.50, 0.75)
)
wald_test(fit)

---

Panel Quantile Autoregressive Distributed Lag Model

Description

Estimate Panel Quantile ARDL (PQARDL) models that combine panel ARDL methodology with quantile regression. Supports Pooled Mean Group (PMG), Mean Group (MG), and Dynamic Fixed Effects (DFE) estimators.

Usage

xtpqardl(
  formula,
  data,
  id,
  time,
  lr,
  tau = c(0.25, 0.5, 0.75),
  p = 1,
  q = 1,
  model = c("pmg", "mg", "dfe"),
  lagsel = NULL,
  pmax = 4,
  qmax = 4,
  constant = TRUE
)

Arguments

formula	A formula specifying the model. The response variable should be in first differences (e.g., d.y ~ d.x1 + d.x2), where short-run dynamics are estimated.
data	A data frame containing panel data with variables specified in the formula and lr argument.
id	Character string specifying the panel (cross-section) identifier variable name.
time	Character string specifying the time variable name.
lr	Character vector of long-run level variable names. The first element should be the lagged dependent variable level (for the error correction term), and remaining elements are the long-run explanatory variables.
tau	Numeric vector of quantiles to estimate, each in (0,1). Default is c(0.25, 0.50, 0.75).
p	Integer specifying the autoregressive lag order for the dependent variable. Default is 1.
q	Integer or integer vector specifying the distributed lag order(s) for explanatory variables. If a single integer, the same lag order is applied to all variables. Default is 1.
model	Character string specifying the estimation method: "pmg" for Pooled Mean Group (default), "mg" for Mean Group, or "dfe" for Dynamic Fixed Effects.
lagsel	Character string for automatic lag selection. If "bic" or "aic", optimal lag orders are selected using the specified criterion. Default is NULL (no automatic selection).
pmax	Maximum p to consider in lag selection. Default is 4.
qmax	Maximum q to consider in lag selection. Default is 4.
constant	Logical. Include a constant term? Default is TRUE.

Details

The PQARDL model extends the standard panel ARDL framework to allow for heterogeneous effects across the conditional distribution of the response variable. The error correction representation is:

$\Delta y_{it} = \rho_i(\tau) \cdot ECT_{i,t-1} + \sum_{j=1}^{p-1} \phi_{ij} \Delta y_{i,t-j} + \sum_{m=0}^{q-1} \theta_{im} \Delta x_{i,t-m} + \varepsilon_{it}(\tau)$

where $ECT_{i,t-1} = y_{i,t-1} - \beta(\tau)' X_{i,t-1}$ is the error correction term, $\rho(\tau)$ is the speed of adjustment (should be negative for convergence), and $\beta(\tau)$ are the long-run cointegrating parameters.

Value

An object of class "xtpqardl" containing:

beta_mg

Matrix of mean group long-run coefficients across quantiles

rho_mg

Vector of mean group ECT speed of adjustment by quantile

halflife_mg

Vector of mean group half-life of adjustment by quantile

sr_mg

Matrix of mean group short-run coefficients

phi_mg

Matrix of mean group AR coefficients (if p > 1)

beta_V

Variance-covariance matrix for beta_mg

rho_V

Variance-covariance matrix for rho_mg

beta_all

Matrix of per-panel long-run coefficients

rho_all

Matrix of per-panel ECT coefficients

halflife_all

Matrix of per-panel half-life values

tau

Vector of estimated quantiles

p

AR lag order used

q

Distributed lag order(s) used

model

Estimation method used

n_obs

Total number of observations

n_panels

Number of panels

valid_panels

Number of successfully estimated panels

depvar

Dependent variable name

lrvars

Long-run variable names

call

The matched call

References

Pesaran MH, Shin Y, Smith RP (1999). "Pooled Mean Group Estimation of Dynamic Heterogeneous Panels." Journal of the American Statistical Association, 94(446), 621-634. doi:10.1080/01621459.1999.10474156

Cho JS, Kim TH, Shin Y (2015). "Quantile Cointegration in the Autoregressive Distributed-Lag Modeling Framework." Journal of Econometrics, 188(1), 281-300. doi:10.1016/j.jeconom.2015.02.030

Bildirici M, Kayikci F (2022). "Uncertainty, Renewable Energy, and CO2 Emissions in Top Renewable Energy Countries: A Panel Quantile Regression Approach." Energy, 247, 124303. doi:10.1016/j.energy.2022.124303

Koenker R, Bassett G (1978). "Regression Quantiles." Econometrica, 46(1), 33-50. doi:10.2307/1913643

Examples

# Load example panel data
data(pqardl_sample)

# Estimate PQARDL model at 25th, 50th, and 75th quantiles
fit <- xtpqardl(
  formula = d_y ~ d_x1 + d_x2,
  data = pqardl_sample,
  id = "country",
  time = "year",
  lr = c("L_y", "x1", "x2"),
  tau = c(0.25, 0.50, 0.75),
  model = "pmg"
)

# View results
summary(fit)

# Wald test for parameter equality across quantiles
wald_test(fit)

---