Asymmetric Portmanteau Test

• The asymmetric portmanteau test is a diagnostic approach that modifies classical Box–Pierce/Ljung–Box tests to capture nonlinear and directional dependencies through transformations like squaring and sign-sensitive functions.
• It integrates multiple methodologies—including squared-residual analysis, mixed autocorrelation tests, and generalized-covariance frameworks—to detect conditional heteroskedasticity and asymmetric volatility effects.
• The approach emphasizes tailored calibration, model-specific adequacy testing, and directional diagnosis, making it effective for assessing ARMA, VARMA, and various GARCH-type models.

Searching arXiv for papers on asymmetric portmanteau tests and related diagnostics. An asymmetric portmanteau test is a residual-based omnibus diagnostic that departs from the classical Box–Pierce/Ljung–Box paradigm by targeting asymmetric, nonlinear, or directional dependence structures that are not exhausted by linear residual autocorrelation. In the literature represented here, the label does not denote a single universal statistic. Instead, it covers several non-equivalent constructions: standard portmanteau statistics applied to squared residuals to detect conditional heteroskedasticity; mixed tests that jointly stack residual and absolute-residual autocorrelations; generalized-covariance tests based on odd or sign-sensitive transforms; model-specific adequacy tests for asymmetric power GARCH classes; and a one-sided cross-correlation test that removes inverse-causality effects when testing weak exogeneity (Mahdi, 2020, Tan et al., 2020, Jasiak et al., 2023, Maïnassara et al., 2018, Maïnassara et al., 2024, Andriollo, 5 Jun 2026).

1. Terminological scope and conceptual distinctions

In this literature, “asymmetric” has several precise meanings. In diagnostic checking for ARMA- and VARMA-type models, it can mean replacing residuals $\hat e_t$ by squared residuals $\hat e_t^2$ so that the test targets dependence in volatility magnitudes rather than linear mean dynamics. In asymmetric volatility models, it can refer to separate positive and negative shock effects entering the conditional scale recursion. In generalized-covariance frameworks, it means choosing odd or sign-sensitive transforms such as $x$, $x^3$, or $\operatorname{sign}(x)|x|^\alpha$ to interrogate skewness-related or leverage-type serial dependence. In the weak-exogeneity setting, asymmetry is directional: the test isolates dependence from past omitted variables to current shocks and excludes the reverse direction from its variance (Mahdi, 2020, Maïnassara et al., 2018, Jasiak et al., 2023, Andriollo, 5 Jun 2026).

Variant	Core object	Null calibration
Squared-residual portmanteau in portes	Classical BP/LB/Hosking/Li–McLeod/Mahdi–McLeod statistics applied to $\hat e_t^2$	Approximate $\chi^2$ or Monte Carlo (Mahdi, 2020)
ALDAR mixed portmanteau	Stacked ACFs of $\hat\eta_t$ and $|\hat\eta_t|$	$\chi^2_{2M}$ (Tan et al., 2020)
GCov asymmetric subtest	Generalized autocovariances of odd/sign-sensitive transforms	$\hat e_t^2$0 with projector-rank df (Jasiak et al., 2023)
Weak-exogeneity asymmetric Portmanteau Test	Corrected one-sided cross-covariances $\hat e_t^2$1	Asymptotically $\hat e_t^2$2 (Andriollo, 5 Jun 2026)
APGARCH and CCC-APGARCH adequacy tests	Squared standardized residual autocovariances or $\hat e_t^2$3 autocovariances	$\hat e_t^2$4 (Maïnassara et al., 2018, Maïnassara et al., 2024)

A recurrent misconception is that there exists a single canonical “Asymmetric Portmanteau Test.” The supplied sources instead show a family of diagnostics with different null hypotheses, different objects of aggregation, and different reference laws. This is explicit in the portes paper, which states that it does not define a separate “Asymmetric Portmanteau Test” statistic; the asymmetry-oriented diagnostic there is simply the standard portmanteau machinery run on squared residuals via squared.residuals = TRUE (Mahdi, 2020).

2. Squared-residual and mixed formulations

The classical univariate Box–Pierce and Ljung–Box statistics are

$\hat e_t^2$5

with $\hat e_t^2$6 the sample autocorrelation at lag $\hat e_t^2$7. In portes, the nonlinear diagnostic is obtained by replacing $\hat e_t^2$8 with $\hat e_t^2$9 throughout, so that the same statistic tests dependence in magnitudes rather than linear autocorrelation. This operationalizes the McLeod–Li idea for detecting ARCH-type effects and volatility clustering. The package applies the same principle to multivariate Hosking, Li–McLeod, and generalized-variance statistics, and supports asymptotic $x$0 calibration or Monte Carlo p-values with innov.dist = c("Gaussian","t","stable","bootstrap") (Mahdi, 2020).

The null hypothesis changes when squared residuals are used. For residual-based tests, adequacy means no remaining linear autocorrelation. For squared-residual tests, adequacy means no remaining nonlinear dependence in conditional variance. In the portes framework, the relevant degrees of freedom for Box–Pierce, Ljung–Box, Hosking, and Li–McLeod are

$x$1

where $x$2 is the series dimension and $x$3 counts autoregressive and moving-average parameters, including seasonal components. For the generalized variance statistic $x$4,

$x$5

The paper emphasizes Monte Carlo p-values when $x$6 is modest, $x$7 is large relative to $x$8, innovations are non-Gaussian or stable, or residuals exhibit conditional heteroskedasticity (Mahdi, 2020).

A distinct mixed construction appears in the asymmetric linear double autoregressive model, ALDAR($x$9). After QMLE, the standardized residuals are

$x^3$0

For lags $x^3$1, the paper defines the residual ACF

$x^3$2

and the absolute-residual ACF

$x^3$3

The mixed portmanteau statistic stacks both vectors and uses a full covariance correction: $x^3$4 Here “mixed” means joint testing of serial dependence in the conditional mean and in the conditional volatility/asymmetric components. The paper reports that for S&P 500 weekly returns, BIC-based selection chose $x^3$5, asymmetry tests strongly rejected equality of positive and negative volatility coefficients, while mixed portmanteau p-values at $x^3$6 were $x^3$7, $x^3$8, and $x^3$9, indicating no lack of fit (Tan et al., 2020).

3. Directional asymmetry: weak exogeneity versus inverse causality

The most literal use of the title “asymmetric Portmanteau test” appears in the weak-exogeneity problem studied in "Causality versus Serial Correlation: an Asymmetric Portmanteau Test" (Andriollo, 5 Jun 2026). The setup involves a shock process $\operatorname{sign}(x)|x|^\alpha$0 and an omitted-variable process $\operatorname{sign}(x)|x|^\alpha$1, with joint past

$\operatorname{sign}(x)|x|^\alpha$2

The null is weak exogeneity: $\operatorname{sign}(x)|x|^\alpha$3 Under stationarity and square integrability, this implies

$\operatorname{sign}(x)|x|^\alpha$4

The paper’s central point is that conventional one-sided Hong-type tests based on

$\operatorname{sign}(x)|x|^\alpha$5

are still “symmetric” in their variance. Their variance incorporates not only the direction relevant for weak exogeneity, from past $\operatorname{sign}(x)|x|^\alpha$6 to current $\operatorname{sign}(x)|x|^\alpha$7, but also “inverse causality,” from past $\operatorname{sign}(x)|x|^\alpha$8 to current $\operatorname{sign}(x)|x|^\alpha$9. The decomposition

$\hat e_t^2$0

isolates a sum-of-squares term $\hat e_t^2$1 preserving the desired ordering and a sum-of-cross-products term $\hat e_t^2$2 whose variance is inflated by inverse causality (Andriollo, 5 Jun 2026).

The proposed correction subtracts the unordered part: $\hat e_t^2$3 This is the asymmetry of the test: the retained cross-products respect the martingale ordering that is consistent with $\hat e_t^2$4 being a martingale difference under $\hat e_t^2$5. After centering and separate variance normalization, the implemented statistic

$\hat e_t^2$6

is asymptotically standard normal under $\hat e_t^2$7. The regularity conditions include conditional homoskedasticity and conditional homokurtosis of $\hat e_t^2$8 given $\hat e_t^2$9, strict stationarity and finite $\chi^2$0-order moments for $\chi^2$1, strong mixing of $\chi^2$2, and a kernel bandwidth satisfying $\chi^2$3 or $\chi^2$4 under weaker dependence (Andriollo, 5 Jun 2026).

The interpretive contrast is explicit. If a symmetric Hong test rejects but the asymmetric test does not, the paper interprets this as likely inverse causality rather than failure of weak exogeneity. If the asymmetric test rejects, past omitted variables help predict current shocks, so weak exogeneity fails. In the empirical application to Economic Policy Uncertainty shocks, both the benchmark and asymmetric tests reject, but at different horizons; augmenting local projections with lagged macro factors changes the estimated inflation response from negative to positive (Andriollo, 5 Jun 2026).

4. Adequacy testing in asymmetric volatility models

For asymmetric power GARCH models, the portmanteau problem is not formulated on raw residual autocorrelations but on squared standardized residual autocovariances. In the APARCH/APGARCH($\chi^2$5) model with unknown power $\chi^2$6,

$\chi^2$7

or equivalently, in split-coefficient form,

$\chi^2$8

The paper defines

$\chi^2$9

and proves

$\hat\eta_t$0

The portmanteau statistic is

$\hat\eta_t$1

Here the estimation effect of all APARCH parameters, including the unknown power, is absorbed into $\hat\eta_t$2 and $\hat\eta_t$3, so no ad hoc degree-of-freedom reduction is required (Maïnassara et al., 2018).

The multivariate analogue in the CCC-APGARCH class uses

$\hat\eta_t$4

with asymmetric positive and negative matrices $\hat\eta_t$5 and $\hat\eta_t$6 entering the power recursion. The paper defines

$\hat\eta_t$7

Under the null,

$\hat\eta_t$8

and similarly for the autocorrelation-based version

$\hat\eta_t$9

The covariance correction now involves $|\hat\eta_t|$0, $|\hat\eta_t|$1, and $|\hat\eta_t|$2, reflecting the QMLE effect in the multivariate asymmetric power recursion (Maïnassara et al., 2024).

Both papers explicitly contrast these diagnostics with classical Ljung–Box testing. In GARCH-type settings, raw residual autocorrelations are not the primary adequacy criterion; the relevant misspecification appears in second-order dependence of standardized residual squares or their multivariate analogue. This is why the test is built from $|\hat\eta_t|$3 in the univariate APGARCH case and from $|\hat\eta_t|$4 in the CCC-APGARCH case (Maïnassara et al., 2018, Maïnassara et al., 2024).

5. Generalized-covariance, rank-sign, and high-dimensional extensions

The GCov-based framework generalizes portmanteau testing from linear autocovariances to nonlinear generalized covariances

$|\hat\eta_t|$5

The full GCov portmanteau statistic is

$|\hat\eta_t|$6

For asymmetry, the paper restricts the transformation set to odd or sign-sensitive functions, producing the asymmetric-only statistic

$|\hat\eta_t|$7

Under GCov estimation and regularity conditions, the df is the rank of the projector $|\hat\eta_t|$8; in the common case of full identification using the same moments for estimation, $|\hat\eta_t|$9. If estimation is performed by ML, QML, or AML instead of GCov, the paper recommends a bootstrap test (Jasiak et al., 2023).

A closely related high-dimensional extension is the shrinkage-regularized NLSD test. It starts from the augmented vector

$\chi^2_{2M}$0

and replaces the potentially ill-conditioned $\chi^2_{2M}$1 by the Ledoit–Wolf shrinkage estimator

$\chi^2_{2M}$2

The resulting statistic

$\chi^2_{2M}$3

has asymptotic $\chi^2_{2M}$4 null law, with $\chi^2_{2M}$5, when $\chi^2_{2M}$6. The paper describes asymmetry as being operationalized by including sign- or state-sensitive transforms alongside linear and magnitude-sensitive ones (Giancaterini et al., 10 Mar 2026).

A different extension, although not labeled an asymmetric portmanteau test in the title, is the center-outward rank- and sign-based VARMA portmanteau procedure. It constructs center-outward ranks and signs from residuals and forms score-based lagged cross-covariances. The resulting statistic is asymptotically $\chi^2_{2M}$7 for sufficiently large $\chi^2_{2M}$8 and is distribution-free under broad innovation densities. Its relevance here is that the paper explicitly emphasizes robustness to skewness, heavy tails, and non-elliptical innovation densities; the method therefore functions as a portmanteau diagnostic robust to innovation asymmetry rather than a test of asymmetric dependence per se (Hallin et al., 2022).

6. Calibration, implementation, and interpretation

Across these constructions, the reference distribution depends on the null being tested. Squared-residual diagnostics in portes are approximately $\chi^2_{2M}$9 with df adjusted for model order, but the paper repeatedly recommends Monte Carlo calibration for heavy tails, stable innovations, small $\hat e_t^2$00, or large $\hat e_t^2$01 (Mahdi, 2020). The ALDAR mixed statistic is $\hat e_t^2$02 after covariance correction (Tan et al., 2020). GCov-based asymmetric subtests are $\hat e_t^2$03 with projector-rank df under GCov estimation and use bootstrap critical values when parameters are estimated by another method (Jasiak et al., 2023). The weak-exogeneity asymmetric statistic is asymptotically standard normal, not chi-square (Andriollo, 5 Jun 2026). APGARCH and CCC-APGARCH adequacy tests return to $\hat e_t^2$04, but only after nontrivial parameter-estimation corrections in the covariance matrix (Maïnassara et al., 2018, Maïnassara et al., 2024).

Lag or bandwidth selection is likewise model-dependent. portes recommends using a range of lags such as $\hat e_t^2$05 and notes that larger $\hat e_t^2$06 interrogates longer-range dependence but increases df and small-sample distortions (Mahdi, 2020). The ALDAR paper uses modest values such as $\hat e_t^2$07 and states that no explicit data-driven rule is imposed (Tan et al., 2020). The GCov paper commonly uses small $\hat e_t^2$08, for example $\hat e_t^2$09 (Jasiak et al., 2023). The weak-exogeneity paper recommends kernel weighting $\hat e_t^2$10 with practical choices $\hat e_t^2$11 (Andriollo, 5 Jun 2026).

Interpretation also differs sharply across variants. In squared-residual diagnostics, rejection indicates remaining variance dynamics such as ARCH/GARCH or volatility clustering, even when linear residual autocorrelations are negligible (Mahdi, 2020). In ALDAR, inspecting $\hat e_t^2$12 and $\hat e_t^2$13 separately locates whether misfit is in the conditional mean or in the volatility/asymmetric component (Tan et al., 2020). In GCov and SR-NLSD settings, significant odd-transform blocks indicate sign-related nonlinear serial dependence, whereas even-transform blocks indicate volatility-type nonlinear dependence (Jasiak et al., 2023, Giancaterini et al., 10 Mar 2026). In the weak-exogeneity setting, rejection of the asymmetric test indicates that omitted variables’ past predicts current shocks; failure to reject when Hong rejects suggests inverse causality rather than a violation of weak exogeneity (Andriollo, 5 Jun 2026).

A plausible implication of this literature is that “asymmetry” in portmanteau testing is best understood as a design principle rather than a unique formula. The design principle may target magnitudes instead of levels, positive and negative shocks separately, odd versus even transforms, or directional predictability rather than bidirectional dependence. The common feature is that each statistic augments or restructures the portmanteau aggregation so that the null distribution remains tractable while the test becomes sensitive to dependence patterns that the classical symmetric residual autocorrelation check was not constructed to detect.