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Forecasting Earnings of Firm's Listed in ASE Using ARIMA
Model
Omar Alsinglawi1, S. AL Wadi1, Mohammad Al Adwan1 and Bassam Bouqaleh2
1Faculty of Management and Finance, The University of Jordan, Jordan
2Golden Meadow Agriculture Co. LLC.
O.alsinglawi@ju.edu.jo
Sadam_alwadi@yahoo.co.uk
Msm_Adwan@ju.edu.jo
bassambouqaleh@yahoo.com
Abstract.
The study aims to estimate and forecast the earnings of the firm's listed in Amman Stock exchange
(ASE), using a time series data of earning per share(EPS) for the period from 1978 till 2016, the
data has been extracted from firms' annual reports, we deployed a wavelet transform (WT) model
to decomposes the data to detect the fluctuations and outliers values, then parameters p, d, and q
are estimated using the ARIMA the accuracy criteria which are MSE and RSME. The results
showed that the ARIMA models as has lowest MASE and RMSE values, Thus, the forecasting
accuracy is high. It is concluded that firms' earnings show slow increasing trend for upcoming 38
financial years.
Key words: Forecasting, WT, ARIMA, Earning per share
INTRODUCTION
Earning naturally refer to after-tax net income, sometimes called the bottom line, or a profit, it's
the main factor of a firm's share price, it indicates whether the business will be profitable and
successful in the long run or not, so it measured a firm performance which is formed based on the
accrual basis of accounting. Earning is an important figure that its used extensively as measure of
firm performance by users of financial information like investors, management, creditors,
customers and all other stakeholders to make decision for future actions, therefore the need to
forecast firms’ future performance is of high deliberation for them.
The main goal of financial reporting is to provide useful information to make informed financial
decisions. Thus, the evaluation of accounting information in forecasting future profitability has
been always studied and presented in financial accounting studies (Oskouei, & Zadeh, 2017).
The aim of this research is to forecast the future firm's earning based on past earnings' pattern by
using Earnings per Share (EPS) as a single figure that summarize the firm's' performance, EPS is
generally considered as the most important factor to determine the share price and the firms' value
(Islam,et,al.,2014). This study is supposed to offer a valuable understanding for the firm's expected
financial performance. Therefore, a more accurate forcasting would be helpful for decision makers
to make a data-driven decision.
Financial data generally are a type of time series data that often show trending and seasonality that
presenting a challenge to develop an effective forecasting model. The issue of how effectively
modeling financial time series data and how to increase the superiority of forecasting is still
unresolved in literatures. Plenty of previous studies used a diverse modeling in forecasting like;
Autoregressive Integrated Moving Average (ARIMA) models is considered as most widely used
approaches in time series forecasting, RIMA model aims to describe the autocorrelations in the
data. The ARIMA framework to forecasting is formerly developed by (Box,et,al.,2015).
The research question addressed in this study is: “Can the use of time series of Earnings based on
Earning per share (EPS) would help to forecast the future direction and movement of firms'
earnings?”
The problem under study in this paper is forecasting the future earnings of firms listed in Amman
Stock Exchange (ASE), therefore we need to achieve the following objectives:
To know the earning's trend of firms listed in ASE.
To select the best ARIMA model firms listed in ASE.
To Forecast earning's trend for firms listed in ASE.
To develop ARIMA model for predicting future earnings for firms listed in ASE.
This paper is organized as follows. Section 2 describes the literature review. Section 3 introduces
the methodology and framework. Sections 4 describes the data descriptions and analytical results.
Finally, Section 5presents our conclusions.
Literature review:
Many investors depend on earnings to understand firms' performance to make their investment
decisions, the basic measurement of earnings is earnings per share (EPS). This metric is considered
as a summary indicator for performance (Schroeder, et, al,2009).
Most of the time the earnings forecasts are based on analysts' prospecting of firms' growth and
profitability; to forecast earnings, stock analysts shape financial models that estimate prospective
revenues and costs, and incorporate other factors like economic growth, currencies and other
macroeconomic factors that influence firms' growth (Ramezani, et, al.,2002). The analysts use
market research reports to get a sense of firms' growth trends. To understand the dynamics of the
firms under consideration.
Time Series forecasting:
The time series is defined as data points indexed in time order, it is a series taken at consecutive
similarly spaced points in time. Therefore, it is ordered and the requirement to maintain this
ordering imposes certain restrictions on its processing. Furthermore, the time series data usually
ordered by factors such as distance but naturally the time factor is the main ordering encounter.
Thus, some groups are referred to as time series. Time series analysis has widely applications in
many fields like; Sales, accounting, economic and stock market forecasting. Time series analysis
is used to recognize the relationship between the attributes of current value to that of its previous
or later values (Mills, et, al.,2011).based on this for financial data like earnings we are interested
in identifying previous values to see how current values get affected, also we are interested to
forecast the future values (Steel,2014).ARIMA model is widely used to analyze time series data
and to understand the impact of past values in forecasting future values. Worthful to say It was
rarely to find in literatures studies that used ARIMA model to forecast future earning based on
time data series of earning per share (EPS), most research work focused on forecasting stock
market indices, like (Junior.et.al.,2014) who assessed the performance of ARIMA model for time
series forecasting of IBOVESPA., they concluded that the ARIMA model can be used for financial
time series data forecasting. A study by (Uko & Nkoro,2012) that analytically compared the
influence of ECM, VAR, and ARIMA models in forecasting inflation of Nigeria, it revealed that
ARIMA is a superior forecaster of inflation at Nigeria.
(Devi, et,al.,2013) has used ARIMA model with its parameters to forecast the NSE – Nifty
Midcap50 companies among them top 4 companies the results were implemented using criterions
like AIC & BIC.(Paul,et,al.,2013) empirically found that the best ARIMA (2,1,2) model for
forecasting based on AIC, SIC AME, RMSE, MAPE in content of SPL data series.(Jaya&
Sundar,2012) used ARIMA model for 19 IT firms and analyzed the market capitalization of the
firms. Authors found that firms are categorized into three trends i.e. companies on an upward,
linear and downward trends. (Shrimal & Prasad,2016) They found the best ARIMA model for
predicting market capitalization using some of mathematical criteria such as RMSE and MAPE,
they applied this model on 21companies.
Why Earning per share (EPS)?
The term earnings per share (EPS) summarize a firm's earnings that is a net income after taxes and
preferred stock dividends. The EPS is evaluated by dividing net income earned in a specific
annually reporting period, by the entire number of shares remaining during the same period, EPS
is most important variable that affect the share's price, it is a key driver of share prices, it is a main
component to compute the price-earnings valuation ratio. EPS is used as a indicator to capture a
firm's profitability per each unit of shareholder ownership called per share., furthermore
profitability can be assessed by prior earnings, current earnings or future projected earnings,
therefore earning per share is widely considered to be the most popular method of quantifying a
firm's profitability, EPS is generally viewed as a more accurate measure and is more commonly
cited (Besley & Brigham,2007).
Methodology:
This section consists of the research framework, the mathematical model that is used to achieve
the purpose of this research and mathematical criteria used.
RESEARCH FRAMEWORK:
FIGURE 1. The flowchart of the paper
ARIMA MODEL:
An Auto regressive (AR) process is a series depends on its lagged values. The AR(p)model is a
regression model which defined as :
Yt = α0 + α1Yt-1 + α2Yt-2+… +αpYt-p
Moving average (MA) model is related if the AR process is not the only mechanism that generates
Y , but it contains past values with its error terms. MA (q) process is defined:
εt=β1εt + β2εt-2 + β3εt-3 + … +βqεt-q
which contains the white noise errors. When Y has both the features of AR and MA, it is called as
ARMA (p, q) process. (Gujarathi, et, al.,2012) ARIMA (Box-Jenkins model) is to classify and
estimate a statistical model which can be explained as having generated the sample data. Since the
financial time series data are type of non-stationary, therefore differencing the series will yield a
stationary time series.
If the financial data becomes stationary when differenced d times, we name the series as I(d).
Consequently, if ARMA(p,q) is applied to a series financial data which is I(d), then the original
time series is ARIMA(p, d, q). The ARIMA methodology proposed that finding the values of p
Financial Time Series Data
forecasting and decomposing
DWT model
Explaining the main feature and co-movement
in the transformed data
ARIMA model
implimentaing forecasting accuracy by ARIMA.
Selection the best model for ARIMA models and discussing the benifets decomposition and
forecasting for the invertors
and q for AR and MA respectively by referring to the correlogram. In MA (q) model, moving
average of order q, ACF Dies Down or Cuts off after lag q while for AR (p), autoregressive of
order p PACF Dies Down or Cuts off after lag p. (Princeton,2008). Model diagnosis can be applied
based on the values of Root mean Square Error (RMSE) and Mean Absolute percent Error
(MAPE).
Wavelet Transform Formula
WT is a mathematical model employed to convert the original observations into a time-scale
domain. The model is very appropriate with the non-stationary data since most of the financial
data are non-stationary. WT can be divided into Discrete Wavelet transform (DWT) and
continuous wavelet transform (CWT). DWT consists of many functions such as Haar, Daubechies,
Maximum overlapping Wavelet transform (MODWT) and others. All of these functions have the
same properties with different applications. In this article, the WT will be presented with its
equation for all functions. For more details please refer to (Daubechies,1992; Chiann &
Morettin,1998; Gençay, et, al., 2002; Al Wadi, 2010)
Wavelets theory is based on Fourier analysis, which represents any function as the sum of the sine
and cosine functions. A wavelet is simply a function of time t that obeys a basic rule, known as
the wavelet admissibility condition ( Gençay, et, al., 2002):
(2)
where is the Fourier transform and a function of frequency f, of. The WT is a
mathematical tool that can be applied to numerous applications, such as image analysis and signal
processing. It was introduced to solve problems associated with the Fourier transform, when
dealing with non-stationary signals, or signals that are localized in time, space, or frequency.
There are two types of wavelets within a given function/family. Father wavelets describe the
smooth and low-frequency parts of a signal, and mother wavelets describe the detailed and high-
frequency components. Equation (3) represents the father wavelet and mother wavelet
respectively, with j=1,2,3,..., J in the J-level wavelet decomposition( Gençay, et, al., 2002):
(3)
where J denotes the maximum scale sustainable by the number of data points and the two types
of wavelets stated above, namely father wavelets and mother wavelets and satisfies:
(4)
time series data, i.e., function f(t), is an input represented by wavelet analysis, and can be built
up as a sequence of projections onto father and mother wavelets indexed by both {k}, k = {0, 1,
2,...} and by{S}=2j, {j=1,2,3,. . .J}.
Analyzing real discretely sampled data requires creating a lattice for making calculations.
Mathematically, it is convenient to use a dyadic expansion, as shown in equation (4). The
expansion coefficients are given by the projections:
(5)
The orthogonal wavelet series approximation to f (t) is defined by:
(6)
and
(7)
The WT is used to calculate the coefficient of the wavelet series approximation in Eq. (6) for a
discrete signal, where and are introducing the smooth and details coefficients
respectively. The smooth coefficients dives the most important features of the data set and the
details coefficients are used to detect the main features in the dataset. For more details about the
WT and its functions please refer to(Wadi,2015;Al-Khazaleh et,al.,2015). When the data pattern
is very rough, the wavelet process is repeatedly applied. The aim of preprocessing is to minimize
the Root Mean Squared Error (RMSE) between the signal before and after transformation. The
noise in the original data can thus be removed. Importantly, the adaptive noise in the training
pattern may reduce the risk of over fitting in training phase. Thus, we adopt WT twice for the
preprocessing of training data in this study.
Accuracy Criteria
This section consists of two subsections. Firstly, we will present the criteria which have been
used to make a fair comparison, and then the framework comparison will be presented with
more details. The researchers have been adopted to compare the performance of the models
within three types of accuracy criteria which are Mean square error (MSE), Root mean squared
error (RMSE) and Mean absolute error (MAE). For more details about the mathematical
model refer to (Aggarwal,et,al.,2008;Wadia,et,al.,2011).
RESULTS
Data Description
In this research, the statistical population includes firms listed in ASE over the time period of
1978-2016,a complete 38 years of time series data for 266 firms listed firms in ASE; because of
variations in the number of listed companies from a year to year, we have used average earning
per share(EPS) as a proxy to the firms' earning, EPS is considered as summary indicator that
explain the firms' performance in term of profitability(Schroeder,et,al,2009).the time series data
were accessed from the ASE and extracted from firms' annual financial reports. The MATLAB
and MINTAB software were used to analyze the data.
Table(1). Data analysis matrix.
Research hypothesis
Independent
Variable
Dependent Variable
Statistics
Modeling and
forecasting EPS
Time
EPS
Histogram, accumulated histogram and
descriptive statistics, WT, ARIMA
TABLE 2. Data Description EPS
Histogram, accumulated histogram and descriptive statistics of the time series are shown in table
(2), which shows a linear path with positive slope; therefore, it is none stationary homogenous
type, characterized by constant changes from one period to another.
Decomposing time series
A time series generally has three components that are a trend, noise and seasonal components.
Decomposition of the time series means separating original time series into these components:
First-Trend: The increasing or decreasing values in any time series.
Second-Seasonal: The repeating cycle over a specific period in any time series.
Third-Noise: The random of values in any time series.
Figure (1) shows the decomposition of based on WT. The decomposition consists of a1, which is
the approximated coefficients used for the proper forecasting and d1 that show the fluctuations of
data. Mathematically, the equation can be represented as S= a1+d1 where S is the original data.
EPS 1978-2016 using WT
Refer to the figure (1) a visual inspection of the time plots shows that EPS data time series has a
trend of random fluctuations this means the data are non-stationary and it is not constant around
mean and variance. This type of non-stationary time series data contains a seasonal trend can be
carried out by spectral Analysis function which is WT. Yield, random trend can be transformed
into a linear trend. Before conducting additional analysis using ARIMA model then the data has
to be discussed with its behaviors. Therefore, d1 is used to clarify the main features and
fluctuations of the time series data, it has been clear that there was much fluctuation we descript
that there is many events that ASE faced during the period under study which can be summarized
as follow:
Observation
Year of
occurrence
Event that affect the EPS
4
1981
Iraq-Iranian Ware, which reflected in excellent firms earning due to
increase firms' exports to Iraq.
11
1988
Releasing ties with Palestine Authority reflected on shrinking in
liquidity in the economy which in turn weaken the firms earning
13
1990
Iraq Kuwait conflict which reflected on enhancing demand on firms'
products and services and support earnings
17
1994
pull back in demand on products and services slowdown in earning,
because of sanctions on Iraq
28
2005
Enhanced liquidity in the economy because of foreign investment from
Gulf investors, Markets get over heated and firms benefited from
Market momentum
31
2008
International financial crisis
34
2011
Syrian conflict
After we have done the decomposing process of time series data, we applied the ARIMA
forecasting process,
TABLE 3..
ARIMA
Average EPS. Jordanian
Dinar
Model
(2,2,1)
MASE
0.7089
RMSE
0.0709
The fitted ARIMA models were diagnosed using MASE and RMSE. Parameter estimation for the
ARIMA models was done using the Gaussian MLE criterion. The ARIMA models fitted based on
the lowest value of MASE (0.7089)and RMSE(0.0709),with a fit ARIMA is ARIMA(2,2,1).
According to the fitted ARIMA model respectively, the best model can be re-written as follows:
Insert The model
Where; represents the value of EPS.
Figure(3)
Figure (3) shows the EPS original time data series from (1978 till 2016) and the forecasted data of
the EPS for the coming forty (years 2017 till 2066), this suggests that the EPS long term trend is
up ward slopping, which gives an indication that ASE firms' long-term profitability is growing.
Furthermore, the EPS as experienced high fluctuation in the past 38 years, in the year of 1991, we
are expecting the EPS value will reach its high past level in the year of 2044.
.
CONCLUSION
In this paper we deployed a WT model to decomposes the data to detect the fluctuation and outlier
values, then we utilized ARIMA model in forecasting firms' future earnings using earning per
share (EPS) time series of firms' listed ASE of the years 1978 to 2016. it is clear that ARIMA
model offers an excellent technique for forecasting any variable like EPS. It is strength lies in its
fitting varieties of different types of time series with any pattern of change. In the process of model
building, the original data is found none stationary then converted to be stationary. An ARIMA
(2,2,1) model is developed for analyzing and forecasting EPS for ASE firms among all of various
tentative ARIMA models as it has lowest BIC values. From the results, it can be observed that
influence R Square value is (95%) high and Mean Absolute Percentage Error is very small for the
fitted model. Therefore, the forecasting accuracy is high. It is concluded that firms' earnings show
slow fluctuations and increasing trend for upcoming seventy-six financial years.
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