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Transcript of 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)
Method: Least Squares
Date: 01/08/12 Time: 16:48
Sample (adjusted): 2/02/1999 12/30/2011
Included observations: 3369 after adjustments
Variable Coefficient Std. Error t-Statistic Prob.
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
Sum squared resid 0.956696 Schwarz criterion -5.323942
Log likelihood 8976.302 Hannan-Quinn criter. -5.326276
F-statistic 2551.351 Durbin-Watson stat 2.003117
Prob(F-statistic) 0.000000
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
Included observations: 4946 after adjustments
Convergence achieved after 35 iterations
MA Backcast: 1/01/1993 1/14/1993
Variable Coefficient Std. Error t-Statistic Prob.
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
Adjusted R-squared 0.017755 S.D. dependent var 0.016005
S.E. of regression 0.015862 Akaike info criterion -5.447964
Sum squared resid 1.242163 Schwarz criterion -5.436125
Log likelihood 13481.81 Hannan-Quinn criter. -5.443812
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F-statistic 12.17300 Durbin-Watson stat 1.991289
Prob(F-statistic) 0.000000
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
Method: Least Squares
Date: 01/08/12 Time: 16:08
Sample (adjusted): 2/18/1999 12/30/2011Included observations: 3357 after adjustments
Convergence achieved after 17 iterations
MA Backcast: 2/01/1999 2/17/1999
Variable Coefficient Std. Error t-Statistic Prob.
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
R-squared 0.031263 Mean dependent var 0.000474
Adjusted R-squared 0.028367 S.D. dependent var 0.017029
S.E. of regression 0.016786 Akaike info criterion -5.333304
Sum squared resid 0.942772 Schwarz criterion -5.313254
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Log likelihood 8962.950 Hannan-Quinn criter. -5.326133
F-statistic 10.79807 Durbin-Watson stat 2.010582
Prob(F-statistic) 0.000000
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
Mean Abs. Percent Error 189.0695
Theil Inequality Coefficient 0.834825
Bias Proportion 0.000000
Variance Proportion 0.700192
Covariance Proportion 0.299808
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
Included observations: 4956
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
Included observations: 3370
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
Included observations: 4955 after adjustments
Convergence achieved after 15 iterations
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
R-squared 0.009715 Mean dependent var 0.000372
Adjusted R-squared 0.009515 S.D. dependent var 0.016002
S.E. of regression 0.015925 Akaike info criterion -5.661515
Sum squared resid 1.256153 Schwarz criterion -5.654948
Log likelihood 14031.40 Hannan-Quinn criter. -5.659213
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
Included observations: 4955
Root Mean Squared Error 0.015922
Mean Absolute Error 0.011422Mean Abs. Percent Error 146.9406
Theil Inequality Coefficient 0.873712
Bias Proportion 0.000802
Variance Proportion 0.766432
Covariance Proportion 0.232766
.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
Included observations: 3369 after adjustments
Convergence achieved after 16 iterations
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
R-squared 0.017134 Mean dependent var 0.000473
Adjusted R-squared 0.016842 S.D. dependent var 0.017018S.E. of regression 0.016874 Akaike info criterion -5.641372
Sum squared resid 0.958727 Schwarz criterion -5.632286
Log likelihood 9507.892 Hannan-Quinn criter. -5.638123
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
Mean Abs. Percent Error 182.1879
Theil Inequality Coefficient 0.862610
Bias Proportion 0.002042
Variance Proportion 0.770079
Covariance Proportion 0.227879
.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