Estimation of the Risk Measure under a Nuisance Autoregression

Abstract: The goal of an experiment is to evaluate the profit, loss, or the amount of a physical entity over a period. The measurements $X_t$ can be influenced by the values measured in the past; hence we describe the situation with an autoregression model, whose autoregression coefficients are generally unknown. The variable of interest is the error term $Z_t$ of the model, which is the increment of $X_t$ with respect to the past, but itself unobservable. The problem is to estimate various quantile functions of $Z$, as the risk measure of the loss or the related economic indicators. We construct an estimate of quantile functions of $Z$ in the situation that the inference is possible only by means of observations $X$. The proposed estimates are based on the R-estimators of autoregression coefficients, combined with the autoregression quantiles.

• The paper develops a plug-in method that combines autoregression quantiles with R-estimation to estimate risk measures of unobservable innovations while removing the first-order effect of nuisance slope estimation.
• The proposed CVaR estimator is root-n consistent under stationarity, smooth innovation densities, and finite fourth moments, and simulations show feasible estimates closely match an oracle using true autoregressive parameters.
• The method performs well at moderate tail levels, but extreme-tail estimation becomes substantially less reliable for heavy-tailed innovations and small samples, highlighting the need for robust inference and broader model validation.

Problem and setting

Jurečková and Picek address the estimation of quantile functionals of the innovation distribution in a $p$-th order autoregression,

$$X_t = \phi_1 X_{t-1} + \cdots + \phi_p X_{t-p} + Z_t,$$

where the slope parameters $\phi = (\phi_1,\ldots,\phi_p)$ are unknown and, consequently, the innovations $Z_t$ — the increments of the monitored quantity with respect to its past — are unobservable. The target of inference is a functional $\mathcal S(Q)$ of the quantile function $Q = F^{-1}$ of $Z$, typically a risk measure such as the Conditional Value-at-Risk (CVaR, expected shortfall), but potentially also energy consumption over a period, water flow, or welfare/inequality indicators. The authors emphasize that ignoring serial dependence can either mask or spuriously indicate trends, a concern documented in the hydrological literature (Yue et al.).

The distributional assumptions are standard: (A1) $F$ has a continuous density positive on its support; (A2) zero mean, finite variance and fourth moment; (A3) stationarity, i.e., all roots of the AR characteristic polynomial lie inside the unit circle.

Methodology: autoregression quantiles combined with R-estimation

The construction rests on two classical components. The first is the $\alpha$-autoregression quantile $\widehat\phi(\alpha)$ of Koul and Saleh, defined as the minimizer of the check-loss criterion $$X_t = \phi_1 X_{t-1} + \cdots + \phi_p X_{t-p} + Z_t,$$0, whose asymptotic representation is uniform on compact subintervals of $$X_t = \phi_1 X_{t-1} + \cdots + \phi_p X_{t-p} + Z_t,$$1. The second is the R-estimator $$X_t = \phi_1 X_{t-1} + \cdots + \phi_p X_{t-p} + Z_t,$$2 of the nuisance slopes, obtained by minimizing the Jaeckel dispersion of ranked residuals with the step score $$X_t = \phi_1 X_{t-1} + \cdots + \phi_p X_{t-p} + Z_t,$$3; Koul–Ossiander and Mukherjee–Bai establish its asymptotic linearity and $$X_t = \phi_1 X_{t-1} + \cdots + \phi_p X_{t-p} + Z_t,$$4-consistency under (A1)–(A3).

The key step is a residual empirical process result showing that replacing true innovations by residuals from the R-estimated slopes is asymptotically negligible uniformly in local shifts around $$X_t = \phi_1 X_{t-1} + \cdots + \phi_p X_{t-p} + Z_t,$$5:

$$X_t = \phi_1 X_{t-1} + \cdots + \phi_p X_{t-p} + Z_t,$$6

This "nuisance elimination" lemma implies that the $$X_t = \phi_1 X_{t-1} + \cdots + \phi_p X_{t-p} + Z_t,$$7-th order statistic of the residuals, $$X_t = \phi_1 X_{t-1} + \cdots + \phi_p X_{t-p} + Z_t,$$8, admits the same first-order representation as if the innovations were observed, converging to $$X_t = \phi_1 X_{t-1} + \cdots + \phi_p X_{t-p} + Z_t,$$9 at rate $\phi = (\phi_1,\ldots,\phi_p)$0. In particular, $\phi = (\phi_1,\ldots,\phi_p)$1 itself estimates the Value-at-Risk at level $\phi = (\phi_1,\ldots,\phi_p)$2.

Estimation of CVaR

Exploiting the identity of Bassett, Koenker and Kordas that $\phi = (\phi_1,\ldots,\phi_p)$3, the paper proposes the plug-in estimator

$\phi = (\phi_1,\ldots,\phi_p)$4

and states a theorem asserting $\phi = (\phi_1,\ldots,\phi_p)$5 under (A1)–(A3). The proof is short, combining the residual-empirical result with Bassett et al.; it relies on the finite fourth-moment assumption for the sample-mean correction term. The same template extends to other linear quantile functionals.

Simulation evidence

The simulation design covers AR(1) with $\phi = (\phi_1,\ldots,\phi_p)$6 and $\phi = (\phi_1,\ldots,\phi_p)$7, and AR(2) with $\phi = (\phi_1,\ldots,\phi_p)$8; sample sizes $\phi = (\phi_1,\ldots,\phi_p)$9; tail levels $Z_t$0; and four innovation laws (Gaussian, standardized Student-$Z_t$1, a scale mixture $Z_t$2, and rare-large-shock contamination). A crucial feature is an infeasible oracle benchmark that uses the true AR slopes, isolating the finite-sample cost of estimating the nuisance autoregression.

The central empirical finding is that the feasible R-based estimator is essentially indistinguishable from the oracle across all settings — differences in bias and RMSE are typically in the third decimal place. For example, at $Z_t$3, $Z_t$4, contamination: RMSE 2.0033 (feasible) versus 1.9986 (oracle). This supports the claim that estimating the nuisance autoregression carries negligible finite-sample cost, and that the dominant error source is the intrinsic difficulty of tail-risk estimation itself.

At $Z_t$5, biases are small throughout (e.g., below 0.11 in absolute value even for mixtures at $Z_t$6). At $Z_t$7, however, performance degrades substantially under heavy tails: for Student-$Z_t$8 at $Z_t$9, bias reaches roughly $\mathcal S(Q)$0 to $\mathcal S(Q)$1 and RMSE exceeds 2.0, improving only gradually ($\mathcal S(Q)$2: RMSE ≈ 1.11). This reflects the well-known slow convergence of extreme-quantile functionals under heavy-tailed laws rather than a defect of the nuisance-estimation strategy, but it does qualify the practical usability of the method at very high tail levels in small samples.

Numerical illustration

The method is applied to daily mean discharge data (2021–2024) from the Czech Hydrometeorological Institute gauge Labský důl on the Labe river, transformed as $\mathcal S(Q)$3 and modeled as AR(1). With an effective sample size of $\mathcal S(Q)$4, the rank-based slope estimate is $\mathcal S(Q)$5, indicating strong persistence. The resulting raw-residual tail-risk estimates are $\mathcal S(Q)$6 and $\mathcal S(Q)$7 on the transformed scale, with exceedances of the estimated VaR$\mathcal S(Q)$8 flagged on the time series plot. The illustration demonstrates feasibility on real data but is not accompanied by any validation against an alternative estimator or out-of-sample assessment.

Limitations and open questions

Several caveats bear directly on the results. First, the asymptotic theory assumes a correctly specified, finite AR order and i.i.d. innovations; heteroskedastic or conditionally dependent innovations fall outside the stated guarantees. Second, the $\mathcal S(Q)$9 claim for the CVaR estimator presumes finite fourth moments of $Q = F^{-1}$0, which is incompatible with the heavy-tailed settings (e.g., $Q = F^{-1}$1 has only three finite moments) where simulations show the largest errors — so the theoretical accuracy guarantee does not strictly cover the most difficult scenarios studied. Third, no inference tools (standard errors, confidence intervals for $Q = F^{-1}$2) are developed here, although the introduction lists them among the goals. Fourth, the choice of the score parameter $Q = F^{-1}$3 and the extension beyond linear quantile functionals are left unexplored. Finally, the real-data illustration involves a single gauge and a fixed AR(1) specification without model diagnostics.

Conclusion

The paper provides a coherent plug-in methodology for estimating quantile functionals of unobservable AR innovations, combining autoregression quantiles with Jaeckel-type R-estimation of the nuisance slopes, and proves root-$Q = F^{-1}$4 consistency for the CVaR estimator under regularity conditions. Its main empirical contribution is demonstrating, via an oracle comparison, that the nuisance autoregression can be estimated at essentially no finite-sample cost, with remaining error driven by tail-level and innovation-tail characteristics. The open issues are the development of accompanying inferential machinery, robustness of the theory to moment violations at extreme tail levels, and validation on broader real-data applications.