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A Gentle Introduction to Conformal Time Series Forecasting
As underlined by the most recent reviews on the subject, the quantification and formalisation of prediction uncertainty is a key challenge in forecasting petropoulos_forecasting_2022. A simple point forecast, $\hat{y}$, which provides a single-value estimate for a quantity $y \in \mathbb{R}$, is inherently limited. For any continuous random variable $Y$, the probability of the outcome being exactly the point forecast is zero (i.e., $\mathbb{P}(Y=\hat{y})=0$). Therefore, the true practical value of forecasting, particularly for robust decision-making and risk assessment, lies not in identifying a single "most likely" value, but in quantifying the full spectrum of possible outcomes. The ultimate objective is to move beyond single-point estimates toward probabilistic forecasts that describe the entire predictive distribution of future quantities gneiting_probabilistic_2014, gneiting_probabilistic_2007
The challenge, however, lies in generating these distributions accurately. Traditionally, uncertainty quantification has relied on strong parametric assumptions, such as imposing a Gaussian (Normal) distribution on the forecast errors elliott_economic_2016. This assumption is frequently violated in real-world applications. Many phenomena, particularly in economics, finance, and anthropogenic systems, do not follow "mild" randomness. They are instead characterised by heavy tails (leptokurtosis), significant skewness, and non-linear dependencies, which the Gaussian distribution fundamentally fails to capture cont_empirical_2001, taleb_black_2007. This mismatch often leads to a drastic underestimation of risk and a false sense of security in model predictions.
In this context, Conformal Prediction (CP) has emerged as a powerful and principled alternative to parametric methods vovk_algorithmic_2023. It provides a distribution-free framework that wraps around any point forecasting model, from simple linear regressions to deep normalising flow models, and calibrates its predictions to produce sets with rigorous, finite-sample predictive coverage guarantees. Let $\{ (X_t, Y_t) \}_{i=1}^T$ be a sample of \(T\) random covariate/response pairs with stationary marginals. Each pair \((X_t, Y_t) = Z_t\) takes values in \(\mathcal{X} \times \mathcal{Y}\), where \(\mathcal{X}\) and \(\mathcal{Y}\) are measurable spaces. Our objective is, for given a miscoverage level $\alpha \in (0,1)$, to construct a prediction set $\mathcal{C}_{1-\alpha}(X_{T+1})$ for a new, unobserved, $Y_{T+1}$ such that the following validity property holds:
where the probability $\mathbb{P}_{\mathrm{tr}}$ is taken over $\{ (X_t, Y_t) \}_{i=1}^T \cup Z_{T+1} $ The methodologies under the CP umbrella provide techniques and algorithms to identify regions endowed with the validity property described above. Two main families of methodologies are available: Full (or Inductive) CP and Split (or Transductive) CP. The two methodologies distinguish themselves according to how they treat the training data, and how intensive is their computational load. Apart from this general distinction, our goal in this paper is not to describe analogies and differences between the two methodologies. The interested reader can refer to the already cited main text on the subject, as well as to several introductions and reviews fontana_conformal_2023,angelopoulos_conformal_2023-5.
We will focus our attention on Split, or Transductive CP (SCP), firstly introduced in papadopoulos_inductive_2002, and analysed with a remarkable level of detail in lei_distribution-free_2018. This choice is due to the specific nature of the algorithms analysed in this review, mainly focused on a Split framework. The recipe is as follows:
First, we partition the data indices into two disjoint sets: a training set \(I_{\mathrm{train}}\) and a calibration set \(I_{\mathrm{cal}}\). Second, we use the training set to fit our model of choice, $\hat{\mu}$, which learns a function from $\mathcal{X}$ to $\mathcal{Y}$. Third, we define a nonconformity score function $s(x, y)$, which quantifies how "strange" or "nonconforming"\footnote{It is in principle possible to define conformity, rather than non-conformity scores, that measure "conformity" with respect to the original data} a given data pair $(x, y)$ is with respect to the training set $\{(x_t,y_t), t\in I_{\mathrm{train}}\}$. For regression problems, traditionally absolute residuals, $s(X_i, Y_i) = |Y_i - \hat{\mu}(X_i)|$ are used, but other choices like quantile-based scores (romano_conformalized_2019) and Density/HDR-based ones (izbicki_cd-split_2022) are possible. The key idea is that higher scores should correspond to data that fits the model poorly. We compute the nonconformity scores for all points in the calibration set, creating a set of "typical" errors: $S_{\mathrm{cal}} = \{s_i \mid i \in I_{\mathrm{cal}}\}$. Fifth, to achieve a target coverage of $1-\alpha$, we find the empirical $(1-\alpha)$quantile of these calibration scores. Specifically, letting $n_{\mathrm{cal}} = |I_{\mathrm{cal}}|$, we compute: \[ \hat{q}_{1-\alpha} = \textit{Quantile}(S_{\mathrm{cal}}, \lceil (1-\alpha)(n_{\mathrm{cal}}+1) \rceil / n_{\mathrm{cal}}) \]. This $\hat{q}_{1-\alpha}$ represents the error threshold that $(1-\alpha)$ of the calibration points did not exceed. Finally, for a new point $X_{T+1}$, we construct the prediction set by inverting the score function. We include all possible values $y \in \mathcal{Y}$ whose nonconformity score is no larger than our threshold $\hat{q}_{1-\alpha}$: \[ \mathcal{C}_{1 - \alpha}(X_{T+1}) = \{ y \in \mathcal{Y} : s(X_{T+1}, y) \leq \hat{q}_{1-\alpha} \} \] When using the absolute residual score, this definition simplifies to the familiar interval: $$\mathcal{C}_{1 - \alpha}(X_{T+1}) = [\hat{\mu}(X_{T+1}) - \hat{q}_{1-\alpha}, \hat{\mu}(X_{T+1}) + \hat{q}_{1-\alpha}].$$
If the data in $I_{\mathrm{cal}}$ and the new test point $(X_{\text{test}}, Y_{\text{test}})$ are exchangeable, this simple procedure provides the powerful guarantee of finite-sample marginal coverage: $\mathbb{P}_{tr}(Y_{T+1} \in \mathcal{C}_{1 - \alpha}(X_{T+1})) \ge 1-\alpha$.
The exchangeability assumption in this context is key This property is the key theoretical mechanism that enables the methods to provide guaranteed finite-sample coverage (i.e., validity) without making any specific parametric assumptions about the underlying data-generating process angelopoulos_theoretical_2025.
In the absence of exchangeability, it is not trivial anymore to establish validity properties. This complication motivates the central questions driving the research in this area: Can we preserve the distribution-free guarantees of CP when predicting non-exchangeable data? How? And, in doing so, what theoretical properties or practical efficiencies must we trade off?
This review synthesizes and unifies the modern approaches developed to solve this problem, mainly in the context of time-series data. We will collectively call this branch of CP, Conformal Forecasting. We aim to move beyond a simple survey of algorithms. Instead, we provide a structured classification of these methods, providing an harmonised notation, and a grouping by the core philosophy they employ to address non-exchangeability.
Our contributions are threefold: (i) a practical and narrative synthesis of these baseline algorithms and their modern variants; (ii) a controlled empirical comparison that maps the validity–efficiency–compute trade-offs of these competing strategies and (iii) a deeper, unified theoretical rework of finite-sample coverage guarantees for standard SCP under checkable, weak-dependence ($\beta$-mixing) conditions; (with full proofs in the Appendixs), providing additional details with respect to the original work oliveira_split_2024.
The review is structured as follows. Section (ref) formalizes how time series data violate the exchangeability assumption through temporal dependence and distribution shifts. Section (ref) presents theoretical guarantees for SCP under weak dependence. Section (ref) reviews the four main families of adaptive conformal forecasting methods: Weighted CP (WCP), EnbPI, Adaptive CP (ACI), and Block CP (BCP), providing pseudocode for each. Section (ref) details and discusses the results of a comprehensive empirical study comparing these methods on simulated data. Finally, Section (ref) concludes with practical recommendations and a discussion of limitations and future research directions.
We argued in the previous section that the good properties of CP are fundamentally based on an exchangeability assumption. We provide a more formal definition below.
This assumption, although weaker than IID, is typically violated in time series settings. By definition, time series data are ordered. This temporal order is not a nuisance but the very structure that carries meaningful information. Such ordering may break exchangeability in several fundamental ways:
It is crucial to distinguish this specific temporally-structured non-exchangeability from other violations encountered in static machine learning. A common case, for instance, is covariate shift. In that setting, data $(X_t, Y_t)$ is often assumed to be independent within the training and test sets, but the marginal distribution of the covariates $P_X$ differs between them (i.e., $P_{X_{\text{train}}} \neq P_{X_{\text{test}}}$), even if the conditional $P_{Y|X}$ remains invariant. Another violation is given by panel data settings, where data points are correlated within a group (e.g., multiple measurements from the same patient or samples from the same batch) but the groups themselves are independent and exchangeable. In these settings, the data index is a nominal label, and the non-exchangeability arises from a latent group structure. In the time-series context, the index $t$ is fundamentally ordinal. The violations are defined by this sequence: temporal dependence relates $Z_t$ to $Z_{t-k}$, and distribution shift makes $P_t$ a function of $t$ itself. This sequential structure, governed by proximity and direction, is the unique challenge of conformal forecasting. As a comment, since the filtration $\mathcal{F}_{t-1}$ is order-dependent, permuting the observations would give a different joint law, thus breaking exchangeability.
To better visualise these concepts, we provide some examples.
We have described above how the defining characteristics of time series data ,namely temporal dependence and distribution shifts ,fundamentally violate the exchangeability assumption required in standard conformal prediction.
Yet, in the presence of mild violations, the validity property of Conformal is only mildly affected, and in specific cases such violation can be computed.
In the Appendix, we present a detailed theoretical analysis of this degradation, following the framework of oliveira_split_2024. We show that under general assumptions of weak dependence (Assumptions (ref)-(ref) in the Appendix), the coverage gap, defined as the non-negative difference $G := \max\left(0, (1-\alpha) - \mathbb{P}(Y \in \mathcal{C}_{1-\alpha}(X))\right)$, representing the shortfall between the nominal coverage $1-\alpha$ and the true coverage probability, is explicitly bounded by a small constant. For any test point $i \in I_{\mathrm{test}}$, the coverage is: \[ \mathbb{P}_{\mathrm{tr}} \left[ Y_i \in \mathcal{C}_{1 - \alpha}(X_i) \right] \geq 1 - \alpha - \underbrace{(\varepsilon_{\mathrm{cal}} + \delta_{\mathrm{cal}} + \varepsilon_{\mathrm{train}})}_{\text{Slack term}} \] These slack terms have intuitive meanings: $(\varepsilon_{\mathrm{cal}}, \delta_{\mathrm{cal}})$ bound the concentration error (how well the $n_{\mathrm{cal}}$ calibration scores represent the "true" error distribution), while $\varepsilon_{\mathrm{train}}$ bounds the decoupling (how much dependence exists between the training set and the test point). See Theorem (ref) in Appendix (ref) for the full statement and proof.
We further show in Theorem (ref) in Appendix (ref) how to derive explicit, non-asymptotic bounds for these slack terms in the specific case of stationary $\beta$-mixing processes (Propositions (ref)-(ref)). The $\beta$-mixing coefficient $\beta(a)$ quantifies how quickly a process "forgets" its past; if $\beta(a)$ decays rapidly, our slack terms become small, and the coverage guarantee $1-\alpha - \eta$ approaches the nominal $1-\alpha$.
The crucial takeaway is that for stationary, weakly dependent processes, standard SCP is approximately valid, and its deviations from validity are indeed very mild. The true problem is non-stationarity (distribution shift), against which these theoretical results offer limited protection. The following methods are designed to explicitly address both strong dependence and, more importantly, distribution shift.
In case we have dependencies that are more severe than the weak cases considered by oliveira_split_2024 several methodologies, with very different philosophical approaches, have appeared in the literature.
These conformal forecasting techniques are designed to restore valid coverage by explicitly handling temporal dependence and distribution shift. We first propose a narrative classification of methods according to their core philosophy, and then provide to state the algorithms in detail. In a nutshell, methodologies can be based on:
We now proceed analysing the different groups one by
The first family of methods adapts to non-exchangeability by challenging the assumption that all calibration points are created equal. If the calibration set is no longer representative, perhaps we can re-weight its elements to prioritize points that are more relevant to the current prediction.
This is formalized by replacing the standard empirical quantile with a weighted empirical quantile. Given non-negative weights \(\{w_i\}_{i\in I_{\mathrm{cal}}}\) for each calibration point, we can normalized to sum to one: $\widetilde w_i = w_i / \sum_j w_j$. The weighted quantile is then defined as the smallest score $t$ that captures at least $1-\alpha$ of the weighted mass: \[ \hat q^{(w)}_{1-\alpha} := \inf\Big\{ \tilde{s} \in \mathbb{R} : \sum_{i\in I_{\mathrm{cal}}} \widetilde w_i\,\mathbf{1}\{s(X_i,Y_i) \le \tilde{s}\} \ge 1-\alpha \Big\}, \] and the resulting prediction set is $\mathcal{C}^{(w)}_{1-\alpha}(x) = \{ y : s(x,y) \le \hat q^{(w)}_{1-\alpha} \}$. Algorithm (ref) summarises the non-exchangeable CP (Nex-CP) framework.
As shown by barber_conformal_2023, the coverage gap of this method is bounded (see Theorem (ref) in the Appendix). The bound is small if large weights $\widetilde{w}_i$ are assigned to calibration points $i$ that are "distributionally similar" to the test point.
The critical design choice, of course, is how to set the weights.
Instead of re-weighting a fixed calibration set, a second class of methods adapts to new test input by actively refreshing the set of residuals $\{\hat{\varepsilon}_i\}^T_{i=1}$, in this case computed without the need of splitting training and calibration. The goal is to ensure the quantile $\hat{q}_{1-\alpha}$ is always based on the most recent, and therefore most relevant, error distribution.
The main representative is EnbPI (Ensemble Batch Prediction Intervals) by xu_conformal_2021. EnbPI avoids the static train/calibration split by using bootstrap ensembles and out-of-bag (OOB) predictions. The procedure is as follows:
EnbPI provides approximate marginal coverage under stationarity and mixing conditions xu_conformal_2021. Its main trade-off is computational: it requires training and storing $B$ models and, in its sequential form, re-calculating OOB residuals, which is significantly more expensive than SCP or WCP.
This approach has been powerful and influential, inspiring several key extensions. A major limitation of EnbPI is that it ignores heteroskedasticity: it assumes all residuals are drawn from the same (shifting) pool. It produces a single interval width for all $X_t$. SPCI (Sequential Predictive Conformal Inference) by xu_sequential_2023 fixes this. It replaces the global residual quantile with a conditional one. Instead of asking "How big are residuals on average?", it asks, "Given the features $X_t$, how big is the residual likely to be?" It uses Quantile Random Forests (QRF) to learn a function $\hat{Q}_{\tau}(\varepsilon \mid X_t)$ that maps features to a residual quantile. This allows the interval to be naturally wider for volatile inputs and tighter for stable inputs, providing a much sharper and more adaptive forecast. This idea was further modernized by SPCI-T lee_transformer_2024, which replaces the QRF module with a state-of-the-art Transformer architecture, allowing the conditional quantile model to capture more complex and long-range temporal dependencies.
The final family of methods takes a different approach. It keeps the indices of the original calibration set $I_{\text{cal}}$ fixed, unlike EnbPI, and does not introduce modifications to the quantile computation, unlike WCP. What it updates online is the target miscoverage level $\alpha$ itself.
The first example of these algorithms is ACI (Adaptive Conformal Inference), proposed by gibbs_adaptive_2021 for sequential settings where the true $Y_t$ is revealed after each prediction. It maintains a running "effective" miscoverage level, $\alpha_t$, which it updates at every time step based on its past performance. It is, in essence, a feedback controller.
At time $t$, the algorithm produces an interval $\mathcal{C}_{1-\alpha_t}(X_t)$ using the current level $\alpha_t$ and the fixed calibration set $\{Z_i,i\in I_{\text{cal}}\}$. After observing the true $Y_t$, it updates the level for the next step using a simple additive rule:
where $\gamma > 0$ is a step-size parameter, or learning rate.
The logic is intuitive. $\alpha$ is the target level, and for each time step, the algorithm observes a binary error $\mathbf{1}\{Y_t \notin \mathcal{C}\}$.
This feedback loop (see Algorithm (ref)) provides a guarantee that is different from the classical CP guarantee: the long-run empirical miscoverage rate is proven to converge to $\alpha$, regardless of the data-generating process (see Theorem (ref) in Appendix B). It can handle arbitrary distribution shifts, as long as feedback is provided.
The main practical challenge of ACI is choosing the step size $\gamma$. A large $\gamma$ (high learning rate) adapts very quickly to shifts but can be unstable and oscillate wildly. A small $\gamma$ is stable and converges smoothly but adapts too slowly to abrupt shifts. Several extensions have been proposed to solve this:
A final, conceptually distinct approach is Block CP (BCP), introduced in chernozhukov_exact_2018. This method redefines the object of randomization, positing that while individual data points $Z_t$ are not exchangeable, entire blocks of data may be (approximately) exchangeable.
The original transductive formulation is computationally intensive, as it requires re-estimating the model for every candidate future and every permutation. This procedure is generally considered "unfeasible" in high-dimensional or functional data settings ajroldi_conformal_2023. A scalable and computationally efficient alternative is the Split-Conformal BCP, which adapts the blocking scheme to the inductive (split) framework ajroldi_conformal_2023, diquigiovanni_distribution-free_2024.
This split-BCP procedure modifies the standard BCP algorithm as follows:
This split-BCP approach loses the exact finite-sample validity of the transductive method. However, it retains robust theoretical guarantees, providing approximate validity and asymptotic exactness under weak dependence conditions (e.g., strong mixing or ergodicity) ajroldi_conformal_2023, diquigiovanni_distribution-free_2024. A significant practical advantage is that, with an appropriate choice of nonconformity score, the set $\mathcal{C}_{1-\alpha}^B$ can often be computed in closed form, avoiding the infeasible search over all possible $y$ ajroldi_conformal_2023, diquigiovanni_distribution-free_2024.
We now empirically compare the main conformal methods (SCP, WCP, ACI, and EnbPI, SCP-Block) on simulated time series data. We evaluate each on test coverage, average interval width, and computational cost. The code can be found in malgorzewicz_intro--cp-for-time-series-forecasting_2025.
We study four canonical processes: two stationary, $\beta$-mixing benchmarks, one non-exchangeable process designed to break the methods, and one heteroscedastic model. We generate $n=900$ pairs $(X_t,Y_t)$, split into train/cal/test of (300, 300, 300), and repeat $R=50$ times. The covariate $X_t = (Y_{t-1},\ldots,Y_{t-p})$ collects past lags.
\paragraph{Base forecaster} To isolate the effect of the conformal post-processing, all methods use the same simple autoregression fit by least squares (AR-LS). The model $\hat Y_t = \langle a, X_t\rangle$ is fit once on the training block $I_{\mathrm{train}}$ and its coefficients $a$ are held fixed. This is a crucial design choice: the forecaster itself does not adapt to the mean shift, forcing the conformal layer to do all the work.
All methods use absolute residual scores $|Y_t-\hat\mu(X_t)|$ with target coverage $1-\alpha=0.9$.
\paragraph{Metrics} For each method on the test block (points 601-900), we compute: (1) Coverage (empirical frequency $\frac{1}{300}\sum \mathbf{1}\{Y_t \in \mathcal{C}_t\}$), (2) Average width ($\frac{1}{300}\sum |\mathcal{C}_t|$), and (3) Wall time.
We plot mean coverage vs. mean width for each process (Figure (ref)) and a single bar chart for average runtime (Figure (ref)). For each process, we additionally plot the associated error bars, using a 95% confidence interval around the mean coverage and mean width.
The empirical results distinguish the performance of the conformal strategies across the different data-generating processes.
In the stationary settings (AR(1), ARMA(1,1), and GARCH(1,1)), the results are consistent. As shown in Figure (ref), most tested methods, including SCP, all WCP variants, all ACI variants, and all EnbPI variants, achieve empirical coverage very close to the nominal $0.9$ target. The notable exception is SCP-block, which visibly under-covers in all three stationary scenarios, failing to reach the nominal target. Among the valid methods, primary differences are in statistical efficiency (interval width). EnbPI consistently produces the widest intervals, a likely result of the variance introduced by its bootstrap-based procedure. In contrast, SCP, WCP, and ACI are more efficient, yielding tighter intervals of comparable widths. For stable, stationary processes, these results suggest the baseline SCP is a sufficient, efficient, and valid method.
The non-stationary setting (Mean-Shift) reveals significant performance disparities. Here, several methods fail. SCP's coverage degrades to approximately 0.84, and SCP-block's coverage falls to \ 0.81-0.84. The WCP-window method also fails, with coverage dropping to 0.81. Their calibrated quantiles, $\hat{q}_{1-\alpha}$, are computed from the pre-shift calibration data ($I_{\mathrm{cal}}$) and are therefore invalid estimates for the post-shift regime. These methods, blind to the model's new systematic error, continue to produce overly narrow and invalid intervals.
In contrast, the other adaptive methods successfully handle the abrupt shift. ACI (all $\gamma$ values), EnbPI (all $s$ values), WCP-exp, and WCP-linear all maintain coverage at or near the nominal 0.9 level. Their success is attributable to their explicit adaptation mechanisms. ACI uses an active feedback loop to widen intervals. EnbPI's sliding window refreshes its residual pool, while WCP-exp and WCP-linear successfully
Classical CP relies on the assumption of exchangeability, which is fundamentally violated by time series data due to temporal dependence and distribution shifts. This review synthesizes and evaluates modern conformal forecasting methods designed to address this limitation.
The findings indicate that methods adapt via four primary mechanisms: reweighting calibration data (WCP), refreshing the residual pool (EnbPI), adapting the target coverage level online (ACI), or blocking the data (SCP-block). The theoretical analysis (Appendix (ref)) confirms that for weakly-dependent, stationary ($\beta$-mixing) processes, standard SCP provides approximately valid coverage. The empirical study supports this, showing that SCP, WCP, ACI, and EnbPI all achieve nominal coverage on stationary data. However, the study also reveals two key failures: (1) SCP-block failed to provide valid coverage even in the simple stationary settings, and (2) standard SCP and some adaptive variants (like WCP-window) fail under an abrupt distribution shift. In such non-stationary settings, methods with explicit recency-focused adaptation (ACI, EnbPI, and WCP-exp/linear) were all shown to successfully restore nominal coverage.
Based on this analysis, the choice of method should be guided by the properties of the data stream and operational constraints.
The review of the theory, while not being exhaustive, serves as a good starting point in order to formalise and systematise the growing body of knowledge on Conformal Forecasting. The simulation study shows few applicative cases, and serves as minimalistic baseline comparison to start comparing different methods. Such minimalism highlight avenues for future research.
First, the use of a simple, fixed AR-LS model was intended to isolate the effect of the conformal layer. In practice, more complex prediction methods (e.g., re-trained ARIMA or neural models) would reduce the magnitude of the residuals, improving the efficiency of all methods and potentially altering their relative performance. In addition, more sophisticated scoring functions could be employed—ones capable of producing not only sharper prediction regions but also asymptotic conditional validity properties (see, e.g., izbicki_cd-split_2022, chernozhukov_distributional_2021)—and evaluated empirically.
Secondly, the empirical evaluation was limited to simple processes. Further testing is required on data with more complex structures, strong seasonality, long memory and multi-step forecast horizons. Thirdly, we mainly focused our attention to the univariate case. The issue of multivariate time series forecasting in a Conformal Setting joins two very new areas of research (see e.g. dheur_unified_2025). all simulation examples are assumed to be stationary. The deviations from exchangeability considered in this analysis are also fairly simple, and an exploration of methods for conformal forecasting data with locally stationary, or plainly nonstationary time series data. Third, the hyperparameter sweeps were coarse; the performance of ACI, EnbPI, WCP, and SCP-block is sensitive to their respective parameters ($\gamma$, $s$, $\rho$, $B$), and a comprehensive optimization study would be required to establish a definitive performance ranking. Finally, this study focused on canonical baselines and did not empirically test more advanced variants, such as learned-weight WCP (e.g., Hop-CPT) or advanced ACI (e.g., AgACI, PID control), which remain promising areas for benchmarking
The outlook for conformal forecasting involves the development of hybrid methods that combine these adaptive strategies. Examples include integrating ACI feedback loop with SPCI conditional quantile estimation, or pairing WCP with weights learned with more advanced models, as well as the identification of fixed or learned weight schemes for more complex dependency stuctures. As data complexity increases, the demand for computationally tractable uncertainty quantification that is robust to non-exchangeability will continue to grow.
Moreover, while our attention has been focused on the classical, interval prediction setting, very interesting and recent results (allen_-sample_2025) pave the way for extensions of Conformal forecasting methods to distributional forecasting.
The Authors would like to thank Prof. Dr. Mathias Trabs and Prof. Rainer Von Sachs for comments on early drafts of this manuscript.