fqardl

• The statistics are OLS statistics of the package's own design (build_ardl_design() for fqardl, build_nardl_design() for fnardl): the overall F test on the lagged levels of y and the regressors (Fov), the t ratio on the lagged level of y (t) and the F test on the lagged levels of the regressors (Find). For fnardl the positive and negative partial sums are rebuilt from x* with the package's own decomposition (starting at zero). The Fourier terms are held fixed in the data generating process.
• The Fourier frequency and the lag orders are re-selected in every replication (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) with the criteria of select_fourier_frequency() and select_optimal_lags() (computed with qr() for speed; a test checks that the choices are identical). The new argument fourier_k of fqardl() and fnardl() fixes the frequency in advance; it is then held fixed in the bootstrap.
• Decision: AND rule. Cointegration only if Fov, t and Find all reject at 5 percent (bootstrap order-statistic critical values); Fov rejecting without t, and Fov and t rejecting without Find, are reported as the degenerate cases of the first and second type. Cointegration is never declared from a single test (no OR rule).
• The quantile ARDL of Cho, Kim and Shin (2015) has no bounds test and no bootstrap for the null of no cointegration, so no quantile-specific bootstrap statistic is computed; the quantile Wald bounds statistics of perform_bounds_test() are descriptive.
• New arguments: fqardl(fourier_k, boot_scheme), fnardl(fourier_k, boot_scheme, seed), bootstrap_bounds_test(reselect, scheme, level, seed). A seed given to the bootstrap functions is used locally and the caller's random number state is restored. fqardl(seed) still calls set.seed() at the start, as before.

Bounds statistics

• fnardl(): the bounds F statistic tested every coefficient whose name ended in _lag1, so with p > 1 or q > 1 it also tested dy_lag1 and the lagged differences of the regressors. It now tests only the lagged level of y and the lagged levels of the regressors, with k equal to the number of level regressors. With p = q = 1 the value is unchanged. Examples with data set.seed(s); n <- 120; x1 <- cumsum(rnorm(n)); x2 <- cumsum(rnorm(n)); y <- 0.5 * x1 + cumsum(rnorm(n, sd = 0.5)); d <- data.frame(y, x1, x2) and fnardl(y ~ x1 + x2, d, decompose = "x1", max_k = 3, max_p = 3, max_q = 3) (selected p = 2, q = 1): s = 1, F 4.9160 becomes 6.1315; s = 2, 2.2814 becomes 2.8463; s = 42, 6.2361 becomes 7.5356. The bounds test also returns t_cv_5, Find_stat and k.
• mtnardl(): same correction (defect F1 of the 2026-10-03 audit); with the data above, s = 2, and mtnardl(y ~ x1, d, max_p = 3, max_q = 3) (selected p = 2, q = 1), F 1.5051 becomes 1.9533; k for the PSS table is the number of level regressors (partial sums and undecomposed variables), a package choice pending verification against Shin, Yu and Greenwood-Nimmo (2014) and Pal and Mitra (2016). The decision is the F-based three-way verdict, as before.

mtnardl (deprecated)

• F2: the multi-threshold decomposition was mathematically wrong. With a nonzero threshold (0 is always added) the regime increments did not sum to the change of the variable (for example, with thresholds -2, 0 and 2 the changes -3, -1, 1, 3 and 0.5 gave regime increments summing to -5, 1, 3, 5 and 2.5; with the single threshold 2 the changes -3 and -1 gave -5 and -3). Thresholds other than 0 now give an informative error in mtnardl() and decompose_multi_threshold(); with threshold 0 (the default) the decomposition is the standard pair of partial sums and results other than the bounds F are unchanged.
• F3: case was ignored in the bounds test and forced to 3 in the lag selection; cases other than 3 now give an error.
• The print and summary methods are registered for class "fqardl_mtnardl" only, so they no longer mask the methods of ardlverse for its "mtnardl" objects when both packages are loaded.

Printing

• summary() of an fnardl object printed the long-run multipliers as NA (the elements of multipliers$long_run_positive and long_run_negative are named <var>.<var>_pos_lag1 and were indexed by <var>). The printout now shows the values; the stored values are unchanged.
• Author names in the documentation and in the headers printed by the Fourier unit root tests are joined with "and" instead of "&"; the help page of get_pss_critical_values() now describes the function as implemented since 1.0.4.

Monte Carlo (size at the 5 percent level)

y and x independent random walks, n = 100, 200 replications, B = 199, default calls (BIC, max_p = max_q = 4, max_k = 3); rejection rates with their Monte Carlo standard errors sqrt(r(1 - r)/200) in parentheses. Before: fqardl 1.0.6 (audit of 2026-10-03, same data seeds).

| Statistic | 1.0.6 | 1.1.0 |

|---|---|---|

| fqardl bootstrap, Fov | 0.085 (0.020) | 0.065 (0.017) |

| fqardl bootstrap, t | 0.120 (0.023) | 0.100 (0.021) |

| fqardl bootstrap, Find | not computed | 0.075 (0.019) |

| fqardl bootstrap, decision | 0.130 (0.024) (F or t) | 0.035 (0.013) (Fov and t and Find) |

| fnardl bootstrap, Fov | not computed | 0.060 (0.017) |

| fnardl bootstrap, t | 0.170 (0.027) | 0.105 (0.022) |

| fnardl bootstrap, Find | not computed | 0.100 (0.021) |

| fnardl bootstrap, decision | none | 0.030 (0.012) (Fov and t and Find) |

| fqardl analytical decision (Fourier) | 0.100 (0.021) | no verdict |

| fnardl analytical decision (Fourier) | 0.475 (0.035) | no verdict |

The table shows the run with engine 1.1.0. A run with engine 1.0.0 on the same data seeds gave identical fqardl rates and, for fnardl, Fov 0.060 (0.017), t 0.120 (0.023), Find 0.105 (0.022), decision 0.035 (0.013); an independent review run gave for fnardl a decision rate of 0.055 (0.016) and t and Find rates of 0.120 (0.023). Across these runs the combined decisions are close to the nominal level (0.030 to 0.055). The individual t tests and the fnardl Find test remain above 5 percent (0.100 to 0.120) with the frequency and lags re-selected in every replication; read them through the combined decision.

Unchanged

• QARDL point estimates, quantile standard errors and covariances, long-run and short-run multipliers, the quantile Wald bounds F and t statistics, quantile_wald_test(), the fnardl and mtnardl coefficients, multipliers and asymmetry tests, the fnardl and mtnardl t statistics, the Fourier frequency and lag selection, and the Fourier unit root tests are identical to 1.0.6 (checked on five simulated data sets).