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Causality versus Serial Correlation: an Asymmetric Portmanteau Test
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Estimating dynamic causal effects through structural models, such as Structural Vector Autoregressions (SVARs), is a central tool in applied macroeconometrics. Following sims1980macroeconomics, identification assumptions allow the innovations of a multivariate time series model to be interpreted as linear functions of the underlying structural shocks. Economists routinely exploit this link by estimating the residuals of a structural model, thus investigating the shocks' propagation through impulse response analysis kilian2017structural. Related arguments extend to univariate approaches, such as local projections dufour1998short, plagborg2021local, jorda2023local, combined with external instruments.
The validity of such analyses hinges critically on the correct specification of the structural dynamics, both for variables included in and external to the model. Structural shocks must be “internally” exogenous to the other current and lagged endogenous variables in the model ramey2016macroeconomic, and this property cannot be undermined by variables omitted from the model. If the latter do interfere, the shocks fail to be “externally” exogenous and no longer represent truly unanticipated movements in the macroeconomic system. As a consequence, the history of the observed internal variables is insufficient to recover the shocks, violating invertibility or fundamentalness lippi1994var, nakamura2018identification.
Existing approaches to assessing this issue have cast it either as a problem of Granger causality testing giannone2006does, forni2014sufficient, plagborg2022instrumental, miranda2023identification or as tests of conditional mean independence chen2017testing. From an econometric standpoint, both approaches can be viewed as addressing dynamic specification testing in the presence of omitted variables. This paper brings these perspectives together by formulating the problem in terms of weak exogeneity, understood here as the property of the structural shocks having zero conditional expectation given the past of both internal and external variables mikusheva2025linear.
The literature on specification testing in the presence of omitted variables can be organized according to whether the practitioner explicitly models the joint dynamics of internal and external variables. When the joint system is estimated, classical specification tests are available (hong2005generalized, escanciano2006generalized). Inference in this case depends on how accurately the interaction between internal and external variables is specified and estimated. Parametric joint modeling is therefore sensitive to misspecification, while strategies based on semi-parametric or nonparametric estimation face the well-known curse of dimensionality and finite-sample problems. When the joint dynamics are left unspecified, practitioners typically rely on nonparametric tests (hong1996testing, lobato2002testing). Despite not requiring augmentation, these tests are designed to detect symmetric forms of dependence. However, weak exogeneity imposes a directional restriction. In our context, this implies that both classes of approaches may produce rejections when there is dependence from past structural shocks to current omitted variables. Since it reflects the causal direction opposite to the one being tested, this paper refers to this dependence as inverse causality: broadly speaking from past structural shocks to present omitted variables rather than from past omitted variables to present shocks. This channel becomes particularly salient in applied macroeconometrics because structural shocks, which capture primitive fluctuations of the macroeconomic system, are expected to influence (external) macro variables over time. As a result, inverse causality can trigger rejections that researchers may mistakenly interpret as evidence against shocks' exogeneity.
This paper addresses these limitations by proposing an asymmetric Portmanteau test that isolates violations of weak exogeneity from inverse causality. The benchmark Portmanteau statistic tests the null of zero cross-correlation between current estimated shocks and lagged omitted variables by aggregating squared sample cross-correlations.\footnote{For univariate processes, hong1996testing defines the Portmanteau statistic as the weighted sum of squared cross-correlation between univariate time series at positive and negative lags, with weights determined by a kernel function. Following hong2001test and bouhaddioui2006generalized, this paper regards as representative of the class of tests based on the serial cross-correlation function its one-sided multivariate formulation. Specifically, the benchmark is the weighted sum of the $\ell_2$ norm of the cross-correlation between the two multivariate processes at positive lags.} A key difficulty is that the quadratic norm introduces a symmetry: the variance of the resulting statistic depends not only on dependence from lagged omitted variables to current shocks, which is relevant under the null of weak exogeneity, but also on dependence from lagged shocks to current omitted variables, that is inverse causality. The newly proposed statistic subtracts out the contribution of this latter channel, leading to a modified statistic that targets only the dependence implied by violations of weak exogeneity. By construction, the procedure avoids parametric modeling of the joint dynamics and remains robust to misspecification of how past shocks affect the present of omitted variables, which is particularly useful when prior knowledge of the interaction between omitted variables and the dynamic system is limited.
The asymptotic distribution of the proposed test statistic is studied under the null that estimated shocks are weakly exogenous. Establishing asymptotic normality requires additional assumptions on the shocks' second and fourth conditional moments that, however, are testable. The second moment condition serves to isolate the contribution of squares to the mean vs. variance of the statistic. In technical terms, this restriction ensures correct centering and scaling of the statistic under the null. The fourth moment condition is mainly used to establish asymptotic normality.\footnote{When relaxing the assumption of independence, hong2001test's footnote 8 briefly discussed the condition of conditional homokurtosis for establishing the asymptotic normality of his testing procedure.} The main asymptotic result is derived for observed processes.\footnote{Extension to settings where shocks and omitted variables are estimated can be found in the Online Appendix.} Under the fixed alternatives of nonzero cross-correlation, this paper proves that the asymmetric Portmanteau achieves equivalent asymptotic power to the benchmark. Since the statistic has inherently limited power against alternatives involving nonlinear non-pairwise dependence, a discussion on possible generalizations of the statistic is provided. Finite-sample properties of the statistics are examined through Monte Carlo simulations, with a summary of the main findings provided in the appendix and the full results reported in the online appendix.
As an empirical application, the paper examines the exogeneity of the baker2016measuring's Economic Policy Uncertainty (EPU) shocks. The analysis controls for economic conditions using the mccracken2016fred's macroeconomic factors. EPU shocks fail the exogeneity test with respect to lagged macro factors. Building on this, the paper revisits diercks2024rains and strengthens their conclusions: when these additional controls are included, the response of inflation to EPU shocks shifts from modestly negative to markedly positive. Combined with contractionary responses in other variables, this evidence suggests that the EPU structural shock operates as a supply-side negative shock, similar to the `expectational' shocks discussed in ascari2023endogenous.
Literature. This paper contributes to three strands of literature. First, it relates to specification testing in dynamic linear models. Early contributions include ljung1978measure, hosking1980multivariate and li1981distribution. Rather than modeling the joint process, haugh1976checking developed a two-step procedure to test for independence between time series: first fitting univariate models, then examining cross-correlations at different lags. hong1996consistent, hong1996testing generalized these tests to all lags using kernel-weighted schemes, with bouhaddioui2006generalized extending the framework to multivariate processes. This paper contributes to haugh1976checking's approach by introducing a new version of the Portmanteau statistic that isolates a specific direction of causality. Second, the paper relates to tests of the martingale difference hypothesis used for specification testing, which indeed require modeling the conditional mean of the joint process durlauf1991spectral, hong2005generalized, escanciano2006generalized. This class of tests can be viewed as extending ljung1978measure's rather than haugh1976checking's. The newly proposed method improves upon these by avoiding the need to model joint conditional means and variances. Third, this paper contributes to the literature on testing invertibility of structural shocks, by providing a new testing strategy.
Outline. Section (ref) introduces the benchmark Portmanteau statistic and shows how inverse causality is incorporated into it. The definition of the asymmetric Portmanteau statistic is in Section (ref). Section (ref) develops the asymptotic theory for the proposed test statistic. Section (ref) presents the empirical application. Section (ref) concludes. Appendixes (ref)-(ref) contain all proofs and the summary of the simulation evidence.
Section (ref) establishes the framework and discusses the class of Portmanteau statistics. Section (ref) defines the asymmetric Portmanteau statistic. Proof of the proposition appears in Appendix (ref).
Let $\{X_{t}, Z_{t}; t=1,..,T\}$ denote two zero-mean multivariate square-integrable jointly stationary processes of respective finite dimensions $d_1,d_2\in\mathbb{N_+}$. Let $\mathcal{I}(t-1)$ be the information set available at period $t-1$ comprising the joint past, $\{X_{s},Z_{s}; s< t\}$. Unless stated otherwise, these processes are standardized.\footnote{$\text{Var}[X_t]=I_{d_1}$ and $\text{Var}[Z_t]=I_{d_2}$. It is relaxed in the Online Appendix.}
Throughout, $X$ represents the structural shock series and $Z$ represents the omitted variables. The null hypothesis of interest is weak exogeneity, defined as the shocks having zero conditional expectation given their own past and the past of omitted variables:
Before introducing the modified test statistic in Eq.((ref)) (Section (ref)), we discuss testing strategies based on squared serial cross-correlations following hong1996testing. The benchmark is the one-sided statistic based on weighted quadratic forms:
for some nonrandom non-negative weights $\{\omega(j)\}$, where $\widehat{\Gamma}_{XZ}(j)$ is the sample cross-correlation between the processes:\footnote{For a clearer exposition of the asymptotic theory, we do not consider the finite-sample corrected cross-correlation functions (i.e., scaled by $T-j$ rather than $T$), as the conclusions remain valid hong2001test, hong2005generalized. For the finite-sample statistics, refer to Appendix (ref).}
The statistic is one-sided ($j>0$) because $\mathcal{H}_0$ concerns a particular direction of causality: the influence of the past of $Z$, $\{Z_s; s< t\}$, on the present $X$, $\{X_t\}$. The quadratic forms, $\{Q(j)\}$, correspond to the squared $\ell_2$-norms of vectorized sample cross-correlation matrices, equivalently their squared Frobenius norms.\footnote{This generalization from univariate to multivariate analysis dates to li1981distribution. See bouhaddioui2006generalized for further discussion. For the equivalence between Euclidean norm and trace, see chapter 4 of lutkepohl1997handbook.} For its connection to kernel estimation of the spectrum, see hong1996consistent.
By construction, the quadratic forms treat causality directions symmetrically. This becomes apparent when decomposing the test statistic: the inner product of cross-correlation matrices generates cross-product terms where $X$ and $Z$ enter symmetrically across different time lags. Formally, by means of some algebra:
The Portmanteau statistic, $\mathcal{T}_\omega$, consists of two components: the “sum of squares”, $\mathcal{T}_{1\omega}$, and the “sum of cross-products”, $\mathcal{T}_{2\omega}$. While the sum of squares preserves temporal ordering with respect to the tested causal direction (from past omitted variables to present shocks), the sum of cross-products incorporates the interaction between two time indexes, $s$ and $t$, thus blending both directions of causality. This symmetry due to the norm suggests that when testing the null, Portmanteau statistics may not effectively distinguish between violations of weak exogeneity and dependence from past shocks to present omitted variables (inverse causality).
The distinction between components matters for understanding asymptotic properties of the testing strategies based on hong1996testing and subsequent work. Intuitively, the sum of cross-products, $\mathcal{T}_{2\omega}$, dominates under the null and thus controls test size, whereas the sum of squares, $\mathcal{T}_{1\omega}$, dominates under the alternatives, and so regulates power.\footnote{ In Proposition (ref), the asymptotic properties of the test under the null are described in detail. For the power of the test, refer to Theorem (ref) and the discussion that follows.}
In a stylized setting, the following proposition clarifies how cross-product terms incorporate both directions of dependence. We impose two simplifying assumptions: i) marginal independence of the processes, i.e., $X_t \perp\!\!\!\perp X_k, Z_t \perp\!\!\!\perp Z_k, t\not=k $; ii) conditional homoskedasticity of the shocks, $\mathbb{E}[||X_t||^2|\mathcal{I}(t-1)]=\mathbb{E}[||X_t||^2]$.
Proposition (ref) demonstrates that under the null hypothesis, the variance of the statistic $\mathcal{T}_\omega$ incorporates, through the cross-products in $\mathcal{T}_{2\omega}$, the dependence from past $X$ to present $Z$ captured by cross-moments of the joint process (i.e., the inverse causality channel). This vanishes when either the processes are independent, $X_t \perp\!\!\!\perp Z_s, \forall t,s$ (strict exogeneity between shocks and omitted variables at all lags/leads), or when a specific time ordering holds, $s>t-j$ (with $t>s$). Two remarks follow. First, in the univariate case $(d_1=d_2=1)$, it is evident that inverse causality operates through the conditional variance of $Z$. Second, the presence of such dependencies in the variances arises even: a) under the restrictive assumption on the univariate processes, namely marginally i.i.d. time series, b) under past independence of shocks, $X_t \perp\!\!\!\perp Z_{s_1}, X_{s_2},$ with $s_1,s_2< t$.\footnote{Past independence would imply conditional homoskedasticity. Similar assumptions to past independence appear in hong2009granger, and candelon2016nonparametric. This condition, together with the assumption of marginally i.i.d., is weaker than statistical independence, $X_t \perp\!\!\!\perp Z_s, \forall s,t $.}
Lemma (ref) in the Appendix (ref) explicitly derives how dependencies from shocks to omitted variables affect the variance of the sum of cross-products, $\mathcal{T}_{2\omega}$, under a general class of DGPs where inverse causality is present and the null of interest holds true.
Motivated by the previous proposition, we introduce a modified statistic that accounts for the following cross-products of unordered time indexes $(s,t)$:
The two formulations are associated with either the inverse causality channel, $\mathcal{C}_{\omega}$, or the corrected causality channel, $\mathcal{T}_{2\omega}^c$. Proposition (ref) motivates removing cross-products whose time ordering prevents a martingale structure, or equivalently retaining the one satisfying $j>|t-s|$ for the unordered indexes $(s,t)$. Since the correction term, $\mathcal{C}_{\omega}$, differences out the influence of inverse causality and so breaks the symmetry, the proposed statistic becomes an asymmetric Portmanteau test. To further motivate the correction term, we offer two perspectives.
The first perspective interprets a subset of the cross-products as bias in the variance of sample covariance estimators, under weak exogeneity and conditional homoskedasticity (Proposition (ref)). Along this heuristic, a jackknife solution would then employ a block-deletion scheme targeting observations associated with the temporal ordering $j\leq |t-s|$ (for a given $j$), precisely those observations linked to the inverse causality channel.
The second perspective provides deeper insight by viewing moments of cross-products as coefficients in predictive regressions. Consider the bivariate process $\{X_t,Z_t\}$ for time indexes $t>s$. After applying the correction, the first moment of remaining cross-products, $\mathcal{T}_{2\omega}^c$, is proportional to coefficients in regressions of the form: $X_tX_s = \sum_{j=t-s+1}^{s-1} \phi_{j}^{(1)}Z_{t-j}Z_{s-j}+e_t$ (for a fixed $s>0$), where $e_t$ is an error term, $Z_{t-j},Z_{s-j}\in\mathcal{I}(t-j)$ and $X_s\not\in\mathcal{I}(t-j)$. Conversely, the first moment of cross-products constituting the correction term, $\mathcal{C}_{\omega}$, is proportional to coefficients in autoregressions: $X_tZ_{t-j} = \sum_{l=1}^s \varphi_{l}^{(1)}X_{l}Z_{l-j}+\epsilon_t$ (for a fixed $j>0$), where $\epsilon_t$ is an error term, $X_{l},Z_{l-j}\in\mathcal{I}(s)$ and $Z_{t-j}\not\in\mathcal{I}(s)$. These coefficients assess distinct implications of the null. By law of iterated expectations, weak exogeneity implies: i) $\mathbb{E}[X_tX_s]=0, \; \forall s<t$; ii) $\mathbb{E}[X_tZ_{t-j}]=0, \; \forall j>0$, captured by the first and second regression sets, respectively. When looking at the second moment of $\mathcal{T}_{2\omega}^c$ and $\mathcal{C}_\omega$, then those are proportional to coefficients of the following regressions:
Conditional homoskedasticity of $X$ has the following implication:\\ $ \phi_{j}^{(2)}\propto \mathbb{E}[(X_t^2X_s^2)M_{Z} (Z_{t-j}^2Z_{s-j}^2)]/\text{Var}[Z_{t-j}^2Z_{s-j}^2]=d_1 \mathbb{E}[M_{Z}Z_{t-j}^2 Z_{s-j}^2]/\text{Var}[Z_{t-j}^2Z_{s-j}^2]$, for an appropriate projection operator, $M_Z \in \mathcal{I}(s)$, following a Frisch–Waugh–Lovell argument. Coefficients $\{\phi_{j}^{(2)}\}$ from the first regression set depend solely on higher moments of the marginal process $\{Z_t\}$. In contrast, coefficients from the second set depend on higher-order moments of the joint process $\{X_t,Z_t\}$. Consistent with Proposition (ref), this distinction motivates the correction term.
Section (ref) establishes the asymptotic properties of the proposed statistic under the null of interest. Section (ref) derives asymptotic properties under a general class of alternatives and discusses settings where the test has limited power. Proofs of the Proposition and Theorems appear in Appendix (ref). A discussion on the finite-sample properties of the statistics can be found in Appendix (ref). In the Online Appendix, the main result of Section (ref) is extended to estimated processes, and the full results of the simulations are reported.
This paper considers the following condition on the weighting scheme in Eq.((ref)):
This assumption is standard in nonparametric spectral density estimation via kernel functions hong2001test. The sequence of integers $M$, growing with sample size $T$, characterizes the kernel estimation window: larger $M$ incorporates more lags when summing cross-correlations in the statistic.\footnote{See the discussion at the end of Section (ref) regarding the smoothing parameter.} Define:
with $\gamma_{t,s}(j,l)=\mathbb E\!\left[\langle Z_{t-j},Z_{s-j}\rangle \langle Z_{t-\ell},Z_{s-\ell}\rangle\right]$, assuming these last moments exist. The first two quantities approximately match the mean and the variance of the proposed statistic under the null hypothesis, scaled by the sample size (Proposition (ref)). The last quantity, $D_{\omega,T}$, represents the asymptotic variance of the benchmark statistic scaled by the sample size, under mutual independence of $X$ and $Z$ hong2001test. Under Assumption (ref) and appropriate conditions on $\{\gamma_{t,s}(j,l)\}$, these quantities are of same order $M$: $\mu_{\omega,T}=O(M)$, $D_{\omega,T}=O(M)$, and $D_{\omega,T}^{(Hete)}=O(M)$. The following proposition characterizes the moments of the asymmetric Portmanteau statistic.
Proposition (ref) formalizes the benefits of the correction term. Under conditional homoskedasticity and conditional homokurtosis of $X$ with respect to the joint past, the first two moments of the asymmetric Portmanteau statistic do not incorporate dependencies running from past $X$ to present $Z$. Specifically, these moments depend only on the weighting scheme, $\{\omega(j)\}$ or, at most, on particular higher moments of the marginal process $Z$, $\{\gamma_{t,s}\}$, rather than on moments of the joint process that would reflect inverse causality. In fact, by breaking the symmetry of the quadratic form, the correction term restores a martingale structure that permits separating variances via law of iterated expectations. Consequently, the variance of the proposed statistic reduces to (nonparametrically) estimating the long-run variance of second-order moments of the process $Z$ (i.e., the cross-products $\{\langle Z_{t-j},Z_{s-j}\rangle\}$).
Propositions (ref)-(ref) highlight a trade-off in terms of restrictions on marginal vs. joint processes: maintaining directional inference while avoiding to specify the dynamics between shocks and omitted variables. This paper prioritizes an agnostic stance toward inverse causality, imposing minimal restrictions on how past shocks influence present omitted variables, at the cost of stronger moment restrictions on the structural shocks themselves. Indeed, these moment restrictions serve to isolate effects rather than causes: they ensure the statistic correctly detects violations of weak exogeneity without constraining the inverse causality channel. Conditional homoskedasticity ensures proper centering by isolating weak exogeneity violations to the mean of cross-products, $\mathcal{T}_{2\omega}^c$, rather than the sum of squares, $\mathcal{T}_{1\omega}$. Conditional homokurtosis bounds the variance of the latter sum, ensuring that the cross-products dominate under the null.
These moment restrictions on structural shocks are testable and offer practical advantages for empirical application. Since structural shocks are estimated rather than observed, conditional moment restrictions can guide towards sharper identification (e.g., hafner2022identification, or related heteroskedastic identification schemes). Under additional parametric assumptions on inverse causality, these conditions could be relaxed.\footnote{These moment restrictions are weaker than mutual independence hong1996testing or past independence candelon2016nonparametric, and comparable to approximate q-dependence hong2005generalized, while being more directly testable.} Notably, analogous conditions have been proposed for testing in Proxy-SVAR frameworks bruns2024testing.
Using the quantities defined in Eq.((ref))-((ref)), let:
where $\mathcal{T}^{(Hete)}$ is the centered and scaled version of the proposed asymmetric Portmanteau statistic, whereas $\mathcal{T}^{(Hong)}$ corresponds to centered and scaled version of the benchmark (e.g., hong1996consistent). To state the dependence conditions used below, define the triangular array: $\Lambda_{s,t}= \sum_{j=t-s+1}^{s-1}\omega(j)X_{s}\langle Z_{t-j},Z_{s-j}\rangle$, with $1\leq s\leq t$, and for $r\geq 1$, define the fourth-order dependence coefficient: \\ $C_{r,4}^{\Lambda}:= \sup_{t\geq 1}\sup_{m=1,2,3}\sup_{\mathcal I_{m,r,t}}\left\|\operatorname{Cov}\left(\Lambda_{i_1,t}\otimes..\otimes\Lambda_{i_m,t},\Lambda_{i_{m+1},t}\otimes..\otimes\Lambda_{i_4,t}\right)\right\|_{F}$,\\ with: $\mathcal I_{m,r,t} =\left\{(i_1,..,i_4):1\leq i_1\leq..\leq i_m<i_{m+1}\leq..\leq i_4\leq t,i_{m+1}-i_m\geq r \right\}.$\\ Additionally, we consider the following assumption on the process $Z$:
Theorem (ref) offers two notable improvements over existing testing strategies.
First, Portmanteau statistics following hong1996consistent, hong1996testing are typically studied under statistical independence. In the presence of inverse causality, benchmark tests based on squared cross-correlations may therefore exhibit size distortions under weak exogeneity, as their higher moments incorporate dependencies from past $X$ to present $Z$ (Proposition (ref)). The asymmetric Portmanteau statistic addresses this issue directly: Theorem (ref) establishes that its asymptotic normality depends essentially on the martingale properties of $X$ with respect to the joint past.
Second, tests for the martingale difference property hong2005generalized, escanciano2006generalized could, in principle, be used to test $\mathcal{H}_0$, but at a high cost. These procedures would require either (i) the joint process $\{X_t,Z_t\}$ to be a martingale difference sequence (stronger than weak exogeneity), or (ii) explicit modeling of the joint conditional mean. The latter approach would involve additional high-level assumptions (e.g., Assumptions A2-A3 in hong2005generalized) whose implications and testability are less transparent than the primitive moment conditions imposed in Theorem (ref). By contrast, the proposed framework avoids modeling the joint dynamics altogether, while maintaining interpretable and testable restrictions on the moments of the structural shocks.
The main cost relative to hong1996testing is a more stringent rate condition on the smoothing parameter $M$, which must diverge slower than $\sqrt{T}$. This slower rate reflects the need to control the variance of the sum of squares under minimal assumptions on the marginal process $Z$ (i.e., finite eighth moments). However, the second part of Theorem (ref) shows that, under mild additional dependence conditions dedecker2007weak, the standard rate $M/T\rightarrow 0$ suffices.
The smoothing parameter $M$ governs the number of lags considered in the test statistic and thus the rate at which the weighted sum of covariance terms converges to a Gaussian limit. For small/finite $M$, the limiting distribution is a weighted sum of chi-squared variables box1970distribution, francq2007multivariate. As $M$ increases with $T$, the sum converges to normality by a standard central limit argument. The choice of $M$ involves a familiar trade-off: sufficiently fast growth ensures asymptotic normality under the null, while sufficiently slow growth preserves power against alternatives (Theorem (ref)). In addition, since the correction term permits separating the variances, the smoothing parameter $M$ also governs the effective window used to estimate the long-run variance of second-order moments of $Z$, thus relating its choice to the standard size-power tradeoff in HAR inference lazarus2021size.
This section establishes the asymptotic power under a general class of alternatives. Let $\kappa_{mrmr,XY}(j,k,l)$ denote the fourth-order cumulant of $\{X_{m,t},Z_{r,t-j},X_{m,t-k},Z_{r,t-l}\}$, where $X_{m,t}$ and $Z_{r,t}$ are the $m^{th}$ and $r^{th}$ entries of $X_{t}$ and $Z_{t}$. We require absolute summability of fourth-order cumulants: $\sum_{m,r=1}^{d_1,d_2}\sum_{j,k,l=-\infty}^\infty \vert \kappa_{mrmr,XZ}(j,k,l)\vert <\infty $.
Theorem (ref) establishes a consistency result: the proposed statistic, properly centered and scaled, converges in probability to the sum of squared cross-correlations across lags (up to a positive scalar). Under fixed alternatives with nonzero cross-correlation, the proposed statistic explodes at rate $(M^{1/2}/T)^{-1}$. Slower growth of $M$ yields faster divergence and higher power, consistent with the discussion concluding Section (ref). Fourth-order stationarity and absolute summability of joint cumulants are standard conditions imposed when studying the power of tests following hong1996testing, as they accommodate a wide class of processes hong1996consistent.\footnote{More general conditions are discussed in lobato2002testing, see pg.731-3.} Two technical remarks clarify the role of fourth-order cumulants in our framework.
First, the proof exploits that under alternatives the sum of squares, $\mathcal{T}_{1\omega}$, stochastically dominates the sum of cross-products, $\mathcal{T}_{2\omega}^c$, as anticipated in Section (ref). This follows from Theorem 6 in hannan1970multiple (pg.210) via an Isserlis-type argument priestley1981spectral, based on establishing $\ell_2$-convergence of covariance estimators to their population counterparts under fourth-order stationarity and absolute summability. These conditions ensure the process is sufficiently close to a multivariate normal (or to a generalized linear process) for mean-square convergence to take effect. Crucially, while fourth-order cumulants are asymptotically negligible under alternatives (Theorem (ref)), they govern the asymptotic behavior under the null (Theorem (ref)). When $X$ is a martingale with respect to higher moments, cumulants drive the distribution of the test statistic under $\mathcal{H}_0$ (Proposition (ref)). Hence, the correction term in Eq.((ref)) specifically targets the subset of cumulants associated with inverse causality.
Second, our asymptotic approach differs fundamentally from escanciano2006generalized. Their strategy first establishes that sample autocovariances converge weakly in $\ell_2$-norm to a Gaussian process under the null of martingale difference (their Theorem 1).\footnote{They generalize autocovariances to measure conditional mean dependence nonparametrically (pg.155). Convergence occurs in the Hilbert space of square-integrable functions; see pg.158–159 for details.} Second, since covariances enter their statistic “squared”, they show their statistic converges in distribution to a weighted sum of independent $\chi^2_1$ variables. In contrast, hong1996consistent focuses on quadratic forms directly, showing that sums of cumulants converge to normality. The distinction is one of convergence order: escanciano2006generalized achieve convergence of covariances before squaring, while hong1996consistent's inferential theory with cross-products is after squaring. Our framework follows the latter approach, underscoring the importance of fourth-order cumulants when inverse causality is left unrestricted, precisely the setting where practitioners lack information about how omitted variables $Z$ interact with the structural shock and the structural dynamics.
The proposed test has no power against uncorrelated but non-martingale processes. This reflects a well-known limitation of Portmanteau tests: nonlinear dependencies from past $Z$ to present $X$ that leave linear associations unaffected cannot be detected. One potential extension addresses this limitation through generalized spectral analysis hong2005generalized. Rather than summing squared covariances between $X$ and $Z$, their approach considers squared covariances between $X$ and the empirical characteristic function of $Z$, thereby capturing nonlinear dependencies. Since their statistic involves quadratic forms hong2005generalized, an analogous correction term can be constructed. Adapting their framework, define:
where, in this instance, $i$ denotes the imaginary unit and $Q: \mathbb{R}\rightarrow \mathbb{R}^+$ a nondecreasing weighting function symmetric about zero. Parallel to Eq.((ref)), the asymmetric version of hong2005generalized's statistic is:
A second limitation concerns the nature of detectable dependencies. Portmanteau statistics capture temporal dependence in a pairwise manner, detecting relationships of the form $\mathbb{E}[X_{t}|Z_{t-j}]$ but not necessarily non-pairwise interactions such as, for instance, \\ $\mathbb{E}[X_{t}|Z_{t-j}, Z_{t-j-1}]$. A standard solution is to augment $Z$ with its first $R$ lags: \\ $\{Z_t^\ddag=(Z_t^\prime,Z_{t-1}^\prime,...,Z_{t-R}^\prime)^\prime\}$, and then testing the null using cross-covariances between $X$ and $Z^\ddag$ dominguez2004consistent, kuan2004new, wang2022testing. Albeit potentially applicable here, this type of solution complicates inference: the variance of the resulting statistic involves substantially more cross-product terms, requiring more elaborate correction terms to difference out the inverse causality effects. Developing such extensions remains an avenue for future research.
Finite-sample properties under the null and fixed alternatives are studied via Monte Carlo experiments, summarized in Appendix (ref). Appendix (ref) also define the finite-sample versions of the statistics that are used in the simulations and empirical application. The full set of results is reported in the Online Appendix.
This section presents an empirical application of the proposed testing procedures. Section (ref) introduces the concept of invertibility or fundamentalness of structural shocks and relates it to the null of interest. Section (ref) studies the exogeneity property of the uncertainty shock of baker2016measuring by revisiting the empirical analysis of diercks2024rains.
In applied macroeconometrics, by a Wold-type of argument, common practice is to assume that the macroeconomic (stationary) multivariate time series, $\{W_{t}\}$, admits a Moving Average (MA) representation driven by mutually orthogonal structural shocks: $ W_t=B(L)\epsilon_t = \sum_{j=0}^{\infty}B_j\epsilon_{t-j}$, with $\epsilon_{t}\sim (0,I)$, where $B(L)$ captures the propagation of the structural disturbances.\footnote{This rationale is supported by a twofold motivation: i) by the Wold Representation theorem, if the time series is covariance-stationary then it admits a MA($\infty$) representation brockwell1987time, with the Wold innovations being the reduced-form residuals of the linear projection of $W$ onto its infinite past; ii) the linear (or linearized) dynamic stochastic economic model, based on the variables $W$, usually admits a VARMA solution, whose structural shocks are assumed to be mutually orthogonal fernandez2007abcs.} When $W$ is causal and invertible, structural shocks can be recovered from current and lagged values of $W$, up to a rotation matrix which governs the instantaneous relationships among the components of $W$, thus in turn motivating the use of SVAR models. Invertibility may however fail when economic agents' information differs from the econometrician's hansen2019two and, in such cases, the MA representation is said to be non-fundamental lippi1994var, nakamura2018identification. In practice, the issue of invertibility spells out as a problem of VAR misspecification, due to omitted variables or insufficient set of lagged controls chen2017testing, miranda2023identification.
To fix ideas, following giannone2006does, partition $W$ into two blocks $W_1$ and $W_2$ of dimensions $d_1$ and $d_2$, with reduced-form residuals $(X_{t},Z_t)^\prime$ and structural shocks $(\epsilon_{1,t},\epsilon_{2,t})^\prime$, where the process $Z$ collects the innovations of the block $W_2$. For simplicity, suppose $B_0=I$, so that structural shocks coincide with reduced-form residuals:
Suppose the econometrician is interested in recovering the structural shocks, $\{\epsilon_{1,t}=X_{t}\}$, but omits from the empirical analysis the block $W_2$ (and so all the linear space spanned by the history of $Z$). The MA representation is fundamental only if $A_{1,2}(L)=0$ holds or, equivalently, whether there is Granger noncausality from the omitted variables $W_2$ to $X$. Vice versa, if $A_{1,2}(L)\not=0$, recovering the structural shock requires the enlarged information set $\{W_{1,t},W_{2,t}\}$ or, equivalently, $\{W_{1,t},Z_{t}\}$.
As the example makes clear, testing fundamentalness reduces to testing Granger non-causality or conditional lagged exogeneity giannone2006does, forni2014sufficient, plagborg2022instrumental, miranda2023identification. Complementarily, chen2017testing show that, when the DGP is a VARMA process generated by non-Gaussian i.i.d. shocks, fundamentalness holds if and only if the reduced-form innovations are m.d.s. chen2017testing. Maintaining this assumption, the shocks $X$ are fundamental or invertible if jointly: i) the structural shocks are non-Gaussian, ii) $\{X_{t}\}$ is m.d.s. with respect to its own past, $\sigma(X_{t-1},...)$, that is invertibility of $W_1$ alone, and iii) $\{X_{t}\}$ is m.d.s. with respect to the past of the omitted innovations, $\sigma(Z_{t-1},...)$. Taken together, these three conditions amount to the structural shocks $X$ having zero conditional mean given the past of both internal and external variables, $\sigma(X_{t-1},Z_{t-1},...)$, which is the null of weak exogeneity of Eq.((ref)).
baker2016measuring construct an index of Economic Policy Uncertainty (EPU) based on the frequency of newspaper articles that contain terms related to uncertainty, economy, and policy. Following their Section IV.D, the EPU structural shocks are estimated at monthly frequency by fitting a VAR(6) to five U.S. time series from Jan. 1985 to Dec. 2019, imposing a Cholesky ordering with the EPU index first, followed by the log S&P 500, federal funds rate, log employment and log industrial production. As noted by diercks2024rains, the estimated shock series is serially uncorrelated.\footnote{The shock series are provided in the replication package of diercks2024rains.} Both the Lilliefors and Jarque--Bera tests reject the null hypothesis of Gaussianity at the 5% level.
The small VAR system, however, arguably does not control for all relevant macroeconomic conditions. This turns out to be critical when questioning the exogeneity of the estimated shock, especially since the EPU index captures “uncertainties related to the economic ramifications of “noneconomic” policy matters [...] both near-term concerns [...] and longer term concerns” baker2016measuring. To assess these properties, this paper tests whether the EPU shock is weakly exogenous with respect to its own past and the past of omitted variables, using both the benchmark and the asymmetric Portmanteau statistics.
For omitted variables, this paper considers the first 8 principal components of large macroeconomic datasets, motivated by the intuition that, under a state space representation of the economic system, these estimated factors should approximate the relevant state variables forni2014sufficient. Specifically, we consider mccracken2016fred's (McK Ng) macroeconomic factors from FRED-MD, a database of 134 monthly U.S. macroeconomic indicators.\footnote{Spanning until Jun. 2021, 8 static factors are estimated by PCA allowing for missing values via EM algorithm stock2002forecasting.}
The left panel of Figure (ref) reports the statistics as the bandwidth $M$ varies. Both procedures reject weak exogeneity at the 5% level, but for different reasons. The benchmark rejects at very short horizons ($M\leq2$). By contrast, the asymmetric Portmanteau statistic rejects only for bandwidths larger than about two years. Hence, while the benchmark suggests strong short-run evidence against weak exogeneity, the proposed statistic points to a rejection mainly driven by medium- and longer-run cycles.
The right panel of Figure (ref) sheds light on this discrepancy by decomposing the benchmark statistic as in Eq. ((ref)), with each component expressed as a share of the absolute value of the centered benchmark statistic, $|T_\omega-\mu_\omega|$. At short horizons, the inverse causality channel, $\mathcal{C}_\omega$, accounts for almost all the magnitude of the benchmark statistic. Therefore, the short-horizon rejections are driven primarily by dependence from past EPU shocks to current omitted factors, rather than by the direction of weak exogeneity, namely dependence from past omitted factors to current EPU shocks. As $M$ increases, the contribution of the inverse channel declines, while the corrected channel becomes relatively more important.
The conclusion is therefore not merely about finite-sample differences. Figure (ref) shows that the benchmark rejections are largely due to inverse causality, whereas the rejections delivered by the asymmetric statistic are tied to the channel targeted by weak exogeneity. This distinction has practical consequences: by relying on the benchmark statistic alone, a practitioner could be misled into focusing on short-lag controls, while neglecting longer-lag dependencies. From this perspective, the decomposition in Eq. ((ref)) is pivotal, since it prevents the short-horizon rejection from being incorrectly interpreted as evidence against weak exogeneity.
In light of these results, the EPU shocks cannot be deemed fundamental or invertible when considering mccracken2016fred's factors. Impulse response analysis can therefore benefit from augmenting the system with such controls. To illustrate this point, I revisit diercks2024rains's analysis of the superadditive effects of uncertainty shocks, focusing on the impulse response of inflation and the stock market to EPU shocks. diercks2024rains estimate the following set of state-dependent local projections:
where $h$ sets the predictive horizon, ranging from 0 to 36 months, $p$ are the lags of the control variables, $\{w_t\}$, and the indicator function takes value 1 if each one of the previous $L$ shocks $\{\epsilon_{unc,t-1},..,\epsilon_{unc,t-L}\}$ has been positive. The state-multiplier coefficient $\beta_{1,h}$ captures the superadditive effect of uncertainty: the impact of a cascade of positive uncertainty shocks is more severe than the isolated sum of them. In their Appendix B.1 (Figure B.2), the shock of interest $\epsilon_{unc}$ is the EPU shock, the outcome variable $y$ is inflation, the number of consecutive positive shocks is 2 ($L=1$), the number of lags for the controls is 6, and the control set follows baker2016measuring. Figure (ref) reproduces their baseline results.
Upon adding lags of mccracken2016fred's macroeconomic factors to the controls in the local projections, the unconditional linear impulse response remains largely unchanged (dashed vs. solid black lines in Figure (ref)), while the state-dependent response, $\{\beta_{1,h}\}_{h=0,...,H}$, becomes positive and more pronounced as the number of factor lags increases. This pattern is consistent with the diagnostics in Figure (ref). Once longer-lag macroeconomic information is included, the superadditive response of inflation to EPU shocks becomes clearer, strengthening diercks2024rains's conclusions: a sequence of consecutive positive uncertainty shocks leads to a marked increase of inflation. This finding connects to two strands of the literature. First, it aligns with fernandez2015fiscal, who show that unexpected changes in fiscal policy uncertainty can lead to increased inflation within a standard New Keynesian model. Second, it relates closely to ascari2023endogenous, who demonstrate that, in a rich DSGE model with firm dynamics, an `expectational' shock that raises short-term inflation expectations results in negative macroeconomic effects, causing inflation to rise while output declines.
Turning to the remaining variables of the system, while the impulse responses of industrial production and short rates are consistent with diercks2024rains (not reported), the notable exception is the stock market. The state-dependent response of the real S&P 500 to consecutive positive EPU shocks becomes significantly more negative at longer horizons (Figure (ref)), consistent with berger2020uncertainty's finding that uncertainty shocks are linked to declines in stock returns (refer to their discussion in Section 5.2).
A comparison across uncertainty shocks reveals an interesting heterogeneity. The inflation response to shocks from the other two considered measures (ludvigson2021uncertainty's and berger2020uncertainty's) is strongly negative, in contrast to the positive inflationary effect documented for EPU shocks. Meanwhile, the response of industrial production is negative across all three shock series, confirming their countercyclical nature diercks2024rains. This pattern suggests that EPU shocks might operate primarily through a supply-side channel, whereas the financial uncertainty and realized volatility uncertainty shocks might be better characterized as demand-side disturbances.
This paper studies specification testing in dynamic linear models in the presence of omitted variables, with its focus on testing weak exogeneity of structural shocks. Portmanteau statistics based on quadratic forms of serial cross-correlations might confound violations of weak exogeneity with dependence running from past shocks to current omitted variables (inverse causality), potentially producing misleading rejections. To address this issue, this paper proposes an asymmetric Portmanteau statistic by introducing a correction term which removes the influence of inverse causality. The proposed statistic isolates the directional restriction implied by weak exogeneity without requiring parametric modeling of the joint dynamics. Under mild restrictions on the conditional moments, the asymmetric Portmanteau statistic is asymptotically normal under the null, and achieves asymptotic power comparable to the benchmark under fixed alternatives. An empirical application revisits the exogeneity of a widely used shock series, baker2016measuring's Economic Policy Uncertainty shocks, and provides evidence against its (weak) exogeneity. By revisiting diercks2024rains, enlarging the information set in light of the previous findings leads to a shift from negative to positive inflation (superadditive) response, together with contractionary effects elsewhere, thus suggesting a supply-side interpretation of the shock series.
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