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Project report On

Study and Forecasting of Financial Time Series DataBy

AMAR SUBHASH PATILAMIT BALKRISHNA DOIFODE

HEENAKAUSHAR INAYATBHAI VHORAMANISHA JAYANTILAL KANANI

PRAMOD BALKRISHNA GHADAGESACHIN KRISHNA RASANKAR

TRUPTI RAMESHBHAI RATHOD

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Introduction

Stock Market

When people talk about the Stock Market, it's no always immediately clear what they're referring to. Is the Stock Market a place? Or is it something different? To many people it is an abstract idea. They buy stocks in "the stock market" without ever leaving the comfort of their computer terminal. But the stock market is indeed a physical place with buildings and addresses, a place you can go visit.

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Current Stock Market

The current "stock market" is comprised of 300,000 computers situated on pro

trader's desks. These computers are networked together using sophisticated protocols.

This level of information sharing makes pricing an almost exact science.

These 300,000 computers are further linked to another 26 million computers

worldwide. These computers are located in banks, small businesses, and large

corporations. These computers comprise the banking networks which make

computerized transactions possible.

Finally, these computers are connected to another 300 million+ computers

which connect and disconnect from the financial markets daily. In New York City

alone, these transactions amount to over $2.2 trillion dollars daily

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Bombay Stock Exchange (BSE)

The Bombay Stock Exchange is known as the oldest exchange in Asia. It traces its history to the 1850s, when stockbrokers would gather under banyan trees in front of Mumbai's Town Hall. The location of these meetings changed many times, as the number of brokers constantly increased. The group eventually moved to Dalal Street in 1874 and in 1875 became an official organization known as 'The Native Share & Stock Brokers Association'. In 1956, the BSE became the first stock exchange to be recognized by the Indian Government under the Securities Contracts Regulation Act.It is the 11th largest Stock Exchange in the world.

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Two main goals of the time series analysis There are two main goals of time series analysis:

(a) identifying the nature of the phenomenon represented by the

sequence of observations

(b) forecasting (predicting future values of the time series variable).

Both of these goals require that the pattern of observed time-

series data is identified

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Identifying Patterns in Time Series Data

Systematic pattern and random noise

Two general aspects of time series patterns

1. Trend Analysis

I. Smoothing

II. Fitting a function

2. Analysis of Seasonality

White Noise

Autocorrelation Correlogram (ACF)

Partial Autocorrelation Correlogram (PACF)

Removing serial dependency

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TIME SERIES MODELS

The task facing the modern time-series econometrician is to develop

reasonably simple models capable of forecasting, interpreting, and testing

hypotheses concerning economic data. . The challenge has grown over time

the original use of time-series analysis was primarily as an aid to forecasting.

As such, a methodology was developed to decompose a series into a trend, a

seasonal, a cyclical, and an irregular component. Uncovering the dynamic

path of a series improves forecast accuracy. . Using the time-series methods,

it is possible to decompose this series into the trend, seasonal, and irregular

components.

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Autoregressive Model

As stationary process is somewhat parsimonious with parameter. But it

is not sufficiently parsimonious similar to the general non-stationary

process. The problem is that there are infinite number of parameters.

What we need is the class of stationary time series model with only finite

parameters preferably small number of parameters. That’s why the simplest

autoregressive (AR) models are used.

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AR (1) Process

Most time series consist of elements that are serially dependent in the

sense that one can estimate a coefficient or a set of coefficients that describe

consecutive elements of the series from specific, time-lagged (previous) elements.

This can be summarized in the equation:

Note that an auto regressive process will only be stable if the parameters are

within a certain range for example if there is only one autoregressive parameter then

it must fall within the interval of -1<a<1. Otherwise, that is, the series would not be

stationary. If there is more than one autoregressive perimeter similar restriction on the

perimeter values can be defined.

titi

p

it xyaay

10

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Moving Average(1) Process

Independent from the autoregressive process, each element in the seriese can also be

affected by the past error that can’t be accounted for by the auto regressive component that is :

yt= a0 +

q

ii

0

There is “ duality “ between the moving average process and the autoregressive process,

that is, the moving average equation above can be rewritten into an autoregressive form.

However , analogous to the stationarity condition described above, this can only be done if the

moving average parameter follow certain condition, that is, if the model is invertible. Otherwise

the series will not be stationary.

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ARMA model

An autoregressive model of order p is conventionally classified as

AR(p). A moving average model with q terms is classified as MA(q). A

combination model containing p autoregressive terms and q moving average

terms is classified as ARMA (p, q).

It is possible to combine a moving average process with a linear

difference equation to obtain an autoregressive moving average model.

Consider the p-th order difference equation:

titi

p

it xyaay

10 ……………… (6)

Now let { xt } be the MA (q) process so from white noise process

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0

q

t ii

X

so that we can write,

yt= a0 +

p

iia

1yt-i +

q

ii

0

We follow the convention of normalizing units so that β0 is always equal to unity. If the

characteristic roots of above equation are all in the unit circle, {yt} is called an autoregressive

moving-average (ARMA) model for yt. The autoregressive part of the model is difference

equation is given by the homogeneous portion of and the moving average part is the xt

sequence.

If the homogeneous part of difference equation contains p lags and the model for xt contains q lags, the model is called an ARMA(p,q) model, i.e., the model with the p AR parameters, q MA parameters and the variance of the error term

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ARIMA model

If the object series is differenced d times to achieve stationarity, the

model is classified as ARIMA (p, d, q), where the symbol "I" signifies

"integrated." An ARIMA(p,0,q) is the same as an ARMA(p, q) model; likewise,

an ARIMA(p,0,0) is the same as an AR(p) model, and an ARIMA(0,0,q) is the

same as an MA(q) model

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Data analysis steps

First of all we have to check whether the given time series is stationary or not, if not then we try to make it stationary by taking difference. Most of the financial time series become stationary by taking first order differentiation.

Identify the parameters p and q of AR( p), MA(q) from the correlogram ACF and PACF. From ACF we can identify the parameter q and from PACF we identify the parameter p.

We now fit the ARIMA (p, d, q) model and estimate the parameters of the model. If the parameters are not significant then we choose another combination of p and q and fit another model and identify the appropriate p and q by trial and error.

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Check the Model Selection Criteria for identifying the best model. Find predicted value for in-sample period and save the residuals.

Check the normality assumption for the model residuals.

If the residual is not normally distributed than detect the influential points

which affect the normality assumption. And try to make it normal. Generate the normal deviate from mean and variance which obtain in above

step . Find the future predicted value from the fitted model replacing the error terms

by generated normal deviate.

Interpret the result.

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Analysis of MRF (MONTHLY) data

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96 101 106 111 116 121 126

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

return

return

SEQUENCE CHART

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Autocorrelation Function (ACF)

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Interpretation (ACF):- Here, we have to examine that the consecutive lags re normally dependent or not, i.e. the first element is closely related to the second, and the second to the third which determine the serial dependency. In the above figure alternating positive and negative decaying, so we can predict that there is an MA model effects are there in the model. In the above figure, we make the assumption lag q of the ARMA(p,q) model. It was found that they were stationary and it has single large spike which is significant. So, we take q=1

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Partial Autocorrelation Function (PACF)

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Interpretation (PACF):- In the above figure, we make the assumption lag p of the ARMA (p,

q) model. It was found that they were stationary and it has single large spike which is significant. So, we take p=1. From the above correlogram we get p=1 and q=1. We fit ARIMA (1, 0, 1) model . We have choosen ARIMA(1,0,1) in which auto regressive perameter( AR) i.e. p is 1 differencing parameter d=0 as our data on return is stationary and moving average perameter MA i.e. q=1. These perameter s are choosen from ACF and PACF graph.

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Time Series Modeler

Model Fit Fit Statistic Mean Stationary R-squared 0.061 R-squared 0.061 RMSE 0.16 MAPE 176.6 MaxAPE 2477.7 MAE 0.10 MaxAE 1.11 Normalized BIC -3.53

Model Description

ARIMA(1,0,1)Model_1returnModel IDModel Type

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ARIMA Model Parameters Estimate SE T Sig. return-Model_1

return No Transformation

Constant 0.03219 0.002311 13.9264 2.08E-27

AR Lag 1 0.858689 0.082952 10.35163 1.28E-18 MA Lag 1 0.998545 0.803012 1.243499 0.215957

Interpretation:-

The above table gives interpretation of ARIMA(1,0,1) from the table it is clear that the estimate of constant is 0.03219, estimate of AR is 0.858689 and estimate of MA is 0.998545. Standared error of constant and coefficient of AR lag1 is very small which is near to zero.

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Therefore these two parameters are statistically significant. It is also see from p-value criteria i.e. if p-value is greater than 0.05 which means that we can’t reject the null hypotheses that estimates are statistically insignificant. It is clear from the above table that the estimates of constant and coefficient of AR lag1 are statistically significant. And the estimate of MA is statistically insignificant. The Computed model is as follows.

𝑅𝑡= 𝛽0 + 𝛽1𝑅𝑡−1 + 𝜀𝑡

𝑅𝑡=0.03219 +0.858689∗𝑅𝑡−1 + 𝜀𝑡 and the predicted value can be find from 𝑅𝑡= 𝛽መ0 + 𝛽መ1𝑅𝑡−1

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Graph for return and predicted values

Interpretation:-

Here we can see from the graph that predicted value obtained

from the model is near to the original value so we can say that

our predictive model is good for the in-sample-data.

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 8 15 22 29 36 43 50 57 64 71 78 85 92 99 106

113

120

127

return

predicted

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K-S test for checking normality assumption

In the K-S test our null hypotheses is : The test distribution is normal. And from the p-value we do not reject our null hypotheses. So, we can say that our model residual is normally distributed.

One-Sample Kolmogorov-Smirnov Test

131-.0037.12701

.059

.059-.057.672.757

NMeanStd. Deviation

Normal Parametersa,b

AbsolutePositiveNegative

Most ExtremeDifferences

Kolmogorov-Smirnov ZAsymp. Sig. (2-tailed)

Noiseresidual fromVAR00003-

Model_1

Test distribution is Normal.a.

Calculated from data.b.

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MRF Forecast plot:-

From the above chart display the forecast value of return, It is easy to see from the graph that the value of the return are in increasing trend and it may continue increasing with time.

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Programe for forecasting:-

b0=0.032

b1=0.858

et=0.998

rt<-vector("numeric")

i=0

for(i in i:12)

{

rt[i]=0.048721335

e<-rnorm(1,0.0265,0.131)

rt[i+1]=b0+(b1)*(rt[i])+(et)*(e)

rt[i]=rt[i+1]

i<-i+1

}

print(rt)

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Forecasted values

[1] 0.459984384 0.003063379 0.266967056 0.071628528 0.215974207 0.115193849

[7] 0.027578631 0.015844679 0.122600279 0.199623276 0.316388738 0.141274116

[13] 0.141274116

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THANK YOU