Identifying Patterns in Financial Markets: Extending the Statistical

Jump Model for Regime Identification∗†

‡
Afşar Onat Aydınhan1 , Petter N. Kolm2 , John M. Mulvey1 , Yizhan Shu1
1
Department of Operations Research and Financial Engineering, Princeton University
2
Courant Institute of Mathematical Sciences, New York University
March 20, 2024

Abstract
Regime-driven models are popular for addressing temporal patterns in both financial market
performance and underlying stylized factors, wherein a regime describes periods with relatively
homogeneous behavior. Recently, statistical jump models have been proposed to learn regimes
with high persistence, based on clustering temporal features while explicitly penalizing jumps
across regimes. In this article, we extend the jump model by generalizing the discrete hidden
state variable into a probability vector over all regimes. This allows us to estimate the probability
of being in each regime, providing valuable information for downstream tasks such as regime-
aware portfolio models and risk management. Our model’s smooth transition from one regime
to another enhances robustness over the original discrete model. We provide a probabilistic
interpretation of our continuous model and demonstrate its advantages through simulations and
real-world data experiments. The interpretation motivates a novel penalty term, called mode
loss, which pushes the probability estimates to the vertices of the probability simplex thereby
improving the model’s ability to identify regimes. We demonstrate through a series of empirical
and real world tests that the approach outperforms traditional regime-switching models. This
outperformance is pronounced when the regimes are imbalanced and historical data is limited,
both common in financial markets.

Keywords: Regime Switching; Temporal Clustering; Statistical Jump Models; Probabilistic Mod-
eling; Times Series; Unsupervised Learning

∗
This article is dedicated to Professor Bill Ziemba, who spent his long career researching and discovering systematic
approaches to improve investment performance by strategies and approaches based on conditional “inefficient” pricing
behavior. His research covers examples in a wide range of markets and instruments, from sporting and gambling
events to traditional financial markets, and even less conventional markets such as wine and Turkish rugs. These
patterns are sometimes identified as anomalies (arbitrage), obtaining an edge (such card counting in blackjack) or
systematic risk-factors (traditional asset classes). With the rise of micro-level data and new data science methods,
the research Bill initiated has continued to grow, marking him as a pioneer explorer.
†
We emphasize that the statistical jump models we study in this article are not related to jump-diffusion models,
a common class of stochastic processes.
‡
Afşar Onat Aydınhan, aydinhan@princeton.edu. Petter N. Kolm, petter.kolm@nyu.edu. John M. Mulvey, mul-
vey@princeton.edu. Yizhan Shu (corresponding author), yizhans@princeton.edu.

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1 Introduction
Regime-switching models have gained significant popularity in economics and finance for their
ability to capture the abrupt but persistent shifts in the dynamics of macroeconomic variables
(Hamilton and Susmel, 1994; Stock and Watson, 1996) and financial markets (Rydén et al., 1998;
Bulla, 2011). These models partition time series data into distinct regimes, where each regime
represents periods of relatively homogeneous behavior, and can persist for extended periods. A
major reason why regime-switching models have become popular is that the regimes inferred by
econometric techniques often possess intuitive interpretations, such as representing different phases
of the business cycle (Hamilton, 1989), periods of low or high volatility (Schwert, 1989), and bull or
bear markets in equities (Pagan and Sossounov, 2003). Thus, regime-switching models align with
changing fundamentals which often can only be interpreted retrospectively (such as the lagged
identification of NBER business cycles), but in a way that enables ex ante real-time forecasting,
providing valuable insights for decision-making (Ang and Timmermann, 2012). By integrating
regime-switching dynamics into their investment strategies, investors can tailor their asset perfor-
mance expectations across regimes and make decisions informed by anticipated regime changes.
A number of studies document improvements associated with employing regime-aware investment
approaches (Elliott et al., 2010; Bulla et al., 2011; Bae et al., 2014; Mulvey and Liu, 2016; Reus
and Mulvey, 2016).
A vast majority of literature on regime-switching models are based on hidden Markov models
(HMMs), hence called Markov regime-switching models. In an HMM, each time series observation
depends only on a hidden (unobserved) state variable at the specific time, and is independent of
past and future hidden states, given the present one. The dynamics of the hidden state sequence,
which represents the regime process, are modeled by a finite-state homogeneous first-order Markov
chain. Statistically, the presence of multiple hidden states in HMMs allows the marginal distri-
bution of each observation to be a mixture model, thereby providing greater flexibility compared
to that of parametric distribution families like the normal- and Student’s t-distributions. Tempo-
rally, the hidden state sequence’s Markovian structure is capable of capturing abrupt jumps and
regime persistence to a certain extent. HMMs and their variants have demonstrated the ability to
reproduce numerous stylized facts of financial return series, including fat tails, volatility clustering,
skewness, and time-varying correlations (Rydén et al., 1998), and have found significant applica-
tions in financial time series analysis, including the modeling of returns (Hardy, 2001), interest
rates (Gray, 1996), high-frequency trading (Cartea and Jaimungal, 2013), and derivatives pricing
(Goutte et al., 2017).
However, real-world applications often pose challenges for HMMs, due to model misspecification,
lack of data, feature selection issues, and computational challenges. It is well-known that the
parameters in HMMs, especially the transition probabilities, are time-varying (Bazzi et al., 2017;
Nystrup et al., 2017), invalidating the Markovian structure imposed on the regime process. Other
stylized facts, such as low signal-to-noise ratio, high persistence of regimes and high imbalance
across different regimes, typically necessitate a time series with an unrealistically long history
for the estimation method to achieve satisfactory statistical accuracy. Such misspecification or
misestimation can lead to instability in the inferred hidden state sequence and a lack of true
persistence (Bulla, 2011). For example, if an impersistent hidden state sequence with unrealistically
rapid switches is utilized as the basis for portfolio allocation, the frequent transitions between
different regimes can incur high trading costs due to excessive turnover, while false regime shift
alarms may further contribute to sub-optimal performance (Nystrup et al., 2020b). Additionally,
minimizing the negative loglikelihood function for HMMs is computationally challenging due to its
highly non-convex and multi-modal characteristics (Andersson et al., 2003).

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 To address the aforementioned challenges, various methods have been proposed for estimating
regime-switching models without relying on precise assumptions about the parametric forms of
conditional or sojourn-time distributions. These methods include trend filtering (Mulvey and Liu,
2016), spectral clustering HMMs (SC-HMMs) (Zheng et al., 2021), and statistical jump models
(JMs) (Nystrup et al., 2020b,a). For brevity in the following, when discussing jump models, we
will omit the term “statistical.”
Among the methods above, jump models are particularly relevant to our study. The jump model
performs clustering on a set of temporal features, while explicitly penalizing jumps across different
regimes in the hidden state sequence. To avoid confusion, we emphasize that the JMs discussed
in our article are not related to jump-diffusion models, a common class of stochastic processes.
We note that the jump model aligns with the concept that investors’ behavioral preferences, such
as sentiment shifts, influence regime changes, such as transitioning between bull and bear periods
(Barberis and Thaler, 2003; Boswijk et al., 2007). Through an extensive simulation study, the article
Nystrup et al. (2020b) establishes the robustness of jump models compared to classical HMMs and
SC-HMMs, especially under conditions commonly encountered in financial applications, such as
small data sizes, imbalanced regimes, and high persistence (corresponding to large diagonal values
in the probability transition matrix of HMMs).
In this article, we introduce the continuous statistical jump models (CJMs), an extension of
the original jump models where the discrete hidden state variable is replaced by a probability
vector over the states (regimes). This extension allows us not only to label the current state
but also to estimate the probability of each time period belonging to a particular state. This
additional information is crucial for various downstream financial applications, such as regime-
aware portfolio construction (Li and Mulvey, 2021), risk management (Uysal and Mulvey, 2021),
and optimal execution (Sawhney, 2020; Li and Mulvey, 2023). The smooth transition in probability
estimation, as opposed to abrupt jumps between regimes, improves the model’s robustness to model
misspecification. Moreover, the continuous model shows reduced sensitivity to hyperparameter
tuning and improved accuracy in detecting rebounds during bear market periods.
Our contribution to the literature on statistical jump models and their empirical application to
financial time series is threefold. First, we introduce a continuous extension of the original jump
model by allowing the discrete hidden state variable to be a probability vector over the regimes. We
propose a jump penalty term for the probability vectors of the continuous model and establish its
connection to the jump penalty in the discrete model through optimal transport. The jump penalty
ensures high persistence in the estimated probability vector, enabling the model to capture the true
persistence in the underlying regime-switching process. Additionally, the specific mathematical
formulation of the jump penalty term leads to a computationally tractable optimization problem
used in estimating the new continuous statistical jump model.
Second, we provide a probabilistic interpretation of the continuous jump model under the as-
sumption that the data is generated from a generalized HMM governed by a Markov process defined
on a probability simplex. We derive the transition kernel of this Markov process from the jump
penalty, capturing the transition probabilities across different regimes. Notably, this interpreta-
tion suggests an additional penalty term referred to as the mode loss, which promotes sparsity
of the hidden probability vector, thereby increasing the model’s ability to provide definitive state
assignments. While Bemporad et al. (2018) suggest incorporating a mode loss penalty to express
preferences over states themselves (in addition to the penalty from state transitions), our contri-
bution lies in deriving the exact form of this term for the continuous jump model.
Third, we conduct an extensive simulation study to evaluate the performance of the continuous
jump model in different settings, including two- and three state models with different levels of
persistence, and misspecified conditional distributions and sojourn times. In our simulations, the

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continuous jump model outperforms the HMM estimated using the Baum-Welch algorithm and
the discrete jump model across most settings, both in terms of state classification and parameter
estimation. In particular, the significant improvement in the area under the Receiver Operating
Characteristic (ROC) curve demonstrates that the continuous model provides more reliable proba-
bility estimates, indicating an ability to better capture the true underlying probabilities and thereby
make more accurate predictions.
This article is organized as follows. In Section 2 we summarize the relevant literature. We
review the original jump model in Section 3. In Section 4, we introduce the new continuous jump
model, analyze its statistical properties, and provide a probabilistic interpretation. Then, in Section
5 we present an extensive empirical simulation study. We provide an application of the new model
by conducting an analysis of the economic regimes of the Nasdaq Index across two distinct time
periods in Section 6. Finally, Section 7 concludes.

2 Related Literature
In time series analysis, it is often valuable to partition the data into consecutive time periods
that exhibit similar characteristics. This allows us to fit separate models to each segment, thereby
capturing segment- or regime-specific characteristics and their changes more effectively. This task
has been extensively studied in various contexts, including cyclical analysis (Bry and Boschan,
1971), change point detection (Aminikhanghahi and Cook, 2017), segmentation (Himberg et al.,
2001), mixture models (Picard et al., 2011), and trend filtering (Kim et al., 2009; Gu and Mulvey,
2021). These approaches provide valuable insights into understanding and modeling the underlying
dynamics of financial time series.
After the seminal work of Rydén et al. (1998), which fitted simple HMMs on daily financial
return series, researchers have proposed numerous extensions to HMMs. By using conditional
distributions with heavy tails (Bulla, 2011) and sojourn time distributions other than memoryless
geometric distributions (Bulla and Bulla, 2006), the resulting models better capture long memory,
exhibit improved persistence of the hidden state sequence, and can reproduce many stylized facts
of financial time series. Nystrup et al. (2017) fit classical HMMs, but in an adaptive way that
allows for time-varying parameters, resulting in similar improved performance. Other extensions
to classical HMMs have been explored, including the utilization of higher-order Markov chains
to model the dynamics of the hidden state sequence (Zhang et al., 2019), the incorporation of
multi-scale hierarchical structures within HMMs (Fine et al., 1998), and the use of HMMs with
distributed state representations (Ghahramani and Jordan, 1995). Classical discrete-time HMMs
suffer from a quadratic increase in the number of parameters as the number of states expands,
often limiting researchers to models with few states. Nystrup et al. (2015) suggest alleviating this
constraint by fitting HMMs in continuous time, leading to linear growth in parameters with states.
While maximum likelihood estimation (MLE) is commonly used to estimate HMMs, gradient-
based optimizations of the likelihood function are computationally difficult due to the non-concavity
resulting from the integration over the hidden state variables. The Baum-Welch algorithm (Baum
et al., 1970), an iterative procedure later recognized as a specialization of the expectation–maximization
(EM) algorithm (Dempster et al., 1977), is typically employed but is only guaranteed to conver-
gence to a local optimum. To address the issue of local optima, Hsu et al. (2012) propose a singular
value decomposition (SVD)-based spectral algorithm with provable guarantees. This algorithm is
specifically designed for HMMs with a large number of hidden states. Bulla and Berzel (2008) pro-
pose a hybrid algorithm that combines the advantages of both direct numerical maximization and
EM algorithms. Additionally, several authors have introduced Bayesian approaches as alternative

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estimation methods (Rydén, 2008; Ebbers et al., 2017).
Recently, there has been a growing interest in clustering-based methods for estimating regime-
switching models as an alternative to the HMM approach. These methods aim to address challenges
such as model misspecification, difficulty in optimizing the likelihood function and sensitivity to
poor initialization. One such approach is the spectral clustering HMM (SC-HMM) proposed by
Zheng et al. (2021), which extends the method introduced by Hsu et al. (2012) to HMMs with
continuous observations. SC-HMM first constructs a feature set based on local observations and
applies spectral clustering (Ng et al., 2001) to group the data into clusters using the spectral
analysis of the Laplacian matrix of pairwise feature similarities. Then it estimates the distributional
parameters within each cluster and computes the empirical transition matrix. Frequently, the SC-
HMM achieves better accuracy and robustness compared to MLE.
Time series segmentation distinguishes itself from general clustering problems by incorporating
temporal information. The jump model, introduced by Nystrup et al. (2020b), addresses this chal-
lenge by applying a clustering algorithm to the set of temporal features proposed by Zheng et al.
(2021), while explicitly penalizing jumps between different regimes through a fixed-cost regulariza-
tion term calibrated to achieve a desired level of persistence. The jump penalty plays a crucial role
in successfully applying clustering techniques for estimating regime-switching models, as it strikes
a balance between fitting the data and incorporating prior assumptions about the persistence of
the state sequence. Notably, the jump model outperforms MLE and SC-HMMs, particularly in
scenarios where regimes exhibit high persistence. Nystrup et al. (2020a) further leverage the re-
cursive inference proposed by Bemporad et al. (2018), enabling the jump model to be applied in
an online fashion. This online capability allows the model to determine the regime to which a
new observation belongs without the need to parse the historical observation sequence. The model
achieves robust out-of-sample performance, while maintaining the persistence of the hidden state
sequence.
The estimation of the jump model is typically performed using a coordinate descent method. To
mitigate the risk of converging to local suboptimal solutions, Lin (2023) proposes a mixed-integer
programming (MIP) based estimation approach. This method effectively addresses the challenge
of local optima and facilitates the integration of additional constraints, thereby improving the
estimation of jump models.
Several extensions and applications of jump models have been developed. Nystrup et al. (2021)
introduce the sparse statistical jump model (SJM), which integrates the framework of feature se-
lection in clustering developed by Witten and Tibshirani (2010) into the existing jump model.
This allows for simultaneous feature selection, parameter estimation, and state classification. To
facilitate hyperparameter tuning and model selection in high-dimensional settings, Cortese et al.
(2023b) extend the generalized information criteria (GIC) for high-dimensional penalized models
to sparse statistical jump models. Cortese et al. (2023a) apply this technique to investigate the
underlying regimes driving cryptocurrency returns. Additionally, Shu et al. (2024) utilize jump
models to construct a regime-aware trading strategy for major U.S. equity indices, demonstrating
its outperformance compared to buy-and-hold strategies.

3 Statistical Jump Models

The statistical jump model framework proposed in Bemporad et al. (2018) incorporates tem-
poral information when fitting multiple models to a time series. Specifically, given an observation
sequence of D (standardized) features Y := {y0 , . . . , yT −1 } with yt ∈ RD for all t, we estimate a

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statistical jump model with K states by solving the optimization problem
T
X −1 T
X −1
arg min l(yt , θst ) + λ 1{st−1 6=st } , (1)
Θ,S t=0 t=1

where Θ := θk ∈ RD : k = 0, . . . , K − 1 are the model parameters with θk ∈ RD for all k, S :=



{s0 , . . . , sT −1 } denotes the state sequence, and λ ∈ R+ is a hyperparameter referred to as the

jump penalty. We emphasize that each hidden state st takes on values in {0, . . . , K − 1} and is
associated with a state-specific model parameter θk . At a high level, we can intuitively understand
the objective function (1) as a compromise between fitting the data using multiple models and
incorporating our prior beliefs about the persistence of the state sequence. After estimating a jump
model, we can infer the transition probability matrix based on the optimal hidden state sequence.
Throughout this article, we employ the scaled squared `2 -distance as the loss function, defined
by l(y, θ) := 12 ky − θk22 . A general quadratic form for the loss function, e.g. squared Mahalanobis
distance, is also applicable if we have prior knowledge of the covariance matrix of the features or if
we intend to learn it from the data (Hallac et al., 2017).

3.1 Features
For financial return series of an individual instrument, the observation sequence consists of
scalar values x0 , . . . , xT −1 ∈ R. However, to enhance model performance, we construct a set of
additional features. Specifically, to facilitate comparison with previous work, we adopt the feature
construction method proposed by Nystrup et al. (2020a), as outlined in Algorithm 1. In turn,
these features are based on those in Zheng et al. (2021), with the distinction of allowing only
backward-looking features to enable easy adaptation for online predictions. These features include
a collection of rolling mean returns and volatilities calculated over different window sizes, essential
for understanding a return series. Specifically, we use two window lengths, w = 6 and 14, resulting
in a total of fifteen standardized features.
While we continue to use the feature sets popularized by Zheng et al. (2021); Nystrup et al.
(2020a), it is crucial to highlight the flexibility inherent in choosing the feature set within the
jump model framework. Features can be derived from various sources, including exogenous time
series such as macro-economic indicators and returns of relevant assets, as demonstrated in Cortese
et al. (2023a). Moreover, features can be specifically tailored for a particular financial market, as
illustrated in Shu et al. (2024). These examples underscore the diverse range of possibilities for
constructing feature sets within the jump model framework.

3.2 Model Fitting

Nystrup et al. (2020b) propose to minimize the objective function in equation (1) using a
coordinate descent algorithm, which alternates between (i) minimizing the model parameters with
a fixed state sequence, and (ii) minimizing the state sequence with fixed model parameters. We
summarize the coordinate descent algorithm in Algorithm 2. The iteration of the algorithm stops
either when the state sequence does not change or when the improvement of the value of the
objective function is smaller than a specified tolerance. The fitting of model parameters is performed
within each cluster, and is computationally tractable when the loss function l is convex with respect
to model parameter θ. For fixed model parameters, one obtains the optimal state sequence using
a dynamic programming (DP) algorithm outlined in Algorithm 3, similar to the Viterbi algorithm
(Viterbi, 1967) for obtaining the maximum a posteriori (MAP) estimate of the most likely hidden
state sequence in an HMM.

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 Algorithm 1: Generate features for the jump models
Input: Univariate time series {x0 , . . . , xT −1 } and window lengths w = 6, 14.
Compute features at each time t:
1. Observation: xt .
2. Absolute change: |xt − xt−1 |.
3. Previous absolute change: |xt−1 − xt−2 |.
Compute features at each time t for w = 6, 14:

4. Centered mean: mean[xt−w+1 , . . . , xt ].

5. Centered standard deviation: std[xt−w+1 , . . . , xt ].
6. Left mean: mean[xt−w+1 , . . . , xt− w2 ].
7. Left standard deviation: std[xt−w+1 , . . . , xt− w2 ].
8. Right mean: mean[xt− w2 +1 , . . . , xt ].
9. Right standard deviation: std[xt− w2 +1 , . . . , xt ].

Output: Fifteen-dimensional feature set.

Given the non-convex nature of the objective function in equation (1), we can only expect
the coordinate descent algorithm to converge to a stationary point, without any guarantee of
finding the global optimum. Wright (2015) examines the use, implementation, and convergence of
coordinate descent algorithms for convex objective functions. Notably, global convergence results
for non-convex objective functions are only obtainable in particular special cases (see, for example,
Bertsekas (1999, Proposition 2.7.1); Attouch et al. (2010); Bolte et al. (2014)). To address the issue
of local optima, we run the algorithm ten times with different starting points, generated by the
K-means++ algorithm (Arthur and Vassilvitskii, 2007), and retain the solution that obtains the
best value of the objective function.

Algorithm 2: Coordinate descent algorithm for the discrete jump model

Input: Time series Y := {y0 , . . . , yT −1 }, number of states K, and jump penalty λ ∈ R+ .
Iterate for i = 1, . . .
P −1  
(a) Fit model parameters Θi = arg minΘ Tt=0 l yt , θsi−1 .
t
nP o
T −1 PT −1
(b) Fit state sequence S i = arg minS t=0 l(yt , θsi t ) + λ t=1 1{st−1 6=st } .

Until S i = S i−1 , or the improvement of the value of the objective function is smaller than
a specified tolerance.
Compute the empirical transition probability matrix.
Output: Model parameters and hidden state sequence

The time complexity of the DP algorithm is O(T K 2 ), which is equivalent to one forward-
backward pass in the E-step of the Baum-Welch algorithm. As discussed later in Section 4.2,
fitting the state sequence with fixed model parameters in the discrete jump model can also be
addressed by solving a relaxed linear programming (LP) problem. In our experience, the DP

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 Algorithm 3: Fit the state sequence for the discrete jump model using dynamic program-
ming
Input: Loss matrix L = (lt,k ) ∈ RT ×K , where each lt,k := l(yt , θk ), and jump penalty
λ ∈ R+ .
Initialize: V (0, k) = l0,k , k = 0, . . . , K − 1.
for t = 1, . . . , T − 1 do


V (t, k) = lt,k + min V (t − 1, j) + λ1{j6=k} , k = 0, . . . , K − 1. (2)
j

end
Compute optimal value v̂ = mink V (T − 1, k).
Compute optimal state ŝT −1 = arg mink V (T − 1, k).
for t = T − 1, . . . , 1 do


ŝt−1 = min V (t − 1, j) + λ1{j6=ŝt } . (3)
j

end
Output: Optimal value v̂ and optimal state sequence {ŝ0 , . . . , ŝT −1 }.

solution is typically faster than calling an LP solver, especially when the number of states K is
relatively low (less than ten).

3.3 Jump Penalty

The hyperparameter λ serves as a control parameter for the fixed-cost regularization term
associated with transitions between different states. Its value reflects our prior assumptions about
the frequency of state transitions. When λ is set to zero and the loss function l corresponds to the
squared `2 -distance, the jump model becomes the K-means algorithm (Lloyd, 1982) which does not
take the temporal order into account. As we increase the value of λ, the number of state transitions
decreases. Eventually, a single state transition becomes so costly that the jump model is reduced
to a single model fitted to the entire time series.
The penalization of jumps is a critical aspect of successfully applying this type of temporal
clustering algorithms to estimate regime-switching models. Techniques for selecting the optimal
hyperparameter λ include cross-validation methods tailored to the specific application (Shu et al.,
2024) and statistical information criteria (Cortese et al., 2023b)1 . While the objective function
in equation (1) applies the same penalty to all jumps between states, it is possible to introduce
state-specific penalties for different types of jumps, such as distinguishing between a jump from
a bull to a bear market and vice versa. To incorporate these penalties, we can define a penalty
matrix Λ ∈ RK×K+ , where each entry Λij ≥ 0 represents the penalty for transitioning from state
i to state j. This penalty matrix allows for a more nuanced treatment of different types of state
transitions.

4 Continuous Statistical Jump Models

In many financial applications, such as portfolio optimization and risk management, the ability
to estimate the probability of each time period belonging to a specific regime is crucial. A proba-
1
The optimal selection of the number of states K can be addressed in a similar manner.

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bilistic perspective provides more informative and valuable insights compared to the hard clustering
approach in the discrete jump model, where a single label is assigned to each time period indicat-
ing the identified regime. A hard clustering approach lacks the ability to capture the uncertainty
associated with regime assignment and the varying probabilities of belonging to different regimes.
For this purpose, we propose extending n the hidden state space from the discrete set {0,o. . . , K − 1}
T K
PK−1
to the probability simplex ∆K−1 := s = (s0 , . . . , sK−1 ) ∈ R : k=0 sk = 1, s ≥ 0 . Then we
let each hidden state vector st be a probability vector over all the states. We use K − 1 as the
subscript of the simplex since its affine dimension is K − 1. The hidden vector sequence is now
T
represented by S := s0 , · · · , sT −1 ∈ RT ×K , which should satisfy the constraint


S ≥ 0, S1K = 1T , (4)

where 1n represents a vector of all ones of length n.

Given an observation sequence Y of (standardized) features, we propose solving the following
minimization problem
T −1 T −1
X λX
arg min L(yt , Θ, st ) + kst−1 − st k21 , (5)
Θ,S 4
t=0 t=1

where Θ represents the model parameters, S is the hidden vector sequence, and the loss function
L is defined as a weighted average of the loss between yt to each cluster center, i.e.
X
L(yt , Θ, st ) = st,k l(yt , θk ) , (6)
k

with l and st,k denoting the scaled squared `2 -distance and the k-th element of st , respectively.
We remark that the factor of 1/4 in the second term of equation (5) ensures that λ is consistent
with that of the discrete model, if we were to reduce a continuous model to a discrete one. Like
the original jump model, the objective (5) fits the data to multiple models while simultaneously
keeping the total number of jumps small. We emphasize that the second term of the the objective
function penalizes transitions between two subsequent probability vectors using the squared `1 -
norm of their difference. The `1 -norm promotes sparsity in the difference between two consecutive
estimated probability vectors, hence encouraging persistence in the inferred state process. In Section
4.2, we provide theoretical justification for the choice of this regularization term for jumps on the
probability simplex.

4.1 Model Estimation

We propose to solve the optimization problem in (5) above by a coordinate descent method
outlined in Algorithm 4, which alternates between fitting (i) the model parameters, and (ii) the
hidden vector sequence, while keeping the other fixed 2 . Specifically, in fitting the model parameters,
we solve a weighted minimization problem for each regime, given by
T
X −1
arg min st,k l(yt , θk ) . (7)
θk t=0

Similarly, when the loss function l is convex with respect to the parameter θk , this optimization
problem is tractable. In the case of scaled squared `2 -distance, we obtain the solution analytically.
2
To address the issue of local optima, just like for the discrete jump model we run the algorithm ten times with
different starting points, generated by the K-means++ algorithm.

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 When fitting the hidden vector sequence, with the loss matrix L ∈ RT ×K , where Lt,k :=
(l(yt , θk )), as the input, the optimization problem over the state sequence is given by
T −1
λX
arg min hS, Li + kst−1 − st k21 , (8)
S: S≥0, S1K =1T 4
t=1
P
where hS, Li := t,k st,k lt,k denotes the inner product between two matrices. As the problem has
a quadratic objective function and linear constraints, it is a quadratic programming (QP) problem.
Although this optimization problem can be solved with a QP solver, we propose an alternative
approximate solution using a DP algorithm, which has a similar structure to the one used in the
discrete jump model and in general offers a shorter running time3 . In particular, we first discretize
the probability simplex. In our implementation, we generate a set of uniformly sampled discretized
vectors c0 , . . . , cN −1 ∈ ∆K−1
 on the probability simplex with a fixed grid size of δ > 0, and stack
them into a matrix C := c0 · · · cN −1 ∈ RK×N . Note

that the total number N of candidate
vectors scales exponentially with K, i.e. N = O d 1δ eK . In contrast to the discrete model, which

selects one of the K states at each time step t, the continuous model chooses one of the N candidate
probability vectors. Furthermore, instead of uniformly penalizing jumps across states with a penalty
term λ, the change between two consecutive estimated probability vectors is penalized using the
squared `1 -norm of the vector difference. We summarize this algorithm in Algorithm 5.
The time complexity of the approximate DP algorithm is O(T N 2 ), in contrast to the discrete
model, where the time complexity is O(T K 2 ). Consequently, the total running time of the con-
tinuous model increases by a factor in the order of hundreds. However, since the discrete model
is exceedingly fast to solve, the time required to solve one continuous jump model remains very
short, typically well under a second on a regular desktop computer. Nevertheless, if tasks such as
rolling fits or fitting with a list of jump penalties are required, we recommend leveraging parallel
processing to expedite the process.

Algorithm 4: Coordinate descent algorithm for the continuous jump model

Input: Time series Y := {y0 , . . . , yT −1 }, number of states K, and jump penalty λ ∈ R+ .
Iterate for i = 1, . . .
P −1
(a) Fit model Θi = arg minΘ Tt=0 L yt , Θ, sti−1 .

nP o
T −1 i, s ) + λ
PT −1 2
(b) Fit state sequence S i = arg minS t=0 L(yt , Θ t 4 t=1 ks t−1 − s t k 1 .

Until S i = S i−1 , or the improvement of the value of the objective function is smaller than
a specified tolerance.
Compute the empirical transition probability matrix.
Output: Model parameters and hidden state sequence

In our experience, the error introduced by the DP approximation is negligible when the grid
size δ is less than approximately 0.05 4 . Employing the approximate DP algorithm to solve the QP
problem offers consistency with the discrete model and potential reductions in running time when
3
The approximate DP algorithm can decrease the running time by approximately five times with an appropriate
grid size, as described below. This reduction is crucial as QP problems need to be solved tens to hundreds of times
for each jump model solution. Another reason for proposing this DP approach is its adaptability to the mode loss
introduced later in Section 4.4.
4
The exact sequence of hidden vectors solved by a QP solver differs very little from the sequence given by the
approximate DP algorithm.

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 Algorithm 5: Fit the state sequence for the continuous jump model using dynamic pro-
gramming
Input: Loss matrix L = (l(yt , θk )) ∈ RT ×K , candidate vectors C ∈ RK×N , and jump
parameter λ ∈ R+ .
Compute the new loss matrix L e = (e lt,i ) := LC ∈ RT ×N .
Compute the jump penalty matrix Λ = (Λi,j ) ∈ RN ×N , where Λi,j := λ4 kci − cj k21 .
Initialize: Ve (0, i) = e l0,i , i = 0, . . . , N − 1.
for t = 1, . . . , T − 1 do
 
lt,i + min Ve (t − 1, j) + Λj,i , i = 0, . . . , N − 1.
Ve (t, i) = e (9)
j≤N −1

end
Compute optimal value v̂ = mini≤N −1 Ve (T − 1, i).
Compute optimal index îT −1 = arg mini≤N −1 Ve (T − 1, i), and optimal hidden vector
ŝT −1 = cîT −1 .
for t = T − 1, . . . , 1 do
 
ît−1 = arg min Ve (t − 1, j) + Λj,ît , ŝt−1 = cît−1 . (10)
j≤N −1

end
Output: Optimal value v̂ and optimal hidden vector sequence {ŝ0 , . . . , ŝT −1 }.

the number N of candidate vectors can be limited to around a hundred. However, since N scales
exponentially with K, if the number K of states is relatively large and requires more candidate
vectors, it is preferable to directly call a QP solver.

4.2 Jump Penalty for Probability Vectors

In this section we justify the use of squared `1 -norm of vector difference used in equation (5)
for the jump penalty on the probability simplex. First, our approach uses the `1 -norm to measure
the difference between two probability vectors. This choice can be interpreted as the Wasserstein
distance induced by the discrete metric in the discrete jump model. Specifically, the jump penalty
Ljump (k, j) := λ1{k6=j} , k, j ∈ {0, . . . , K − 1} defines the discrete metric on the space {0, . . . , K − 1}
(Munkres, 2000). A natural way to lift this metric to the probability simplex is via optimal transport,
where the cost of moving one unit of resources between two points is some power of their distance.
The resulting distance between probability vectors is the Wasserstein distance. Remarkably, under
the ground metric Ljump , the Wasserstein distance is exactly the scaled `1 -norm (Peyré and Cuturi,
2019, Remark 2.26), also known as the total variation distance (Levin et al., 2017, Proposition 4.2),
defined by
1
TV(p, q) := kp − qk1 , p, q ∈ ∆K−1 . (11)
2
Therefore, the use of the scaled `1 -norm as a distance on the probability simplex in the continuous
jump model naturally inherits the discrete metric on {0, . . . , K − 1} implicit in the discrete jump
model. Furthermore, as mentioned in Section 3.3, when considering a state-specific penalty matrix
Λ ∈ RK×K
+ in the discrete model, the use of the Wasserstein distance with the ground cost matrix
chosen as Λ provides a natural choice for the corresponding jump penalty on the probability simplex.
This connection allows for a consistent formulation of the penalty terms across both models.

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 Secondly, however, if we directly use the scaled `1 -norm itself as the jump penalty in (5), the
continuous model will be reduced to exactly a discrete jump model. The reason is that, when fitting
the hidden vector sequence with fixed model parameters, the optimization problem becomes an LP,
namely
T −1
λX
arg min hS, Li + kst−1 − st k1 . (12)
S: S≥0, S1K =1T 2
t=1

We remark that in contrast to (8), in the formula above we apply a factor of λ2 to the `1 -norm
to make it consistent with the discrete jump model. As per the fundamental theorem of linear
programming (Bertsimas and Tsitsiklis, 1997, Theorem 2.7), the minimum of (12) is attained at
an extreme point of the feasible set, which corresponds to a corner of the simplex. At the corners,
the probability estimation for a specific state is either 0 or 1, indicating a definitive assignment to
a single state. Consequently, the optimal hidden vector sequence represents a hard clustering of
the observations, reducing the model to a discrete one.5
As a result, we choose to use the squared `1 -norm of the vector difference as the final jump
penalty on the simplex ∆K−1 , which transforms (12) into the QP problem (8). When K is relatively
small and the number of candidate probability vectors is limited to a few hundred, we can expedite
computations by approximately solving this QP using dynamic programming with Algorithm 5,
yielding negligible numerical discrepancies. However, when the number of states are larger, it is
preferable to directly call a QP solver.
Using other powers of the `1 -norm of the vector difference, i.e. Ljump (st−1 , st ) ∝ kst−1 − st kα1 ,
is also a valid option. This introduces a mixed norm penalty (Kowalski, 2009), allowing for the
promotion of structured sparsity. However, in our empirical experiments, we did not find significant
differences when using α strictly greater than two. Therefore, we use α = 2 in our empirical work
in this article to maintain the QP form.

4.3 Probabilistic Interpretation

In this section, we establish a probabilistic interpretation of the continuous jump model. Our
aim is to provide a framework for understanding the model’s behavior based on probabilistic and
statistical principles. Unlike traditional HMMs where the state sequence S follows a finite-state
Markov chain, in the continuous jump model it follows a Markov process on the probability simplex.
We assume that the sequence (Y , S) is generated from a generalized hidden Markov model
(GHMM), where the emission distribution of yt is a mixture model and the hidden vector st
represents the prior probabilities of the observation belonging to each mixture component. Formally,
we make the following assumptions:

1. (Hidden Markov process) The hidden vector sequence S follows a Markov process on the
probability simplex, with initial distribution π : ∆K−1 → R+ , and transition kernel K :
∆K−1 × ∆K−1 → R+ , i.e.
−1
TY
p(S) = π(s0 ) × K(st−1 , st ) . (13)
t=1

2. (The conditional likelihood of Y ) Given the hidden vector sequence S, the observation yt
5
We remark that the LP problem (12) is a tight relaxation for fitting the hidden state sequence S in the discrete
jump model (1) when the model parameters Θ are held fixed, a task previously addressed by the DP algorithm 3.

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 only depends on st , i.e.
−1
TY
p(Y |S, Θ) = p(yt |Θ, st ) , (14)
t=0
where p(·|Θ, s) is the emission distribution specified next.
3. (Emission distribution) p(·|Θ, s) follows a mixture model, where all of the K components
belong to the same parametric family {p(·|θ) : θ ∈ RD }. The k-th component has parameter
θk and weight sk , i.e.
K−1
X
p(y|Θ, s) = sk p(y|θk ) . (15)
k=0

To establish the correspondence of the probabilistic assumptions mentioned above with the
objective function (5) of the continuous jump model, it is convenient to express the objective in its
general form, following the notation of Bemporad et al. (2018)
T
X −1 T
X −1
J(Θ, S) = L(yt , Θ, st ) + Linit (s0 ) + Ljump (st−1 , st ) , (16)
t=0 t=1

where L(yt , Θ, st ) = k st,k l(yt , θk ) is the same as that in (5), Linit is a penalty term for the initial
P
state s0 , which we choose to be zero in our model, indicating that we do not hold any specific view
about the initial state. Ljump (st−1 , st ) represents the jumpP penalty, for which we have justified our
−1 jump
choice of λ4 kst−1 − st k21 . We denote by L(S) := Linit (s0 ) + Tt=1 L (st−1 , st ) the penalty on the
hidden vector sequence.
The subsequent propositions extend the result in Bemporad et al. (2018) to continuous jump
models. The first proposition states that, if (Y , S) follows a GHMM, and in (16), the loss function
l, initial penalty Linit , and jump penalty Ljump correspond to the negative log-likelihood of the
conditional distribution, initial distribution, and transition kernel, respectively, then maximizing a
lower bound of the joint log-likelihood of (Y , S) is equivalent to minimizing the objective function
J in the continuous jump model.
Proposition 1. Suppose assumptions 1 through 3 above are satisfied, and define the terms in the
loss function J as
l(y, θ) := − log p(y|θ),
Linit (s0 ) := − log π(s0 ), (17)
jump
L (st−1 , st ) := − log K(st−1 , st ).
Then maximizing a lower bound of the joint likelihood p(Y , S|Θ) is equivalent to minimizing the
objective function J with respect to Θ and S.
Conversely, given an objective function J, there is a statistical interpretation of it. In particular,
the following proposition states that if certain assumptions are satisfied, including the use of prob-
ability density functions p(y|θ) and p(S), then minimizing the objective function J is equivalent
to maximizing a lower bound of the joint likelihood.
Proposition 2. Define the probability density functions
e−l(y,θ)
p(y|θ) := , (18a)
ν
e−L(S)
p(S) := , (18b)
η

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where Z
ν := e−l(y,θ) dy is assumed to be independent of θ, and
Z
η := e−L(S) dS
∆T (19)
K−1
Z Z −1
TY
−Linit (s0 ) jump (s
= ··· e e−L t−1 ,st )
dsT −1 · · · ds0 .
∆T
K−1 t=1

Suppose assumptions 2 and 3 above are satisfied with (18a) being the parametric family, and (18b)
is the density function for S. Then minimizing the objective function J with respect to Θ and S is
equivalent to maximizing a lower bound of the joint likelihood p(Y , S|Θ).

We remark that the assumption that ν does not depend on θ in the proposition above is satisfied
if, for example, l(y, θ) = g(y − θ).

4.4 Mode Loss

Inspired by the probabilistic interpretation of the continuous jump model in the previous section,
we introduce another penalty on each individual hidden vector st−1 , called mode loss. Specifically,
given a jump penalty Ljump in the loss function J, according to Equation (18b) in Proposition 2,
the vector sequence S can be associated with a likelihood function
−1
TY
init (s jump (s
p(S) ∝ e−L(S) = e−L 0)
e−L t−1 ,st )
. (20)
t=1

Then, for S to follow a Markov process, the transition kernel K(st−1 , st ) needs to be proportional
jump (s
to e−L t−1 ,st ) , with a normalization constant given by

Z
jump (s
C(st−1 ) = e−L t−1 ,st )
dst . (21)
∆K−1

Therefore, the transition kernel can be expressed as

1 jump (s
K(st−1 , st ) = e−L t−1 ,st )
. (22)
C(st−1 )

Conversely, Proposition 1 suggests that the appropriate jump penalty for this transition kernel is

Lejump (st−1 , st ) := − log K(st−1 , st ) (23)

jump
=L (st−1 , st ) + log C(st−1 ) . (24)

Following Bemporad et al. (2018), we refer to the additional penalty term as the mode loss, defined
as Z
jump (s
L mode
(st−1 ) := log C(st−1 ) = log e−L t−1 ,st )
dst . (25)
∆K−1

The full optimization problem with the mode loss becomes

T
X −1 T
X −1 T
X −1
arg min L(yt , Θ, st ) + Ljump (st−1 , st ) + Lmode (st−1 ) . (26)
Θ,S t=0 t=1 t=1

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When the mode loss Lmode (st−1 ) is large, then the model penalizes staying in the same state vector
at time t. In contrast, with no mode loss this penalty is not present. With the jump penalty
Ljump (st−1 , st ) ∝ kst−1 − st k21 , we observe that the vectors at the corners of the simplex (e.g.
(1, 0, . . . , 0)) have a small mode loss. They represent probability vectors with a high confidence
of the current regime. On the other hand, the vectors around the center of the simplex (e.g.
(1/K, . . . , 1/K)), which represent probability vectors with a low confidence, have a large mode
loss. Hence, the mode loss promotes sparsity in the state vectors, pushing the vectors to those with
only a few non-zero elements. In empirical experiments, we observe that the mode loss penalty
improves predictive performance, particularly during periods of state transitions. Therefore, if
sparsity in the estimated probability vector is desirable for a particular application, we advise to
include the mode loss in the objective function. However, a QP solver cannot handle the form of
(25), and we must resort to the DP approximation approach. As we discretize the simplex, we can
approximate the integration in (25) by
 X   X 
mode 1 −Ljump (ci ,cj ) 1 −Λi,j
L (ci ) ≈ log e = log e (27)
N N
j j

= logsumexp(−Λi,· ) + const , (28)

P 
where Λ ∈ RN ×N is the penalty matrix in Algorithm 5, and logsumexp(x) := log j exj . It

is straightforward to incorporate these changes into Algorithm 5. In particular, we modify the

matrix Λ by first computing Lmode (ci ), which is the logsumexp of the negative of the i-th row of Λ
(neglecting the constant), and then adding this value to the i-th row to obtain the modified jump
penalty matrix Λ.
e

5 Simulation Study
We conduct a simulation study to compare the performance of the HMM estimated using the
Baum-Welch algorithm, the discrete jump model, and the continuous jump model with and without
the mode loss. In our unsupervised problem setting, we simulate data from well- and misspecified
two- and three-state models where the true hidden state sequences and model parameters are
known, thereby allowing us to evaluate the accuracy of different approaches in identifying the state
sequences and in recovering the model parameters.
When evaluating classification accuracy in the context of financial time series, it is important to
consider the high imbalance among regimes, where the majority of periods are in the bull (normal)
regime rather than the bear regime. As a result, relying solely on overall accuracy frequently
leads to inflated scores that do not accurately reflect a model’s performance. To address this, we
employ three additional metrics: accuracy per class, balanced accuracy (BAC), and the area under
the Receiver Operating Characteristic curve (ROC-AUC). Accuracy per class measures the true
positive rate for each individual class, while BAC reflects the average accuracy across all the classes,
giving equal weight to each class irrespective of its proportion in the dataset (Brodersen et al., 2010).
Breaking down the accuracy by class allows for a more comprehensive evaluation of the model’s
performance across different regimes. While the first two metrics, accuracy per class and BAC,
are derived from the estimated label sequence, ROC-AUC is based on the estimated probability
and assesses the model’s ranking ability. Specifically, ROC-AUC represents the probability that a
classifier assigns a higher estimated probability to a randomly selected positive instance compared
to a randomly selected negative instance (Bradley, 1997). Unlike BAC, ROC-AUC can only be
computed for the sequences where all of the states are present.

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 In the case of the discrete jump model, which performs hard clustering, we treat the estimated
labels as probabilities of either zero or one when computing ROC-AUC. Notably, with two regimes,
ROC-AUC is then equal to BAC. For the continuous jump model, which estimates probability
vectors, we assign the predicted regime to be the one with the highest estimated probability.
For HMMs, whether using true or estimated parameters, the predicted label sequence is obtained
through the Viterbi algorithm, while the estimated probability refers to the smoothing posterior
probability calculated using the forward-backward algorithm (Rabiner, 1989). In clustering-like
problems, the cluster structure remains invariant to permutations of label symbols. Therefore, we
select the permutation that yields the highest overall accuracy.

5.1 Two-State HMM

Following Zheng et al. (2021), we simulate data from the two-state Gaussian HMM

yt |st ∼ N µst , σs2t ,

(29)

where the state sequence {st } follows a first-order Markov chain and the state-conditional param-
eters and transition probability matrix are given by

µ1 = .0123, µ2 = −.0157,
σ1 = .0347, σ2 = .0778,
  (30)
.9629 .0371
P = .
.2102 .7899

In this two-state HMM, the first state can be interpreted as a bullish market regime, while the second
state represents a bearish market regime. We note that Hardy (2001) estimated these parameters
from the monthly returns of an equity market index. We choose the starting distribution of s0 as
the stationary distribution of the Markov chain.
To simulate observation sequences with different state persistence, we construct daily and weekly
versions of the model above. Assuming a geometric Brownian motion for the return process and a
continuous-time Markov chain for the hidden state process, the following relationship between the
parameters for two time scales t1 and t2 holds

µs (t1 )/t1 = µs (t2 )/t2 , σs2 (t1 )/t1 = σs2 (t2 )/t2 ,
(31)
P (t1 )1/t1 = P (t2 )1/t2 ,

where the last equation is expressed in terms of matrix exponentials. In our empirical work, we use
t = 1, 5, 20 for the daily, weekly and monthly time scales, respectively. We remark that simulated
daily data exhibits the highest level of persistence but also has the lowest signal-to-noise ratio 6 .

5.1.1 Selecting λ
Figure 1 displays the average BAC of the three jump models as a function of the penalty
parameter for 1024 simulated sequences each of length 1000 using the daily parameters in (30).
Consistent with previous findings (Nystrup et al., 2020a,b), the discrete model achieves the highest
BAC at λ = 102 . Interestingly, the continuous model, although it can be reduced to a discrete model
with the same λ, achieves the best performance at λ = 103 and demonstrates better robustness to
6
Here, the signal-to-noise ratio measures the separability among clusters, and is calculated as the ratio of the
distance between two cluster centers to the volatility (Balakrishnan et al., 2017).

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the hyperparameter. The BAC plots for sequences with different lengths exhibit similar patterns,
except that the overall level of BAC is higher for longer sequences. While we omit the details,
we find that the discrete and continuous models exhibit optimal λ values of around 50 and 102 ,
respectively, at the weekly scale. At the monthly scale, both models perform best with a smaller
value of λ of around one. These findings imply that in less persistent data, optimal performance is
attained by imposing a smaller jump penalty.

Figure 1: Balanced accuracy of the jump models as a function of the penalty parameter for 1024
simulated sequences each of length 1000. discrete, cont, and contM denote the discrete and contin-
uous jump models without and with mode loss, respectively.

5.1.2 Daily Scale

Table 1 presents the mean and standard deviation (in parenthesis) of parameter estimates and
accuracy scores of 1024 simulations of length T = 250, 500, 1000, 2000 from the daily two-state
HMM estimated with HMM, and the discrete and continuous jump models with and without mode
loss. The bold values represent the best parameter estimates and highest accuracy scores. For the
daily scale, the sequence lengths we consider correspond approximately to 1, 2, 4, and 8 years,
respectively. It is worth noting that, similar to Nystrup et al. (2020b), we retain all the sequences
without filtering out those that only contain a single regime, aiming to better emulate the real-world
scenarios.
We estimate the HMMs with Baum-Welch algorithm implemented in the Python package
hmmlearn. Inline with common practise to ensure numerical stability, for the transition proba-
bilities we incorporate a Dirichlet prior distribution of order K where all parameters are set to

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1+10−8 . Additionally, we initialize the estimation by the K-means++ algorithm, similar to the
initialization approach we use for the jump models. In the continuous jump model when fitting the
hidden vector sequence through the coordinate descent algorithm, we employ the approximate DP
approach and discretize the probability simplex uniformly with a grid size of 0.01 for all two-state
models.
When examining the accuracy of parameter estimation, a notable distinction between HMM
and jump models is in the estimation of transition probabilities. Specifically, HMM significantly
overestimates the transition probabilities, often by tens of multiples compared to their true values,
especially for shorter time series. In addition, even as the length of the time series increases, the
convergence of the estimated HMM to the true value is very slow. In contrast, jump models demon-
strate the ability to capture the true persistence in the state sequence, even for shorter time series,
thereby significantly outperforming the HMMs in terms of BAC and ROC-AUC. The performance
of the HMMs are on par with the jump models only when the time series are longer, e.g. more than
1000 observations. Notably, the considerably larger standard deviation of the estimated transi-
tion probabilities of the HMMs highlights a lack of robustness, despite the estimated models being
well-specified. These findings are consistent with previous studies (Nystrup et al., 2020a,b).
While the MLE of HMMs is consistent,
 √  Bickel et al. (1998) demonstrate that the convergence
rate to the true parameters is O 1/ T , where T is the length of the time series. Based on our
empirical results, this implies that achieving accurate parameter estimates would require an obser-
vation sequence of several multiples of 2000, which is challenging in practical financial applications
with daily return series. Recent theoretical work (Yang et al., 2017; Balakrishnan et al., 2017) on
the statistical properties of Baum-Welch algorithms provides insights into the slow convergence rate
observed in practice. This slow rate can be attributed to the stylized facts observed in financial
time series, including a low signal-to-noise ratio due to significant overlap between states, high
persistence, and large imbalances of observations from different regimes.
Comparing the discrete and continuous jump models, we observe that the continuous model
achieves a similar BAC to the discrete model, while significantly improving the ROC-AUC. This
indicates that, in addition to providing commensurate labeling results, the continuous model also
offers reliable probability estimates for each regime. This gives us confidence in applying the
estimated probabilities to downstream tasks such as constructing regime-aware portfolios.
When comparing the two versions of the continuous model, we notice that the inclusion of
the mode loss slightly improves the model’s performance in small sample sizes. This supports the
notion that the mode loss promotes sparsity and persistence in the estimated probability vector.
However, with sufficient data, the impact of the mode loss becomes less apparent.

5.1.3 Weekly Scale

Table 2 presents the mean and standard deviation (in parenthesis) of parameter estimates and
accuracy scores of 1024 simulations of length T = 50, 100, 250, 500 from the weekly two-state HMM
estimated with HMM, and the discrete and continuous jump models with and without mode loss.
For the weekly scale, the sequence lengths we consider correspond approximately to 1, 2, 5, and 10
years, respectively.
By going from the daily to the weekly scale, the number of time series observations decrease by a
factor of five, making it significantly more challenging to fit an HMM. Despite having the equivalent
of about ten years of weekly observations when T = 500, the estimated transition probabilities
in the HMMs deviates significantly from the true ones, resulting in a lack of persistence of the
state sequence. In contrast, the jump models capture the true persistence of the underlying state
sequence. The performance difference between the discrete and continuous models is similar to the

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 µ1 µ2 σ1 σ2 γ12 γ21 Accuracy 1 Accuracy 2 BAC ROC-AUC
250
true 0.0006 (0.0000) -0.0008 (0.0000) 0.0078 (0.0000) 0.0174 (0.0000) 0.0021 (0.0000) 0.0120 (0.0000) 0.9872 (0.0893) 0.8481 (0.2895) 0.9177 (0.1456) 0.9905 (0.0577)
HMM 0.0005 (0.0043) -0.0013 (0.0074) 0.0078 (0.0018) 0.0104 (0.0061) 0.1960 (0.2731) 0.4525 (0.3961) 0.8410 (0.2182) 0.7642 (0.3132) 0.8026 (0.2007) 0.8477 (0.2397)
discrete 0.0006 (0.0008) -0.0003 (0.0031) 0.0080 (0.0015) 0.0140 (0.0048) 0.0031 (0.0051) 0.0182 (0.0266) 0.9217 (0.1514) 0.8123 (0.2758) 0.8670 (0.1537) 0.8778 (0.1533)
cont 0.0006 (0.0008) 0.0001 (0.0022) 0.0082 (0.0016) 0.0118 (0.0044) 0.0034 (0.0048) 0.0125 (0.0233) 0.8782 (0.1667) 0.8175 (0.2955) 0.8478 (0.1704) 0.9391 (0.1481)
contM 0.0006 (0.0008) 0.0001 (0.0024) 0.0081 (0.0016) 0.0121 (0.0046) 0.0042 (0.0049) 0.0169 (0.0273) 0.8670 (0.1616) 0.8434 (0.2558) 0.8552 (0.1593) 0.9380 (0.1500)
500
true 0.0006 (0.0000) -0.0008 (0.0000) 0.0078 (0.0000) 0.0174 (0.0000) 0.0021 (0.0000) 0.0120 (0.0000) 0.9955 (0.0154) 0.8685 (0.2495) 0.9320 (0.1238) 0.9915 (0.0591)
HMM 0.0006 (0.0021) -0.0008 (0.0065) 0.0077 (0.0009) 0.0130 (0.0057) 0.1159 (0.2046) 0.2931 (0.3711) 0.9102 (0.1622) 0.8228 (0.2827) 0.8665 (0.1731) 0.9229 (0.1926)
discrete 0.0006 (0.0005) -0.0005 (0.0028) 0.0079 (0.0005) 0.0148 (0.0043) 0.0022 (0.0021) 0.0163 (0.0193) 0.9457 (0.1020) 0.8294 (0.2407) 0.8876 (0.1355) 0.9023 (0.1284)
cont 0.0006 (0.0005) -0.0002 (0.0022) 0.0080 (0.0005) 0.0136 (0.0044) 0.0031 (0.0022) 0.0147 (0.0157) 0.9017 (0.1355) 0.8598 (0.2146) 0.8808 (0.1388) 0.9575 (0.1266)
contM 0.0006 (0.0005) -0.0002 (0.0023) 0.0080 (0.0005) 0.0137 (0.0044) 0.0027 (0.0021) 0.0153 (0.0192) 0.9178 (0.1228) 0.8529 (0.2205) 0.8854 (0.1371) 0.9531 (0.1234)
1000
true 0.0006 (0.0000) -0.0008 (0.0000) 0.0078 (0.0000) 0.0174 (0.0000) 0.0021 (0.0000) 0.0120 (0.0000) 0.9957 (0.0123) 0.9049 (0.1819) 0.9503 (0.0905) 0.9956 (0.0321)

19
HMM 0.0006 (0.0006) -0.0004 (0.0039) 0.0077 (0.0006) 0.0155 (0.0043) 0.0491 (0.1486) 0.1341 (0.2739) 0.9656 (0.1060) 0.8905 (0.2006) 0.9280 (0.1249) 0.9616 (0.1414)
discrete 0.0006 (0.0003) -0.0007 (0.0023) 0.0079 (0.0003) 0.0163 (0.0031) 0.0019 (0.0013) 0.0154 (0.0162) 0.9744 (0.0606) 0.8566 (0.1840) 0.9155 (0.1003) 0.9212 (0.0970)
cont 0.0006 (0.0003) -0.0005 (0.0019) 0.0079 (0.0003) 0.0156 (0.0033) 0.0023 (0.0016) 0.0137 (0.0108) 0.9539 (0.0969) 0.8816 (0.1662) 0.9177 (0.1059) 0.9710 (0.1011)
contM 0.0006 (0.0003) -0.0006 (0.0019) 0.0079 (0.0003) 0.0157 (0.0033) 0.0021 (0.0014) 0.0141 (0.0120) 0.9636 (0.0783) 0.8763 (0.1719) 0.9199 (0.1037) 0.9674 (0.0938)
2000
true 0.0006 (0.0000) -0.0008 (0.0000) 0.0078 (0.0000) 0.0174 (0.0000) 0.0021 (0.0000) 0.0120 (0.0000) 0.9965 (0.0061) 0.9212 (0.1230) 0.9588 (0.0612) 0.9975 (0.0157)
HMM 0.0006 (0.0003) -0.0010 (0.0037) 0.0078 (0.0004) 0.0166 (0.0028) 0.0266 (0.1040) 0.0738 (0.1971) 0.9819 (0.0733) 0.8946 (0.1773) 0.9382 (0.1096) 0.9693 (0.1185)
discrete 0.0006 (0.0002) -0.0008 (0.0016) 0.0079 (0.0002) 0.0171 (0.0018) 0.0017 (0.0009) 0.0136 (0.0111) 0.9899 (0.0190) 0.8717 (0.1388) 0.9308 (0.0707) 0.9314 (0.0702)
cont 0.0006 (0.0002) -0.0007 (0.0014) 0.0079 (0.0002) 0.0168 (0.0020) 0.0021 (0.0015) 0.0140 (0.0091) 0.9795 (0.0607) 0.8926 (0.1143) 0.9360 (0.0688) 0.9768 (0.0628)
contM 0.0006 (0.0002) -0.0007 (0.0014) 0.0079 (0.0002) 0.0168 (0.0019) 0.0020 (0.0011) 0.0141 (0.0095) 0.9846 (0.0391) 0.8885 (0.1219) 0.9366 (0.0685) 0.9712 (0.0641)

Table 1: Mean and standard deviation (in parenthesis) of parameter estimates and accuracy scores of 1024 simulations of length T =
250, 500, 1000, 2000 from the daily two-state HMM estimated with HMM, and the discrete and continuous jump models with and without mode

Electronic copy available at: https://ssrn.com/abstract=4556048

loss (discrete, cont, and contM ). The bold values represent the best parameter estimates and highest accuracy scores.
daily results above. Specifically, while the discrete and continuous jump models obtain comparable
BACs, the continuous jump models exhibit better ROC-AUCs. Notably, due to the reduced level
of persistence, all models exhibit lower accuracy in detecting the bear market regime compared to
their daily counterparts.

5.1.4 Monthly Scale

Table 3 presents the mean and standard deviation (in parenthesis) of parameter estimates and
accuracy scores of 1024 simulations of length T = 60, 120, 250, 500 from the monthly two-state
HMM estimated with HMM, and the discrete and continuous jump models with and without mode
loss. For the monthly scale, the sequence lengths we consider correspond approximately to 2, 10,
20 and 40 years, respectively.
Consistent with the empirical results at the daily and weekly scales, the jump models achieve
higher BAC and more accurate parameter estimates when the number of observations is limited.
Due to the relatively low persistence of monthly data, the estimates of the transition probabilities
for the HMMs with a sequence length of 500 are the best. In comparison to the results in Nystrup
et al. (2020b), our jump models do not achieve a higher BAC than the true model. This suggests
that the outperformance reported in Nystrup et al. (2020b) is likely due to that some of their
features are forward-looking. We emphasize that as all of our features are backward-looking, our
model can be applied in an online forcasting fashion while still achieving a BAC close to that of
the true model. Additionally, the jump models demonstrate higher accuracy for the bear regime
than the HMMs.
It is worth noting that while the ROC-AUCs of the HMMs are significantly higher than those
of the jump models once the number of observations exceed 250, there is an important difference.
The labels and probabilities from HMMs are calculated using two different algorithms, the Viterbi
algorithm and the forward-backward algorithm. As a result, they may not be consistent with
each other, meaning that the regime with the maximum estimated probability from the forward-
backward algorithm may not align with the label from the Viterbi algorithm. In contrast, our
continuous jump model ensures consistency between the labels and estimated probabilities, as we
assign the label corresponding to the regime with the highest estimated probability.

5.2 Three-State HMM

Several studies suggest HMMs with more than two states can provide a better fit to financial
return series, based on information criteria such as the Bayesian Information Criterion (BIC) (Bulla,
2011; Nystrup et al., 2015; Dias et al., 2015; Cortese et al., 2023a,b). As the number of states
increases, the estimation of HMMs becomes considerably more challenging due to the quadratic
growth of parameters in the number of states.
To assess the performance as the number of states increases, we follow Zheng et al. (2021) by
augmenting the two-state model with a third intermediate state. This augmentation results in a
model with the following parameters
µ1 = .0123, µ2 = .0000, µ3 = −.0157,
σ1 = .0347, σ2 = .0500, σ3 = .0778,

.9629 .0185 .0186
 (32)
P = .0618
 .8764 .0618 .
.1051 .1050 .7899

The parameters can be transformed to different scales using the formulas in equation (31).

20

Electronic copy available at: https://ssrn.com/abstract=4556048

 µ1 µ2 σ1 σ2 γ12 γ21 Accuracy 1 Accuracy 2 BAC ROC-AUC
50
true 0.0031 (0.0000) -0.0039 (0.0000) 0.0174 (0.0000) 0.0389 (0.0000) 0.0103 (0.0000) 0.0582 (0.0000) 0.9727 (0.1179) 0.6019 (0.4176) 0.7873 (0.1932) 0.9421 (0.1387)
HMM 0.0032 (0.0155) -0.0029 (0.0260) 0.0159 (0.0051) 0.0164 (0.0127) 0.2775 (0.2908) 0.6081 (0.3652) 0.7533 (0.2248) 0.5371 (0.3276) 0.6452 (0.1688) 0.6592 (0.2504)
discrete 0.0029 (0.0046) 0.0001 (0.0103) 0.0180 (0.0045) 0.0250 (0.0110) 0.0140 (0.0207) 0.0436 (0.0531) 0.8049 (0.2002) 0.6593 (0.3702) 0.7321 (0.1917) 0.7377 (0.1992)
cont 0.0028 (0.0040) 0.0008 (0.0074) 0.0185 (0.0047) 0.0232 (0.0093) 0.0182 (0.0192) 0.0408 (0.0582) 0.7431 (0.1708) 0.7173 (0.3397) 0.7302 (0.1907) 0.8087 (0.2358)
contM 0.0029 (0.0043) 0.0008 (0.0082) 0.0183 (0.0048) 0.0235 (0.0098) 0.0201 (0.0198) 0.0471 (0.0587) 0.7498 (0.1674) 0.7095 (0.3302) 0.7296 (0.1923) 0.7981 (0.2210)
100
true 0.0031 (0.0000) -0.0039 (0.0000) 0.0174 (0.0000) 0.0389 (0.0000) 0.0103 (0.0000) 0.0582 (0.0000) 0.9822 (0.0639) 0.6273 (0.3816) 0.8048 (0.1842) 0.9512 (0.1235)
HMM 0.0028 (0.0094) -0.0031 (0.0243) 0.0167 (0.0044) 0.0220 (0.0144) 0.2085 (0.2717) 0.5116 (0.3813) 0.8178 (0.2089) 0.5616 (0.3352) 0.6897 (0.1872) 0.7220 (0.2722)
discrete 0.0030 (0.0027) -0.0009 (0.0097) 0.0180 (0.0029) 0.0282 (0.0112) 0.0097 (0.0086) 0.0424 (0.0459) 0.8527 (0.1545) 0.6585 (0.3392) 0.7556 (0.1877) 0.7743 (0.1853)
cont 0.0029 (0.0027) -0.0002 (0.0077) 0.0181 (0.0029) 0.0269 (0.0102) 0.0149 (0.0106) 0.0408 (0.0311) 0.8052 (0.1581) 0.7202 (0.3082) 0.7627 (0.1870) 0.8394 (0.1974)
contM 0.0029 (0.0027) -0.0001 (0.0079) 0.0182 (0.0029) 0.0269 (0.0103) 0.0135 (0.0102) 0.0406 (0.0335) 0.8175 (0.1538) 0.7082 (0.3176) 0.7629 (0.1886) 0.8263 (0.1974)
250
true 0.0031 (0.0000) -0.0039 (0.0000) 0.0174 (0.0000) 0.0389 (0.0000) 0.0103 (0.0000) 0.0582 (0.0000) 0.9857 (0.0286) 0.6924 (0.2893) 0.8390 (0.1415) 0.9668 (0.0810)

21
HMM 0.0031 (0.0024) -0.0035 (0.0176) 0.0175 (0.0021) 0.0310 (0.0129) 0.0964 (0.2054) 0.2943 (0.3431) 0.9268 (0.1371) 0.6379 (0.3107) 0.7824 (0.1809) 0.8403 (0.2386)
discrete 0.0032 (0.0017) -0.0033 (0.0083) 0.0178 (0.0016) 0.0336 (0.0088) 0.0125 (0.0086) 0.0532 (0.0289) 0.8987 (0.1238) 0.7144 (0.2243) 0.8065 (0.1317) 0.8143 (0.1281)
cont 0.0031 (0.0016) -0.0026 (0.0073) 0.0181 (0.0016) 0.0327 (0.0086) 0.0119 (0.0083) 0.0457 (0.0235) 0.8886 (0.1330) 0.7259 (0.2281) 0.8073 (0.1377) 0.8627 (0.1452)
contM 0.0032 (0.0017) -0.0028 (0.0074) 0.0180 (0.0016) 0.0330 (0.0087) 0.0130 (0.0091) 0.0503 (0.0255) 0.8900 (0.1308) 0.7261 (0.2200) 0.8080 (0.1316) 0.8445 (0.1321)
500
true 0.0031 (0.0000) -0.0039 (0.0000) 0.0174 (0.0000) 0.0389 (0.0000) 0.0103 (0.0000) 0.0582 (0.0000) 0.9879 (0.0165) 0.7549 (0.2009) 0.8714 (0.0986) 0.9777 (0.0434)
HMM 0.0031 (0.0013) -0.0045 (0.0127) 0.0174 (0.0012) 0.0360 (0.0090) 0.0416 (0.1287) 0.1640 (0.2468) 0.9665 (0.0888) 0.7150 (0.2421) 0.8408 (0.1351) 0.9209 (0.1638)
discrete 0.0032 (0.0011) -0.0042 (0.0064) 0.0179 (0.0010) 0.0364 (0.0060) 0.0096 (0.0056) 0.0529 (0.0220) 0.9419 (0.0785) 0.7222 (0.1618) 0.8321 (0.0912) 0.8324 (0.0910)
cont 0.0031 (0.0011) -0.0034 (0.0057) 0.0181 (0.0010) 0.0354 (0.0058) 0.0090 (0.0054) 0.0457 (0.0175) 0.9357 (0.0846) 0.7394 (0.1594) 0.8375 (0.0935) 0.8899 (0.0969)
contM 0.0030 (0.0010) -0.0036 (0.0058) 0.0182 (0.0010) 0.0357 (0.0057) 0.0082 (0.0044) 0.0455 (0.0187) 0.9428 (0.0742) 0.7289 (0.1664) 0.8359 (0.0944) 0.8739 (0.0984)

Table 2: Mean and standard deviation (in parenthesis) of parameter estimates and accuracy scores of 1024 simulations of length T =
50, 100, 250, 500 from the weekly two-state HMM estimated with HMM, and the discrete and continuous jump models with and without mode

Electronic copy available at: https://ssrn.com/abstract=4556048

loss (discrete, cont, and contM ). The bold values represent the best parameter estimates and highest accuracy scores.
 µ1 µ2 σ1 σ2 γ12 γ21 Accuracy 1 Accuracy 2 BAC ROC-AUC
60
true 0.0123 (0.0000) -0.0157 (0.0000) 0.0347 (0.0000) 0.0778 (0.0000) 0.0371 (0.0000) 0.2101 (0.0000) 0.9821 (0.0410) 0.4389 (0.3465) 0.7105 (0.1669) 0.8820 (0.1674)
HMM 0.0124 (0.0109) -0.0172 (0.0613) 0.0332 (0.0082) 0.0397 (0.0285) 0.2287 (0.2799) 0.6212 (0.3520) 0.8181 (0.1815) 0.4459 (0.3115) 0.6320 (0.1654) 0.6415 (0.2450)
discrete 0.0125 (0.0082) -0.0031 (0.0225) 0.0343 (0.0059) 0.0558 (0.0201) 0.0678 (0.0467) 0.1325 (0.0644) 0.7517 (0.1478) 0.6498 (0.2886) 0.7007 (0.1605) 0.7075 (0.1599)
cont 0.0125 (0.0083) -0.0033 (0.0223) 0.0342 (0.0059) 0.0558 (0.0198) 0.0674 (0.0471) 0.1316 (0.0631) 0.7524 (0.1475) 0.6477 (0.2916) 0.7000 (0.1615) 0.7068 (0.1609)
contM 0.0125 (0.0083) -0.0032 (0.0224) 0.0342 (0.0058) 0.0557 (0.0199) 0.0679 (0.0473) 0.1322 (0.0646) 0.7512 (0.1463) 0.6450 (0.2903) 0.6981 (0.1611) 0.7048 (0.1605)
120
true 0.0123 (0.0000) -0.0157 (0.0000) 0.0347 (0.0000) 0.0778 (0.0000) 0.0371 (0.0000) 0.2101 (0.0000) 0.9829 (0.0267) 0.4696 (0.2650) 0.7263 (0.1282) 0.8969 (0.1002)
HMM 0.0120 (0.0069) -0.0260 (0.0548) 0.0345 (0.0060) 0.0560 (0.0293) 0.1412 (0.2159) 0.4881 (0.3414) 0.8913 (0.1526) 0.4577 (0.2721) 0.6745 (0.1453) 0.7267 (0.2230)
discrete 0.0131 (0.0063) -0.0079 (0.0201) 0.0347 (0.0045) 0.0630 (0.0172) 0.0580 (0.0407) 0.1402 (0.0523) 0.7969 (0.1353) 0.6252 (0.2213) 0.7111 (0.1199) 0.7117 (0.1198)
cont 0.0132 (0.0062) -0.0080 (0.0201) 0.0346 (0.0045) 0.0630 (0.0173) 0.0584 (0.0410) 0.1399 (0.0517) 0.7960 (0.1351) 0.6269 (0.2208) 0.7114 (0.1185) 0.7120 (0.1183)
contM 0.0132 (0.0063) -0.0081 (0.0205) 0.0346 (0.0045) 0.0629 (0.0172) 0.0578 (0.0405) 0.1389 (0.0511) 0.7967 (0.1357) 0.6304 (0.2184) 0.7135 (0.1166) 0.7141 (0.1164)
250
true 0.0123 (0.0000) -0.0157 (0.0000) 0.0347 (0.0000) 0.0778 (0.0000) 0.0371 (0.0000) 0.2101 (0.0000) 0.9836 (0.0186) 0.5105 (0.1842) 0.7470 (0.0894) 0.9120 (0.0583)

22
HMM 0.0125 (0.0033) -0.0258 (0.0365) 0.0343 (0.0032) 0.0694 (0.0205) 0.0665 (0.1098) 0.3350 (0.2599) 0.9539 (0.0886) 0.5080 (0.2182) 0.7310 (0.1084) 0.8483 (0.1473)
discrete 0.0136 (0.0043) -0.0122 (0.0139) 0.0349 (0.0031) 0.0650 (0.0129) 0.0520 (0.0312) 0.1570 (0.0433) 0.8368 (0.1041) 0.6391 (0.1515) 0.7380 (0.0789) 0.7380 (0.0789)
cont 0.0136 (0.0042) -0.0121 (0.0138) 0.0349 (0.0031) 0.0650 (0.0130) 0.0523 (0.0313) 0.1574 (0.0437) 0.8360 (0.1044) 0.6405 (0.1509) 0.7382 (0.0795) 0.7382 (0.0794)
contM 0.0136 (0.0044) -0.0121 (0.0138) 0.0349 (0.0031) 0.0650 (0.0130) 0.0525 (0.0315) 0.1576 (0.0437) 0.8356 (0.1050) 0.6403 (0.1508) 0.7380 (0.0795) 0.7380 (0.0795)
500
true 0.0123 (0.0000) -0.0157 (0.0000) 0.0347 (0.0000) 0.0778 (0.0000) 0.0371 (0.0000) 0.2101 (0.0000) 0.9842 (0.0135) 0.5194 (0.1291) 0.7518 (0.0627) 0.9135 (0.0377)
HMM 0.0123 (0.0022) -0.0151 (0.0254) 0.0346 (0.0025) 0.0739 (0.0154) 0.0554 (0.1050) 0.2785 (0.1928) 0.9678 (0.0670) 0.5136 (0.1810) 0.7407 (0.0895) 0.8810 (0.1108)
discrete 0.0138 (0.0029) -0.0134 (0.0095) 0.0351 (0.0022) 0.0663 (0.0092) 0.0473 (0.0234) 0.1668 (0.0356) 0.8597 (0.0710) 0.6301 (0.1047) 0.7449 (0.0523) 0.7449 (0.0523)
cont 0.0137 (0.0028) -0.0134 (0.0096) 0.0351 (0.0022) 0.0663 (0.0091) 0.0471 (0.0231) 0.1666 (0.0352) 0.8600 (0.0702) 0.6302 (0.1053) 0.7451 (0.0517) 0.7451 (0.0517)
contM 0.0138 (0.0028) -0.0134 (0.0096) 0.0351 (0.0021) 0.0663 (0.0092) 0.0471 (0.0233) 0.1664 (0.0356) 0.8600 (0.0707) 0.6294 (0.1057) 0.7447 (0.0524) 0.7447 (0.0524)

Table 3: Mean and standard deviation (in parenthesis) of parameter estimates and accuracy scores of 1024 simulations of length T =
60, 120, 250, 500 from the monthly two-state HMM estimated with HMM, and the discrete and continuous jump models with and without mode

Electronic copy available at: https://ssrn.com/abstract=4556048

loss (discrete, cont, and contM ). The bold values represent the best parameter estimates and highest accuracy scores.
 As the ROC curve is typically defined for binary classification problems, in the case of the three-
state model, we compute the ROC-AUC as the unweighted average area under the ROC curves for
all possible pairwise comparisons among the states (Hand and Till, 2001). This allows us to assess
the model’s ranking ability in a multi-state setting and provide a comprehensive evaluation of its
performance across different class combinations.
Table 4 presents the mean and standard deviation (in parenthesis) of parameter estimates and
accuracy scores of 1024 simulations of length T = 500, 1000, 2000 from the daily three-state HMM
estimated with HMM, and the discrete and continuous jump models with and without mode loss.
We do not show results for a sequence length of 250 as it is insufficient for accurately estimating a
persistent model. We only present results for the daily scale, as the conclusions are consistent across
weekly and monthly parameters. For the continuous jump model, we discretize the probability
simplex using a uniform grid with a size of 0.05.
The addition of a third state significantly increases the difficulty of estimating the HMMs. Even
for the longest time series of 2000 observations, it is not possible to capture the true persistence
accurately. In contrast, we observe that the jump models obtain the best parameter estimates for
almost all of the twelve model parameters. When comparing the discrete and continuous jump
models, we note that the continuous model achieves an improvement of 2% and 4% in BAC and
ROC-AUC, respectively. This validates the effectiveness of the continuous model, as it allows the
representation of hidden states as continuous probabilities, contributing to increased accuracy.
The BAC of all models increases slowly with the length of the time series, underscoring the
difficulty of precisely fitting models with more than two states. However, compared to the results
in Nystrup et al. (2020b), our jump models achieve higher accuracy even when the feature set is
limited to solely backward-looking features.

5.3 Misspecified Conditional Distributions

Rather than Markovian mixtures of Gaussian distributions, Bulla (2011) uses t-distributions
for the state-conditional distributions in the HMMs. They demonstrate that the t-HMM is able to
capture stylized facts of financial time series more effectively than the Gaussian HMM, particularly
in terms of the slow decay of the autocorrelation function of absolute or squared returns. Moreover,
the augmented mass around the mean and the fat tails of the t-distributions contribute to increased
persistence in the estimated state sequence and enhance the model’s robustness to outliers.
To examine the robustness of different models to misspecified conditional distributions, we
generate observation sequences from t-HMMs, where each emission t-distribution has the same
mean and covariance as specified in the daily parameter (30), with a degree of freedom of five,
based on Bulla (2011)’s empirical estimates on daily return series of stock indices. The transition
probability matrix of the state sequence remains unchanged.
Table 5 presents the mean and standard deviation (in parenthesis) of parameter estimates and
accuracy scores of 1024 simulations of length T = 250, 500, 1000, 2000 from the daily two-state t-
HMM estimated with HMM, and the discrete and continuous jump models with and without mode
loss. In our experiments, both the true and estimated HMMs assume a Gaussian state-conditional
distributions, allowing us to investigate the model’s sensitivity to misspecification. Examining the
parameter estimates, we observe a pronounced overestimation of transition probabilities in the
HMMs compared to the well-specified scenario. The presence of fat tails in the t-distributions leads
to an increased number of switches between the two states for the misspecified Gaussian HMM.
In contrast, the jump models exhibit no difficulty in capturing the low values of the transition
probabilities, thereby accurately capturing the persistence of the state sequence. In terms of BAC,
we note a degradation of 5-8% for HMMs due to the misspecification of the conditional distributions,

23

Electronic copy available at: https://ssrn.com/abstract=4556048

 µ1 µ2 µ3 σ1 σ2 σ3 γ12 γ13 γ21
500
true 0.0006 (0.0000) 0.0000 (0.0000) -0.0008 (0.0000) 0.0078 (0.0000) 0.0112 (0.0000) 0.0174 (0.0000) 0.0009 (0.0000) 0.0010 (0.0000) 0.0032 (0.0000)
HMM 0.0009 (0.0061) 0.0012 (0.0086) -0.0017 (0.0073) 0.0078 (0.0023) 0.0082 (0.0039) 0.0100 (0.0054) 0.2601 (0.3376) 0.2120 (0.3009) 0.3909 (0.3968)
discrete 0.0005 (0.0015) -0.0000 (0.0029) -0.0007 (0.0031) 0.0085 (0.0027) 0.0115 (0.0033) 0.0145 (0.0042) 0.0023 (0.0093) 0.0020 (0.0069) 0.0076 (0.0161)
cont 0.0005 (0.0011) 0.0000 (0.0023) -0.0008 (0.0024) 0.0084 (0.0021) 0.0116 (0.0029) 0.0147 (0.0039) 0.0017 (0.0071) 0.0016 (0.0073) 0.0093 (0.0487)
contM 0.0006 (0.0012) -0.0000 (0.0023) -0.0007 (0.0024) 0.0084 (0.0021) 0.0117 (0.0030) 0.0148 (0.0038) 0.0015 (0.0052) 0.0015 (0.0058) 0.0084 (0.0458)
1000
true 0.0006 (0.0000) 0.0000 (0.0000) -0.0008 (0.0000) 0.0078 (0.0000) 0.0112 (0.0000) 0.0174 (0.0000) 0.0009 (0.0000) 0.0010 (0.0000) 0.0032 (0.0000)
HMM 0.0007 (0.0033) 0.0008 (0.0073) -0.0010 (0.0062) 0.0084 (0.0027) 0.0093 (0.0038) 0.0117 (0.0053) 0.2127 (0.3264) 0.1505 (0.2788) 0.3367 (0.4006)
discrete 0.0006 (0.0014) -0.0000 (0.0034) -0.0007 (0.0029) 0.0084 (0.0022) 0.0122 (0.0033) 0.0155 (0.0038) 0.0015 (0.0052) 0.0017 (0.0054) 0.0065 (0.0132)
cont 0.0006 (0.0009) 0.0000 (0.0024) -0.0006 (0.0023) 0.0083 (0.0019) 0.0121 (0.0030) 0.0153 (0.0035) 0.0011 (0.0034) 0.0011 (0.0026) 0.0054 (0.0151)
contM 0.0006 (0.0008) 0.0000 (0.0024) -0.0006 (0.0023) 0.0084 (0.0019) 0.0121 (0.0030) 0.0155 (0.0033) 0.0012 (0.0034) 0.0012 (0.0034) 0.0052 (0.0139)
2000
true 0.0006 (0.0000) 0.0000 (0.0000) -0.0008 (0.0000) 0.0078 (0.0000) 0.0112 (0.0000) 0.0174 (0.0000) 0.0009 (0.0000) 0.0010 (0.0000) 0.0032 (0.0000)
HMM 0.0005 (0.0022) 0.0008 (0.0072) -0.0003 (0.0051) 0.0083 (0.0018) 0.0105 (0.0041) 0.0125 (0.0048) 0.1329 (0.2858) 0.1120 (0.2575) 0.2290 (0.3704)
discrete 0.0006 (0.0007) -0.0001 (0.0034) -0.0009 (0.0027) 0.0082 (0.0013) 0.0128 (0.0031) 0.0165 (0.0026) 0.0009 (0.0023) 0.0011 (0.0020) 0.0056 (0.0099)
cont 0.0006 (0.0005) -0.0002 (0.0021) -0.0007 (0.0021) 0.0081 (0.0010) 0.0124 (0.0026) 0.0163 (0.0025) 0.0008 (0.0019) 0.0008 (0.0015) 0.0055 (0.0158)
contM 0.0006 (0.0006) -0.0001 (0.0022) -0.0008 (0.0021) 0.0081 (0.0011) 0.0124 (0.0026) 0.0163 (0.0025) 0.0009 (0.0020) 0.0009 (0.0016) 0.0048 (0.0088)
γ23 γ31 γ32 Accuracy 1 Accuracy 2 Accuracy 3 BAC ROC-AUC
500

24
true 0.0037 (0.0000) 0.0058 (0.0000) 0.0062 (0.0000) 0.9557 (0.1685) 0.7223 (0.3969) 0.7654 (0.3480) 0.8145 (0.1590) 0.9667 (0.0662)
HMM 0.2695 (0.3320) 0.3199 (0.3735) 0.2922 (0.3513) 0.6284 (0.3051) 0.4472 (0.3521) 0.5452 (0.3660) 0.5403 (0.1077) 0.6626 (0.1611)
discrete 0.0059 (0.0150) 0.0091 (0.0168) 0.0091 (0.0171) 0.8446 (0.2650) 0.6677 (0.3842) 0.6558 (0.3201) 0.7227 (0.1463) 0.7560 (0.1189)
cont 0.0059 (0.0327) 0.0071 (0.0130) 0.0084 (0.0297) 0.8671 (0.2477) 0.7046 (0.3786) 0.6480 (0.3651) 0.7399 (0.1467) 0.7968 (0.1274)
contM 0.0060 (0.0440) 0.0075 (0.0171) 0.0072 (0.0182) 0.8645 (0.2560) 0.7072 (0.3722) 0.6564 (0.3577) 0.7427 (0.1497) 0.7909 (0.1298)
1000
true 0.0037 (0.0000) 0.0058 (0.0000) 0.0062 (0.0000) 0.9703 (0.1017) 0.7465 (0.3560) 0.8129 (0.2843) 0.8432 (0.1325) 0.9828 (0.0328)
HMM 0.2671 (0.3474) 0.2474 (0.3617) 0.2516 (0.3265) 0.7114 (0.3099) 0.4029 (0.3858) 0.5460 (0.4026) 0.5534 (0.1224) 0.6812 (0.1968)
discrete 0.0078 (0.0179) 0.0090 (0.0136) 0.0081 (0.0138) 0.8866 (0.2370) 0.6296 (0.3796) 0.6941 (0.2814) 0.7368 (0.1495) 0.7932 (0.1180)
cont 0.0063 (0.0362) 0.0069 (0.0116) 0.0073 (0.0172) 0.8909 (0.2215) 0.6553 (0.3721) 0.7101 (0.2920) 0.7521 (0.1452) 0.8303 (0.1195)
contM 0.0064 (0.0243) 0.0072 (0.0115) 0.0070 (0.0145) 0.8878 (0.2294) 0.6576 (0.3748) 0.7123 (0.2923) 0.7526 (0.1455) 0.8291 (0.1205)
2000
true 0.0037 (0.0000) 0.0058 (0.0000) 0.0062 (0.0000) 0.9841 (0.0439) 0.8013 (0.2734) 0.8568 (0.2133) 0.8807 (0.1066) 0.9867 (0.0284)
HMM 0.2954 (0.3698) 0.2051 (0.3571) 0.2474 (0.3331) 0.8185 (0.2668) 0.3810 (0.4022) 0.5275 (0.4033) 0.5756 (0.1413) 0.7150 (0.2064)

Electronic copy available at: https://ssrn.com/abstract=4556048

discrete 0.0072 (0.0141) 0.0079 (0.0103) 0.0070 (0.0096) 0.9487 (0.1520) 0.6156 (0.3863) 0.7107 (0.2520) 0.7584 (0.1549) 0.8173 (0.1177)
cont 0.0042 (0.0101) 0.0064 (0.0082) 0.0053 (0.0084) 0.9484 (0.1295) 0.6676 (0.3558) 0.7270 (0.2534) 0.7810 (0.1458) 0.8614 (0.1131)
contM 0.0050 (0.0333) 0.0064 (0.0077) 0.0060 (0.0181) 0.9480 (0.1376) 0.6638 (0.3608) 0.7283 (0.2464) 0.7800 (0.1448) 0.8603 (0.1120)

Table 4: Mean and standard deviation (in parenthesis) of parameter estimates and accuracy scores of 1024 simulations of length T = 500, 1000, 2000
from the daily three-state HMM estimated with HMM, and the discrete and continuous jump models with and without mode loss (discrete, cont,
and contM ). The bold values represent the best parameter estimates and highest accuracy scores.
a 4% degradation for the discrete jump model, and only a 2% degradation for the continuous jump
model. Consequently, the continuous jump model outperforms all other models in terms of both
BAC and ROC-AUC for time series of at least 500 observations.

5.4 Misspecified Sojourn-Time Distributions

As another extension of HMMs, Bulla and Bulla (2006) propose hidden semi-Markov models
(HSMM) to model financial return series. HSMM replaces the geometrically distributed sojourn
time, implicitly assumed by the Markovian structure in HMMs, with a negative binomial distribu-
tion. By allowing for more flexible sojourn time distributions, they show that the model achieves
improved performance in reproducing the long memory of return series.
To examine the models’ performance under misspecified sojourn time distributions, we sample
observation sequences from HSMMs. Following the approach in Nystrup et al. (2020a), we use
shape parameters of n1 = 0.1 for state 1 and n2 = 0.06 for state 2 in the negative binomial
distribution. These values are based on estimates from Bulla and Bulla (2006) on daily stock
indices. We calibrate the other probability parameters in the negative binomial distribution to
match the expected sojourn time of the HMM with the parameters in (30).
Table 6 presents the mean and standard deviation (in parenthesis) of parameter estimates and
accuracy scores of 1024 simulations of length T = 250, 500, 1000, 2000 from the daily two-state
HSMM estimated with HMM, and the discrete and continuous jump models with and without
mode loss. Due to the increased number of both very short and prolonged sojourns from the
negative binomial distribution, the estimated HMM struggles to obtain reasonable estimates of the
transition probabilities, and thus underperforms the jump models in terms of BAC and ROC-AUC.
In contrast, the jump models exhibit better robustness to misspecified sojourn time distributions.
The continuous jump model achieves a comparable BAC score to the discrete model, with a slight
improvement in ROC-AUC.

6 An Application to the Nasdaq Index

In this application, we focus on the daily log-returns of the Nasdaq Composite index, a market-
cap weighted index based on the approximately 3,000 companies listed on Nasdaq. We consider two
distinct time periods: (i) 1996-2005 and (ii) 2013-2022, each spanning 10 years. By choosing these
periods of contrasting market conditions, with the former encompassing the rise and burst of the
dot-com bubble and the latter representing a relatively calm period, we aim to gain insights into the
performance of the models under different market conditions. Considering the lack of persistence
of HMMs in predicting regimes in the equity market shown in previous studies (Nystrup et al.,
2020a,b), we opt to explore the three jump models in this application. To select the jump penalty
parameter λ, we leverage the optimal values derived from the simulation study in Section 5.1 on
daily two-state HMMs. Specifically, for the discrete jump model, we choose λ = 102 , while for the
continuous jump models, we select λ = 103 .

6.1 The 1996-2005 Time Period

Table 7 presents the estimated parameters for the three jump models with two states applied
to the Nasdaq Composite index from 1996 through 2005. The parameters include the mean return
and volatility under each regime, and the resulting empirical transition probabilities. Consistent
with previous literature, we observe that the first and second states are characterized by positive

25

Electronic copy available at: https://ssrn.com/abstract=4556048

 µ1 µ2 σ1 σ2 γ12 γ21 Accuracy 1 Accuracy 2 BAC ROC-AUC
250
true 0.0006 (0.0000) -0.0008 (0.0000) 0.0078 (0.0000) 0.0174 (0.0000) 0.0021 (0.0000) 0.0120 (0.0000) 0.9839 (0.0576) 0.8073 (0.3069) 0.8956 (0.1511) 0.9766 (0.0805)
HMM 0.0005 (0.0085) -0.0005 (0.0181) 0.0070 (0.0041) 0.0142 (0.0065) 0.1942 (0.2333) 0.5392 (0.3739) 0.8597 (0.1889) 0.6921 (0.3165) 0.7759 (0.1710) 0.8426 (0.2059)
discrete 0.0005 (0.0026) -0.0003 (0.0061) 0.0082 (0.0041) 0.0158 (0.0080) 0.0040 (0.0149) 0.0360 (0.0539) 0.9272 (0.1472) 0.7039 (0.3413) 0.8156 (0.1713) 0.8150 (0.1790)
cont 0.0006 (0.0012) 0.0001 (0.0026) 0.0083 (0.0031) 0.0124 (0.0050) 0.0031 (0.0058) 0.0157 (0.0261) 0.8799 (0.1649) 0.7511 (0.3391) 0.8155 (0.1823) 0.9036 (0.1833)
contM 0.0006 (0.0010) -0.0000 (0.0030) 0.0082 (0.0022) 0.0133 (0.0051) 0.0026 (0.0051) 0.0158 (0.0295) 0.9001 (0.1630) 0.7211 (0.3609) 0.8106 (0.1906) 0.8940 (0.1852)
500
true 0.0006 (0.0000) -0.0008 (0.0000) 0.0078 (0.0000) 0.0174 (0.0000) 0.0021 (0.0000) 0.0120 (0.0000) 0.9877 (0.0228) 0.8330 (0.2559) 0.9103 (0.1269) 0.9848 (0.0448)
HMM 0.0006 (0.0007) -0.0000 (0.0141) 0.0067 (0.0027) 0.0164 (0.0054) 0.1194 (0.1434) 0.4191 (0.3545) 0.9191 (0.1279) 0.7086 (0.2838) 0.8139 (0.1549) 0.8955 (0.1590)
discrete 0.0006 (0.0009) -0.0002 (0.0060) 0.0080 (0.0022) 0.0169 (0.0076) 0.0025 (0.0057) 0.0313 (0.0453) 0.9507 (0.0983) 0.7390 (0.3112) 0.8448 (0.1634) 0.8550 (0.1604)
cont 0.0006 (0.0005) -0.0002 (0.0024) 0.0080 (0.0008) 0.0141 (0.0049) 0.0023 (0.0019) 0.0162 (0.0307) 0.9157 (0.1254) 0.8023 (0.2784) 0.8590 (0.1600) 0.9285 (0.1565)
contM 0.0006 (0.0005) -0.0003 (0.0029) 0.0079 (0.0008) 0.0146 (0.0052) 0.0027 (0.0019) 0.0201 (0.0265) 0.9262 (0.1091) 0.7963 (0.2708) 0.8613 (0.1536) 0.9219 (0.1496)
1000
true 0.0006 (0.0000) -0.0008 (0.0000) 0.0078 (0.0000) 0.0174 (0.0000) 0.0021 (0.0000) 0.0120 (0.0000) 0.9882 (0.0142) 0.8669 (0.1956) 0.9276 (0.0970) 0.9875 (0.0478)

26
HMM 0.0006 (0.0003) -0.0004 (0.0171) 0.0067 (0.0006) 0.0174 (0.0037) 0.0622 (0.0830) 0.2857 (0.3090) 0.9587 (0.0471) 0.7425 (0.2672) 0.8506 (0.1380) 0.9303 (0.0994)
discrete 0.0006 (0.0003) -0.0006 (0.0062) 0.0079 (0.0006) 0.0189 (0.0093) 0.0018 (0.0011) 0.0278 (0.0404) 0.9827 (0.0354) 0.7635 (0.2735) 0.8731 (0.1370) 0.8748 (0.1362)
cont 0.0006 (0.0003) -0.0006 (0.0028) 0.0079 (0.0005) 0.0168 (0.0061) 0.0022 (0.0014) 0.0169 (0.0175) 0.9615 (0.0811) 0.8201 (0.2282) 0.8908 (0.1254) 0.9513 (0.1123)
contM 0.0006 (0.0003) -0.0006 (0.0030) 0.0079 (0.0005) 0.0170 (0.0062) 0.0020 (0.0012) 0.0179 (0.0195) 0.9699 (0.0631) 0.8127 (0.2349) 0.8913 (0.1238) 0.9398 (0.1178)
2000
true 0.0006 (0.0000) -0.0008 (0.0000) 0.0078 (0.0000) 0.0174 (0.0000) 0.0021 (0.0000) 0.0120 (0.0000) 0.9885 (0.0098) 0.8902 (0.1230) 0.9394 (0.0608) 0.9911 (0.0297)
HMM 0.0006 (0.0002) -0.0003 (0.0153) 0.0068 (0.0004) 0.0178 (0.0025) 0.0418 (0.0449) 0.2227 (0.2352) 0.9697 (0.0199) 0.7491 (0.2334) 0.8594 (0.1195) 0.9421 (0.0734)
discrete 0.0006 (0.0002) -0.0009 (0.0047) 0.0079 (0.0004) 0.0189 (0.0082) 0.0016 (0.0008) 0.0185 (0.0241) 0.9894 (0.0124) 0.7955 (0.2112) 0.8925 (0.1045) 0.8926 (0.1044)
cont 0.0006 (0.0002) -0.0008 (0.0026) 0.0079 (0.0003) 0.0176 (0.0055) 0.0018 (0.0009) 0.0144 (0.0126) 0.9842 (0.0320) 0.8434 (0.1639) 0.9138 (0.0837) 0.9655 (0.0731)
contM 0.0006 (0.0002) -0.0009 (0.0026) 0.0079 (0.0003) 0.0177 (0.0053) 0.0017 (0.0009) 0.0145 (0.0126) 0.9858 (0.0287) 0.8386 (0.1664) 0.9122 (0.0851) 0.9550 (0.0791)

Table 5: Mean and standard deviation (in parenthesis) of parameter estimates and accuracy scores of 1024 simulations of length T =
250, 500, 1000, 2000 from the daily two-state t-HMM estimated with HMM, and the discrete and continuous jump models with and without

Electronic copy available at: https://ssrn.com/abstract=4556048

mode loss (discrete, cont, and contM ). The bold values represent the best parameter estimates and highest accuracy scores.
 µ1 µ2 σ1 σ2 γ12 γ21 Accuracy 1 Accuracy 2 BAC ROC-AUC
250
true 0.0006 (0.0000) -0.0008 (0.0000) 0.0078 (0.0000) 0.0174 (0.0000) 0.0021 (0.0000) 0.0120 (0.0000) 0.9451 (0.2083) 0.7223 (0.4123) 0.8337 (0.1932) 0.9075 (0.1849)
HMM 0.0002 (0.0076) -0.0014 (0.0083) 0.0084 (0.0032) 0.0102 (0.0059) 0.2760 (0.3198) 0.4715 (0.3771) 0.7870 (0.2727) 0.6767 (0.3512) 0.7318 (0.1967) 0.7303 (0.2834)
discrete 0.0005 (0.0016) -0.0002 (0.0026) 0.0086 (0.0028) 0.0138 (0.0045) 0.0036 (0.0073) 0.0165 (0.0275) 0.8843 (0.2252) 0.7281 (0.3634) 0.8062 (0.1981) 0.7811 (0.2213)
cont 0.0004 (0.0017) 0.0001 (0.0020) 0.0094 (0.0036) 0.0118 (0.0045) 0.0037 (0.0063) 0.0113 (0.0186) 0.8389 (0.2254) 0.7191 (0.3682) 0.7790 (0.2126) 0.8120 (0.2542)
contM 0.0005 (0.0013) -0.0001 (0.0020) 0.0089 (0.0031) 0.0129 (0.0045) 0.0034 (0.0067) 0.0109 (0.0200) 0.8600 (0.2280) 0.7212 (0.3743) 0.7906 (0.2068) 0.8083 (0.2447)
500
true 0.0006 (0.0000) -0.0008 (0.0000) 0.0078 (0.0000) 0.0174 (0.0000) 0.0021 (0.0000) 0.0120 (0.0000) 0.9565 (0.1785) 0.7270 (0.4029) 0.8418 (0.1993) 0.9316 (0.1566)
HMM 0.0000 (0.0059) -0.0006 (0.0068) 0.0084 (0.0027) 0.0119 (0.0057) 0.2256 (0.2887) 0.3892 (0.3775) 0.8295 (0.2536) 0.7012 (0.3521) 0.7654 (0.2212) 0.7727 (0.3019)
discrete 0.0005 (0.0015) -0.0003 (0.0024) 0.0086 (0.0027) 0.0142 (0.0044) 0.0033 (0.0083) 0.0163 (0.0263) 0.9046 (0.1952) 0.7554 (0.3482) 0.8300 (0.1917) 0.8280 (0.2004)
cont 0.0004 (0.0013) -0.0000 (0.0018) 0.0090 (0.0031) 0.0128 (0.0045) 0.0031 (0.0060) 0.0120 (0.0240) 0.8680 (0.1988) 0.7593 (0.3470) 0.8137 (0.2051) 0.8600 (0.2238)
contM 0.0005 (0.0015) -0.0001 (0.0020) 0.0089 (0.0031) 0.0129 (0.0046) 0.0047 (0.0123) 0.0146 (0.0216) 0.8642 (0.2009) 0.7771 (0.3298) 0.8207 (0.2021) 0.8559 (0.2216)
1000
true 0.0006 (0.0000) -0.0008 (0.0000) 0.0078 (0.0000) 0.0174 (0.0000) 0.0021 (0.0000) 0.0120 (0.0000) 0.9674 (0.1578) 0.7251 (0.3988) 0.8463 (0.2009) 0.9454 (0.1384)

27
HMM 0.0006 (0.0051) 0.0001 (0.0060) 0.0083 (0.0024) 0.0126 (0.0055) 0.1949 (0.2892) 0.3448 (0.3766) 0.8482 (0.2642) 0.7361 (0.3499) 0.7921 (0.2199) 0.8112 (0.2782)
discrete 0.0005 (0.0011) -0.0004 (0.0022) 0.0082 (0.0022) 0.0143 (0.0043) 0.0019 (0.0048) 0.0157 (0.0262) 0.9230 (0.1761) 0.7653 (0.3454) 0.8442 (0.2011) 0.8465 (0.2046)
cont 0.0005 (0.0009) -0.0003 (0.0017) 0.0084 (0.0022) 0.0133 (0.0044) 0.0020 (0.0042) 0.0106 (0.0161) 0.8939 (0.1772) 0.7704 (0.3471) 0.8322 (0.2104) 0.8713 (0.2201)
contM 0.0005 (0.0009) -0.0003 (0.0018) 0.0084 (0.0022) 0.0134 (0.0045) 0.0023 (0.0038) 0.0124 (0.0179) 0.8898 (0.1869) 0.7803 (0.3329) 0.8350 (0.2119) 0.8657 (0.2229)
2000
true 0.0006 (0.0000) -0.0008 (0.0000) 0.0078 (0.0000) 0.0174 (0.0000) 0.0021 (0.0000) 0.0120 (0.0000) 0.9894 (0.0717) 0.7689 (0.3660) 0.8792 (0.1831) 0.9588 (0.1250)
HMM 0.0002 (0.0040) -0.0005 (0.0044) 0.0081 (0.0018) 0.0136 (0.0051) 0.1429 (0.2447) 0.2602 (0.3489) 0.8950 (0.2141) 0.7505 (0.3509) 0.8227 (0.2212) 0.8319 (0.2754)
discrete 0.0006 (0.0007) -0.0004 (0.0020) 0.0080 (0.0013) 0.0149 (0.0040) 0.0012 (0.0028) 0.0126 (0.0185) 0.9617 (0.1017) 0.7751 (0.3412) 0.8684 (0.1870) 0.8714 (0.1868)
cont 0.0006 (0.0005) -0.0004 (0.0015) 0.0080 (0.0013) 0.0143 (0.0042) 0.0019 (0.0022) 0.0096 (0.0116) 0.9082 (0.1538) 0.8128 (0.3048) 0.8605 (0.2017) 0.8958 (0.2092)
contM 0.0006 (0.0006) -0.0004 (0.0016) 0.0080 (0.0014) 0.0144 (0.0042) 0.0023 (0.0030) 0.0125 (0.0144) 0.9176 (0.1496) 0.8178 (0.2939) 0.8677 (0.1885) 0.8964 (0.1922)

Table 6: Mean and standard deviation (in parenthesis) of parameter estimates and accuracy scores of 1024 simulations of length T =
250, 500, 1000, 2000 from the daily two-state HSMM estimated with HMM, and the discrete and continuous jump models with and without

Electronic copy available at: https://ssrn.com/abstract=4556048

mode loss (discrete, cont, and contM ). The bold values represent the best parameter estimates and highest accuracy scores.
mean return and low volatility, and negative mean return high volatility, respectively. Thus, we
interpret the first state as the bull regime and the second state as the bear regime.
We observe that the estimated mean returns and volatilities are similar for the three models.
However, the continuous jump model yields higher estimates of the transition probabilities com-
pared to the discrete model. This stems from the continuous model’s ability to accurately capture
smooth transitions in the estimated probabilities. We emphasize that the estimated transition
probabilities from all three models are significantly lower than those reported in previous articles
that consider broader equity indexes (Hardy, 2001; Nystrup et al., 2020b). This is attributed to
the composition of the Nasdaq index, which is predominantly comprised of technology companies.

µ0 µ1 σ0 σ1 γ01 γ10
discrete 0.1148% -0.1528% 1.2160% 2.6879% 0.0012 0.0025
cont 0.1029% -0.1879% 1.2891% 2.8616% 0.0021 0.0063
contM 0.0995% -0.1750% 1.2862% 2.8549% 0.0021 0.0062

Table 7: Estimated mean return and volatility under each of the two regimes and the transition
probabilities on the daily log-returns of the Nasdaq Index from 1996 to 2005.

Figure 2 displays the regimes estimated by the three jump models. Notably, the discrete jump
model identifies the dot-com crash as a single bear period, while the continuous jump models are
able to detect two rebound periods. The ability to capture rebounds is a result of the smooth
transition within the probability simplex in the continuous models. We underscore that the mode
loss penalty, when included, effectively smooths the probability curve, thereby eliminating minor
fluctuations with amplitudes lower than approximately 30%. This smoothing results in more per-
sistent state probabilities. In certain applications, maintaining unchanged probability estimation
can eliminate the necessity for portfolio rebalancing, ultimately leading to decreased turnover and
reduced transaction costs.

6.2 The 2013-2022 Time Period

Table 8 presents the estimated parameters for the three jump model from 2013 to 2022. The
estimated values of the transition probabilities provide support for that the continuous jump models
yield slightly higher estimates due to their smoother state transitions. Moreover, the inclusion of
the mode loss penalty appears to further enhance the persistence of the estimated probabilities.
Similarly to the 1996-2005, we interpret the first and second states as bull and bear regimes,
respectively.

µ0 µ1 σ0 σ1 γ01 γ10
discrete 0.0829% -0.1336% 0.9828% 2.3885% 0.0014 0.0051
cont 0.0903% -0.1383% 0.9531% 2.3065% 0.0029 0.0117
contM 0.0868% -0.1386% 0.9686% 2.3490% 0.0019 0.0077

Table 8: Estimated mean return and volatility under each of the two regimes and the transition
probabilities on the daily log-returns of the Nasdaq Index from 2013 to 2022.

Figure 3 depicts the behavior of the three jump models during the relatively calm market
period from 2013 to 2022. The discrete jump model successfully identifies the three major bear

28

Electronic copy available at: https://ssrn.com/abstract=4556048

Figure 2: Regimes (in yellow) of the Nasdaq Index (in blue) from 1996 to 2005 as estimated by the
three jump models.

29

Electronic copy available at: https://ssrn.com/abstract=4556048

market periods: in 2019, the 2020 COVID-19 pandemic, and the Fed’s interest rate hikes in 2022.
However, the continuous jump models also capture other market downturns during this period,
e.g. the drawdown around 2016. This ability to detect and respond to smaller market downturns
can aid in active risk management even in periods of relative market stability and highlights the
advantage of the continuous jump model in identifying varying market conditions.

7 Conclusions
In this article, we extended the statistical jump model of Nystrup et al. (2020b,a) by generalizing
the discrete hidden state variable into a probability vector over all regimes. The continuous jump
model, in contrast to its discrete counterpart, allows for smooth regime transitions and provides a
more robust framework for estimating regime probabilities. We provided a probabilistic interpre-
tation of the new model and demonstrated its advantages through simulations and real-world data
experiments. Specifically, our simulation studies and an application to the Nasdaq Index demon-
strate that the continuous jump model offers advantages over competing methods such as hidden
Markov models and discrete jump models. The advantages of the new approach become particu-
larly pronounced when dealing with relatively short, imbalanced, and highly persistent time series,
making it well-suited for a broad range of applications in finance such as regime-aware portfolio
construction and risk management.
It would be beneficial for future research to integrate continuous jump models into portfolio
strategies, allowing investors to align their allocations with regime-specific insights and performance
expectations.

Statements and Declarations

The authors declare that they have no conflict of interest.

30

Electronic copy available at: https://ssrn.com/abstract=4556048

Figure 3: Regimes (in yellow) of the Nasdaq Index (in blue) from 2013 to 2022 as estimated by the
three jump models.

31

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A Appendix: Proofs
Proof of Proposition 1. From the probabilistic assumptions, the joint likelihood is

p(Y , S|Θ) = p(Y |S, Θ)p(S) (33)

−1
TY −1
TY
= p(yt |Θ, st ) × π(s0 ) × K (st−1 , st ) . (34)
t=0 t=1

Thus
T
X −1 T
X −1
log p(Y , S|Θ) = log p(yt |Θ, st ) + log π(s0 ) + log K (st−1 , st ) . (35)
t=0 t=1

For the first term, by Jensen’s inequality we have

K−1
!
X
log p(yt |Θ, st ) = log stk p(yt |θk ) (36)
k=0
K−1
X
≥ stk log p(yt |θk ) . (37)
k=0

Inserting (37) into (35), we obtain

X T
X −1
log p(Y , S|Θ) ≥ stk log p(yt |θk ) + log π(s0 ) + log K (st−1 , st ) . (38)
t,k t=1

From equation (17), it follows that the above right hand side is precisely the negation of the
objective function J. Therefore minimizing J is equivalent to maximizing a lower bound of the
joint log-likelihood function.

Proof of Proposition 2. From the previous proof, we know that

T
X −1
log p(Y , S|Θ) = log p(yt |Θ, st ) + log p(S) (39)
t=0
X
≥ stk log p(yt |θk ) + log p(S) (40)
t,k

Inserting (18) into (40), we obtain

 
X X
st,k (−l(yt , θk ) − log ν) − L(S) − log η = −  st,k l(yt , θk ) + L(S) + const (41)
t,k t,k

= − J + const . (42)

Thus, minimizing J is equivalent to maximizing a lower bound of the log-likelihood p(Y , S|Θ).

32

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