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American Journal of  Applied 
Statistics and Economics (AJASE)

Novel Neural Network State Switching Models for Returns Predicting with Regime 
Switching: A Monte Carlo’s Simulation Approach

Oluwasegun Agbailu Adejumo1*, Omorogbe Joseph Asemota1, Samuel Olayemi Olanrewaju1

Volume 4 Issue 1, Year 2025
ISSN: 2992-927X (Online)

DOI: https://doi.org/10.54536/ajase.v4i1.5514
https://journals.e-palli.com/home/index.php/ajase

Article Information ABSTRACT

Received: June 10, 2025

Accepted: July 17, 2025

Published: September 05, 2025

This study proposes new neural network state switching models that will improve the regime 
predictive performance of  nonlinear time series returns, in small sample size contexts. In 
this study, we developed four new neural network state switching models namely Recurrent 
Neural Network State Switching Model (RNNSSM), Generalized Regression Neural 
Network State Switching Model (GRNNSSM), Radial Basis Function Network State 
Switching Model (RBFSSM) and Multilayer Perceptrons State Switching Model (MLPSSM). 
The study presents comparative results of  the estimations of  the novel models alongside 
the traditional Markov state switching model as well as the regime prediction performances 
of  the models using the Akaike Information Criterion (AIC), Log-likelihood and prediction 
accuracy measures under a Monte Carlo simulation study. Evidence from the models’ 
estimation results particularly the AIC and Log-likelihood statistics established the novel 
Multilayer Perceptrons State Switching Model (MLPSSM) as the parsimonious model for 
the simulated returns at small sample sizes. Results from the models’ regime predictions 
evaluation, precisely the RMSE and MAE, evidently affirmed the novel MLPSSM model 
as superior in its ability to predict or forecast market returns, particularly the bull and bear 
regimes, at small sample sizes. Therefore, this study concludes that the novel MLPSSM is 
the best-fitted model for market returns at small sample sizes with excellent ability of  market 
returns’ regimes/states (i.e., bull and bear) prediction. This study recommends adoption of  
the novel Multilayer Perceptrons State Switching Model in modelling and regime predictions 
of  time series returns.

Keywords

Markov Process, MLP, Neural 
Network, Regime Switching, 
Returns

1 Department of  Statistics, Faculty of  Science, University of  Abuja, Abuja, Nigeria
* Corresponding author’s e-mail: agbailuoa@gmail.com

INTRODUCTION 
Forecasting returns is essential in financial modeling and 
decision-making. As a measure of  volatility in time series, 
returns exemplify a stochastic process that indicates 
how much variables fluctuate over time. Accurate return 
prediction is vital for assessing risks in key economic 
activities such as value at risk, asset pricing, and exchange 
rate management (Liao et al., 2020; Adejumo et al., 2020; 
Roy & Sarkar, 2024). In finance, returns exhibit three 
important features: clustering property (Cont, 2007), 
asymmetry (Nelson, 1992), and nonlinearity (Maheu & 
McCurdy, 2002). Researchers often encounter difficulties 
when modeling and interpreting modern time series data 
from diverse fields because traditional assumptions—such 
as linearity, normality, and stationarity—are frequently 
inadequate. The origins of  returns modeling trace back 
to Engle (1982), and Bollerslev (1986), who introduced 
discrete-time GARCH and stochastic return processes 
for autoregressive conditional heteroskedasticity. 
However, as noted by Hamilton (1989), Nguyen et al. 
(2014), Aliyu and Wambai (2018), Adejumo et al. (2020), 
Xiuqin et al. (2023), Al-Sulaiman (2024) and others, 
standard GARCH and stochastic models cannot fully 
capture all key features of  financial markets. This results 
in complex nonlinear dynamics and irregular regime 
shifts. A promising approach to address these challenges 
is switching state modeling. Before switching state 
modeling was used in returns forecasting, the Hidden 

Markov Model (HMM) was the primary approach. The 
HMM is a bivariate discrete-time process, denoted as 
{St,Yt}(t≥0), where {St} is an underlying Markov chain, and 
{Yt} is a sequence of  independent random variables. The 
conditional distribution of  Yt depends only on St. Since 
St is hidden, only the stochastic process Yt is observable. 
In other words, the process’s state is not directly visible, 
but the output, which depends on the state, is observable. 
Therefore, all statistical inference must be based solely 
on Yt, as St cannot be directly observed (Rydén 2008). 
An HMM has a distinctive dependence structure, which 
is useful when analyzing financial time series. To better 
understand this dependence, it is illustrated here (Figure 
1) with a graphical model.

Figure 1: The HMM’s Dependence Structure

According to Figure 1, the distribution of  a variable S(t+1) 
conditional on the history of  the process S0,S1,…,St, is 
determined only by the value of  the preceding variable 
St. This is all according to the Markov property, where 
future events are completely independent of  the past, 



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depending only on the present state. In addition, the 
distribution of  Yt conditionally on the past observations 
Y0,Y1,…,Y(t-1) and the past values of  the state, S0,S1,…
,St, is determined by St only (Rydén 2008). Putting 
this into mathematical terms, we get f(S(t+1)│St,S(t-

1),…,S1)=f(S(t+1)│St) and f(Yt│S(t-1),S(t-2),…S1)=f(Yt|St.
Cconsequently, Markov State Switching Model (MSSM) 
was introduced by Hamilton to address the hidden states 
weakness of  the HMM (Hamilton, 2005). To describe the 
MSSM, the number of  states (or regimes) is assumed N, 
i.e. St Ω={1,…,N}. This implies that; for instance the log 
returns of  financial time series are drawn from N distinct 
normal distributions, depending on what state the HMM 
is currently in. This resolve to below models:
Y(t=1)=μ1+ϵt; where ϵt~(N, σ2

1) for state 1
Y(t=2)=μ2+ϵt; where ϵt~(N,σ2

2) for state 2
⋮       (1)
Y(t=N)=μN+ϵt; where ϵt~(N,σ2

N) for state N
By implication, when the state of  the HMM for time t is 
1, then the expectation of  the dependent variable is μ1 
with variance of  innovations σ2

1, similarly when the state 
of  the HMM for time t is 2, then the expectation of  the 
dependent variable is μ2 with variance of  innovations σ2

2 
and so on. Since the underlying Markov chain is hidden 
one cannot observe what state the HMM is in directly, but 
only deduce its operation through the observed behaviour 
of  Yt. In order to attain the probability law governing the 
observed data Yt a probabilistic model of  what causes 
the change from state St=i to state St=j is required. This 
can be specified using the transition probabilities of  an 
N state HMM; ρ(i,j)=Pr (St=j│S(t-1)=i) i,j∈Ω={1,2,…,N}. 
The transition probability i.e. ρ(i,j)=Pr (St=j│S(t-1)=i)=Pr 
(St=j} S(t-1)=i,S(t-2)=k,…,S1=l) is dependent of  the past 
only through the value of  the most recent state. This is 
one of  the central points of  the structure of  a Markov 
regime switching model, i.e. the switching of  the states 
of  the underlying HMM is a stochastic process itself. 
The state-switching modelling approach has proven to be 
more reliable in this aspect in that it models all observed 
salient features of  returns. The approach has documented 
the distinctiveness and forecasting capabilities of  regime 
switching models against the commonly used GARCH 
models. Thus, as a result its application has widely 
increased over time.
Furthermore, Machine Learning (ML) techniques also 
possess high capabilities in modelling all observed salient 
features of  financial markets’ returns. ML methods 
have been used in many other fields for many years. 
For example, Chen et al. (2019) exploit ML method in 
the estimation of  stochastic discount factor and Gu et al. 
(2020) showed superior performance of  ML models for 
empirical asset pricing. Also, is Christensen et al. (2023) 
employed ML in returns forecasting of  index stocks. 
Similarly, Nelson et al. (2017), Shah et al. (2018), Yao et 
al. (2018), and Sunny et al. (2020) studied stock market 
returns based on the machine learning model. ML models 
especially the long-short term memory (LSTM) and 
recurrent neural network (RNN) types have been famous 

for most time series data. The machine learning models 
are well-known for predicting the time series data without 
considering the much assumptions of  the parameters. 
However, despite both approaches significant values to 
returns and regime forecasting, they both suffer from 
either a severe modelling misspecification or a lack of  
effective identification of  meaningful stochastic regimes 
especially for small sample sizes. One way to handle 
these problems is the switching state space modelling 
approach. Thus, there is a need for a well-designed 
modelling approach that allows the disturbance to be 
realistically represented, and at the same time does not 
lead to over frequently switching. Given the criticality 
of  such regime identification, not only for its economic 
implications but also for the profound comprehension of  
underlying phenomena, this study aims to propose new 
neural network state switching models that will improve 
the regime predictive performance of  nonlinear time 
series returns, in small sample size contexts. The specific 
objectives include to:

ⅰ Develop novel Neural Network State Switching 
Models (using Markov algorithms) for time series returns 
predicting in distinct regimes; 

ⅰⅰ Examine the estimations of  the Novel Neural 
Network State Switching Models towards modelling time 
series market returns;

ⅰⅰⅰ Compare the predicting performances of  the novel 
Neural Network State Switching Models using prediction 
accuracy measures under a Monte Carlo simulation study.

LITERATURE REVIEW
Regime-switching model is an unusual case of  a more 
general framework called hidden Markov Model 
(Zucchini & MacDonald, 2009). Regime-switching 
models were early presented to econometric literature 
by Hamilton (1989) and have become very prevalent 
particularly in applied works. Applications of  regime 
switching models range over a broad range of  research 
areas, such as modeling shifts in inflation, exchange 
rates and interest rates (Piger 2013), Altug and Bildirici 
(2010), changes in government policy (Valente, 2003; 
Owyang & Ramey, 2004; Sims & Zha 2006) and shifts 
in exchange rate (Bekaert & Hodrick, 1992; Bollen et 
al., 2000). The regime-switching modelling approach 
provides a completely new approach to the modeling of  
financial returns which it conceives as a multiplicative, 
hierarchically structured process (Frommel et al., 2005).
Over the years, there have been several extensions to 
the state switching modelling by introducing nonlinear 
structures such include Chow and Zhang (2013), 
Johnson et al. (2024), Dong et al. (2020), and Farnoosh 
et al. (2021). The regime switching state space model 
was proposed by Chow and Zhang (2013), the model 
adopts a pre-specified nonlinear transition function. The 
Johnson et al. (2024) model called SVAE, parameterizes 
the emission function by neural networks, while the 
transition function remains linear. The Dong et al. 
(2020) switching non-linear dynamical systems (SNLDS) 



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extension model parameterizes both the emission and 
transition functions with nonlinear neural networks. The 
deep switching autoregressive factorization (DSARF) 
proposed by Farnoosh et al. (2021), approximates high-
dimensional time series with a multiplication of  latent 
factors and latent weights, where the latent weights are 
modeled by a nonlinear autoregressive model, switched 
by a Markov chain of  discrete latent variables. Most of  
the above-mentioned studies assumed that the discrete 
latent variables only influence the transition of  state zt. 
The discrete switching variables in the SLDS are assumed 
to be Markov, i.e. dt depends on dt-1 only. The recurrent 
SLDS (rSLDS) proposed in Linderman et al. (2017) and 
Becker-Ehmck et al. (2020) extends the open-loop Markov 
dynamics and makes dt depending on the hidden state zt-1. 
In Nassar et al. (2018), a tree structure prior is imposed 
on the switching variables of  rSLDS, where the dynamics 
of  the switching variables behave similarly in the same 
sub-trees. The deep Rao-Blackwellised Particle Filter 
proposed in Kurle et al. (2020) also allow dt to depend 
on zt-1. The SNLDS model Dong et al. (2020) extended 
the open-loop Markov dynamics by making dt depends 
on last observations. Such recurrent structures serve as a 
presence of  disturbance to the switching dynamics.
Recently, Mari and Mari (2023) introduced a regime-
switching model to analyze how market prices change 
over time. Their model features a mean-reverting 
diffusion process for the basic regime, while the 
alternate regime is driven by predictions from a deep 
neural network trained on market log-returns. To 
estimate the model using market data, they proposed a 
statistical method based on simulated moments. Xiuqin 
et al. (2023) developed the deep switching state space 
model (DSSSM), a new framework designed to tackle 
challenges in modeling, inference, and understanding 
stochastic processes. The DSSSM aims to deliver accurate 
forecasts and detect hidden regimes that carry significant 
economic implications and deepen understanding of  
market dynamics. It employs discrete latent variables 
for regimes and continuous ones for random influences, 
combining an RNN with a nonlinear switching state 
space model to capture nonlinear dependencies and 
regime shifts driven by a Markov process. Xiuqin et al. 
demonstrated DSSSM’s effectiveness through forecasting 
tests on various simulated and real datasets across sectors 
such as healthcare, economics, traffic, meteorology, and 
energy. Later, Antulov-Fantulin et al. (2024) proposed a 
new method for regime detection using a deep learning 
architecture called the gated recurrent straight-through 
unit (GRSTU). Their extensive simulations showed that 
the GRSTU outperformed traditional statistical jump 
models, especially in regime classification on smaller 
datasets, while performing comparably on larger datasets.
From the above, it is clear that many studies have focused 
on modeling nonlinear time series to address issues like 
significant modeling misspecification or difficulty in 
identifying meaningful stochastic regimes. As a result, 

the field of  deep learning, especially recurrent neural 
networks with gate structures such as the Long-Short 
Term Memory (LSTM), Gated Recurrent Unit (GRU), 
Transformers, and temporal convolution networks, 
has become the new standard for modeling complex 
nonlinear dependencies. However, the small size of  
real-world data samples and, more importantly, the 
stochastic nature of  regime switching, make traditional 
deep learning methods computationally challenging. In 
other words, these methods require large sample sizes 
for reliable estimation, which is often unrealistic because 
many disciplines do not have large volumes of  time series 
data. In light of  these challenges, Dong et al. (2020), 
Farnoosh et al. (2021), and Xiuqin et al. (2023) developed 
Switching Non-Linear Dynamical Systems (SNLDS), 
Deep Switching Autoregressive Factorization (DSARF), 
and Deep State Switching Model (DSSM), respectively, to 
better address issues of  misspecification and the difficulty 
in identifying meaningful stochastic regimes in nonlinear 
time series. Nonetheless, computing the feasibility of  
these models, especially with small sample sizes, remains 
a challenge. Therefore, this research aims to propose new 
neural network-based regime switching models that can 
improve regime prediction for nonlinear time series data, 
such as returns, particularly when working with small 
sample sizes.

MATERIALS AND METHODS
The Novel Neural Network State Switching Models 
To improve the regime prediction performance of  
famous state switching model described in equation 
(1) for nonlinear time series data, especially returns, 
in small sample size contexts, this study integrates 
deep neural networks these include; Recurrent Neural 
Network (RNN), Radial Basis Function Network (RBF), 
Generalized Regression Neural Network (GRNN) and 
Multilayer Perceptrons (MLP) with Markov two-State 
Switching modelling technique. The proposed neural 
network state switching modeling approach combines 
the strengths of  deep neural networks (i.e. efficient in 
high complexity dataset) and Markov two-state switching 
capabilities of  ensuring both interpretability and 
predictability of  two financial state of  market returns (i.e. 
bull and bear states). The models are developed in two 
phases.

1st Phase: Generative Network using RNN, GRNN, 
RBF and MLP
Given time series dataset of  Yt, the generative network 
procedure for the dataset include the following: 

ⅰ Define of  training and testing dataset of  Yt i.e. by 
setting training dataset at time step tk, and testing dataset 
at time step t(t-k)

ⅰⅰ At time step t_k, either RNN, GRNN, RBF or MLP 
is used to process the input data (i.e. training dataset) such 
as ht~fh (yt ) where f_h is the function of  RNN, GRNN, 
RBF or MLP.



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2nd Phase: Integration of  Generative Network ht into 
the Markov Two-State Switching Model 
Subsequently, the generative network ht is integrated 
in the Markov two-state switching model alongside the 
defined training dataset to develop Recurrent Neural 
Network State Switching Model (RNNSSM), Generalized 
Regression Neural Network State Switching Model 
(GRNNSSM), Radial Basis Function Network State 
Switching Model (RBFSSM) and Multilayer Perceptrons 
State Switching Model (MLPSSM). This would give us the 
following model to work with:
Y(tk~ht)=gt=c(st)+Β1(g(t-1)-c(st-1))+Β2 (g(t-2)-c(st-2)) +ϵt (2)
where
c(st)=c0 S0t+c1 S1t+c2 S2t 
σ2

(st)=σ2
1 S1t+σ2

2 S2t
ϵt~i.i.d(0,σ2

(st))
c(st) is the state dependent mean, σ2

(st) is state dependent 
variance and the coefficients are Β1or Β2; which could be 
different for different subsamples. The proposal will be 
to model the state St as the outcome of  an unobserved 
two-state Markov chain with St independent of  ϵt for all t.
The transitions of  the St, are presumed to be ergodic and 
intricate first order Markov-process. This means impacts 
of  earlier observation(s) for the gt and state(s) is/are 
completely captured in the recent gt state(s) observations 
as represented in (3); 
ρij=Prob(St=j⁄S(t-1) =i) ∀i, j=1, 2∑2

(i=1) ρij =1             (3)
Matrix P captures the probability of  switching which is 
known as a transition matrix;
       
                 (4)

The second element of  the numerator is simply the 
previously mentioned initial probability of  the Markov 
chain, i.e. Pr (S1=i)=πi, and it will henceforth be denoted 
as such. One can at this point notice that α(i,1) is the 
normalized value of  the product between the initial 
probability and the conditional probability function f(Y1 
|S1 =i), and can therefore be written as follows:

where P11+P21=1, and P12+P22=1 
The nearer the probability ρij is to one the longer it takes 
to shift to the next regime.
Consider the model given by equation (2), i.e. a Markov 
regime-switching model with 2-regimes. The estimation 
will be performed using Hamilton’s filter, where the 
main idea is to calculate each state’s filter probabilities by 
making inferences on each state’s unknown probabilities 
based on the available information. When the filter 
probabilities are obtained, we have the probabilities 
one needs for calculating the log likelihood of  the 
model. Subsequently, the estimation of  the model Filter 
Probabilities are discussed.
The model’s filter probabilities are calculated by utilizing 
the model’s iterative relations by means of  recursion. This 
can be done using a combination of  the relation between 
observations and hidden states, and the endogenous 
relation between hidden states. Begin from the starting 
value in our recursion, i.e. with the probability of  being in 
state i at time t = 1:

Now, assume that we know the filter probability at time 
t-1, namely α(i,t-1). Following the same strategy as for t=1 
leads to the following recursion:

Data Source: Simulation Setups
To demonstrate the main idea behind the developed neural 
network state switching models towards enhancement of  
the prediction level of  nonlinear time series data such 
as market returns particularly in small sample sizes, we 
consider a two states (regimes) model demonstrating the 
Bull and Bear regimes of  daily market returns. 
Y(t=1)=μ1+ϵt; where ϵt~(N,σ2

1)   for state 1
Y(t=2)=μ2+ϵt; where ϵt~(N,σ2

2)   for state 2

Parameters Settings
μ1= 0.01; is the mean returns of  bull regime, 
μ2= -0.02; is the mean returns of  bear regime, 
σ1= 0.05; is the standard deviation of  bull regime of  the 
market returns,
σ2= 0.1; is the standard deviation of  bear regime of  the 
market returns,
matrix(c(0.9,0.1,0.2,0.8)= is the transition matrix for the 
aforementioned regimes,
t = n is categories of  small sample sizes sets at 30, 50, 
70, and 100.
Number of  replication r = 1000times.

Models’ Estimation Procedure and Prediction 
Performance Evaluation
Prior to the estimation of  proposed neural network state 
switching models (i.e. RNNSSM, GRNNSSM, RBFSSM 
and MLPSSM) against the traditional Markov State 
Switching Model (SSM), the simulated market returns (Yt) 
are tested for nonlinear modelling suitability. Nonlinear 
models are employed where the financial system suggests 
nonlinearity in the system (Adejumo et al., 2020; Mendy 
& Widodo, 2018). We utilized the most widely used tests 
known as BDS test by Brock, Dechert and Scheinkman. 
Also, the simulated returns were tested for stationarity 
and presence of  volatility using Augmented Dickey-
Fuller (ADF) test and ARCH test respectively.
Also, this study utilized the frequently used model 
selection principle - Akaike Information Criterion (AIC). 
AIC = T ln(residual sum of  squares) + 2n, where T is 



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the numeral of  operational observations, while n is 
the number of  parameters to be evaluated. The log-
likelihood of  the fitted model was also utilized. Hence, 
the outperform model is one with the smallest AIC value 
and highest log-likelihood. 
Frequent error measures are available for model prediction 
evaluation; we evaluate the prediction ability of  the proposed 
models against the Markov State Switching Model by means 
of  several performance measures such as Root Mean Square 
Error (RMSE) and Mean Absolute Error (MAE).

 

RESULTS AND DISCUSSION
Simulated Data Presentation and Preliminary 
Assessment
This section presents and discusses the summary 
statistics and some time series features of  simulated 
market returns data. Figure 2 and Fig 3 reveal the time 
series trend plots and the regimes plots of  the simulated 
daily market returns. Explicitly, Fig 2 depicts the time 
series and regimes plots of  simulated daily market returns 
of  sample sizes 30 and 50 while Fig 3 shows the time 
series and regimes plots of  simulated daily market returns 
of  sample sizes 70 and 100. As observed, all the sample 
sizes’ time series trend plots depict relative stationarity i.e. 
constant means, variances and autocorrelation however 
relative to some external factors or trend. Also, the time 
series plots depict clustering feature within the plots 
indicating presence of  volatility.

where At is the actual value in time t, and Ft is the prediction 
value in time t. The models’ performances would be assessed 
using the earlier described Monte Carlos Simulation Study 
under four categories of  small sample sizes.

Figure 2: Time Series Plots of  Simulated Daily Market Returns of  Sample Sizes 30 and 50

Figure 3: Time Series Plots of  Simulated Daily Market Returns of  Sample Sizes 70 and 100



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Furthermore, Table 1 depicts the summary statistics, 
stationarity test, non-linearity test and volatility tests of  the 
simulated market returns data over four (4) categories of  
small sample sizes. According to the table, the simulated 
market returns returned average returns of  -0.0085±0.06, 
0.0092±0.06, 0.0047±0.077 and 0.0153±0.06 for sample 
size 30, 50, 70 and 100 respectively. Based on the aforesaid, 
it can be observed that the simulated data depicts a relative 
uniform variation (i.e. standard deviation at around 0.06, 
which is close to induced 0.05) of  the simulated returns 
across the sample sizes. Also, consequent to the observed 
features in the time series plots, the simulated data were 
tested for stationarity, non-linearity and presence of  
volatility. 

According to the ADF test results, all the returns series as 
expected are stationary at level. As observed in the BDS 
test results, the p-values depict significant at 0.05 level 
suggesting the non-acceptance of  linearity assumption 
for the simulated returns series across all the sample sizes. 
Similarly, the ARCH test p-values results reveal significant 
at 0.05 level suggesting the non-acceptance of  assumption 
of  no presence of  volatility in the simulated returns series 
across all the sample sizes. Based on the foregoing, is quite 
evident that the simulated returns at all the sample sizes 
significantly exhibit non-linearity and volatility features. 
Hence, the simulated returns data necessitate non-linear 
and volatility models such as the State Switching Model 
and novel neural network state switching models.

Table 1: Summary Statistics and Preliminary Tests of  the Simulated Data
 30 50 70 100
Summary Statistics
Mean -0.0085 0.0092 0.0047 0.0153
Median -0.0150 0.0053 0.0063 0.0174
Min. -0.1203 -0.1470 -0.1935 -0.1567
Max. 0.1387 0.1349 0.2717 0.1699
Std. Dev. 0.0604 0.06199 0.0772 0.0618
Preliminary Tests
ADF Test I(0) 0.00* I(0) 0.00* I(0) 0.00* I(0) 0.00*
BDS Test 0.0302* 0.0310* 0.0386* 0.0309*
ARCH Test 0.0068* 0.0272* 3.21e-06* 4.70e-06*

Note: * denotes significant at 0.05 level
Source: Researchers’ Compilations from R-Output

Models Estimations
This section presents and discusses the estimations of  
the novel models and the traditional SSM. The models 
were fitted for training dataset that consisted of  80% 
of  the simulated returns at different sample sizes. The 
diagnosis of  the goodness of  fit of  the estimated models 
for the return series depicts, the Q–statistics p-values 
greater than 0.05, indicating that there is no statistically 
significant trace of  dependency or autocorrelation left in 
the squared standardized residuals, indicating that all the 

estimated models are adequately specified. Table 2 present 
the summary of  models’ estimations for the simulated 
market returns at a sample size of  30. According to Table 
2, among the estimated models the novel MLPSSM and 
RBFSSM returned with the least AIC of  -314.6393 and 
-115.7039, respectively as well as higher log-likelihood 
of  161.3196 and 61.8519 respectively. Thus, MLPSSM 
returns to be the most parsimonious model among the 
estimated SSMs.

Table 2: Summary of  Models Estimations at 30 Sample Size
 SSM RNNSSM RBFSSM GRNNSSM MLPSSM 
Regime 1
 

Intercept 0.0461* -0.4385 589.59* 0.0087 -1437.70 
Training_Network - rnn= 0.7062* rbf=-1.18e+07* grnn=1.1443 mlp=1443.53*

Regime 2
 

Intercept -0.0626* 0.2849 -922.40* 0.0754 -1443.05 
Training_Network - rnn=-0.4932 rbf=1.84e+7* grnn=0.6202 mlp=1448.90*

Transition 
Prob.

State 1 State 2 0.3484 0.6578 0.3461 0.3138 0.3423 0.6648 0.000 0.7007 0.9468 0.2294 
State 2 State 1 0.6516 0.3422 0.6539 0.6862 0.6577 0.3352 1.000 0.2992 0.0532 0.7705

AIC -64.31498 -59.7527 -115.7039 -56.3426 -314.6393
logLik 34.1575 33.8764 61.8519 32.1713 161.3196

Note: * denotes significant at 0.05 level
Source: Researchers’ Compilations from R-Output



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In addition, Table 2 depicts the transition probabilities 
for the two identified states (namely bull and bear). 
The transition probabilities for MLPSSM model show 
that there is a high probability that the market returns’ 
system remains in the same state hence implying limited 
switches in the state or regime. The results also indicate 
that the MLPSSM has an 95% probability of  staying in 
the bull state and a 5% probability of  switching to the 

bear state. When the system is in a bear state, it has a 
77% probability of  remaining in bear state and lower 
probability of  23% to switch to bull state. The transition 
probability results highlighted shows that only extreme/
great events can switch the returns between states i.e. 
state 1(bull phase) to state 2 (bear phase), (Figure 4). It 
further indicates that not any of  the state is lasting since 
all transition probabilities are below one.

Figure 4: Models’ Transition Probabilities of  Returns at Sample Size 30

Furthermore, Table 3 depicts the summary of  models’ 
estimations for the simulated market returns at sample size 
of  50. The table reveals MLPSSM and RBFSSM returned 
with the least AIC values of  -543.7162 and -190.5718, 
respectively as well as higher log-likelihood values of  
275.8581 and 99.2859 respectively. Similarly, the results 
returned MLPSSM with the lowest AIC and highest log-
likelihood values as the most parsimonious model among 
the estimated SSMs at sample size 50. Additionally, Table 
3 illustrates the transition probabilities between the two 
identified states: bull and bear. 
The transition probabilities for the MLPSSM model 
suggest a low likelihood of  the market return system 

remaining stable, indicating frequent transitions between 
bull and bear states. Specifically, the results show that 
the MLPSSM has a 32% chance of  remaining in the bull 
state and a 68% chance of  transitioning to the bear state. 
In a bear state, there is a 38% probability of  staying in 
that state, while the chance of  switching to the bull state 
is lower at 62%. The highlighted transition probability 
results indicate that minor events can trigger changes in 
returns between states, such as from state 1 (bull phase) 
to state 2 (bear phase) or the other way around (see Figure 
5). This suggests that neither state is permanent since all 
transition probabilities are less than one.



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Table 3: Summary of  Models Estimations at 50 Sample Size
 SSM RNNSSM RBFSSM GRNNSSM MLPSSM 
Regime 1
 

Intercept 0.0032 0.0851 -505.86* 0.0342 -4218.13* 
Training_Network  - rnn= -0.1698 rbf= 2.02e+07* grnn=-0.1689 mlp=4231.02*

Regime 2
 

Intercept -0.0036 -0.3103 -555.89* 0.0295 -4167.55* 
Training_Network  - rnn=0.6503 rbf= 2.22e+07* grnn=-1.4262 mlp=-4180.3*

Transition 
Prob.

State 1 State 2 0.7489 0.2135 0.5193 0.9275 0.7376 0.7779 0.522 0.9997 0.3202 0.3813 
State 2 State 1 0.2511 0.7865 0.4807 0.0725 0.2624 0.2221 0.478 0.0003 0.6798 0.6187

AIC -110.943 -108.51 -190.5718 -98.6980 -543.7162
logLik 57.4715 58.2549 99.2859  53.3490 275.8581

Note: * denotes significant at 0.05 level
Source: Researchers’ Compilations from R-Output

Figure 5: Models’ Transition Probabilities of  Returns at Sample Size 50

Moreover, Table 4 shows the summary of  models’ 
estimations for the simulated market returns at sample 
size of  70. The table reveals MLPSSM and RBFSSM 
returned with the least AIC values of  -659.263 and 
-320.2873 respectively as well as higher log-likelihood 
values of  333.6315 and 164.1436 respectively. Similarly, 
the results returned MLPSSM with lowest AIC and 
highest log-likelihood as the most parsimonious model 
among the estimated SSMs at sample size 70. 

Subsequently, Table 4 depicts the transition probabilities 
for the two identified states (namely bull and bear). The 
transition probabilities for MLPSSM model show that 
there is a high probability that the market returns’ system 
remains in the same state hence implying low or limited 
switches in the bull state and bear state. Explicitly, the 
results indicate that the MLPSSM has an 86% probability 
of  staying in the bull state and a 14% probability of  
switching to the bear state. When the system is in a bear 



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state, it has a 65% probability of  remaining in bear state 
and lower probability of  35% to switch to bull state. The 
transition probability results highlighted shows that only 
extreme/great events can switch the returns between 

states i.e. state 1(bull phase) to state 2 (bear phase) or 
vice versa, (see Figure 6). These results therefore, indicate 
that no any of  the state is lasting since all transition 
probabilities are below one.

Table 4: Summary of  Models Estimations at 70 Sample Size
 SSM RNNSSM RBFSSM GRNNSSM MLPSSM 
Regime 1
 

Intercept 0.0330* 0.0509 4.32+02* 0.1226* -12577.37* 
Training_Network - rnn= -0.0493 rbf= -1.7e+07* grnn=-6.6674 mlp=12597.9

Regime 2
 

Intercept -0.1010* 0.0469* 3.36e+02* 0.0117 -12094.81 
Training_Network - rnn=-0.3404* rbf= -1.34e+07* grnn=-4.0712 mlp=12114.5

Transition 
Prob.

State 1 State 2 0.8561 0.6539 0.9024 0.7165 0.5097 0.1724 0.0079 0.5004 0.8610 0.6472 
State 2 State 1 0.1439 0.3461 0.0976 0.2835 0.4903 0.8276 0.9921 0.4996 0.1390 0.3528

AIC -140.2999 -153.0316 -320.2873 -133.6245  -659.263
logLik 72.1499 80.51579 164.1436  70.8122  333.6315

Note: * denotes significant at 0.05 level
Source: Researchers’ Compilations from R-Output

Figure 6: Models’ Transition Probabilities of  Returns at Sample Size 70



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To conclude, Table 5 presents the summary of  models’ 
estimations for the simulated market returns at sample 
size of  100. Table 5 depicts MLPSSM and RBFSSM 
returned with the least AIC values of  -594.7593 and 
-387.7917 respectively as well as higher log-likelihood 

values of  301.3796 and 197.8958 respectively. Similarly, 
the results returned MLPSSM with lowest AIC and 
highest log-likelihood as the most parsimonious model 
among the estimated SSMs at sample size 100.

Table 5: Summary of  Models Estimations at 100 Sample Size
 SSM RNNSSM RBFSSM GRNNSSM MLPSSM 
Regime 1
 

Intercept 0.0395* -0.1989 6.13+02* -0.2097* -114443.95* 
Training_Network  - rnn= 0.6103* rbf= -2.5e+07* grnn=3.6865* mlp=114617.88

Regime 2
 

Intercept -0.0100 0.1085* -8.82e+02*  0.0395 -94142.78 
Training_Network  - rnn=-0.2464 rbf= 3.53e+07*  grnn=-0.1143 mlp=9428.86

Transition 
Prob.

State 1 State 2 0.5108 0.7283 0.3991 0.3916 0.6214 0.7046 1.4e-07 0.1696 0.5705 0.5839 
State 2 State 1 0.4892 0.2717 0.6009 0.6084 03786 0.2954 0.9999 0.8304 0.4295 0.4160

AIC -223.3181 -226.2772 -387.7917 -218.4201 -594.7593
logLik 113.659 117.1386 197.8958 113.2101 301.3796

Note: * denotes significant at 0.05 level
Source: Researchers’ Compilations from R-Output

Figure 7: Models’ Transition Probabilities of  Returns at Sample Size 100

Subsequently Table 5 depicts the transition probabilities 
for the two identified states (namely bull and bear). The 
transition probabilities for MLPSSM model show that 
there is a fair probability that the market returns’ system 
remains in the same state hence implying almost equal 
chances of  state switching. Explicitly, the results indicate 
that the MLPSSM has an 57% probability of  staying in 
the bull state and a 43% probability of  switching to the 

bear state. When the system is in a bear state, it has a 58% 
probability of  remaining in bear state and lower probability 
of  42% to switch to bull state. The transition probability 
results highlighted shows that either weak or extreme/
great events can switch the returns between states i.e. state 
1(bull phase) to state 2 (bear phase) or vice versa, (Figure 
7). These results also, indicate that no any of  the state is 
lasting since all transition probabilities are less than one.



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Consequently, Figures 8 and 9 present line plots for 
each of  the models’ AIC and Log-likelihood estimates, 
respectively. As observed from the figures, two out of  
four introduced novel models i.e. RBFSSM and MLPSSM 
outperformed the traditional SSM and the rest introduced 
models by returning with lower AIC and higher log-
likelihood estimates across all four small sample sizes 
examined. Furthermore, according to Figure 8, the novel 
MLPSSM returned the least AIC estimates across all four 
small sample sizes examined. Similarly, as observed in 
Figure 9, the novel MLPSSM returned the highest Log-
likelihood estimates across all four small sample sizes 
examined. Also, it is important to note that the AIC 
estimate inclines to increase and Log-likelihood estimate 
tends to decline when the sample size is 100. This result 
suggests that the novel MLPSSM is best fitted for sample 
sizes less than 100. Thus, the novel MLPSSM returned 
as the most parsimonious model among the estimated 
SSMs. Evidence from the AIC and log-likelihood line 

plots established Multilayer Perceptrons State Switching 
Model (MLPSSM) as the best-fitted model for the 
simulated returns at all the considered small sample sizes.

Evaluation of  Models’ Regime Prediction Performances
This section presents and discusses the performance 
of  the fitted models for the training dataset. Each of  
the model performances was assessed based on in-
sample predictions of  the testing dataset at 20% of  each 
sample size, i.e., 30, 50, 70, and 100. Root Mean Square 
Error (RMSE) and Mean Absolute Error (MAE) were 
employed to assess the prediction accuracy errors of  
the models. Figure 10 depicts the models’ prediction 
performances results. According to the figure, the novel 
MLPSSM returned the least RMSE and MAE for the in-
sample predictions across all four sample sizes examined. 
Therefore, our introduced novel model, i.e., MLPSSM, 
returned as the optimal model for regime predictions of  
the stimulated returns.

Figure 8: Models’ AIC Estimates across the Sample Sizes

Figure 9: Models’ Loglikelihood Estimates across the Sample Sizes

Figure 10: In-Sample Prediction Performances of  the Fitted Models 



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Furthermore, Table 6 presents the MLPSSM in-sample 
daily regime mean predictions of  20% of  the simulated 
market returns sample sizes. The table reveals a stable or 
static regime means across all the sample sizes considered. 
Explicitly, for sample sizes 30 and 50, the novel MLPSSM 
predicted respectively a bear regime mean of  -0.00746 and 
-0.00068 across the days predicted at each scenario. The 
predicted bear regime in sample size 30 and 50 returned 
high transition probability (i.e. 0.77 and 0.62) to bull 
regime. This implies the predicted bear regimes in sample 
size 30 and 50 has little probability to remain in the state, 
i.e., weak events can switch the bear regime to bull regime. 
On the other hand, for sample sizes 70 and 100, the novel 
MLPSSM predicted respectively a bull regime mean of  
+0.007425 and +0.01948 across the days predicted at 
each scenario. The predicted bull regime in sample size 
70 returned a very low transition probability (i.e. 0.14) to 
bear regime while the predicted bull regime in sample size 
100 returned a relatively moderate transition probability 
(i.e. 0.43) to bear regime. This implies the predicted bull 
regimes in sample size 70 has a great probability to remain 
in the state, i.e., only extreme events can switch the bull 
regime to bear regime. Meanwhile, for sample size 100, 
the result implies that the predicted bull regimes has a 
relatively fair or moderate probability of  remaining in the 
bull state, i.e. not too weak or too extreme events can 
switch the bull regime to bear regime.

Table 6: MLPSSM 20% In-Sample Daily Regime Mean Predictions of  Small Sample Sizes Considered
30 
P_12=0.23,P_22=0.77

50 
P_12=0.38,P_22=0.62

70 
P_11=0.86,P_21=0.14

100 
P_11=0.57,P_21=0.43

1 -0.00746 -0.00068 +0.007425 +0.01948
2 -0.00746 -0.00068 +0.007425 +0.01948
3 -0.00746 -0.00068 +0.007425 +0.01948
4 -0.00746 -0.00068 +0.007425 +0.01948
5 -0.00746 -0.00068 +0.007425 +0.01948
6 -0.00746 -0.00068 +0.007425 +0.01948
7 -0.00068 +0.007425 +0.01948
8 -0.00068 +0.007425 +0.01948
9 -0.00068 +0.007425 +0.01948
10 -0.00068 +0.007425 +0.01948
11 +0.007425 +0.01948
12 +0.007425 +0.01948
13 +0.007425 +0.01948
14 +0.007425 +0.01948
15 +0.01948
16 +0.01948
17 +0.01948
18 +0.01948
19 +0.01948
20 +0.01948

Note: + and - denote Bull and Bear Regime respectively
Source: Researchers’ Compilations from R-Output

CONCLUSIONS
This paper centers on introducing novel Neural Network 
State Switching Models using the combination of  
strengths of  deep learning neural networks and traditional 
Markov state switching approaches for regime predictions 
of  time series returns. The paper shows the comparative 
results of  the estimations of  the novel models as well as 
the regime prediction performances of  the novel models 
using prediction accuracy measures under a Monte Carlo 
simulation study. 
Evidence from the models’ estimation results particularly 
the AIC and Log-likelihood statistics established the 
Novel Multilayer Perceptrons State Switching Model 
(MLPSSM) as the best fitted model (against the traditional 
Markov state switching models and other introduced 
neural network state switching models) for the simulated 
returns at small sample sizes especially for sample sizes 
lower than 100. 
Moreover, results from the models’ regime predictions 
evaluation precisely the RMSE and MAE evidently 
affirmed the novel MLPSSM model as superior in its 
ability to predict or forecast market returns particularly 
the bull and bear regimes at small sample sizes. Base on 
the aforementioned, this study therefore concludes that 
the novel MLPSSM is best fitted model for market returns 
at small sample sizes with excellent ability of  market 
returns’ regimes/states (i.e. bull and bear) prediction.



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In view of  the aforementioned, this study recommends 
adoption of  the novel Multilayer Perceptrons State 
Switching Model in modelling and regime predictions of  
time series returns. 

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