Package 'fqardl'

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 [email protected]

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

y	Dependent variable (no missing values).
X	Matrix of regressors in levels.
fourier	Matrix of Fourier terms (from generate_fourier_terms) used for the observed data and as fixed deterministic terms of the bootstrap data generating process.
p	Lag order for y, as in build_ardl_design.
q	Lag order for X, as in build_ardl_design.
tau	Not used since 1.1.0 (the statistics are OLS statistics); kept for compatibility.
case	Model case; only 3 is supported.
n_boot	Number of bootstrap replications.
verbose	Print a progress message.
reselect	NULL (hold the frequency and lags fixed) or a list with elements max_k, max_p, max_q, criterion and optionally k (a frequency fixed by the user in advance) to re-run the selection on every bootstrap sample.
scheme	"bvz" or "mcnown"; see Details.
level	Significance level of the decision.
seed	Optional seed for the bootstrap; the caller's random number state is restored afterwards.

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

---

Compute Asymmetric Multipliers

Description

Compute Asymmetric Multipliers

Usage

compute_asymmetric_multipliers(nardl_result, decompose, x_names)

Arguments

nardl_result	NARDL estimation results
decompose	Decomposed variable names
x_names	Original variable names

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

qardl_results

List of QARDL estimation results

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

qardl_results	List of QARDL results for each quantile
x_names	Names of independent variables
tau	Vector of quantiles

Value

List with long-run and short-run multiplier matrices

---

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

x	Numeric vector
thresholds	Threshold values; only 0 is supported.

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

data	Data frame
variables	Variables to decompose

Value

List with positive and negative components

---

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

y	Dependent variable
X	Independent variables
fourier	Fourier terms
p	Lag for dependent variable
q	Lag for independent variables
tau	Quantile (0 < tau < 1)
case	Model case (1-5)

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

y	Time series
model	Model specification
k	Fourier frequency
p	Number of lags

Value

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

---

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

formula	A formula of the form y ~ x1 + x2 + ...
data	A data frame containing the time series variables
decompose	Vector of variable names to decompose into positive/negative
max_p	Maximum lag for dependent variable
max_q	Maximum lag for independent variables
max_k	Maximum Fourier frequency
criterion	Information criterion ("AIC", "BIC", "HQ")
case	Model case; only 3 (unrestricted intercept, no trend) is supported.
bootstrap	Perform the recursive bootstrap bounds test (see Details).
n_boot	Number of bootstrap replications
verbose	Logical. Print progress messages (default: TRUE)
fourier_k	Optional Fourier frequency fixed by the user in advance; NULL (default) selects it from 1:max_k and the bootstrap re-selects it in every replication.
boot_scheme	"bvz" (default) or "mcnown"; see bootstrap_bounds_test.
seed	Optional seed for the bootstrap; the caller's random number state is restored afterwards.

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

y	Time series vector
max_k	Maximum Fourier frequency
max_lag	Maximum number of lags for ADF
criterion	Lag selection criterion

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

y	Numeric vector of time series data
model	Model specification: "c" (constant), "ct" (constant + trend)
max_freq	Maximum Fourier frequency to test (default: 3)
max_lag	Maximum lag for ADF (default: NULL, auto-select)
criterion	Lag selection criterion ("AIC", "BIC", "t-sig")
verbose	Logical. Print progress messages (default: TRUE)

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

y	Numeric vector of time series data
model	Model specification: "c" (constant), "ct" (constant + trend)
max_freq	Maximum Fourier frequency (default: 3)
verbose	Logical. Print progress messages (default: TRUE)

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

y	Time series
name	Optional name for the series
max_freq	Maximum Fourier frequency
verbose	Logical. Print progress messages (default: TRUE)

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

formula	A formula of the form y ~ x1 + x2 + ...
data	A data frame containing the time series variables
tau	Numeric vector of quantiles to estimate (default: c(0.25, 0.5, 0.75))
max_p	Maximum lag for dependent variable (default: 4)
max_q	Maximum lag for independent variables (default: 4)
max_k	Maximum Fourier frequency to test (default: 3)
criterion	Information criterion for lag selection ("AIC", "BIC", "HQ")
case	Model case; only 3 (unrestricted intercept, no trend) is supported.
bootstrap	Logical, perform the recursive bootstrap bounds test (bootstrap_bounds_test).
n_boot	Number of bootstrap replications (default: 1000)
seed	Random seed, set with set.seed() at the start (the quantile regression standard errors and the bootstrap use random numbers).
verbose	Logical. Print progress messages (default: TRUE).
fourier_k	Optional Fourier frequency fixed by the user in advance. NULL (default) selects it from 1:max_k; the bootstrap then re-selects it in every replication.
boot_scheme	Bootstrap scheme, "bvz" (Bertelli, Vacca and Zoia 2022, default) or "mcnown" (the Fov null for all statistics, McNown, Sam and Goh 2018, applied to the conditional ECM); see bootstrap_bounds_test.

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:

coefficients	Estimated coefficients for each quantile
long_run	Long-run multipliers
short_run	Short-run multipliers
optimal_k	Optimal Fourier frequency
optimal_lags	Optimal lag structure
bounds_test	Wald bounds statistics by quantile (no verdict with Fourier terms; see Details)
bootstrap	Result of bootstrap_bounds_test when bootstrap = TRUE
diagnostics	Model diagnostics

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

obj	FNARDL object
file	Output file path (HTML or PDF)
horizon	Horizon for dynamic multipliers
verbose	Logical. Print completion message (default: TRUE)

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

n	Sample size (number of observations)
k	Fourier frequency (integer >= 1)
cumulative	If TRUE, includes all frequencies from 1 to k

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

---

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

formula	A formula of the form y ~ x1 + x2 + ...
data	Data frame with time series
decompose	Variables to decompose
thresholds	Named list of threshold values for each variable; only 0 is supported since 1.1.0.
max_p	Maximum lag for dependent variable
max_q	Maximum lag for independent variables
criterion	Information criterion ("AIC", "BIC", "HQ")
case	Model case; only 3 (unrestricted intercept, no trend) is supported.
verbose	Logical. Print progress messages (default: TRUE)

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)

---

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

qardl_results	List of quantile estimation results from estimate_qardl. The test is computed for every element.
n	Sample size.
k	Number of regressors (excluding the dependent variable).
case	Model case. Only case = 3 is supported; see Details.

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

---

Plot Cumulative Multipliers

Description

Plot Cumulative Multipliers

Usage

plot_cumulative_multipliers(obj, variable, horizon = 20)

Arguments

obj	FNARDL object
variable	Variable to plot
horizon	Number of periods

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

obj	FNARDL object
variable	Variable to plot
horizon	Number of periods

Value

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

---

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

obj	FQARDL object
horizons	Number of periods for persistence profile

Value

ggplot object

---

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

x	An object of class "fnardl"
type	Type of plot
variable	Variable to plot
horizon	Horizon for dynamic multipliers
...	Additional 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

x	An object of class "fqardl"
type	Type of plot: "coefficients", "multipliers", "3d", "heatmap", "residuals"
variable	Variable name for coefficient plots
...	Additional arguments passed to plotting functions

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

x	An object of class "fqardl_mtnardl".
...	Not used.

Value

Does not return; always an error.

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Quantile Wald Test for Coefficient Constancy

Description

Tests whether coefficients are constant across quantiles.

Usage

quantile_wald_test(qardl_results, coef_name)

Arguments

qardl_results	List of QARDL results
coef_name	Name of coefficient to test

Value

List with test results

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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

y	Dependent variable vector
X	Matrix of independent variables
max_k	Maximum Fourier frequency to test
criterion	Information criterion ("AIC", "BIC", "HQ")

Value

A list containing:

optimal_k	The optimal Fourier frequency
ic_values	Information criterion values for each k
criterion	The criterion used

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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

y	Dependent variable
X	Matrix of independent variables
fourier	Fourier terms matrix
max_p	Maximum lag for dependent variable
max_q	Maximum lag for independent variables
criterion	Information criterion

Value

List with optimal lags

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Test for Asymmetry (Wald Test)

Description

Test for Asymmetry (Wald Test)

Usage

test_asymmetry(nardl_result, decompose)

Arguments

nardl_result	NARDL estimation results
decompose	Decomposed variables

Value

List of Wald test results for each variable

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