Predictability of Stock Indices

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Project Report on predictability

of

Stock Indices

Internal Guide:External Guide:

Prof V K VasalK P Sharda

DFS, Delhi UniversityADM, LIC India

Submitted by:

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Rishabh Tambi

2436, MFC-II

Acknowledgement

I would like to express my deep and sincere gratitude to my

supervisor, Professor V K Vasal for his detailed and

constructive comments, and for his important support throughout

this work. His wide knowledge, understanding, encouragement

and personal guidance have provided a good basis for the present

report.

I would also like to thank my external supervisor , Mr. K P

Sharda for his support and guidance throughout the work. His

guidance have been a good support for the report.

Rishabh Tambi

MFC-II

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Index

I. Introduction.. 4

II. Past Literature Review .

5

III. Methodology

..6-8

IV. Results

..9-37

V. Summary..

..38

VI. Bibliography

..39

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Introduction

Central to investors and policy makers dealing with emerging equity marketsis the knowledge of how efficiently those markets incorporate market

information into security prices. Specifically, what is the empirical validity of

the random walk hypothesis (RWH) in these markets? We would try to find out

whether various stock indices are predictable or not .If markets turn out to be

predictable than we would like to also analyze the amount of predictability

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which can be done in various indices. We have taken Sensex and BSE 500

index from Indian market to test for their predictability.

The principal tools for testing the RWH in emerging markets are the Lo Mac

Kinlay (1988) variance ratio (VR) test, ARMA, GARCH tests. In this study, we

have used Variance ratio test, ARMA and GARCH models to know about theamount of predictability of various indices. Variance ratio test states that

index is predictable or not and then ARMA & Garch models tells us about the

amount of predictability for various indices. It is the aim of this report to make

a complementary contribution to this important issue relating to the

predictability of stock indices in Indian Market.

Past Literature Review

Random walk properties of stock indices have long been a prominent topic in

the study of stock returns (see summers, 1986; Fama and French, 1988; Lo

and Mac Kinlay, 1988; Liu and He, 1991; Malkiel, 2005). Several studies

attempt to address the RWH in emerging markets, with mixed results. Butler

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and Malaikah (1992) report evidence of inefficiency in the Saudi Arabian stock

market, but not in the Kuwaiti market. El-Erian and Kumar (1995) find the

Turkish and the Jordanian markets to be inefficient.

Abraham, Seyyed, and Alsakran (2002) examine the random walk in three Gulf

markets (Saudi Arabia, Kuwait, and Bahrain) and find that the stock markets ofSaudi Arabia and Bahrain, but not Kuwait, are efficient. Using Wrights (2000)

non parametric VR tests, Bugak and Brorsen (2003) find evidence against the

random walk in the Istanbul stock exchange.

Among other emerging markets, Barnes (1986) reports that the Kuala Lumpur

Stock market is inefficient. Panas (1990) reports that market efficiency cannot

be rejected for the Greek market while Urrutia (1995) rejects the RWH for the

markets of Argentina, Brazil, Chile, and Mexico.

In contrast, Ojah and Karemera (1999) find that RWH holds in Argentina,Brazil, Chile, and Mexico. Grieb and Reyes (1999) reexamine the random walk

properties of in Brazil and Mexico using the VR test and conclude that the

index returns in Mexico exhibit mean reversion and a tendency toward a

random walk in Brazil.

Alam, Hasan, and Kadapakkam (1999) examine five Asian markets

(Bangladesh, HongKong, Malaysia, SriLanka, and Taiwan) and conclude that all

the index returns follow a random walk with the exception of SriLanka. Darrat

and Zhong (2000) and Poshakwale (2002) reject the RWH for the Chinese and

Indian stock markets, respectively.

Hoque, Kim, and Pyun (2007) test the RWH for eight emerging markets in Asia

using Wrights (2000) rank and sign VR tests and find that stock prices of most

Asian developing countries do not follow a random walk with the possible

exceptions of Taiwan and Korea.

Methodology

Nonparametric VR tests in the study of the RWH in emerging markets, VR tests

have been by far the most widely used econometric tools since the pioneering

work of Lo and Mac Kinlay (1988). A potential limitation of the LoMac Kinlay-

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type (1988) VR tests is that they are asymptotic tests, so their sampling

distributions infinite samples are approximated by their limiting distributions.

An assumption underlying the VR tests is that stock returns are at least

identically, if not normally, distributed and that the variance of the random

walk increments in a finite sample is linear in the sampling interval.

If the hypothesis is rejected, there is a high probability that the time series is

non linear or has chaotic characteristics.

Index levels can be determined from index returns, so here basis of our report

is index returns and then index levels can be determined from index returns.

Index returns are indicator of index level.

As seen from the past studies of Bugak and Brorsen, 2003; O. M. Al-

Khazali,2007 ; R. K. Mishra,2011 the principal tools for testing the RWH in

Stock indices are ARMA, GARCH, E GARCH tests. These tests can easily beperformed in EViews. So, mainly here we will perform these tests.

We will first take daily closing data for index (Sensex and BSE 500) to be

checked. After that we will normalize the data by taking natural log of closing

data and subtracting it from natural log of previous day closing data, so that

variation between them can be reduced. After that we will check about the

predictability of index by variance ratio test. If test hypothesis is rejected, than

index is predictable. If test results turn out that index is predictable, then we

will go for ARMA model to check about the amount of predictability by this

model. If results are not satisfactory, then we will go for BDS test .By results ofBDS test, we will take decision regarding going for ARCH Models. If BDS test is

rejected, then we can go for Garch /EGarch models to predict the index.

Flow Chart of methodology

7Desired

characteristics

not obtained

Preparation of Index Data

Determi

ne

characte

ristics ofdata


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Data

We have collected data of various indices in Indian market through BSE site

and through prowess databases. We worked on daily return data from 1st Jan

1993 to 31st Dec 2011 of BSE Sensex and 1st Feb 1999 to 31st dec, 2011 of BSE

500 indices. We then run various random walk tests on the data collected to

find out whether data is martingale or not. Martingale means that data is

8

Null hypothesisRejected

Null hypothesis

accepted

AIC value small and variables

AIC value

small and

variables

significant

Z Statistics significant

Z- Statistics

insignificant

AIC value small and

variables insignificant

AIC value

small&variables

significant

Null Hypothesis

Rejected

Null Hypothesis

accepted

Desired Characteristics obtained

Statistic tests cant

Run VR

test to

test RWH

on Data

Index follows a random

walk and prediction is n

possible

ndex does not follow a

random walk and

prediction is possible

Unit Root

Test is

done

Correlogram is

made &by

Results of it,

ARMA Model is

made

This model c

predict our in

Index cannot be predicted by

this model have to go on

higher models

PerformBDS Test

Other Linear mode

are required for

Make ARCH

Family models

like GARCH

This model can

predict our inde

Have to go on further

Higher models of Prediction

Make data

Stationary


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purely following random walk and old data does not contain any memory for

future data. That means if data is martingale, then it is difficult to predict and

forecast future data.

The return series of the index exhibits significant levels of skewness and

kurtosis. The skewness of the return series for BSE 500 is negative whereasthat of Sensex is positive .The negative skewness implies that the index

returns are flatter to the left compared to the normal distribution and positive

skewness vice versa . The kurtosis reported indicates that the index return

distributions have sharp peaks compared to a normal distribution. Jarque Bera

statistics confirm the significant non normality of returns.

Process

We first find out log normal return of daily data on index to be checked.

Log Normalized Return = Ln (Pt) - Ln (Pt-1)

Pt = Closing Point of index on t day

Pt-1 = Closing point of Index on t-1 day

This has been done so that data values does not differ in large absolute values

as we know that index like sensex has closing value ranging from 1000 to

21000 . So to get good econometric model, we normalized the data, so thatvalues does not differ by large values.

After that we performed variance ratio test on obtained data to find out that

data is martingale or not. This will suggest us about the nature of data

whether old data has memory for current data or not. So now by result of

variance ratio test, we can state about the predictability of index data.

Now we can perform various tests like ARMA, GARCH etc. to forecast about the

model which can closely be related to current data of index to find out that

which model closely fits with the data.

Descriptive Statistics of Data

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Sensex

0

200

400

600

800

1,000

1,200

1,400

1,600

-0.10 -0.05 0.00 0.05 0.10 0.15

Series: RESID

Sample 1/01/1993 12/30/2011

Observations 4955

Mean -0.000172

Median 0.000225

Maximum 0.156536

Minimum -0.111974

Std. Dev. 0.015914

Skewness 0.000316

Kurtosis 8.327294

Jarque-Bera 5859.301

Probabil ity 0.000000

Residuals of sensex data are positively skewed and

its kurtosis value is 8.32, by large Jarque-Bera value

of residuals, we can say that residuals show non

normality and we can go for higher test.

BSE 500

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0

200

400

600

800

1,000

1,200

-0.10 -0.05 0.00 0.05 0.10 0.15

Series: RESID

Sample 2/01/1999 12/30/2011

Observations 3369

Mean -0.000573

Median 0.000430

Maximum 0.143053

Minimum -0.115562

Std. Dev. 0.016862

Skewness -0.343433

Kurtosis 8.556076

Jarque-Bera 4399.600

Probabil ity 0.000000

Residuals of BSE 500 data are negatively skewed

and its kurtosis value is 8.55, also by large Jarque-

Bera value of residuals, we can say that residuals

show non normality and we can go for higher test.

Variance Ratio Test (to check that data is

martingale or not)

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The question of whether asset prices are predictable has long been the subject

of considerable interest. One popular approach to answering this question, the

Lo and MacKinlay (1988, 1989) overlapping variance ratio test, examines the

predictability of time series data by comparing variances of differences of the

data (returns) calculated over different intervals. If we assume the data follow

a random walk, the variance of a -period difference should be multiple of the

variance of the one-period difference. Evaluating the empirical evidence for or

against this restriction is the basis of the variance ratio test.

Now we have performed Variance ratio test in EViews for Sensex and BSE 500indices .we have taken hypothesis as:

Null Hypothesis: Index is a martingale

Alternate Hypothesis: Index is not martingale

Significance level: 0.05

Variance Ratio Test in EViews

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The Output combo determines whether we wish to see our test output

in Table or Graph form. The Data specification section describes the

properties of the data in the series.The Test specification section

describes the method used to compute test.The Compute using

combo, which defaults to Original data, instructs EViews to use theoriginal Lo and MacKinlay test statistic based on the innovations

obtained from the original data.

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Variance Ratio Test Result for Sensex

Null Hypothesis: SENSEX is a martingale

Date: 01/08/12 Time: 11:19

Sample: 1/01/1993 12/30/2011

Included observations: 4955 (after adjustments)

Heteroskedasticity robust standard error estimates

User-specified lags: 2 4 8 16

Joint Tests Value Df Probability

Max |z| (at period 2)* 16.22327 4955 0.0000

Individual Tests

Period Var. Ratio Std. Error z-Statistic Probability

2 0.568789 0.026580 -16.22327 0.0000

4 0.273353 0.045887 -15.83542 0.0000

8 0.137399 0.066272 -13.01611 0.0000

16 0.070194 0.091073 -10.20950 0.0000

*Probability approximation using studentized maximum modulus with

parameter value 4 and infinite degrees of freedom

Test Details (Mean = -5.6916575218e-07)

Period Variance Var. Ratio Obs.

1 0.00046 -- 4955

2 0.00026 0.56879 4954

4 0.00013 0.27335 4952

8 6.3E-05 0.13740 4948

16 3.2E-05 0.07019 4940

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-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

2 4 8 16

Variance Ratio Statistic

Variance Ratio 2*S.E.

Variance Ratio Statistic for SENSEX with Robust 2*S.E. Bands

Now clearly probability value in joint test comes out to be 0.0000

which states null hypothesis gets rejected at both 5% and 1% level ofsignificance and data is not martingale. Therefore values in data do

consist of memory of old data. Therefore we can predict index by

various models and can forecast future values.

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Variance Ratio Test Result for BSE 500

Null Hypothesis: BSE_500 is a martingale

Date: 02/26/12 Time: 10:26Sample: 2/01/1999 12/30/2011

Included observations: 3369 (after adjustments)

Heteroskedasticity robust standard error estimates

User-specified lags: 2 4 8 16

Joint Tests Value df Probability

Max |z| (at period 2)* 14.10390 3369 0.0000

Individual Tests

Period Var. Ratio Std. Error z-Statistic Probability

2 0.559842 0.031208 -14.10390 0.0000

4 0.278258 0.055678 -12.96282 0.0000

8 0.142236 0.081565 -10.51637 0.0000

16 0.074607 0.112381 -8.234438 0.0000

*Probability approximation using studentized maximum modulus with

parameter value 4 and infinite degrees of freedom

Test Details (Mean = -1.21810132421e-05)

Period Variance Var. Ratio Obs.

1 0.00050 -- 3369

2 0.00028 0.55984 3368

4 0.00014 0.27826 3366

8 7.1E-05 0.14224 3362

16 3.7E-05 0.07461 3354

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0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

2 4 8 16

Variance Ratio Statistic

Variance Ratio 2*S.E.

Variance Ratio Statistic for BSE_500 with Robust 2*S.E. Bands

Now clearly p value comes out to be 0.000 which states that null

hypothesis gets rejected at both 5% and 1% level of significance and

data is not martingale. Therefore values in data do consist of memory

of old data. Therefore we can predict index by various models and can

forecast future values.

INFERENCE: Variance Ratio test states that both indices

(sensex and BSE 500) are predictable and they both stronglyreject the null hypothesis of being martingale, so we can

predict both the indices by various models.

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Unit root test

This test is being done to check stationarity of data; we require

stationary data for Arma estimation, so first we will first check

that our data is stationary or not.

Null Hypothesis: Index has a unit root

Alternate Hypothesis: Index does not has a unit root

Significance Level: 0.05

For data to be stationary null hypothesis has to be rejected.

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Unit Root Test for Sensex

Null Hypothesis: SENSEX has a unit root

Exogenous: Constant

Lag Length: 0 (Automatic - based on SIC, maxlag=31)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -63.32467 0.0001

Test critical values: 1% level -3.431488

5% level -2.861928

10% level -2.567019

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation

Dependent Variable: D(SENSEX)

Method: Least Squares

Date: 01/08/12 Time: 16:46

Sample (adjusted): 1/04/1993 12/30/2011

Included observations: 4955 after adjustments

Variable Coefficient Std. Error t-Statistic Prob.

SENSEX(-1) -0.894802 0.014130 -63.32467 0.0000

C 0.000333 0.000226 1.473141 0.1408

R-squared 0.447396 Mean dependent var -5.69E-07

Adjusted R-squared 0.447284 S.D. dependent var 0.021406

S.E. of regression 0.015914 Akaike info criterion -5.442779

Sum squared resid 1.254439 Schwarz criterion -5.440152

Log likelihood 13486.49 Hannan-Quinn criter. -5.441858

F-statistic 4010.014 Durbin-Watson stat 1.993829

Prob(F-statistic) 0.000000

So null hypothesis gets rejected in unit root test. This states that

sensex data is stationary and we can go for ARMA estimation.

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Unit Root Test for BSE 500

Null Hypothesis: BSE_500 has a unit root

Exogenous: Constant

Lag Length: 0 (Automatic - based on SIC, maxlag=28)

t-Statistic Prob.*

Augmented Dickey-Fuller test statistic -50.51090 0.0001

Test critical values: 1% level -3.432103

5% level -2.862200

10% level -2.567165

*MacKinnon (1996) one-sided p-values.

Augmented Dickey-Fuller Test Equation

Dependent Variable: D(BSE_500)


Date: 01/08/12 Time: 16:48

Sample (adjusted): 2/02/1999 12/30/2011



BSE_500(-1) -0.861478 0.017055 -50.51090 0.0000

C 0.000406 0.000291 1.397246 0.1624

R-squared 0.431092 Mean dependent var -1.22E-05

Adjusted R-squared 0.430923 S.D. dependent var 0.022345S.E. of regression 0.016856 Akaike info criterion -5.327576





So null hypothesis for BSE 500 gets rejected in unit root test. This

states that data is stationary and we can go for ARMA estimation.

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Correlogram of Sensex

This is mainly used to identify AR and MA terms in ARMA

estimation.

Here some values of ACF and PACF is significant as compared to other values ,

major spikes are coming in first , sixth , ninth , ten lags so we will take ar(1),

ar(6), ar(9), ar(10), ma(1), ma(6), ma(9), ma(10) terms in arma equation.

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Correlogram of BSE 500

Here some values of ACF and PACF is significant as compared to other values ,

major spikes are coming in first , fourth , sixth , ninth , tenth , thirteen lags so

we will take ar(1), ar(4), ar(6), ar(9), ar(10), ar(13), ma(1), ma(4), ma(6),

ma(9), ma(10), ma(13) terms in arma equation.

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Model Forecasting Types

Now that we know that both indices are predictable, we will try to

forecast the index by using various models in EViews.

As seen from the past studies of Bugak and Brorsen, 2003; O. M. Al-

Khazali,2007 ; R. K. Mishra,2011, the principal tools for testing the RWH in

Stock indices are ARMA, GARCH, E GARCH tests. These tests can easily be

performed in EViews. So, mainly here we will perform these tests.

The various models which we will be trying are:

1.) ARMA Model

2.) ARCH Family Models

ARMA Theory

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ARMA (autoregressive moving average) models are generalizations of

the simple AR model that use three tools for modeling the serial

correlation in the disturbance:

The first tool is the autoregressive, or AR, term. The AR (1) model

uses only the first-order term, but in general, it may use additional,

higher-order AR terms. Each AR term corresponds to the use of a

lagged value of the residual in the forecasting equation for the

unconditional residual. An autoregressive model of order, AR (p) has

the form.

U(t) = r(1)u(t-1) + r(2)u(t-2) +..+ r(p)u(t-p) + e(t)

The second tool is the MA, or moving average term. A moving

average forecasting model uses lagged values of the forecast error toimprove the current forecast. A first order moving average term uses

the most recent forecast error; a second-order term uses the forecast

error from the two most recent periods, and so on. An MA (q) has the

form:

U(t)= e(t) + v(1)e(t-1) + v(2)e(t-2) ++ v(q) e(t-q)

The autoregressive and moving average specifications can be

combined to form an ARMA (p, q) specification:

Ut = r(1)u(t-1) + r(2)u(t-2) ++ r(p)u(t-p) + e(t) + v(1)e(t-1) +

v(2)e(t-2) + + v(q) e(t-q)

In ARMA forecasting, we assemble a complete forecasting model by

using combinations of the three building blocks described above.

We have used single step moving average in ARMA forecasting so that

model remains simplified and can be better understood.

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Forecasting Method in ARMA

We have a choice between Dynamic and Static forecast methods.Dynamic calculates dynamic, multi-step forecasts starting from the

first period in the forecast sample. In dynamic forecasting, previously

forecasted values for the lagged dependent variables are used in

forming forecasts of the current value.This choice will only be

available when the estimated equation contains dynamic components,

e.g., lagged dependent variables or ARMA terms.

Static calculates a sequence of one-step ahead forecasts, using the

actual, rather than forecasted values for lagged dependent variables, ifavailable.In addition, in specifications that contain ARMA terms, we

can set the Structural option, instructing EViews to ignore any ARMA

terms in the equation when forecasting. By default, when our equation

has ARMA terms, both dynamic and static solution methods form

forecasts of the residuals. If we select Structural, all forecasts will

ignore the forecasted residuals and will form predictions using only the

structural part of the ARMA specification.

Output: We can choose to see the forecast output as a graph or anumerical forecast evaluation, or both. Forecast evaluation is only

available if the forecast sample includes observations for which the

dependent variable is observed.

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ARMA Forecasting Result for Sensex

Dependent Variable: SENSEX

Method: Least SquaresDate: 01/08/12 Time: 16:14

Sample (adjusted): 1/15/1993 12/30/2011


Convergence achieved after 35 iterations

MA Backcast: 1/01/1993 1/14/1993


C 0.000381 0.000246 1.549163 0.1214

AR(1) -0.033486 0.086754 -0.385991 0.6995

AR(6) -0.218765 0.082848 -2.640571 0.0083

AR(9) 0.009212 0.116309 0.079202 0.9369

AR(10) -0.307348 0.087197 -3.524747 0.0004

MA(1) 0.135130 0.085907 1.572968 0.1158MA(6) 0.169216 0.081561 2.074724 0.0381

MA(9) 0.030033 0.115302 0.260474 0.7945

MA(10) 0.355058 0.087362 4.064228 0.0000

R-squared 0.019344 Mean dependent var 0.000382





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Inverted AR Roots .84+.31i .84-.31i .52-.68i .52+.68i

-.00-.93i -.00+.93i -.52-.68i -.52+.68i

-.85+.32i -.85-.32i

Inverted MA Roots .85-.31i .85+.31i .52+.71i .52-.71i

-.02-.93i -.02+.93i -.55-.69i -.55+.69i

-.87+.30i -.87-.30i

Akaike info criterion is coming -5.44, which is good as lower the

AIC value, better is the model fit but here in ARMA model all

variables are insignificant, so there may be some nonlinear

relationship present in data.

Arma Forecast Model for Sensex

-.06

-.04

-.02

.00

.02

.04

.06

1994 1996 1998 2000 2002 2004 2006 2008 2010

SENSEXF 2 S.E.

Forecast: SENSEXF

Actual: SENSEX

Forecast sample: 1/01/1993 12/30/201

Adjusted sample: 1/15/1993 12/30/20

Included observations: 4946Root Mean Squared Error 0.015848

Mean Absolute Error 0.011417

Mean Abs. Percent Error 155.6666

Theil Inequality Coefficient 0.867555

Bias Proportion 0.000000

Variance Proportion 0.755708

Covariance Proportion 0.244292

We can infer from forecast model that root mean square error is

1.584%. Root mean square error of 1.584% means that forecasted

values can be predicted at a maximum error of 1.584%.

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ARMA Forecasting Result for BSE 500

Dependent Variable: BSE_500


Date: 01/08/12 Time: 16:08

Sample (adjusted): 2/18/1999 12/30/2011Included observations: 3357 after adjustments


MA Backcast: 2/01/1999 2/17/1999


C 0.000467 0.000399 1.171028 0.2417

AR(1) 0.210507 0.102465 2.054421 0.0400

AR(6) 0.055305 0.086885 0.636533 0.5245

AR(9) 0.156133 0.145528 1.072871 0.2834

AR(10) -0.118975 0.103382 -1.150823 0.2499

AR(13) 0.138227 0.085016 1.625891 0.1041

MA(1) -0.070322 0.104051 -0.675844 0.4992MA(6) -0.100956 0.088437 -1.141558 0.2537

MA(9) -0.117499 0.145681 -0.806550 0.4200

MA(10) 0.161917 0.095586 1.693930 0.0904

MA(13) -0.103646 0.086914 -1.192522 0.2331





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Inverted AR Roots .89 .75-.38i .75+.38i .54+.70i

.54-.70i .12+.89i .12-.89i -.35+.77i

-.35-.77i -.54+.56i -.54-.56i -.86-.25i

-.86+.25i

Inverted MA Roots .84 .73+.34i .73-.34i .53-.72i

.53+.72i .09-.87i .09+.87i -.39+.73i

-.39-.73i -.49-.57i -.49+.57i -.86+.25i

-.86-.25i

Akaike info criterion is coming -5.33 , which is good as lower the

AIC value , better model fit but here in ARMA model some

variables are insignificant , so there may be some nonlinear

relationship present in data.

Arma Forecast Model for BSE 500

-.08

-.06

-.04

-.02

.00

.02

.04

.06

.08

99 00 01 02 03 04 05 06 07 08 09 10 11

BSE_500F 2 S.E.

Forecast: BSE_500F

Actual: BSE_500

Forecast sample: 2/01/1999 12/30/2011

Adjusted sample: 2/18/1999 12/30/2011

Included observations: 3357

Root Mean Squared Error 0.016758Mean Absolute Error 0.011778






Root mean square error is 1.675%. Root mean square error of 1.675%

means that forecasted values can be predicted at a maximum error of

1.675%.

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BDS Test

This test is applied to a series of estimated residuals to

check whether the residuals are independent and

identically distributed. The residuals from an ARMA model

will be tested to see if there is any non linear dependence

in the series after the linear ARMA model is fitted. Null

hypothesis of BDS test being there is linear dependence in

series.

After this test, if we found some non linear dependencethen we can go for ARCH family models as they are mostfrequently used models in financial markets and they havebeen used in other papers (O. M. Al-Khazali etal, TheFinancial Review42 (2007)303317; R.K.Mishra etal,Review of Financial Economics20 (2011)96104) to predictthe indices in financial markets.

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BDS Test result for Sensex

BDS Test for RESID

Date: 01/09/12 Time: 12:27

Sample: 1/01/1993 12/30/2011


Dimension BDS Statistic Std. Error z-Statistic Prob.

2 0.023111 0.001232 18.75670 0.0000

3 0.044565 0.001955 22.79769 0.0000

4 0.059627 0.002324 25.65820 0.0000

5 0.068188 0.002418 28.19865 0.0000

6 0.071682 0.002328 30.78959 0.0000

Raw epsilon 0.020839

Pairs within epsilon 17267499 V-Statistic 0.703302

Triples within epsilon 6.55E+10 V-Statistic 0.537996

Dimension C(m,n) c(m,n) C(1,n-(m-1)) c(1,n-(m-1)) c(1,n-(m-1))^k

2 6349988. 0.517581 8627095. 0.703186 0.494471

3 4810302. 0.392241 8623378. 0.703167 0.347677

4 3727274. 0.304052 8619459. 0.703131 0.244425

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5 2940904. 0.240001 8615474. 0.703090 0.171813

6 2357819. 0.192494 8612136. 0.703102 0.120812

Here z- Statistic are significant, so there is some non linearitydependence present in series, Therefore we will go for ARCHfamily of models to predict index as they are most frequentlyused models in financial forecasting.

BDS Test result for BSE 500

BDS Test for RESID

Date: 01/09/12 Time: 12:28

Sample: 2/01/1999 12/30/2011


Dimension BDS Statistic Std. Error z-Statistic Prob.

2 0.029291 0.001598 18.33139 0.0000

3 0.057532 0.002538 22.66649 0.0000

4 0.078215 0.003021 25.88649 0.0000

5 0.089849 0.003148 28.53889 0.0000

6 0.095245 0.003035 31.37797 0.0000

Raw epsilon 0.021487

Pairs within epsilon 7987415. V-Statistic 0.703727

Triples within epsilon 2.07E+10 V-Statistic 0.541597

Dimension C(m,n) c(m,n) C(1,n-(m-1)) c(1,n-(m-1)) c(1,n-(m-1))^k

2 2972662. 0.524276 3989159. 0.703552 0.494985

3 2298977. 0.405702 3986495. 0.703500 0.348171

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4 1829772. 0.323093 3983885. 0.703457 0.244878

5 1483129. 0.262040 3981191. 0.703399 0.172191

6 1223639. 0.216322 3978594. 0.703359 0.121077

Here z-Statistic are significant , so there is some non linearitydependence present in series, Therefore we will go for ARCHfamily of models to predict index as they are most frequentlyused models in financial forecasting.

GARCH Specifications

In developing an GARCH model, we will have to provide three distinct

specificationsone for the conditional mean equation, one for the

conditional variance, and one for the conditional error distribution.

The GARCH (q, p) Model

Higher order GARCH models, denoted GARCH (q, p), can be estimated

by choosing either q or p greater than 1 where q is the order of the

autoregressive GARCH terms and p is the order of the moving average

ARCH terms.

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We are using GARCH (1, 1) model here as it covers all ate ARCH

models with up to infinity lags. So rather than using an ARCH model

with many lags, we are using GARCH (1, 1) model.

GARCH Model in EViews

In the dependent variable edit box, we entered the specification of the

mean equation. We can enter the specification in list form by listing

the dependent variable followed by the regressors. We should add the

C to our specification if we wish to include a constant. If we have a

more complex mean specification, we can enter our mean equation

using an explicit expression.If your specification includes an ARCH-M

term, we should select the appropriate item of the combo box.To

estimate one of the standard GARCH models as described above,

select the GARCH/TARCH entry in the Model combo box.

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Garch Model for Sensex

Dependent Variable: SENSEX

Method: ML - ARCH (Marquardt) - Normal distribution

Date: 01/08/12 Time: 19:29

Sample (adjusted): 1/04/1993 12/30/2011



Presample variance: backcast (parameter = 0.7)

GARCH = C(3) + C(4)*RESID(-1)^2 + C(5)*GARCH(-1)

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Variable Coefficient Std. Error z-Statistic Prob.

C 0.000775 0.000163 4.744733 0.0000

SENSEX(-1) 0.128803 0.013622 9.455267 0.0000

Variance Equation

C 6.88E-06 7.24E-07 9.499339 0.0000

RESID(-1)^2 0.125949 0.006637 18.97682 0.0000

GARCH(-1) 0.852471 0.006858 124.3066 0.0000






Durbin-Watson stat 2.037897

Here Akaike info criterion value is -5.66 which is low and all

variable values are also significant, so this is a good model to

predict Sensex.

GARCH Forecast Model of Sensex

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-.15

-.10

-.05

.00

.05

.10

.15

94 96 98 00 02 04 06 08 10

SENSEXF 2 S.E.

Forecast: SENSEXF

Actual: SENSEX

Forecast sample: 1/01/1993 12/30/2011

Adjusted sample: 1/04/1993 12/30/2011


Root Mean Squared Error 0.015922

Mean Absolute Error 0.011422Mean Abs. Percent Error 146.9406





.000

.001

.002

.003

.004

94 96 98 00 02 04 06 08 10

Forecast of Variance

Inference: This is forecasted model of Sensex by Garch method. Here

it states that root mean square error is coming out to be around

1.59%. It means that sensex can be predicted at an error of 1.59%.

Garch model for BSE 500

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Dependent Variable: BSE_500

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Method: ML - ARCH (Marquardt) - Normal distribution

Date: 01/08/12 Time: 19:31

Sample (adjusted): 2/02/1999 12/30/2011



Presample variance: backcast (parameter = 0.7)

GARCH = C(3) + C(4)*RESID(-1)^2 + C(5)*GARCH(-1)

Variable Coefficient Std. Error z-Statistic Prob.

C 0.001172 0.000213 5.508114 0.0000

BSE_500(-1) 0.129917 0.017668 7.353387 0.0000

Variance Equation

C 7.37E-06 8.50E-07 8.678975 0.0000

RESID(-1)^2 0.160595 0.010647 15.08312 0.0000

GARCH(-1) 0.821913 0.010347 79.43243 0.0000


Adjusted R-squared 0.016842 S.D. dependent var 0.017018S.E. of regression 0.016874 Akaike info criterion -5.641372



Durbin-Watson stat 1.981550

Here Akaike info criterion value is -5.64 which is low and all

variable values are also significant, so this is a good model to

predict Sensex.

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GARCH Forecast Model of BSE 500

-.15

-.10

-.05

.00

.05

.10

.15

99 00 01 02 03 04 05 06 07 08 09 10 11

BSE_500F 2 S.E.

Forecast: BSE_500F

Actual: BSE_500

Forecast sample: 2/01/1999 12/30/2011

Adjusted sample: 2/02/1999 12/30/2011Included observations: 3369

Root Mean Squared Error 0.016869

Mean Absolute Error 0.011776






.000

.001

.002

.003

.004

.005

99 00 01 02 03 04 05 06 07 08 09 10 11

Forecast of Variance

Inference: This is Garch forecasted model of Sensex. Here it states

that root mean square error is coming out to be around 1.68%.It

means that BSE 500 can be predicted with an error of 1.68%.

Conclusion

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Akaike info

criterion

Sensex BSE 500

ARMA -5.44 -5.33

GARCH -5.66 -5.64

This study has examined the time series behavior of spot

price based daily returns of equity indices for Indian market

by using tests of independence, nonlinearity. In short,

consistent with the findings of many previous studies, for

example Abhyankar etal.(1995,1997) among others, results

of this study reveal that there is a strong evidence of

nonlinear dependence

in daily increments of all equity indices analyzed. The

existing nonlinearity in the data seems to be multiplicative in

nature. This implies that nonlinearity is transmitted only

through the variance of the process.

More precisely, the results of variance ratio test suggest that

the null hypothesis of random walk is strongly rejected for

both the return series. It appears, therefore, that daily

increment in stock returns are highly auto-

correlated. Also by BDS test We can say that there is some

nonlinearity present in data , therefore GARCH model will be

used to predict the indices.

Clearly GARCH model has smaller Akaike info criterion value

as compared to ARMA model. Also variable values are

significant in GARCH model and not all variable values are

significant in ARMA model. Therefore we will use GARCH

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(also other higher ARCH family models) model to predict

both Sensex and BSE 500 indices.

Bibliography

1. Bugak and Brorsen Report, 2003

2. Belaire Franch and Opong Report, 2005b

3. Hoque, Kim, and Pyun Report, 2007.4. O. M. Al- Khazali etal , The Financial Review42(2007)303317

5. R. K. Mishra etal , Review of Financial Economics20(2011)96104

6. BSE Website

7. EViews Software User Guide I

8. EViews Software User Guide II

9. Basic Econometrics by Damodar N Gujarati

10. Wikipedia.com

Predictability of Stock Indices

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