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ELK ASIA PACIFIC JOURNAL OF MARKETING AND RETAIL MANAGEMENT
ISSN 2349-2317 (Online); DOI: 10.16962/EAPJMRM/issn. 2349-2317/2015; Volume 8 Issue 1 (2017)
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ARIMA/GARCH (1,1) MODELLING AND FORECASTING FOR A GE STOCK
PRICE USING R
Varun Malik
Dyal Singh College
University of Delhi India
varunmalikphy@gmail.com
ABSTRACT
This article attempts to present a basic method of time series analysis, modelling and forecasting
performance of ARIMA, GARCH (1,1) and mixed ARIMA - GARCH (1,1) models using historical daily
close price downloaded through the yahoo finance website from the NASDAQ stock exchange for GE
company (USA) during the period of 2001 to 2014. This paper also presents a brief analysis technique
introduction to R to build up graphing, simulating and computing skills to enable one to see models in
economics in a unified way. The great advantage of R compiler is that it is free, extremely flexible and
extensible. It uses data that can be downloaded from the internet, and which is also available in different
R packages. This article provides discuses in modeling and forecasting briefly and simply. This paper
provides short details the R command lines and output. This article is written to be useful for learning time
series analysis on basic different levels as well as a research purpose for beginners who beginning the
analysis of time series data in the various scientific and statistical research approaches. ARIMA/GARCH
(1,1) model is applied to observed the forecasting values of low and high stock price (in USD) for GE
company. The results obtained in this paper are based on the work of [10].
Keywords: ARIMA/GARCH models, time series models, forecasting, R
INTRODUCTION
In time series are analyzed to understand
behavior of the past data points and to
predict the future values on the basis of
past values, enabling analysis's or
decision makers to make properly
informed decisions for others. A time
series analysis quantifies the main
observation findings from data and the
random variable. This reason, combined
with improved computing, technical and
statistical ideas, have made time series
methods widely applicable in scientific
and statistical research approach in
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ISSN 2349-2317 (Online); DOI: 10.16962/EAPJMRM/issn. 2349-2317/2015; Volume 8 Issue 1 (2017)
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governments and private sectors. In most
branches of scientific and statistical
research department in private and
government sectors, there are variables
measured approaches sequentially in
time. Financial/Banking sector record
interest rates as well as exchange rates
each day. The government statistics
department compute and analyze the
country’s GDP on a yearly basis and
other economic data. The weather
department publishes day to day
temperatures and air velocity diagram for
capital cities and in rural areas from
around the world. Meteorological
department record weather parameters at
many different sites with different
instrument such as Weather radar and
optical rain gauge meter and etc. When
such variable is measured sequentially in
time over or at a regular interval, known
as the sampling interval, the resulting
data come from a time series.
Observations that have been collected
over regular sampling intervals from a
historical time series. In this paper, we
give a basic but useful computational and
statistical approach in which the
historical stock price series are treated as
realizations of sequences of random
variables.
A sequence of random variables
published at regular sampling intervals is
sometimes referred to as a discrete-time
stochastic process, though the shorter
name time series model is often
preferred. The theory of stochastic
processes is vast and may be studied
without necessarily fitting any models
over time series data. However, our aim
is to more applied and directed towards
model fitting and forecasting the data
using R computational techniques.
The main features of various time series
are to detect the trends and seasonal
variations that can be modelled
deterministically with respect to
mathematical functions of time. But,
another important feature of most of the
time series is that observations close
together in time tend to be correlated.
Some of the methodology in a time series
analysis is focused on explaining this
correlation factor and the main features
in the data using appropriate statistical
models and descriptive methods. Once an
accurate model is observed and fitted to
data values, then researcher used the
model to forecast future values, or
generate simulations, to guide planning
decisions and future prediction. Fitted
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ISSN 2349-2317 (Online); DOI: 10.16962/EAPJMRM/issn. 2349-2317/2015; Volume 8 Issue 1 (2017)
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models are also used as a basis for
statistical tests.
Finally, a fitted statistical model provides
a concise and informative summary of
the main characteristics of a time series,
which can often be essential for
researcher and scientists and financial
analysis. Sampling intervals may be
differ in their relation to the data. The
data may have been aggregated (for
example, the number of foreign
passengers reaching per day/month/year)
or sampled (as in a daily/weekly/monthly
basis time series of trade share prices). If
data are sampled, the sampling interval
must be short enough for the time series
to provide a very close approximation to
the original continuous signal when it is
interpolated. In a volatile share market,
close of historical prices may not suffice
for interactive trading but will usually be
adequate to show the nature of trends
and movement of the stock market price
over several years [1] [2] [3]
The objective of this paper is to provide
a procedure of modelling and
forecasting method in terms of
ARIMA/GARCH modelling for
researchers by means of statistical R
applications. Furthermore, we have
shown how to use R to find these stock
price prediction. We begin with some
basic thoughts about how to store and
process time series data using R
software.
Despite the fact that the auto regressive
integrated moving average (ARIMA)
technique is powerful and flexible also
but it is not able to handle the volatility
and nonlinearity that are present in the
data series. Some previous studies
showed that generalized autoregressive
conditional heteroskedatic (GARCH)
models are used in time series
forecasting to handle volatility in the data
series. [5][6][8].
In the next section we will discuss the
methodology and data preparation. In
particular, Time series and R analysis
packages have been discussed in brief
in sections 3, we have discussed
stationary, non stationary and estimation
of linear trend (GLM) and ACF and
PACF plots respectively in section 4.
Then, in section 5, selection of ARIMA
model and in section 6 GARCH (1,1)
have been discussed. In section 7, we
have obtained the ARIMA/ GARCH
(1,1) model performance for GE stock
price. Finally, we conclude this paper in
section 8.
ELK ASIA PACIFIC JOURNAL OF MARKETING AND RETAIL MANAGEMENT
ISSN 2349-2317 (Online); DOI: 10.16962/EAPJMRM/issn. 2349-2317/2015; Volume 8 Issue 1 (2017)
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METHODOLOGY AND DATA
T is assumed that you have installed R
software on your computer/laptop or
machine, and it is suggested that you
work through the examples, making sure
your output agrees with the results. If
you do not have R, then it can be
installed free of charge from the Internet
site www.r-project.org. It is also
recommended that you have some
familiarity with the basics statistical
packages of R, [1].
In this analysis, we used some of the
time series and forecasting packages
such as zoos, xts, ts, astsa, fts, and
forcast. For representing irregularly
spaced time series, the packages
timeSeries, zoo and xts are mostly used
in time series analysis. In these packages,
timeSeries objects are the core data
objects. But this timeSeries objects are
not frequently used as zoo and xts
objects for representing time series data.
A very flexible time series class is zeileis
ordered observations (zoo) created by
Achim Zeileis and Gabor Grothendieck
and available in the package zoo on
CRAN , [1][10][11][12].
In this study, A Regularly spaced time
series structure, data are arranged with a
fixed interval of time, can be represented
as under the packages ts. we used
historical stock price data over the
period 2001 to 2015 and stored the data
in the .csv file .The function read.csv()
comes to read data from .csv file which
stored on your computer and laptop .
Notice that the first row contains the
names of the columns namely Date, the
date information is in the first column
with the format dd/mm/YYYY, stock
price in the second column namely close
.The first step of identification is to check
the occurrence of a trend in data series
movement by plotting time series which
is as shown in Figure 1 . From the
plotting, it can be seen that the data time
series does not vary in a fixed level or
not which indicates that the series is non
stationary and stationary in both mean
and variance, as well as exhibits an
nature of trends. Time series are shown
in figure 1 as well as figure 2.
General Electric ,GE, is an
American multi national company
incorporated in New York USA. The
company operates through the following
segments i.e., Power & Water, Oil and
Gas, Aviation, Healthcare, Transportatio
n and Capital which cater to the needs of
services, Medical devices, Life
I
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ISSN 2349-2317 (Online); DOI: 10.16962/EAPJMRM/issn. 2349-2317/2015; Volume 8 Issue 1 (2017)
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Sciences, Pharmaceutical, Automotive, S
oftware
Development and Engineering industries
, [1][10][11][12]. (Refer Fig. 1)
WORKING WITH TIME SERIES
DATA
The native R classes suitable for storing
time series data include vector, matrix ,
data.frame, and ts objects. But the types
of data that can be stored in these objects
are narrow; furthermore, the methods
provided by these representations are
limited in research and analysis scope.
There exist specialized objects that deal
with more general representation of time
series data as zoo, xts, or time Series
objects, available from packages of the
same name. It is not necessary to create
time series objects for every time series
analysis problem, but more sophisticated
analyses require time series objects. You
could calculate the mean or variance of
time series data represented as a vector in
R, but if you want to perform a seasonal
decomposition using decompose, you
need to have the data stored in a time
series object. .[1][10][11][12]
In the following examples, we assume
you are working with zoo,ts
forcast,timeseries ,stats objects and etc.
because we think there are the most
widely used packages. Before using
statistical objects in r software, we need
to install and load the appropriate
statistical and forcasting package (if you
have already installed it, you only need
to load it) using the appropriate
command, [1][2][3][9].
ESTIMATING A LINEAR TREND
Consider the Stock price series is shown
in Figure 2.The data are mean stock close
price index from 2001 to 2014. In
particular data are deviations , measured
in USD. We note that an apparent
upward trend in the series during this
period. A simple kind of generated series
might be a collection of uncorrelated
random variables, wt with mean 0 and
finite variance σ2w. The time series
generated from uncorrelated variables is
used as a model for noise in statistical
research purpose, where it is called white
noise; The designation white originates
from the analogy with white light and
indicates that all possible periodic
oscillations are present with equal
strength.
Now We express simple linear regression
to estimate that trend by fitting the model
over time series xt = β1 + β2t + wt,t =
ELK ASIA PACIFIC JOURNAL OF MARKETING AND RETAIL MANAGEMENT
ISSN 2349-2317 (Online); DOI: 10.16962/EAPJMRM/issn. 2349-2317/2015; Volume 8 Issue 1 (2017)
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2001..2014. where β1 and β2 are
regression coefficient wt is random error
or noise . In general, it is necessary for
time series data to be stationary, so
averaging lagged products over time, as
in above paragraph. With time series
data, it is the dependence between the
values of the series that is important to
measure; we must, at least, be able to
estimate autocorrelations with precision.
Also, the stock price series shown in
Figure 2 contains some evidence of a
trend over time. The first step in
modelling time index data is to convert
the stationary time series. In order to
convert non stationary series to
stationary , differencing method can be
used in which the series is lagged 1 step
and subtracted from original series.
(Refer Fig. 2)
The first difference is denoted as ∆xt = xt
−xt−1 The first difference of data are also
shown in figure 3, produces different
result than removing trend by de-
trending via regression. The differenced
series does not contain the long middle
cycle observed in de-trended time series.
Over differencing can cause the standard
deviation to increase. First difference is
an example of a linear filter to eliminate
a trend and second difference can
eliminate a quadratic trend and so on. For
other information, De-trend, difference,
log and difference of log series of data
are plotted in figure 3.The differencing
technique is an important component of
the ARIMA model of Box and Jenkins
(1970), [14][15] . (Refer Fig. 3)
ACF and PACF plots
CF and PACF are the core of ARIMA
modelling. The box Jenkins method
provides a way to identify an ARIMA
model according to autocorrelation and
partial autocorrelation graph of the series
as shown in figure 4 (panel a, b, c and d).
The parameters of ARIMA consist of
three components, namely
Autoregressive parameter (p), Number of
differencing (d), and Moving average
parameters (q) In order to to identify
ARIMA model we need to follow these
basic steps are mentioned below:
Step 1:- If ACF cut off after lag n and
PACF dies down, then identify the order
of MA (q) in ARIMA (0, d, q) model
Step 2:- if ACF dies down and PACF cut
off after lag n then identify AR (p) in
ARIMA (p, d, 0) model
Step 3:- if ACF and PACF die down
means, then we get mixed ARIMA (p, d,
A
ELK ASIA PACIFIC JOURNAL OF MARKETING AND RETAIL MANAGEMENT
ISSN 2349-2317 (Online); DOI: 10.16962/EAPJMRM/issn. 2349-2317/2015; Volume 8 Issue 1 (2017)
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q) model, time series needs to
differencing (d).
ACF and PCF are a primary tool for
clarifying the relations that may occur
within and between time series at various
lags. In the beginning of fitting ARIMA
model, the idea of model
parameterization as possible yet still be
capable of explaining the series (i.e., p
and q should be 3 or less, or the total
number of parameters should be less than
3 in view of Box-Jenkins method) based
on figure 4. (Refer Fig. 4)
The more parameters the greater noise
that can be introduced into the model and
hence standard deviation. Classical
regression is often insufficient for
explaining all of the interesting dynamics
of a time series. The ACF and PACF of
the residuals of the simple linear
regression fit of the data reveals
additional structure in the data that the
regression did not capture. Instead, the
introduction of correlation as a
phenomenon that may be generated
through lagged linear relations leads to
proposing the autoregressive (AR) model
and moving average (MA) models.
Adding non stationary models to the mix
leads to the autoregressive integrated
moving average (ARIMA) model
popularized in the landmark work by
Box and Jenkins (1970), [14] [15].
ARIMA MODEL
ARIMA is one type of models in the
Box-Jenkins model. The Box - Jenkins
methodology includes four iterative steps
of model identification, parameter
estimation, diagnostic checking and
forecasting. In identification step, data
transformation is required to make the
series stationary. The stationary process
is a foundation in building an ARIMA (p,
d, q) model. When the observed time
series presents trends and non seasonal
behavior, data transformation and
differencing are applied to the data series
in order to stabilize variance and to
remove the trend before an ARIMA
model is applied. However, In order to
understand to the brief mechanism of
ARIMA (p, d, q) models are also capable
of modelling a wide range of seasonal
data and non seasonal data. A seasonal
ARIMA model is formed by including
additional seasonal terms in the ARIMA
models. In brief, The model is written as
follows taken from [16]:
ELK ASIA PACIFIC JOURNAL OF MARKETING AND RETAIL MANAGEMENT
ISSN 2349-2317 (Online); DOI: 10.16962/EAPJMRM/issn. 2349-2317/2015; Volume 8 Issue 1 (2017)
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Where am = number of periods per
season. We use the uppercase notation
for the seasonal parts of the model, and
lowercase notation for the non-seasonal
parts of the model, [16].
The seasonal part of the model consists
of terms that are very similar to the non-
seasonal components of the model, but
they involve back shifts of the seasonal
period. For example, an ARIMA (p, d,
q) (P, D, Q) 4 model (without a
constant) is for quarterly data (m=4) and
can be written as [16]
Hence , An ARIMA (p, d, q) (P,D,Q)
process can be fitted to data using the R
function Arima with the parameters
order set to c(p, d, q) based on ACF and
PACF plots. In considering the
appropriate orders for an ARIMA
model, restrict attention to the seasonal
lags. The modelling procedure is almost
the same as for non-seasonal data,
except that we need to select seasonal
AR and MA terms as well as the non-
seasonal components of the model. In
another way, The R function, auto.
Arima function can help to select all
three parameters, p, d, and q, and
predict and forecast function can also
help in forecasting the future values. It
may be useful to have an automatic
method for selecting an ARIMA model
parameters and forecasting, [2] [3] [4]
[5]. (Refer Fig. 5) In addition to Box-
Jenkins method, the autocorrelation
function (ACF) and the partial
autocorrelation function (PACF) of the
sample stock price data are used to
identify the order of ARIMA (p, d, q)
model. The ordered model then is
statistically checked whether it
accurately describes the series or not.
The model fits well if the P-value of its
parameter is statistically significant, as
well as its residuals are generally small,
randomly distributed, and contain no
useful information, where at this point,
the model can be used for forecasting.
Akaike Information Criterion (AIC)
provides another way to check and
identify the ARIMA (p.d, q) model.
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ISSN 2349-2317 (Online); DOI: 10.16962/EAPJMRM/issn. 2349-2317/2015; Volume 8 Issue 1 (2017)
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AIC is mentioned as shown in table 1.
According to table 1 the p-values for all
parameters are less than 0.05, indicating
that they are statistically significant. In
addition, p-value of the Box- Ljung test
is greater than 0.05, and so we cannot
reject the hypothesis that the
autocorrelation of residuals is different
from 0. The model thus adequately
represents the residuals. Due to the
scope of this paper, we used the
performance GARCH (1,1) with
ARIMA (2,1,2).
According to this procedure, the model
with lowest AICc will be selected.
When perform time series analysis in R,
the program will provide AICc as part
of the result. Based on AICc, we should
select manually ARIMA (2,1,2) model.
The significancy of ARIMA residuals
and QQ plot are shown in figure5 and
figure 6. In figure 6, the red line
indicates the fitted line and blue line
presents the predicted values for 12
months ahead. (Refer Table 1)
In addition, the Ljung - Box test also
provides a different way to double check
the model. Basically, Ljung- Box is a
test of autocorrelation in which it
verifies whether the autocorrelations of
a time series are different from 0. In
other words, if the result rejects the
hypothesis, this means the data is
independent and uncorrelated;
otherwise, there still remains serial
correlation in the series and the model
needs more modification. The
procedure includes observing residual
plot and its ACF and PACF diagram,
and check Ljung- Box result. If ACF
and PACF of the model residuals show
no significant lags, the selected model is
appropriate, [14] [15]. (Refer Fig. 6)
Using ARIMA(2,1,2) as selected model,
the mathematical equation for such
model as follows :
Based on ARIMA (2,1,2), High and
low price corresponding to month,
indicating with numeric value over the
line as shown in figure 6a. In the figure,
X axis indicates the forecast low price
and y axis presents the predicted high
prices corresponding to month
indicating on a curved line. (Refer Fig.
6a)
ELK ASIA PACIFIC JOURNAL OF MARKETING AND RETAIL MANAGEMENT
ISSN 2349-2317 (Online); DOI: 10.16962/EAPJMRM/issn. 2349-2317/2015; Volume 8 Issue 1 (2017)
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GARCH(1,1) MODEL
Although ACF and PACF of residuals
have no significant lags, the time series
plot of residuals shows some cluster of
volatility in figure 5. It is important to
note that ARIMA is a method to linear
model the data and the forecast width
remains constant because the model does
not reflect recent changes or incorporate
new information. In other words, it
provides best linear forecasts for the
series, and thus plays little role in
forecasting model nonlinearly. In order
to model volatility, ARCH/GARCH (1,1)
method is used. Firstly, check if residual
plot (figure 5) displays any cluster of
volatility. Next, observe the squared
residual plot (figure 5). If there are
clusters of volatility, ARCH/GARCH
(1,1) should be used to model the
volatility of the series to reflect most
recent changes and fluctuations in the
series. Finally, ACF and PACF of
squared residuals will help confirm if the
residuals are not independent and can be
predicted. As mentioned earlier, a strict
white noise can- not be predicted either
linearly or nonlinearly while the general
white noise might not be predicted
linearly yet done so nonlinearly. If the
residuals are strict white noise, they are
independent with zero mean, normally
distributed, and ACF and PACF of
squared residuals displays no significant
lags, [10].
According to the plots of squared
residuals are shown in figure 5. The
squared residuals plot shows clusters of
volatility at some points in time. ACF
seems to die down. PACF cuts off after
lag 10 even though some remaining lags
are significant. The residuals therefore
show some patterns that might be
modeled. ARCH/GARCH (1,1) is
necessary to model the volatility of the
series. As indicated by its name, this
method concerns with the conditional
variance of the series. Followings are the
based on the plots of squared residuals: .
The squared residuals plot shows
clusters of volatility at some points in
time. ACF seems to die down. PACF
cuts off after lag 10 even though some
remaining lags are significant The
residuals therefore show some patterns
that might be modeled. ARCH/GARCH
(1,1) is necessary to model the volatility
of the series. Noted that we fit
ARCH/GARCH (1,1) to the residuals
from the ARIMA model selected
previously, not to the original series or
differences log series because we only
ELK ASIA PACIFIC JOURNAL OF MARKETING AND RETAIL MANAGEMENT
ISSN 2349-2317 (Online); DOI: 10.16962/EAPJMRM/issn. 2349-2317/2015; Volume 8 Issue 1 (2017)
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want to model the noise of ARIMA
model. ARIMA 1 year forecasting and
QQ plot are shown in figure 6., [5] [6]
[7] [8]. (Refer Fig. 7 or 7a)
Mathematical equation for GARCH(1,1)
model is
ARIMA GARCH(1,1)
PERFORMANCE
There is two-stage procedure in the
proposed combined model of ARIMA
and GARCH(1,1). In the first stage, the
best of the ARIMA models is used to
modelled the linear data of time series
and the residual of this linear model will
contain only the nonlinear data as shown
in figure 6 and 6a . In the second stage,
the GARCH(1,1) is used to modelled the
nonlinear patterns of the residuals. This
combined model which combines an
ARIMA model with GARCH(1,1) error
components is applied to analyze the
univariate series and to predict the values
of approximation series
[9][10][11][12][13].
In order to estimate the validity of mixed
model, ARIMA forecast obtained using
R methodology and then add conditional
variance to ARIMA forecast as shown in
figure 6 and 6a. The Log price as well
as low and high values at the 95 %
confidence level are also plotted as
shown in figure 7 which gives an
important result with high and low values
for future purpose. The conditional
variances plotted in figure 7, which
reflects the volatility of the time series
over the entire period from 2001 to 2014.
. High volatility in close price is closely
related to a period where the stock price
tumbled. In order to make the final check
on the model is to provide at Q-Q Plot of
residuals of ARIMA-ARCH (1,1) model
as shown in figure 7. Q-Q plot is plotted
directly using the R command to check
the normality of the residuals. The plot
shows that residuals seem to be roughly
normally distributed, although some
points remain off the line. However,
compared to residuals of ARIMA model
in figure 6, those of mixed model are
more normally distributed. [10]
The mathematical equation to complete
a model of ARIMA (2,1,2) as well as
ARCH (1,1) is written below
( Yt - Yt-1 ) - ht = 0.1390 (Yt-1-Yt-2) +
0.8438(Yt-2-Yt-3) - 0.0420 εt-1 + εt-2+εt
+0.0002159 + 0.1553540ε2t- 1 + 404955595 ε2
t-2
ht = 0.0002159 + 0.1553540ε2t- 1 + 404955595 ε2
t-2
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ISSN 2349-2317 (Online); DOI: 10.16962/EAPJMRM/issn. 2349-2317/2015; Volume 8 Issue 1 (2017)
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Note that the R compiler will exclude
constant when fitting ARIMA for series
needed differencing. [10]
It is noted that the 95% confidence
intervals of ARIMA (2,1,2) are wider
than that of the combined model ARIMA
(2,0,2) – ARCH (1,1). ARIMA /GARCH
forecasting tells us that the share price
moves between 22.19 to 34.04USD.
This is because the latter reflects and
incorporate recent changes and volatility
of stock prices by analyzing the residuals
and its conditional variances (the
variances affected as new information
comes in) [10].
CONCLUSION
In this work, time domain method is a
useful technique to analyze the financial
time series for predicting the stock price
to making money. There are some basic
points in forecasting based on combined
ARIM-ARCH/GARCH (1,1) model that
need to take into account. Firstly,
ARIMA (p, d, q) model focused on
analyzing time series linearly and it does
not reflect recent changes as new
information is available in the data.
Therefore, in order to more accurate the
model, researches need to incorporate
new data and estimate parameters again
for forecasting. [10]
The variance of stock price in ARIMA
model is unconditional variance and
remains constant. ARIMA is applied for
stationary series and therefore, non-
stationary series should be transformed.
Additionally, ARIMA and GARCH
models are often used together, namely
ARIMA/GARCH (1,1) model.
ARCH/GARCH (1,1) is a method to
measure the volatility of the series, or,
more especially, to model the noise term
of ARIMA model. ARCH/GARCH (1,1)
incorporates new information and
analyses the series based on conditional
variances where users can forecast future
values with up-to-date information to
making money. The forecast interval for
the mixed model is closer than that of
ARIMA-only model.
ACKNOWLEDGEMENTS
We would like to thank the Dyal Singh
College, University of Delhi for
providing the computational facility
during the course of this work.
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REFERENCES
Norman Matlo, The art of R
programming: tour of statistical
software design, No Starch
Press, Inc., ISBN-13: 978-1-
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Michael J. Crawley, The R
Book, John Wiley Sons Ltd,
ISBN-13: 978-0-470-51024-
7,2007
Robert H. Shumway David S.
Staffer, Time Series Analysis
and Its Applications, Third
edition, DOI 10.1007/978-1-
4419-7865-3,2011
Gergely Darczi ,Michael
Puhle,Edina Berlinger,Pter
Cska,Dniel Havran,Mrton
Michalet-
Sky, Zsolt Tulassay, Kata Vradi,
Agnes Vidovics-Dancs1
Introduction to R for
Quantitative
Finance,www.packtpub.com,
ISBN 978-1-78328-093-3.
[5] Christian P. Robert George
Casella, Introducing Monte
Carlo Methods with R, Springer,
ISBN 978-1-4419-1575-7,2010
Y. Cohen J.Y. Cohen, Statistics
and Data with R: An applied
approach through examples,
John Wiley Sons, Ltd. ISBN:
978-0-470-75805,2008.
[7] Ruey S. Tsay, Analysis of
Financial Time Series, Second
Edition, A John Wiley & Sons,
Inc., Publications,
ISBN-13 978-0-471-69074-0,
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[8] Ghulam Ali ,EGARCH, GJR-
GARCH, TGARCH,
AVGARCH, NGARCH,
IGARCH and APARCH
Models for Pathogens at Marine
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version), 2051- 5065 (online)
Scienpress Ltd, 2013.
[9] Paul S.P. Cowpertwait,
Andrew V. Metcalfe,
Introductory Time Series with R,
Springer Dordrecht Heidelberg
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London, New York, DOI
10.1007/978-0-387- 88698-5,
2009.
[10] Ly Pham, Time Series
Analysis with ARIMA –
ARCH/GARCH model in R, L-
Stern Group, 2013
[11] Varun Malik, Ranjit Kumar,
Abid Hussain, Rather
BAYESIAN ESTIMATION OF
GARCH
Coefficientsof Inr/Usd Exchange
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Finance And Risk
Management, Issn 2349-2325
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2325/2015; Volume 7 Issue 1
(2016)
[12] John C. Hull, Options,
Futures and Other Derivatives,
Seventh Ed., Pearson – Prentice
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[13] J. D. Cryer and K. -S.
Chang, Time Series Analysis
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[14] G M Ljung and G E P Box ,
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65, 2, pp. 297 - 303,1978.
[15] Box, G. E. P., and D. A.
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Moving average time series
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1526.
[16] Rob J
Hyndman,GeorgeAthansopoulos,
www.google.com/fpp
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LIST OF FIGURES:
Figure 1: Time series plot
Figure 2: Time series deviations shown with fitted linear trend line
The given hint helps to produce figure as same as figure 1:
>>data <- read.csv("filename.csv")
>>datat_ts <-ts (data$close, start=c(YYYY,MM), end=c(YYYY,MM) ,frequency = N)
>>plot(data_ts, type="l", ylab = "data", xlab = "Time",main ="Time series")
The given hint helps to produce figure look likes figure 2:
>>plot(data_ts,type="l",ylab = " close price", main="Time Series with trend")
>>newtime<-time(data_ts,start=c(YYYY),end=c(YYYY),frequency=N)
>>fit<-(reg=glm(data_ts ~time(data_ts),na.actio=NULL))
>>abline(fit,col='blue',lty=1:10)
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ISSN 2349-2317 (Online); DOI: 10.16962/EAPJMRM/issn. 2349-2317/2015; Volume 8 Issue 1 (2017)
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Figure 3: De trended (panel a), differenced ( panel b), log series(panel c) and
differenced log series(panel d) shown in figure
Figure 4: ACF and PACF plots
The given procedure helps to create figure look likes figure 3:
>>fit<-(reg=glm(data_ts~time(data_ts),na.actio=NULL))
>>par(mfrow=c(2,2))
>>plot(resid(fit),main="(a) detrended series",ylab="")
>>plot(diff(meandata),type="l",main="(b)differenced series",ylab="")
>>plot(log(meandata),main="(c)log series",ylab="")
>>plot(diff(log(meandata)),main="(d)diff.log series",ylab="")
The given procedure helps to create figure look likes figure 4:
par(mfrow=c(3,2))
acf(data_ts,main =" (a) ACF original series")
pacf(data_ts,main ="(b)PACF original series",ylab="")
acf(resid(fit),main ="(c) ACF detrended series",ylab="")
pacf(resid(fit),main ="(d) PACF detrended series",ylab="")
acf(diff(data_ts),main="(e) ACF differenced series",ylab="")
pacf(diff(data_ts),main ="(f) PACF differenced series",ylab="")
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ISSN 2349-2317 (Online); DOI: 10.16962/EAPJMRM/issn. 2349-2317/2015; Volume 8 Issue 1 (2017)
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Figure 5: acf and pacf arima residuals
Figure 6: ARIMA residuals
The given R codes helps to create figure look likes figure 5:
>>fit<-(reg=lm(data_ts~time(data_ts),na.actio=NULL))
>>difff<-diff(log(data_ts))
>>f405 <-Arima(difff,seasonal=list(order=c(2,0,2),period=12),include.drift=FALSE)
>>plot(f202$residuals,lag.max=100,main=" arima residual (a)",ylab="")
>>plot((f202$residual^2),lag.max=100,main="arima Squared Residuals(b)",ylab ="")
>>acf(f202$residuals,lag.max=100,ylim=c(-1,1),main="acf arima residual (c)",ylab="")
>>pacf(f202$residuals,lag.max=100,ylim=c(-1,1),main="pacf arima residual(d)",ylab="")
>>acf((f202$residual^2),lag.max=100,main="acf Squared Residuals(e)",ylim=c(-1,1),ylab="")
>>pacf((f202$residual^2),lag.max=100,main="pacf Squared Residuals (f)",ylim=c(- 1,1),ylab="")
The given R codes helps to create figure look likes figure 6:
f410<-Arima(logdata,seasonal=list(order=c(2,0,2),period=6),include.drift=F)
par(mfrow=c(1,2))
plot(forecast(f202),main=" forcast model",ylim=c(3.1,3.6))
lines(fitted(f202),col="red")
qq=f202$residuals
qqnorm(qq,main='ARIMA residuals')
qqline(qq)
ELK ASIA PACIFIC JOURNAL OF MARKETING AND RETAIL MANAGEMENT
ISSN 2349-2317 (Online); DOI: 10.16962/EAPJMRM/issn. 2349-2317/2015; Volume 8 Issue 1 (2017)
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Figure 6a: Forcast price for 2015
Figure 7: ARIMA GARCH(1,1) residuals
The given R command lines helps to create figure look likes figure 7:
>>par(mfrow=c(2,2))
>>plot(log(data_ts),type='l',main='Data')
>>arima202<-Arima(logdata,seasonal=list(order=c(2,0,2),period=6),include.drift=TRUE)
>>fit202<-fitted.values(arima203)
>>low<-fit202-1.76*sqrt(ht.arch11)
>>high<-fit202+1.76*sqrt(ht.arch11)
>>plot(data_ts,type='l',main='Log price,Low,High')
>>lines(low,col='red')
>>lines(high,col='blue')
>>ht.arch11<-arch11$fit[,1]^2 #use 1st column of fit
>>plot(ht.arch11,main='Conditional variances')
>>archres<-res.f202/sqrt(ht.arch11)
>>qqnorm(archres,main='ARIMA-GARCH Residuals')
>>qqline(archres)
ELK ASIA PACIFIC JOURNAL OF MARKETING AND RETAIL MANAGEMENT
ISSN 2349-2317 (Online); DOI: 10.16962/EAPJMRM/issn. 2349-2317/2015; Volume 8 Issue 1 (2017)
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The output of Garch(1,1) model as shown in figure7a:
Figure7a: GARCH(1,1) summary
ELK ASIA PACIFIC JOURNAL OF MARKETING AND RETAIL MANAGEMENT
ISSN 2349-2317 (Online); DOI: 10.16962/EAPJMRM/issn. 2349-2317/2015; Volume 8 Issue 1 (2017)
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LIST OF TABLES:
Table 1: AIC values for (ARIMA, GARCH)
Arima(pdq) AIC ARCH (p, q) * AIC P value*
(Ljung - Box test)
Arima
(000)
-
759.48
Arch (00) Not done Not done
Arima
(101)
-
757.40
Arch (01) -798.0746 0.6248
Arima
(201)
-
755.75
Arch (02) -797.000 0.9112
Arima
(301)
-
754.41
Arch (03) -789.859 0.8964
Arima
(102)
-
756.10
Arch (04) -782.745 0.9118
Arima
(202)
-
760.85
Arch (05) -776.335 0.8237
Arima
(302)
-
757.08
Arch (06) -768.186 0.7979
Arima
(103)
-
754.69
Arch (07) -763.061 0.8378
Arima
(203)
-
755.23
Arch (08) -755.636 0.8032
Arima
(303)
-
757.08
Arch (09) -801.266 0.7877