fqardl 1.0.3

This is a correctness release. Results change. Anyone who has run version 1.0.2 should re-run their analysis before relying on it.

The defects below came to light after a question from Prof. Chi-Lu (Edward) Peng, National Kaohsiung University of Science and Technology, who noticed that the bounds-test t statistic was positive at every quantile when it should be negative. Our thanks to him for reporting it.

Corrected

The estimated equation had an error-correction right-hand side and a levels left-hand side

build_ardl_design(), estimate_nardl() and estimate_mtnardl() assembled a Pesaran, Shin and Smith conditional error-correction design on the right-hand side, containing the lagged levels y_{t-1} and x_{t-1} together with the differences Delta y_{t-j} and Delta x_{t-j}, while setting the left-hand side to y_t in levels rather than Delta y_t. The equation actually estimated was therefore the conditional levels ARDL.

Hence the coefficient reported as the error-correction term was phi = 1 + rho rather than rho. Since rho lies in (-1, 0) under cointegration, the reported coefficient was positive and close to unity, with a large positive t ratio at every quantile.

The defect propagated:

• Long-run multipliers carried the wrong sign. compute_multipliers() applies -theta / phi, which is valid only when the left-hand side is Delta y. In a Monte Carlo with a true long-run coefficient of +0.80, version 1.0.2 returned a median of -0.471 and the wrong sign in 100 percent of replications. Version 1.0.3 returns +0.791 with the correct sign in 100 percent of replications.
• The bounds test manufactured cointegration out of the unit root in y. On the package’s own macro_data, version 1.0.2 reported F = 65.57 against a 5 percent upper bound of 4.38 and concluded “Evidence of cointegration”. Version 1.0.3 reports F = 2.34, below the 5 percent lower bound of 3.79, and concludes no cointegration.
• fnardl() and mtnardl() were affected identically. On oil_gdp_data, fnardl() previously reported an error-correction coefficient of +0.894 with t = +27.08 and F = 246.07; it now reports -0.106 with t = -3.21 and F = 4.47.

The “F statistic” was not a Wald statistic

perform_bounds_test(), perform_nardl_bounds_test() and perform_mtnardl_bounds() computed mean(t^2) over the terms matching _lag1$, that is, the arithmetic mean of the squared marginal t ratios. That ignores the covariances among the level coefficients, has no distribution theory, and was being compared against tables that Pesaran, Shin and Smith simulated for a genuine Wald statistic.

All three now compute the Wald form of PSS equation (21), W = (Rb)' (R V R')^{-1} (Rb), with F = W / (k+1) for Case III.

The inconclusive region was not reported

The bounds procedure has three outcomes, not two. When the statistic falls between the I(0) and I(1) bounds, inference is inconclusive without knowing the cointegration rank of the regressors. Version 1.0.2 collapsed this into a binary verdict. All bounds routines now return a three-way decision.

The bounds test is now computed at every quantile

perform_bounds_test() located tau = 0.5, computed one statistic there, and reported it as though it were general. It now returns a per-quantile table in $summary, with the median quantile promoted to the top level so that code written against the 1.0.2 return shape keeps working.

case was accepted and never used

fqardl() validated case and then ignored it: the design matrix was always 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, and case = 3 and case = 5 returned identical coefficients. Any case other than 3 now raises an error. Correct case handling changes the regressors, not the table, and is scheduled for 2.0.0.

Standard errors were unnamed, so lookups returned NA silently

In estimate_qardl() the coefficient vector was renamed to the design column names while the standard-error and t-statistic vectors were not. Any lookup such as std_errors["y_lag1"] returned NA without warning. All three vectors are now named consistently.

Critical-value table

The Case III tables are stated explicitly and checked by a test. The 5 percent column of Tables CI(iii) and CII(iii) has been verified against the published table for every k from 1 to 10. The 1 and 10 percent columns of CI(iii) are inherited from earlier versions and have not been independently verified; the 1 and 10 percent I(1) bounds of CII(iii) are returned as NA rather than guessed. Only the 5 percent level is used in the decision rule.

Smaller fixes

• mtnardl()$bounds_test$t_stat was NULL; it is now returned.
• perform_mtnardl_bounds() used a hard-coded pair of critical values regardless of the number of level terms; it now reads the PSS table for the actual k.

Known limitations, documented rather than silently shipped

Each of these now raises a warning at the point of use. All are scheduled for 2.0.0.

1. bootstrap_bounds_test() does not generate pseudo-samples under the null. The McNown, Sam and Goh (2018) procedure requires each of the three statistics to have its own restricted residuals, and requires the level series to be built by accumulation, y*_t = y*_{t-1} + Delta y*_t. Neither is done. The p-values are indicative only.
2. quantile_wald_test() assumes independence across quantiles. Quantile-regression estimates from the same sample have covariance proportional to min(tau_i, tau_j) - tau_i tau_j, which is strictly positive, so the correct denominator is smaller than the one used. The test is conservative and under-rejects constancy. Fixing it requires the joint cross-quantile covariance, which this release does not compute.
3. Quantile-specific bounds verdicts have no distribution theory. The PSS critical values were simulated for the conditional mean. Treat the per-quantile table as descriptive.
4. Deterministic cases I, II, IV and V are not available.

Verification

This release ships verify_fqardl_103.R, sixteen checks, all passing, including:

• the design’s left-hand side equals Delta y_t;
• the package F statistic equals a hand-computed Wald statistic to machine precision;
• on a design with a true adjustment speed of -0.35 and a true long-run coefficient of +0.80, the package recovers -0.371 and +0.791, with a negative error-correction t ratio in 100 percent of replications;
• the shipped Case III critical values match Tables CI(iii) and CII(iii) at 5 percent for every k from 1 to 10.

fqardl 1.0.0

Initial CRAN Release (2026-02-25)

New Features

FQARDL (Fourier Quantile ARDL)

• fqardl() - Main function for Fourier Quantile ARDL estimation
• Automatic selection of optimal Fourier frequency (k*)
• Automatic lag selection using BIC, AIC, or HQ criteria
• Estimation across multiple quantiles (τ)
• Long-run and short-run multiplier computation
• PSS bounds test for cointegration
• Bootstrap cointegration testing
• Publication-ready plots

FNARDL (Fourier Nonlinear ARDL)

• fnardl() - Main function for asymmetric ARDL with Fourier terms
• Variable decomposition into positive/negative partial sums
• Wald tests for long-run and short-run asymmetry
• Dynamic multiplier computation (positive vs negative shocks)
• Cumulative multiplier visualization
• Asymmetry comparison plots

Helper Functions

• fourier_adf() - Fourier ADF unit root test
• generate_fourier_terms() - Generate Fourier trigonometric terms
• decompose_variables() - Decompose variables for NARDL
• perform_bounds_test() - PSS bounds testing
• bootstrap_bounds_test() - Bootstrap cointegration test

Visualization

• Coefficient plots across quantiles
• Dynamic multiplier plots
• Cumulative multiplier plots
• Persistence profile plots
• 3D coefficient surfaces
• Heatmaps
• Asymmetry comparison plots

References

• Pesaran, M. H., Shin, Y., & Smith, R. J. (2001). Bounds testing approaches to the analysis of level relationships. Journal of Applied Econometrics, 16(3), 289-326.

• Shin, Y., Yu, B., & Greenwood-Nimmo, M. (2014). Modelling asymmetric cointegration and dynamic multipliers in a nonlinear ARDL framework. In Festschrift in Honor of Peter Schmidt (pp. 281-314). Springer.

• Enders, W., & Lee, J. (2012). A unit root test using a Fourier series to approximate smooth breaks. Oxford Bulletin of Economics and Statistics, 74(4), 574-599.

• Cho, J. S., Kim, T., & Shin, Y. (2015). Quantile cointegration in the autoregressive distributed-lag modeling framework. Journal of Econometrics, 188(1), 281-300.

Acknowledgments

• Dr. Merwan Roudane for the original Stata implementation
• Rufyq Elngeh (رفيق النجاح) for supporting this development