Finance, Markets and Valuation

DOI:

10.46503/ZVBS2781

Corresponding author

Javier Oliver Muncharaz

Received: 2 Oct 2020

Revised: 15 Nov 2020

Accepted: 19 Dec 2020

Finance, Markets and

Valuation

ISSN 2530-3163.

Finance, Markets and Valuation Vol. 6, Num. 2 (Julio-Diciembre 2020), 137–148

Comparing classic time series models and the LSTM

recurrent neural network: An application to S&P 500

stocks

Comparativa de los models clásicos de series temporales

con la red neuronal recurrente LSTM: Una aplicación a las

acciones del S&P 500

Javier Oliver Muncharaz

Departamento de Economía y Ciencias Sociales, Universidad Politécnica de Valencia. Valencia,

España. Email: jaolmun@ade.upv.es

JEL: G12; G17; C45

Abstract

In the ﬁnancial literature, there is great interest in the prediction of stock prices. Stock prediction is nec-

essary for the creation of dierent investment strategies, both speculative and hedging ones. The ap-

plication of neural networks has involved a change in the creation of predictive models. In this paper,

we analyze the capacity of recurrent neural networks, in particular the long short-term recurrent neural

network (LSTM) as opposed to classic time series models such as the Exponential Smooth Time Series

(ETS) and the Arima model (ARIMA). These models have been estimated for 284 stocks from the S&P 500

stock market index, comparing the MAE obtained from their predictions. The results obtained conﬁrm a

signiﬁcant reduction in prediction errors when LSTM is applied. These results are consistent with other

similar studies applied to stocks included in other stock market indices, as well as other ﬁnancial assets

such as exchange rates.

Keywords: Recurrent Neural Network; Long short-term neural network; S&P 500; Arima

Resumen

En la literatura ﬁnanciera existe un gran interés por la predicción de precios bursátiles que es necesario

para la creación de diferentes estrategias de inversion, tanto especulativas como de cobertura. La apli-

cación de las redes neuronales ha supuesto un cambio en la creación de modelos de predicción. En este

trabajo se analiza la capacidad que tienen las redes neuronales recurrentes, en concreto la long short-

term recurrent neural network (LSTM) frente a modelos de series temporales clásicos como el Exponen-

tial Smooth Time Series (ETS) y el modelo Arima (ARIMA). Para ello se ha estimado dichos modelos para

284 acciones pertenecientes al índice bursátil S&P 500, comparando el MAE obtenido de sus prediccio-

nes, con el modelo LSTM. Los resultados obtenidos conﬁrman una reducción importante de los errores

de predicción. Estos resultados son coincidentes con otros estudios similares aplicados a acciones de

otros índices bursátiles así como a otros activos ﬁnancieros como los tipos de cambio.

Cómo citar este artículo: Oliver Muncharaz, J. (2020) Comparing classic time series models and the

LSTM recurrent neural network: An application to S&P 500 stocks. Finance, Markets and Valuation

6(2), pp. 137–148.

137

Finance, Markets and Valuation Vol. 6, Num. 2 (Julio-Diciembre 2020), 137–148

Keywords: Redes neuronales recurrentes; Long short-term neural network; S&P 500; Arima

1 Introduction

The prediction of stock returns has been widely studied in the ﬁnancial literature. One of the

main objectives is the construction of stock portfolios. In some cases, these portfolios are

constructed by applying dierent optimisation algorithms based on the classic Markowitz

model (García, González-Bueno, & Oliver, 2015). Some optimisation algorithms, such as the

NSGA-II, allow the construction of these portfolios taking into account more than the two classic

return-risk dimensions (W. Chen, Zhang, Mehlawat, & Jia, 2021; García, González-Bueno, Oliver,

& Tamoši

unien

e, 2019). Other algorithms, such as genetic algorithms together with heuristic

techniques allow good solutions to be found in an NP-hard problem (Ahn, Lee, Ryou, & Oh, 2020;

García, Guijarro, & Oliver, 2018). Other studies try to ﬁnd relations between sustainability and

stock market portfolios or indexes. Thus, for example, Arribas, Espinós-Vañó, García, and Oliver

(2019) analyse the composition and selection of responsible companies for the construction

of portfolios and investment funds, while Espinós-Vañó, García, and Oliver (2018) analyse the

so-called sustainable stock market indexes to determine whether they can be an investment

alternative to traditional stock market indexes.

Stock prediction is also applied to speculate with the price evolution of ﬁnancial assets.

Predictive models can be classiﬁed into linear and non-linear models. To the ﬁrst group belong

time series models such as autoregressive integrated moving average, exponential smooth

models and generalized autoregressive conditional heteroskedasticity, among others. Neural

networks are assigned to a second group. Given that the stock returns are not stationary and

present long-term dependence (Barkoulas & Baum, 1996), this second group of models have

shown to obtain more accuracy and a reduction of errors prediction with respect to linear

models.

In the last few decades, neuronal networks have evolved from the initial models of Hebb,

which proposed the ﬁrst rules of learning processes between neurons, Widrow and Ho with

their model Adaline, and Rosenblatt, which proposed the well-known perceptron. There are

many works related to the prediction of stock prices and their trends applying models based

on artiﬁcial neuronal networks. In Qiu and Song (2016), for example, the authors present the

analysis of the prediction or the direction of stock price index for the Nikkei 225 stock market

index. In this case, they use a backpropagation neural network with two dierent types of inputs

to determine what kind of information improves results. The result suggests that the network

is able to select those variables that are suitable for the model. Moghaddam, Moghaddam, and

Esfandyari (2016) apply the same type of network for the prediction of the Nasdaq stock index

using historical short term data as inputs (between four and nine days). In García, Guijarro,

Oliver, and Tamoši

unien

e (2018) the authors apply a neural network to predict the trend of the

German Dax-30 stock index.

In recent years, recurrent neural networks have been widely used for the analysis of time

series with high time dependency, such as stock returns. These types of networks are based

on Rumelhart’s work on error backpropagation (Rumelhart, Hinton, & Williams, 1986) and

Hopﬁeld’s networks (Hopﬁeld & Tank, 1985). Some works make a comparative analysis between

some already known neural networks against the recurrent neural networks as for example

in Saad, Prokhorov, and Wunsch (1998). The authors compare the time delay neural network

(TDNN), probabilistic neural network (PNN) and a recurrent neural network to predict short-term

stock trends. The conclusion is that the TDNN has the capability to dynamically incorporate past

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Finance, Markets and Valuation Vol. 6, Num. 2 (Julio-Diciembre 2020), 137–148

data to internal recurrence, and is the most powerful network among those analysed, although

it presents certain complexity regarding its implementation and memory requirement.

On the other hand, Yoshihara, Fujikawa, Seki, and Uehara (2014) compare RNN with another

machine learning methodology, the support vector machine (SVM) model. They analyse trend

market prediction on Nikkei companies. In this case, in addition to incorporating numerical

inputs into the models, information on dierent economic and ﬁnancial events is introduced

(news reported by newspapers). The advantage of deep learning is its ability to automatically

construct dierent features from data as well as pattern recognition. The results from RNN were

compared with SVM, and present the lowest error rate. This study shows that recurrent models

are more eective in capturing past events that are signiﬁcant with respect to long-term eects.

Other studies also conﬁrm these results, like the paper by Rather, Agarwal, and Sastry

(2015) who analyse 25 stock returns from Bombay stock exchange indicating that this model is

capable of capturing non-linear patterns more eiciently than classical models. In this case it is

concluded that the RNN learning process improves as it needs to look for smaller weights. da

Silva, Spatti, Flauzino, Liboni, and dos Reis Alves (2016) analyse stocks in the Bovespa index,

which is an alternative for decision making in the ﬁnancial stock markets. RNN has also been

applied to other types of ﬁnancial assets. Ye (2017) focuses on forecasting exchange rate using

gradient descent method or hidden layer in the process learning for recurrent neural networks.

Recurrence in neurons causes a speed up the weights update as well as convergence. This

conﬁrms the reliability and stability of neural networks.

There are dierent types of recurrent neural networks. The Long Short-Term Recurrent

Neural Networks are the most powerful time dynamic neural network (Staudemeyer & Roth-

stein Morris, 2019). This type of network has been applied in many areas. Among others, in

the text translation (Datta, David, Mittal, & Jain, 2020; Nowak, Taspinar, & Scherer, 2017), large

vocabulary speech recognition (Li & Wu, 2015), medicine diagnostic (Choi, Schuetz, Stewart,

& Sun, 2016; Gao, Zhang, Lu, & Wang, 2019), traic control in cities or its environmental im-

pact (Awan, Minerva, & Crespi, 2020). Also for forecasting economics and ﬁnancial time series

(Siami-Namini, Tavakoli, & Namin, 2018).

In Section 2, the well-known time series models, Arima and Exponential Smoothing, are

described. This is followed by a more in-depth description of Recurrent Neural Networks (RNN)

and the particular Long Short-Term Recurrent Neural Network (LSTM). Section 3 describes the

main results obtained in the work. Finally, Section 4 summarizes the main conclusions.

2 Data and Methods

In this section, the models used in this work are discussed. First, the well-known time series

models, Exponential smoothing and Autoregressive moving average model, are presented in

the summary form. Next, the recurrent neural network model is described in more detail, in

particular, the long-short term memory recurrent neural network model.

2.1 Exponential smoothing model

The exponential smoothing model was proposed at the end of the ‘50s (Brown, 1959; Holt, 2004;

Winters, 1960). In this type of model, past observations are weighted in a way that declines

exponentially the further back in time.

In other words, more recent observations are associated with a greater weight while older

observations have lower weights.

The model can be expressed as follow:

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ˆy

t +1

= α y

+ α (1 − α)y

t −1

+ α (1 − α)

t −2

+ . . . (1)

Where

α ∈ [0, 1]

is the smoothing parameter. Thus, the prediction at instant

t + 1

is a

weighted average of the observations of

. The degree of the weightings decrease is controlled

by the parameter α.

2.2 Autoregressive moving average model

The Autoregressive moving average model (ARMA) was introduced by Box, Jenkins, and Reinsel

(1970). It is one of the classic models that analyses time series, and one of the most used in

ﬁnancial literature. These models can become stationary by dierencing. Generally, in most

economic and ﬁnancial time series, a single dierentiation is enough to make the series station-

ary and to be able to apply ARIMA models where the “I” represents the level of dierentiation

(integration) of the series.

Having a time series

where

represents the time index, the

AR M A(p, q)

model is

expressed as:

= α

t −1

+ α

t −2

+ . . . + α

t −p

− θ



t −1

− θ



t −2

− . . . − θ



t −q

+ 

(2)

Where α and θ are estimated coeicients and  are white noise errors.

The

AR M A(p, q)

model is built as the combination of two processes. The ﬁrst is the au-

toregressive process (AR), which tries to predict the variable using a linear combination of past

values of this variable. An autoregressive model of order

, represents the number of lagged

variable. On the other hand, the moving average (MA) part gives a prediction of the variable

from a moving average model on past prediction errors. The order

of this process represents

the number of delays over the prediction errors used in the model.

2.3 Recurrent neural network: LSTM network

Recurrent neural network (RNN) is a network that has backward connections between neurons,

which are generally referred to as global recurrent networks. This type of model presents some

stability problems in the training process, so it requires complex learning algorithms as well

as increasing the training time. Local networks models are global feedforward networks. In

this case, a structure of dynamic models of neurons is designed to build a feedback network, in

which the connections between these neuron models are strictly feedforward as in the case of

Multilayer Perceptron (MLP).

Figure 1 shows an example of the connections between the dierent layers and neurons in a

recurrent neural network. It can be seen how the output obtained in one layer serves as an input

for the neurons of layers located in a previous process. Each recurring unit formed by dierent

neurons computes at each time step an output

which is time dependent on the current

process. In the next time step, the neuron receives a new input vector

and additionally

incorporates the output obtained previously into this vector (

t −1

). The latter is called the

recurrent input. In this way, the neuron computes the output vector from the input vector (

)

and the recurrent input (

t −1

) using an activation function

. This activation function can be

of the linear, sigmoidal or tanh type, although the last type of function is oen used for time

series problems.

= θ

(

W × x

, U × y

t −1

)

(3)

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Figure 1. Recurrent Neural Network representation

Source: Ciaburro and Venkateswaran (2017)

Where

and

are weight matrix that multiply the input vector and the recurrent input

vector.

The simplest recurrent neural network, called "vanilla", presents as recurrent inputs only

a single output obtained in a previous time step. When the net uses the previous outputs as

a new input, the net can remember learned previous data. This process is important for the

learning long short-dependencies.

The importance of inputs and recurrent inputs in the net depends on their corresponding

weight matrix. During the learning process, the net adjusts the weights to improve the pre-

diction, taking into account the calculated error (bacpropagation process). However, while

in a feed forward network the backpropagation process goes back through the hidden layers,

for recurrent neural network it is also necessary to adjust the weights of previous time steps

(time adjusting). In this type of network, if the sequence is long, there may be a problem in

the learning process, since with each prediction the whole way backwards must be covered

again. To avoid this, a split of the dierent sequences is made. In this way, the backpropagation

process should only go backwards the length of the subsequence. But in this case, the neural

network is only able to determine short dependencies. This is the so-called vanishing gradient

problem, in which the further back the sequence is regressed the less important it is in the

current prediction, and therefore cannot adequately capture long term dependencies.

The Long Short-Term Neural Network (LSTM) is a more complex kind of recurrent neural

network as it is able to capture long-term dependencies. This kind of neural network was

proposed by Hochreiter and Schmidhuber (1997) as an evolution of simple RNN. This network

can propagate activations over long periods to process dierent sequences that include long

distance dependencies (Kelleher, 2019). This network solves the vanishing gradient problem. In

this case, the recurrent unit is modiﬁed in blocks, which works like a normal recurrent unit, to

which an additional cell and several gates are added. The gates control the ﬂow of information

within the recurrent unit (block). In this way it is determined which information is more revealing

to improve the prediction and which is not at each of the time steps. These gates (

) are deﬁned

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Type of gate Role

Forget Eliminate neuron

Update Importance of the past

Relevance Drop previous information

Output Which information is used

Table 1. Role Gates

Source: Author’ elaboration

as follow:

τ = σ

(

W × x

+ U × y

t −1

+ b

)

(4)

Where

and

are gate speciﬁc coeicients and

is the common used sigmoid function

(while in the input activation function use a tanh function). In the LSTMrecurrent neural network,

four gates are used, each with a dierent function. Table 1 summarizes the role function that

one of them has.

The forget gate is used to erase a neuron or not, and therefore, forget the information. On

the other hand, the update gate indicates what is the past to be taken into account now. The

relevance gate deﬁnes what information from the past is relevant to incorporate as input to

the neuron. Finally, the output gate selects the information that is useful for the neuron in the

actual prediction. Each of these four gates uses dierent weight matrices and are calculated

individually during the learning process. In short, these gates control the ﬂow of information in

each neuron so that it is useful in predicting at each time step.

3 Results

In this work, it is intended to conﬁrm the eiciency of a LSTM Neural Network, as opposed to

some classic models applied to time series. In this case, it is going to be compared with an

Exponential Smooth Time Series model and an ARIMA model. For this purpose, each of them

has been applied to a sample of 284 stocks from the S&P 500 index with daily data from the last

20 years. The sample has been divided into 70% for the estimation and training processes and

30% for its validation.

For the Exponential Smooth Time Series and ARIMA model, the number of dierentiations

needed to obtain stationary time series, which is required in this type of models, has been

taken into account. As with many economic time series (McCabe & Tremayne, 1995), only one

integration was necessary to achieve stationarity.

For the Exponential Smooth Time Series model, the AIC (Akaike) crystals have been used to

select the appropriate delays in each case. In the case of the ARIMA models, Phillips Perron’s

criteria has been used to determine the delays of both the autoregressive part and the moving

averages.

In the case of the LSTM Neural Network model, the series has been standardised, both for

training and for testing. There are many works that verify that standardisation improves the

learning process in neural networks such as Lachtermacher and Fuller (1995); Shen, Zhang, Lu,

Xu, and Xiao (2020); Zhang, Patuwo, and Hu (1998).

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Finance, Markets and Valuation Vol. 6, Num. 2 (Julio-Diciembre 2020), 137–148

Figure 2. Example LSTM parameters for 3M stock

Source: Author’ elaboration

Figure 2 shows the type of network applied to each of the actions. As already indicated, it is

a recurrent long short-term neural network. In this case, the processes the sequence of vectors

using a LSTM layer of input data. This model presents the dierent layers in a sequential way.

As the model needs to know in the ﬁrst layer the type of input it should expect, (since in the

rest of layers it is inferred automatically), the number of samples per gradient updates is one.

That is, the number of batch size of the inputs for the layer is one. The dimension of the output

space of this ﬁrst layer is ﬁve. This parameter is subject to tuning. The output layer is full-fully

connected (dense layer), it has been conﬁgured with a batch size of one and one unit.

The three models have been estimated for each stock with the corresponding appropriate

delays according to the criteria already indicated. For the evaluation of the eiciency of each

model, the Mean Absolute Error (MAE) has been calculated on the predicted observations.

Figure 3 shows a boxplot with the errors of each model for the 3M action. In the 284 stocks

analysed, two important issues have been observed. Firstly, the Exponential Smooth Time

Series model and the ARIMA model present similar MAE. For example, in the case of 3M the MAE

obtained in the ﬁrst model was 0.8847, while for the ARIMA model it was 0.8857. It is possible

that by applying other exponential models such as Holt-Winters’ that could relatively improve

the ARIMA model (Maria & Dezsi, 2011). Secondly, even in the case of using other exponential

models that improve the ARIMA model, they are far removed from the results obtained from

MAE for the LSTM model. For example, for 3M the MAE obtained was 0.1823, that is, 79% less

error than the other models.

Table 2 describes the main statistics on the distribution of the MAE obtained in the total

number of actions analysed and for each of the models. On the one hand, it can be seen that

the classic time series models (ETS and ARIMA) present a higher MAE for all the quantiles of

the sample in comparison with the LSTM model, as it was already advanced in the previous

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Finance, Markets and Valuation Vol. 6, Num. 2 (Julio-Diciembre 2020), 137–148

Figure 3. MAE comparison model for 3M stock

Source: Author’ elaboration

example. These results are consistent with other studies such as Siami-Namini et al. (2018)

that compare these three models for various stock market indexes such as Nasdaq, Nikkei,

Hang Seng with monthly data. The results suggest that the LSTM model obtains, on average, a

reduction in prediction error of between 84 and 87 percent. Baughman, Haas, Wolski, Foster,

and Chard (2018) compare the ARIMA model with the LSTM for Amazon stock, obtaining an

error reduction of 95%.

However, four stocks have been detected in which the MAE obtained by the LSTM model

is superior to any of the other two models (Table 3). Each of these stocks has been analyzed

in detail to detect if this result is due to some kind of error in the sample. In all four cases,

there are a suicient number of observations (several thousand). Neither have any missing

or anomalous data been detected. Likewise, the quoted prices have been visually contrasted

without apparently detecting any errors. The ﬁrst two shares, LDOS and IRM, are listed on the

NYSE, while CTSH and CHTR are listed on the NASDAQ. It can therefore be concluded that the

resultsobtained for these four shares are plausible. However, the LSTM model has outperformed

the classical time series models in 98.59% of the sample analysed, so recurrent neural networks

are a good alternative for time series prediction in general, and for stocks and stock indices in

particular. Abdoli (2020) analyses the Tehran Stock Exchange conﬁrming the results obtained in

the present work, where the LSTM outperforms ARIMA model, in terms of error of accuracy.

4 Conclusions

In this work, the eiciency of the Long short-termn recurrent neural network has been analysed

in comparison with other time series models. The main conclusion that can be drawn is that

there is a large reduction in the prediction error of more than 85%, which is in line with previous

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Finance, Markets and Valuation Vol. 6, Num. 2 (Julio-Diciembre 2020), 137–148

ETS ARIMA LSTM

Min. 0.0626 0.1441 0.0009

1st Qu. 0.3842 0.3866 0.0565

Median 0.5678 0.5766 0.1289

Mean 0.8177 0.8320 0.2199

3rd Qu. 0.8374 0.8397 0.2708

Max. 9.6813 9.7285 3.1888

Sd. 1.0555 1.0693 0.3000

Table 2. Descriptive Statistics

Source: Author’ elaboration

TICKER ETS ARIMA LSTM

LDOS 0.6080 0.6118 0.9766

IRM 0.2709 0.2701 0.5742

CTSH 0.3843 0.3853 0.9812

CHTR 2.6014 2.6120 3.1888

Table 3. Descriptive Statistics

Source: Author’ elaboration

results from other studies on other ﬁnancial assets. Recurrent neural networks in general,

and the LSTM in particular, may be an alternative to consider in the creation of stock price

prediction models. However, to conﬁrm these results, this analysis should be extended to other

aspects such as the application of a larger number of fully connected intermediate layers or the

application of tuning of other network parameters.

On the other hand, other authors have proposed other types of neural networks that seem to

oer very eicient alternatives, as well. In M, E.A., Menon, and K.P. (2018) the authors compare

several linear time series models (ARIMA) with non-linear models such as ARCH, GARCH and

Neural Networks. In this case, they apply two types of recurrent neural networks, one LSTM

model and the other Convolutional Neural Network. This network is applied to ﬁve stocks of

the National Stock Exchange (NSE) of India. The results suggest that the Convolutional Neural

Network outperforming the other models, even against the LSTM model. In the same line, the

works of Y. Chen, Wei, and Huang (2018)applying a Convolutional model to the prediction of the

stock market in Mainland China incorporating related corporations’ information to create more

accuracy in predictions are presented. In addition, other works propose the use of a hybrid

model between the Convolutional Neural Network and the recurrent neuronal network LSTM

(Kim & Kim, 2019).

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