Package {fqardl}

Type:	Package
Title:	Fourier ARDL Methods: Quantile, Nonlinear, Multi-Threshold & Unit Root Tests
Version:	1.1.0
Date:	2026-10-03
Maintainer:	Muhammad Alkhalaf <muhammedalkhalaf@gmail.com>
Description:	Implementation of ARDL methods for cointegration analysis with structural breaks and asymmetric effects. Includes: (1) Fourier Quantile ARDL (FQARDL), quantile regression with Fourier approximation following the quantile ARDL of Cho, Kim and Shin (2015) <doi:10.1016/j.jeconom.2015.05.003>; (2) Fourier Nonlinear ARDL (FNARDL), asymmetric cointegration with partial sum decomposition following Shin, Yu and Greenwood-Nimmo (2014) <doi:10.1007/978-1-4899-8008-3_9>; (3) a deprecated Multi-Threshold NARDL (MTNARDL) function; (4) Fourier unit root tests, ADF and KPSS tests with Fourier terms following Enders and Lee (2012) <doi:10.1016/j.econlet.2012.04.081> and Becker, Enders and Lee (2006) <doi:10.1111/j.1467-9892.2006.00478.x>. Features automatic lag and frequency selection, Wald bounds statistics with the tables of Pesaran, Shin and Smith (2001) <doi:10.1002/jae.616> (no verdict with Fourier terms, for which the tables are not valid), recursive bootstrap bounds tests following Bertelli, Vacca and Zoia (2022) <doi:10.1016/j.econmod.2022.105987> and McNown, Sam and Goh (2018) <doi:10.1080/00036846.2017.1366643>, Wald tests for asymmetry, multiplier computation, and visualizations.
License:	GPL-3
URL:	https://github.com/muhammedalkhalaf/fqardl
BugReports:	https://github.com/muhammedalkhalaf/fqardl/issues
Encoding:	UTF-8
LazyData:	true
Depends:	R (≥ 3.5.0)
Imports:	quantreg (≥ 5.0), ggplot2 (≥ 3.0.0), tidyr, gridExtra, stats
Suggests:	testthat (≥ 3.0.0), knitr, rmarkdown, plotly, covr
RoxygenNote:	7.3.3
Config/testthat/edition:	3
NeedsCompilation:	no
Packaged:	2026-10-03 16:33:34 UTC; root
Author:	Muhammad Alkhalaf [aut, cre, cph]
Repository:	CRAN
Date/Publication:	2026-10-10 08:00:02 UTC

Fourier ARDL Methods for R

Description

Comprehensive implementation of advanced ARDL methodologies for cointegration analysis with structural breaks and asymmetric effects.

Main functions:

• fqardl: Fourier Quantile ARDL estimation

• fnardl: Fourier Nonlinear ARDL with asymmetry

• mtnardl: Multi-Threshold NARDL (deprecated, to be removed in 2.0.0; use ardlverse::mtnardl)

• fourier_adf_test: Fourier ADF unit root test

• fourier_kpss_test: Fourier KPSS stationarity test

• perform_bounds_test: Wald bounds statistics (no analytical verdict with Fourier terms)

• bootstrap_bounds_test: recursive bootstrap bounds test

Author(s)

Muhammad Alkhalaf contact@rufyqelngeh.com

References

Shin, Y., Yu, B., and Greenwood-Nimmo, M. (2014). Modelling Asymmetric Cointegration and Dynamic Multipliers in a Nonlinear ARDL Framework.

Enders, W., and Lee, J. (2012). The flexible Fourier form and Dickey-Fuller type unit root tests.

Pesaran, M. H., Shin, Y., and Smith, R. J. (2001). Bounds testing approaches to the analysis of level relationships.

Bootstrap Bounds Test

Description

Recursive bootstrap bounds test for the Fourier ARDL model estimated by fqardl. Rewritten in 1.1.0.

Usage

bootstrap_bounds_test(
  y,
  X,
  fourier,
  p,
  q,
  tau = NULL,
  case = 3,
  n_boot = 1000,
  verbose = FALSE,
  reselect = NULL,
  scheme = c("bvz", "mcnown"),
  level = 0.05,
  seed = NULL
)

Arguments

Details

The pseudo-samples are generated by the shared recursive bootstrap engine (file ardl_boot_engine.R, engine version 1.1.0, copied verbatim from the package 'ardlverse'). Under each null hypothesis the restricted conditional error correction model (ECM) for \Delta y is estimated by OLS together with a marginal model for \Delta x; y^* and x^* are then generated recursively from resampled, recentred residuals, with the Fourier terms held fixed as deterministic regressors.

scheme = "bvz" (default)

Bertelli, Vacca and Zoia (2022): a separate restricted model for each of the three statistics, a marginal VECM for \Delta x, residuals recentred after each draw, initial values from a random block of the data (the levels of the regressors and partial sums continue from the block).

scheme = "mcnown"

The Fov null for all statistics (McNown, Sam and Goh 2018, Steps 1-8) applied to the conditional ECM (package choice; MSG write the y equation in unconditional form): the null of the overall F test generates the samples for all statistics, the \Delta x equation is unrestricted (it includes y_{t-1}), residuals mean-centred once, the first observations as initial values.

The three statistics are computed from fqardl's own design (build_ardl_design) estimated by OLS: the overall F test on the lagged levels of y and x (Fov), the t ratio on the lagged level of y (t) and the F test on the lagged levels of x (Find). The same function is applied to the data and to every bootstrap sample, so the bootstrap distribution refers to the reported statistics. When reselect is given, the Fourier frequency (unless fixed) and the lag orders are chosen again on every bootstrap sample with select_fourier_frequency and select_optimal_lags. Re-selection is a package choice: Bertelli, Vacca and Zoia (2022, step 5(c)) re-estimate the unrestricted model on each bootstrap sample but do not discuss re-selection. In the package's Monte Carlo, holding a data-selected frequency fixed made the test oversized.

The decision combines the three tests with the AND rule: cointegration only if Fov, t and Find all reject at level level; Fov rejecting without t, or Fov and t without Find, are reported as the degenerate cases of the first and second type. Critical values are order statistics of the bootstrap distributions.

The statistics are OLS (conditional mean) statistics. Cho, Kim and Shin (2015) give no bounds test and no bootstrap for the null of no cointegration in the quantile ARDL model, and no quantile-specific bootstrap statistic is computed. Versions up to 1.0.6 regressed a new Gaussian random walk on fixed regressors at the first quantile and declared cointegration when F or t rejected; that procedure was oversized and has been removed.

Value

A list with the observed statistics (orig_F, orig_t, orig_Find), the bootstrap distributions (boot_F, boot_t, boot_Find; failed replications are NA), the p-values (p_value_F, p_value_t, p_value_Find), the critical values cv at 10, 5, 2.5 and 1 percent, the decision code decision and its text label, and the engine output engine.

References

Bertelli, S., Vacca, G. and Zoia, M. (2022). Bootstrap cointegration tests in ARDL models. Economic Modelling, 116, 105987. doi:10.1016/j.econmod.2022.105987

McNown, R., Sam, C.Y. and Goh, S.K. (2018). Bootstrapping the autoregressive distributed lag test for cointegration. Applied Economics, 50(13), 1509-1521. doi:10.1080/00036846.2017.1366643

Cho, J.S., Kim, T. 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.2015.05.003

Bootstrap NARDL Test

Description

Recursive bootstrap bounds test for the FNARDL model (rewritten in 1.1.0). The regressors x are generated recursively and the partial sums are rebuilt from x* with decompose_variables; the Fourier terms are fixed deterministic regressors; the statistics (Fov, t, Find) are OLS statistics of fnardl's own design; the Fourier frequency (unless fixed) and the lag orders are re-selected in every replication when reselect is given (a package choice); the decision is the AND rule. See bootstrap_bounds_test for the schemes. Up to 1.0.6 this function regressed a new Gaussian random walk on fixed regressors and reported the t test only, without a warning.

Usage

bootstrap_nardl(
  y,
  x,
  x_names,
  decompose,
  fourier,
  p,
  q,
  case = 3,
  n_boot = 1000,
  reselect = NULL,
  scheme = c("bvz", "mcnown"),
  level = 0.05,
  seed = NULL
)

Arguments

Value

A list as returned by bootstrap_bounds_test; p_value is the p-value of the t test (as in 1.0.6).

Build ARDL Design Matrix

Description

Constructs the design matrix for ARDL estimation including lagged dependent and independent variables.

Usage

build_ardl_design(y, X, fourier, p, q)

Arguments

Value

List with y and X design matrices

Build MTNARDL Regressor Matrix

Description

Build MTNARDL Regressor Matrix

Usage

build_mtnardl_regressors(data, x_names, decompose, decomposed, regime_names)

Build the FNARDL Design Matrix

Description

Conditional ECM design of estimate_nardl: intercept, lagged level of y, lagged differences of y, lagged levels of the regressors, current and lagged differences of the regressors, Fourier terms.

Usage

build_nardl_design(y, X, fourier, p, q)

Arguments

Value

List with y (the differenced dependent variable), X (design matrix), n and level_names (the lagged-level columns of the regressors).

Build NARDL Regressor Matrix

Description

Build NARDL Regressor Matrix

Usage

build_nardl_regressors(data, x_names, decompose, decomposed_data)

Arguments

Value

Matrix of regressors

Compute Asymmetric Multipliers

Description

Compute Asymmetric Multipliers

Usage

compute_asymmetric_multipliers(nardl_result, decompose, x_names)

Arguments

Value

List with long-run and short-run asymmetric multipliers

Compute Model Diagnostics

Description

Computes various diagnostic statistics for the QARDL models.

Usage

compute_diagnostics(qardl_results)

Arguments

Value

List of diagnostics for each quantile

Compute Long-run and Short-run Multipliers

Description

Calculates the long-run and short-run multipliers from QARDL estimates.

Usage

compute_multipliers(qardl_results, x_names, tau)

Arguments

Value

List with long-run and short-run multiplier matrices

Compute Regime-Specific Multipliers

Description

Compute Regime-Specific Multipliers

Usage

compute_regime_multipliers(model_result, decompose, regime_names)

Multi-Threshold Decomposition

Description

Decomposes a variable into partial sums of its positive and negative changes (threshold 0), starting at zero in the first observation.

Since 1.1.0 thresholds other than 0 give an error: the multi-threshold allocation of earlier versions was wrong (for example, with thresholds -2, 0 and 2 a change of -1 was split into pieces that did not sum to -1). Use ardlverse::mtnardl() for several thresholds.

Usage

decompose_multi_threshold(x, thresholds)

Arguments

Value

List with regime components and names

Decompose Variables into Positive and Negative Changes

Description

Decomposes time series into cumulative positive and negative partial sums.

Usage

decompose_variables(data, variables)

Arguments

Value

List with positive and negative components

Estimate Fourier ADF for given k

Description

Estimate Fourier ADF for given k

Usage

estimate_fadf(y, k, model, max_lag, criterion)

Estimate Fourier KPSS

Description

Estimate Fourier KPSS

Usage

estimate_fkpss(y, k, model)

Estimate MTNARDL Model

Description

Estimate MTNARDL Model

Usage

estimate_mtnardl(y, X, p, q, case)

Estimate NARDL Model

Description

Estimate NARDL Model

Usage

estimate_nardl(y, X, fourier, p, q, case = 3)

Arguments

Value

List with estimation results

Estimate Quantile ARDL Model

Description

Estimates the Quantile ARDL model for a given quantile tau.

Usage

estimate_qardl(y, X, fourier, p, q, tau, case = 3)

Arguments

Value

List with estimation results

F-test for Linearity in Fourier ADF

Description

Tests H0: gamma1 = gamma2 = 0 (no Fourier terms needed)

Usage

fadf_f_test(y, model, k, p)

Arguments

Value

List with the F statistic, its critical values (Enders and Lee, 2012, Table 1b) and p_value = NA (the distribution is non-standard)

Fit Fourier ADF Model

Description

Fit Fourier ADF Model

Usage

fit_fadf_model(dy, y_lag1, sin_term, cos_term, p, model)

Fourier Nonlinear ARDL Estimation

Description

Estimates the Fourier Nonlinear ARDL (FNARDL) model that combines: - Nonlinear ARDL for asymmetric effects (Shin et al., 2014) - Fourier approximation for smooth structural breaks

Usage

fnardl(
  formula,
  data,
  decompose = NULL,
  max_p = 4,
  max_q = 4,
  max_k = 3,
  criterion = c("BIC", "AIC", "HQ"),
  case = 3,
  bootstrap = FALSE,
  n_boot = 1000,
  verbose = TRUE,
  fourier_k = NULL,
  boot_scheme = c("bvz", "mcnown"),
  seed = NULL
)

Arguments

Details

The positive and negative partial sums start at zero in the first observation (decompose_variables). The regression always contains Fourier terms, for which the Pesaran, Shin and Smith (2001) bounds are not valid: since 1.1.0 bounds_test reports the F, t and Find statistics with the bounds labelled as the bounds for the model without Fourier terms, not valid with Fourier terms, and no analytical verdict. With bootstrap = TRUE the recursive bootstrap generates x* and y*, rebuilds the partial sums from x*, holds the Fourier terms fixed, re-selects the Fourier frequency (unless fourier_k is given) and the lag orders in every replication (a package choice; Bertelli, Vacca and Zoia 2022 do not discuss re-selection), and combines the three tests with the AND rule (cointegration only if Fov, t and Find all reject).

Value

An object of class "fnardl"

References

Shin, Y., Yu, B. and Greenwood-Nimmo, M. (2014). Modelling asymmetric cointegration and dynamic multipliers in a nonlinear ARDL framework. In Festschrift in Honor of Peter Schmidt, 281-314. Springer, New York. doi:10.1007/978-1-4899-8008-3_9

Bertelli, S., Vacca, G. and Zoia, M. (2022). Bootstrap cointegration tests in ARDL models. Economic Modelling, 116, 105987. doi:10.1016/j.econmod.2022.105987

Examples

result <- fnardl(
  formula = gdp ~ oil_price + exchange_rate,
  data = macro_data,
  decompose = c("oil_price"),  # Decompose oil price into + and -
  max_k = 3,
  verbose = FALSE
)
summary(result)
plot(result, type = "asymmetry")

Fourier ADF Unit Root Test

Description

Performs the Fourier Augmented Dickey-Fuller test for unit roots with smooth structural breaks.

Usage

fourier_adf(y, max_k = 3, max_lag = 8, criterion = c("BIC", "AIC"))

Arguments

Value

The result of fourier_adf_test with a constant, with the additional fields optimal_k and t_statistic.

Fourier ADF Test

Description

Tests for unit roots allowing for smooth structural breaks using Fourier approximation. Implements Enders and Lee (2012) methodology.

Usage

fourier_adf_test(
  y,
  model = c("c", "ct"),
  max_freq = 3,
  max_lag = NULL,
  criterion = c("AIC", "BIC", "t-sig"),
  verbose = TRUE
)

Arguments

Value

Object of class "fadf" with test results

References

Enders, W., and Lee, J. (2012). The flexible Fourier form and Dickey-Fuller type unit root tests. Economics Letters, 117(1), 196-199.

Examples

set.seed(123)
y <- cumsum(rnorm(200))  # Random walk
result <- fourier_adf_test(y, model = "c", max_freq = 3)
print(result)

Fourier KPSS Test

Description

Tests for stationarity allowing for smooth structural breaks. Implements Becker, Enders and Lee (2006) methodology.

Usage

fourier_kpss_test(y, model = c("c", "ct"), max_freq = 3, verbose = TRUE)

Arguments

Value

Object of class "fkpss" with test results

References

Becker, R., Enders, W., and Lee, J. (2006). A stationarity test in the presence of an unknown number of smooth breaks. Journal of Time Series Analysis, 27(3), 381-409.

Complete Unit Root Analysis

Description

Performs both Fourier ADF and Fourier KPSS tests for comprehensive unit root analysis.

Usage

fourier_unit_root_analysis(y, name = "Series", max_freq = 3, verbose = TRUE)

Arguments

Value

List with results from both tests and joint conclusion

Fourier Quantile ARDL Estimation

Description

Estimates the Fourier Quantile Autoregressive Distributed Lag (FQARDL) model. This methodology extends QARDL by incorporating Fourier trigonometric terms to capture smooth structural breaks without prior knowledge of break timing.

Usage

fqardl(
  formula,
  data,
  tau = c(0.25, 0.5, 0.75),
  max_p = 4,
  max_q = 4,
  max_k = 3,
  criterion = c("BIC", "AIC", "HQ"),
  case = 3,
  bootstrap = FALSE,
  n_boot = 1000,
  seed = NULL,
  verbose = TRUE,
  fourier_k = NULL,
  boot_scheme = c("bvz", "mcnown")
)

Arguments

Details

Inference on cointegration. The regression always contains Fourier terms, for which the Pesaran, Shin and Smith (2001) bounds are not valid. Since 1.1.0 bounds_test reports the Wald F and t statistics at each quantile with the PSS bounds labelled as the bounds for the model without Fourier terms, and gives no analytical verdict. Use bootstrap = TRUE: the recursive bootstrap of bootstrap_bounds_test with OLS conditional ECM statistics, the Fourier frequency (unless fourier_k is given) and the lag orders re-selected in every replication (a package choice), and the AND decision rule. The quantile-specific statistics have no published null distribution (Cho, Kim and Shin 2015) and are descriptive.

Value

An object of class "fqardl" containing:

References

Pesaran, M.H., Shin, Y. and Smith, R.J. (2001). Bounds testing approaches to the analysis of level relationships. Journal of Applied Econometrics, 16(3), 289-326. doi:10.1002/jae.616

Cho, J.S., Kim, T. 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.2015.05.003

Bertelli, S., Vacca, G. and Zoia, M. (2022). Bootstrap cointegration tests in ARDL models. Economic Modelling, 116, 105987. doi:10.1016/j.econmod.2022.105987

Examples

data(macro_data)
result <- fqardl(gdp ~ inflation + interest_rate, 
                 data = macro_data,
                 tau = c(0.25, 0.5, 0.75),
                 max_k = 3)
summary(result)
plot(result)

Generate FNARDL Report

Description

Generates a comprehensive report with all plots and tables.

Usage

generate_fnardl_report(
  obj,
  file = "fnardl_report.html",
  horizon = 20,
  verbose = TRUE
)

Arguments

Value

No return value, called for side effects (generates output or plots)

Generate Fourier Trigonometric Terms

Description

Creates sine and cosine terms for Fourier approximation of structural breaks. Based on Enders and Lee (2012) methodology.

Usage

generate_fourier_terms(n, k, cumulative = FALSE)

Arguments

Details

The Fourier terms are computed as:

sin(2\pi k t / T)

cos(2\pi k t / T)

where t is the time index and T is the sample size.

Value

A matrix with sine and cosine columns

Get Fourier ADF Critical Values

Description

Get Fourier ADF Critical Values

Usage

get_fadf_critical_values(n, model, k)

Get Fourier KPSS Critical Values

Description

Get Fourier KPSS Critical Values

Usage

get_fkpss_critical_values(model, k, n = 100)

PSS (2001) Critical Values, Case III

Description

Table CI(iii) for the F statistic and Table CII(iii) for the t statistic, unrestricted intercept and no trend. Values are the 1 percent, 5 percent and 10 percent points, each with an I(0) lower and an I(1) upper bound.

In Table CII the I(0) bound does not depend on k.

Usage

get_pss_critical_values(k, case = 3)

Arguments

Value

A list with F_lower, F_upper, t_lower, t_upper and significance.

References

Pesaran, M.H., Shin, Y. and Smith, R.J. (2001), Tables CI(iii) and CII(iii).

Simulated Macroeconomic Data with Structural Break

Description

A simulated quarterly dataset containing GDP, inflation, and interest rate with a structural break.

Usage

macro_data

Format

A data frame with 100 rows and 5 variables: gdp, inflation, interest_rate, oil_price, exchange_rate.

Source

Simulated data for package demonstration

Examples

data(macro_data)
head(macro_data)

Multi-Threshold NARDL Estimation (deprecated)

Description

Deprecated. fqardl::mtnardl() is deprecated and will be removed in fqardl 2.0.0; use ardlverse::mtnardl(). The first call in a session gives a warning. The conventions differ: the lag order p of ardlverse::mtnardl() equals p - 1 here (here p counts the lagged level of y together with the p - 1 lagged differences), and the thresholds are a single numeric vector there (a named list here).

Estimates the NARDL model with the changes of each decomposed regressor split at threshold 0 into positive and negative partial sums.

Usage

mtnardl(
  formula,
  data,
  decompose = NULL,
  thresholds = NULL,
  max_p = 4,
  max_q = 4,
  criterion = c("BIC", "AIC", "HQ"),
  case = 3,
  verbose = TRUE
)

Arguments

Details

Changes in 1.1.0.

• The bounds F statistic uses only the lagged level of y and the lagged levels of the regressors (partial sums and undecomposed variables), with k equal to the number of level regressors. Up to 1.0.6 every coefficient whose name ended in _lag1 entered the test, including dy_lag1 and the lagged differences, and the table was read at the wrong k. When the selected p or q exceeds 1 the F statistic, the bounds and the verdict therefore differ from 1.0.6.

• Thresholds other than 0 give an error: the multi-threshold decomposition of earlier versions did not allocate the changes correctly (the regime pieces did not sum to the change of the variable).

• Only case = 3 is supported; other cases give an error instead of being silently replaced by case 3.

• The returned object has class c("fqardl_mtnardl", "mtnardl") and the print and summary methods are registered for "fqardl_mtnardl" only, so that they no longer mask the methods of ardlverse.

The bounds are those of Pesaran, Shin and Smith (2001), Table CI(iii), at k equal to the number of level regressors. This choice of k for partial sums is a package choice, pending verification against Shin, Yu and Greenwood-Nimmo (2014) and Pal and Mitra (2016).

Value

Object of class c("fqardl_mtnardl", "mtnardl")

References

Shin, Y., Yu, B. and Greenwood-Nimmo, M. (2014). Modelling asymmetric cointegration and dynamic multipliers in a nonlinear ARDL framework. In Festschrift in Honor of Peter Schmidt, 281-314. Springer, New York. doi:10.1007/978-1-4899-8008-3_9

Examples

result <- suppressWarnings(mtnardl(
  formula = gdp ~ oil_price,
  data = oil_gdp_data,
  decompose = "oil_price",
  max_p = 2, max_q = 2,
  verbose = FALSE
))
summary(result)

Newey-West Variance Estimator

Description

Newey-West Variance Estimator

Usage

newey_west_variance(resid, bandwidth)

Simulated Oil Price and GDP Data with Asymmetric Effects

Description

A simulated quarterly dataset where GDP responds asymmetrically to oil price changes.

Usage

oil_gdp_data

Format

A data frame with 200 rows and 3 variables: date, gdp, oil_price.

Source

Simulated data for package demonstration

Examples

data(oil_gdp_data)
head(oil_gdp_data)

Wald Bounds Test for Cointegration

Description

Computes the Pesaran, Shin and Smith (2001) bounds test from an estimated conditional error correction model, at every quantile supplied.

The F statistic is the Wald statistic for the joint null that all lagged level coefficients are zero, divided by the number of restrictions:

W = (Rb)' (R V R')^{-1} (Rb), \qquad F = W / m

with m = k+1 in Cases I, III and V and m = k+2 in Cases II and IV. The t statistic is the ratio on the lagged level of the dependent variable, which is negative under cointegration.

Usage

perform_bounds_test(qardl_results, n, k, case = 3)

Arguments

Details

Only Case III is supported. In versions up to 1.0.2 the case argument was accepted and never used: the design matrix was always that of Case III (unrestricted intercept, no trend), while the critical values were switched. Cases 4 and 5 therefore compared a Case III model against Case IV and Case V tables. Rather than continue to report that, this release raises an error for any case other than 3. Proper case handling requires changing the design matrix, not the table, and is scheduled for 2.0.0.

Fourier caveat. The PSS critical values are tabulated for the model without Fourier terms. With Fourier terms in the regression the null distributions change and the PSS (2001) bounds do not apply with Fourier terms (Monte Carlo evidence; in a Monte Carlo with n = 100 and independent random walks the 5 percent decision of fqardl 1.0.6 rejected in 10 percent of samples, the fnardl decision in 47.5 percent). Since 1.1.0, whenever the design contains Fourier terms (columns named sin_* or cos_*, which is always the case for models estimated by fqardl and fnardl), no analytical verdict is given: decision says so, bounds_valid is FALSE, and the bounds are returned only for reference, labelled as the bounds for the model without Fourier terms, not valid with Fourier terms. Use the recursive bootstrap (bootstrap = TRUE in fqardl, see bootstrap_bounds_test); when fqardl or fnardl is called without the bootstrap, their decision text adds this advice.

Quantile caveat. The PSS critical values were simulated for the conditional mean. Applying them quantile by quantile has no distribution theory behind it (Cho, Kim and Shin 2015 give no bounds test for the null of no cointegration), so the quantile statistics are descriptive.

Value

A list with one element per quantile plus a summary data frame. Each per-quantile element contains F_stat, t_stat, the critical value bounds, bounds_valid, bounds_note and decision. Without Fourier terms the decision is three-way and reports the inconclusive region; with Fourier terms it states that no analytical verdict is available.

References

Pesaran, M.H., Shin, Y. and Smith, R.J. (2001). Bounds testing approaches to the analysis of level relationships. Journal of Applied Econometrics, 16(3), 289-326. doi:10.1002/jae.616

Cho, J.S., Kim, T. 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.2015.05.003

MTNARDL Bounds Test

Description

Wald bounds F statistic on the lagged level of y and the lagged levels of the regressors, with k equal to the number of level regressors (package choice, pending verification against Shin, Yu and Greenwood-Nimmo 2014 and Pal and Mitra 2016). Corrected in 1.1.0: earlier versions included every coefficient whose name ended in _lag1 (also dy_lag1 and the lagged differences of the regressors).

Usage

perform_mtnardl_bounds(model_result, n, k, case)

Arguments

Value

A list with F_stat, t_stat, k, the 5 percent bounds and the F-based decision.

NARDL Bounds Test

Description

Wald bounds statistics of the FNARDL model. The overall F statistic tests the lagged level of y and the lagged levels of the regressors (partial sums and undecomposed variables); k is the number of level regressors. Up to 1.0.6 the F statistic also included the lagged differences dy_lag1 and d_*_lag1 whenever p or q exceeded 1.

Usage

perform_nardl_bounds_test(nardl_result, n, k, case)

Arguments

Details

With Fourier terms in the regression (always the case for models estimated by fnardl) the PSS (2001) bounds do not apply (Monte Carlo evidence): no analytical verdict is given and the PSS bounds are returned only for reference, labelled as the bounds for the model without Fourier terms, not valid with Fourier terms. Use the recursive bootstrap (fnardl(..., bootstrap = TRUE)).

Value

A list with F_stat, t_stat, Find_stat, k, the 5 percent bounds cv_5 and t_cv_5, bounds_valid, bounds_note and decision.

Plot FNARDL Results

Description

============================================================================= Visualization Functions for FNARDL Dynamic Multiplier Plots and Asymmetry Analysis ============================================================================= Plot FNARDL Results

Usage

## S3 method for class 'fnardl'
plot(
  x,
  type = c("asymmetry", "dynamic", "cumulative", "comparison"),
  variable = NULL,
  horizon = 20,
  ...
)

Arguments

Value

No return value, called for side effects (generates output or plots)

Plot FQARDL Results

Description

Creates various diagnostic and result plots for FQARDL models.

Usage

## S3 method for class 'fqardl'
plot(
  x,
  type = c("coefficients", "multipliers", "3d", "heatmap", "residuals"),
  variable = NULL,
  ...
)

Arguments

Value

A ggplot object or plotly object for 3D plots

Plot method for fqardl mtnardl objects

Description

fqardl provides no plot for its (deprecated) mtnardl objects. This method stops with an informative error instead of letting plot() dispatch, through the inherited class "mtnardl", to a method of another package that expects a different object.

Usage

## S3 method for class 'fqardl_mtnardl'
plot(x, ...)

Arguments

Value

Does not return; always an error.

Plot 3D Surface of Coefficients

Description

Plot 3D Surface of Coefficients

Usage

plot_3d_surface(obj, variable = NULL)

Arguments

Value

plotly object

Plot Asymmetry Bar Chart

Description

Plot Asymmetry Bar Chart

Usage

plot_asymmetry(obj, variable = NULL)

Arguments

Plot Coefficients Across Quantiles

Description

Plot Coefficients Across Quantiles

Usage

plot_coefficients(obj, variable = NULL)

Arguments

Value

ggplot object

Plot Cumulative Multipliers

Description

Plot Cumulative Multipliers

Usage

plot_cumulative_multipliers(obj, variable, horizon = 20)

Arguments

Value

No return value, called for side effects (generates output or plots)

Plot Dynamic Multipliers

Description

Plot Dynamic Multipliers

Usage

plot_dynamic_multipliers(obj, variable, horizon = 20)

Arguments

Value

No return value, called for side effects (generates output or plots)

Plot Heatmap of Coefficients

Description

Plot Heatmap of Coefficients

Usage

plot_heatmap(obj)

Arguments

Value

ggplot object

Plot Long-run and Short-run Multipliers

Description

Plot Long-run and Short-run Multipliers

Usage

plot_multipliers(obj)

Arguments

Value

ggplot object

Plot Persistence Profile

Description

Plots the persistence profile showing the adjustment path to long-run equilibrium after a shock.

Usage

plot_persistence(obj, horizons = 20)

Arguments

Value

ggplot object

Plot Positive vs Negative Comparison

Description

Plot Positive vs Negative Comparison

Usage

plot_pos_neg_comparison(obj)

Arguments

Plot Residual Diagnostics

Description

Plot Residual Diagnostics

Usage

plot_residuals(obj)

Arguments

Value

ggplot object

Quantile Wald Test for Coefficient Constancy

Description

Tests whether coefficients are constant across quantiles.

Usage

quantile_wald_test(qardl_results, coef_name)

Arguments

Value

List with test results

Select Optimal Fourier Frequency

Description

Selects the optimal Fourier frequency k based on information criteria. Tests all frequencies from 1 to max_k and selects the one minimizing the chosen criterion.

Usage

select_fourier_frequency(y, X, max_k = 3, criterion = c("BIC", "AIC", "HQ"))

Arguments

Value

A list containing:

Select Optimal Lags for MTNARDL

Description

Select Optimal Lags for MTNARDL

Usage

select_mtnardl_lags(y, X, max_p, max_q, criterion)

Select Optimal Lag Structure

Description

Selects optimal lag orders for ARDL model using grid search over all combinations of p and q.

Usage

select_optimal_lags(
  y,
  X,
  fourier,
  max_p,
  max_q,
  criterion = c("BIC", "AIC", "HQ")
)

Arguments

Value

List with optimal lags

Test for Asymmetry (Wald Test)

Description

Test for Asymmetry (Wald Test)

Usage

test_asymmetry(nardl_result, decompose)

Arguments

Value

List of Wald test results for each variable

Test Regime Asymmetry

Description

Test Regime Asymmetry

Usage

test_regime_asymmetry(model_result, decompose, regime_names)