Stationarity of NSE indices: a Bayesian approach.

Kumar, Jitendra

Abstract

National Stock Exchange (NSE) is the largest exchange market in India in terms of turnover and declaring sixteen indices in reference of different sectors and company profile. Time series approach is most popular way of analyzing the economic series. Stationarity of time series is an important issue before estimating the parameters of data generating process. In this paper, stationarity under Bayesian framework is tested for NSE closing index. While modeling the indices and applying Bayesian unit root tests, the possibility of presence of linear/nonlinear trend and structural break is also explored.

1. INTRODUCTION

National Stock Exchange (NSE) is the largest Indian exchange in terms of turnover and declaring sixteen indices in reference of different sectors and company profile like S & P CNX Nifty, CNX Nifty Junior, S & P 500 Nifty Midcap 50, S & P CNX dirty etc. The NSE index a stochastic process and changing in regular manner as the market policy, international market and companies profile changes. Data generating processes which are time dependent and NSE indices were regularly analyzed by market researchers to know the market conditions as the time passed with a particular exchange. The growth of indices is much impressive as it become old and time series is a better way for the analysis of indices values because of its popularity.

For modeling data generating process of various economic time series, the box-Jenkins approach (see Box and Jenkins, 1970) uses various stochastic processes like autoregressive (AR), moving average (MA), mixed ARMA or autoregressive integrated moving (ARIMA). A number of researches' are analyzing these in different ways and stationary or non-stationary under consideration of with or without seasonality is an important issue because of its use, applicability and popularity.

The time series may be non-stationary because of time trend or due to unit root. The presence of unit root in economic time series has profound implication for both statistical analysis and economic theorizing. The use of data characterized by unit root may be a cause of misleading conclusion see Dickey and Fuller (1997, 1981), Perron (1989, 1990, 1992), Sims (1988), Sims and Uhling (1991), Madala and Kim (1998) Hassegawa et al. (2000) and Chaturvedi and Jitendra (2005, 2007), Jitendra and Shukla (2009), Jitendra at el (2010), Rishi at el (2010) etc. If an economic time series contains a unit root, the shocks to it will not dissipate but instead will have permanent effect and lead to far reaching implications in policy making.

A long economic time series showing shift in trend is called structural changes in the intercept and/or slope parameter of the trend component or error variance. Such structural changes may occur due to the changes in national and international economic policies, due to the behavior of market players, integration of two or more economies like European Union etc. However, the inferences are taken if such models as structural changes in the parameters are not taken into account may be misleading. For detailed discussion about the unit root hypothesis one can refer Christiano (1988), Zivot and Andrews (1992), Andrews (1993), Bai (1996), and Lin and Yang (1999). Chaturvedi and Jitendra (2005).

The present paper considers Bayesian analysis of NSE closing indices for the period April 2004 to March 2009. The data is taken from the online data source of National Stock Exchange. Since classical unit root tests like Dickey-Fuller test are often more biased towards the acceptance of unit root hypothesis and suffer from size distortion, Bayesian approach is used for testing the difference stationarity against trend stationarity.

2. STATIONARY PROCESSES AND UNIT ROOT

Let the index values starting from April 2004 in month t follows with AR(1) process with disturbances [u.sub.t]:

[Index.sub.t] = Trend + [Index.sub.t-1] + [u.sub.t] (t = 1, 2, ......, T) (1)

Where [u.sub.t] = [[rho].sub.ut-1] + [[epsilon].sub.t] and is the autoregressive parameter and is the disturbances term. In the above model, if the autoregressive parameter, model is said to possess a unit root. If an Index series has a unit root, the past shocks will have permanent effect and the usual estimation and testing procedure cannot be applied. Further, the variance of the series explodes to infinity.

If the trend is free from time, it is called intercept trend. The time trend may be linear or non-linear; the non-linearity of time trend is incorporated by partial time trend, which is a combination of linear and non-linear functions of time.

It is a usual phenomenon to observe structural break in trend component for a long time series, the model is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

While applying unit root tests, if structure breaks are not taken into account, it is possible that even a stationary time series may show presence of unit root. Perron (1989, 1990) developed the classical test for testing the unit root hypothesis against structural break with one known break point and tabulated the percentage points of augmented Dickey-Fuller test. Silvapulle and Maddock (1992) tabulated the percentage points of Lagrange multiplier test proposed by Kwaitkowski et al. (1992). Silapulle (1995) considers the ADF and LM unit root test for the time series having two break points and applied the results to extended Nelson-Plosser macroeconomic time series. Chaturvedi and Kumar (2007) derived the posterior odds ratio for a AR(1) time series having break in trend component and studied the impact of misspecification of break as linear time trend by simulated data.

Sometimes break occurs due to the variance in error and if break in variance is taken into account the model is as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

The classical tests are largely based on asymptotic justification that parameters are constant and often lead to low power of the test, particularly in finite samples. On the other hand, Bayesian approach is free from such problem. Therefore, it provides a more convenient and formal framework. In Bayesian framework, the decision of difference stationarity against the alternative of trend stationarity taken by the use of posterior odds ratio, which is the prior odds ratio of two hypotheses multiplied by the ratio of predictive densities under null hypothesis of difference stationarity and predictive density under the alternative of trend stationarity. If the posterior odd's ratio is less than one, we reject the null hypothesis of unit root, otherwise accept it.

For testing the unit root hypothesis, the posterior odds ratio is used as reported by Jitendra et al. (2010) and Rishi et al. (2010), using following notations.

c([lambda],[rho]) = [T.sub.B] [(1 - [rho]).sup.2] + T - [T.sub.B]/[lambda] [(1 - [rho]).sup.2] + (1 - [[rho].sup.2])

M([rho]) = [T.sub.B] [(1 - [rho]).sup.2] + (1 - [[rho].sup.2]),

N([rho]) = (T - [T.sub.B] [(1 - [rho]).sup.2] + (1 - [[rho].sup.2]),

R([rho]) = (1 - [rho]) [[T.sub.B].summation over (t=1)] ([y.sub.t] - [[rho]y.sub.t-1]) + [y.sub.0](1 - [[rho].sup.2]),

P([rho]) = (1 - [rho]) [T.summation over (t=[T.sub.B]+1)] ([y.sub.t] - [[rho]y.sub.t-1]) + [y.sub.0](1 - [[rho].sup.2]),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

The following theorem was reported

Theorem 1: The posterior odds ratio, denoted by [[beta].sub.1], for testing the unit root hypothesis in which break occur at known break point [T.sub.B] with prior odds ratio [theta]/(1-[theta), is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

Theorem 2: The posterior odds ratio, denoted by [[beta].sub.2], for testing the unit root hypothesis [H.sub.0] : [rho] = 1 against [H.sub.2] : [rho] < 1, for the model with single known break point in intercept and prior odds ratio [[theta].sub.0]/(1-[[theta].sub.0]), is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

Theorem 3: The posterior odds ratio, denoted by [[beta].sub.3], for testing the unit root hypothesis [H.sub.0]: [rho] = 1, [lambda] = 1 against [H.sub.3] : [rho] < 1, [lambda] = 1 for the model without break and prior odds ratio [[theta].sub.0]/(1-[[theta].sub.0]), is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Theorem 4: The posterior odds ratio for testing the unit root hypothesis for a time series model involving partial linear time trend with prior odds ration [p.sub.0]/(1-[P.sub.0]) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

3. EMPIRICAL ANALYSIS OF NATIONAL STOCK EXCHANGE INDICES

The NSE declares fourteen Indices based on different sectors and funds as S & P Nifty, S & P Defty, CNX Nifty Junior, CNX IT, S & CNX 500, Bank Nifty, CNX Midcap, CNX 100, CNX Infrastructure, Nifty Midcap 50, S & P ESG India Index, CNX Reality, S & P CNX 500 Shariah and S & P CNX Nifty Shariah. The average closing price of declared indices for the period April 2004 to March 2009 from the online data source of National Stock Exchange is considered. Monthly closing values are shown in figure I and summary statistics of different indices are given in table 1.

[FIGURE 1 OMITTED]

The Indices values are assumed to follow the following model

[Index.sub.t] = Intercept(1-[rho]) + [rho] [Index.sub.t-1] + [[epsilon].sub.t] (9)

and stationarity of closing index value is concluded on the basis of autoregressive coefficient. If [rho] = 1, series is difference stationary and if [rho] [member of] = {(a,1);a >-1}, series is non-trend stationary. When only intercept is considered in trend, we found that indices series are difference stationary, whereas estimated value of [rho] is less than 1.

For a long time series, occurrence of break in trend component is common behavior, which happens due to the performance of companies listed in index, change in government market policy, shifting of international market strategy. Let us assume that there is structural break in intercept. As the current recession starts in late year 2007, we can assume the time this structural break in the series as December 2007.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (10)

Indices series CNX Realty, S&P CNX 500 Shariah and S&S CNX Nifty Shariah are non-trend stationary and rest are difference stationary.

When, the analysis of indices is carried out under consideration of break in variance (using model 3) at same break point, the series are found to be non-trend stationary. The value of posterior odds ratio also decreases for all index series. The values of posterior odds ratio are reported in table given below. The main disadvantage by the use of these models is that the estimated value of autoregressive coefficient is less than one, but unit root hypothesis is accepted.

In all the above three models, time trend are not taken and for a time series, time is an important variable which influences the change on the index value. Let Index follows the AR(1) time series model with linear time trend

[Index.sub.t] = Intercept + slope t + [rho] [Index.sub.t-1] + [u.sub.t] (11)

Using this model the analysis is worked out and posterior odds ratio is evaluated. The value of which is more than one in most of closing series and less than one in the series as Bank Nifty, Nifty Midcap 50 and S&P ESG India index. These series are trend stationary and rests are difference stationary. The estimated value of coefficient of determination is also good.

In the figure shows the closing series, see that trend is not simply linear therefore the analysis is continued under consideration of quadratic time trend. The series follows the model

[Index.sub.t] = [Intercept.sup.[rho]] + [Linear.sup.[rho]] t + [quadratic.sup.[rho]] [t.sup.2] + [rho] [Indext.sub.t-1] + [[epsilon].sub.t] (12)

Table 3 showed the posterior odds ratio, coefficient of determination, least square estimate of autoregressive parameter and coefficient of quadratic time trend and their standard error. The coefficient of determination is reduced for all the cases as compared to linear time trend. The unit root hypothesis is rejected for all the cases and we can conclude that closing series are trend stationary. The estimated value of autoregressive parameter being less than one followed the testing conclusion.

The posterior odds ratio is less than one in all cases so all the series are trend stationary. In modeling the closing index, the non-linear trend component is incorporated in terms of partial time trend g(t). Different forms of g(t) are taken into consideration. Some popular exponential and logarithmic combinations of t have also been taken. Let index follows the time series model

[Index.sub.t] = [Intercept.sup.[rho]] + [Linear.sup.[rho]] t + [Partial.sup.[rho]] g(t)+[rho] [Index.sub.t-1] + [[epsilon].sub.t] (13)

Table 5-8 provides posterior odds ratio, estimated value of coefficient of autoregressive parameter and coefficient under consideration of non-linear time trend (i) g(t) = t*log(t), (ii) g(t) = log(t), (iii) g(t) = t*exp(t) and (iv) g(t) = exp(t). On the basis of posterior odds ratio, it is concluded that S&P CNX Defty, Bank Nifty and CNX Infrastructure are trend stationary and rest are difference stationary when g(t) = t*g(t). The coefficient of determination increases for the index series CNX Realty, S&P CNX 500 Shariah and S&P CNX Nifty Shariah in comparison to quadratic trend. When non-linear time trend is g(t) = log(t), all index are concluded as trend stationary but [R.sup.2] decreases except CNX Realty, S&P CNX 500 Shariah and S&P CNX Nifty Shariah. The non-linearity is also taken into account in the form of g(t) = t*exp(t) and g(t) = exp(t). S&P CNX Defty is difference stationary and rest are trend stationary but coefficient of determination decreases.

The present paper explored the unit root test to know whether NSE indices are stationary or not under Bayesian framework. While modeling the NSE indices, inclusion of non-linear time trend coefficient of determination is decreasing in comparison to linear time trend. The maximum coefficients of determination is achieved with quadratic time trend in comparison to trends g(t)= t*log(t), log(t), t*exp(t) and exp(t) for the series S&P CNX Nifty, S&P CNX Defty, CNX Nifty Junior, CNX IT, S&P CNX 500, Bank Nifty, CNX Midcap, CNX 100, CNX Infrastructure, Nifty Midcap 50, S&P ESG India Index and CNX Realty with g(t)=log(t), S&P CNX 500 SHARIAH and S&P CNX NIFTY SHARIAH with g(t)=t*log(t) with compared to other non-linear time trend. When linear time trend is taken into account, the estimated value of autoregressive coefficient is found to be more than one except for Bank Nifty, Nifty Midcap 50 and S&P ESG India Index, but indices are concluded difference stationary. The non-linearity is taken into account in the form of time function g(t) and it is concluded that the NSE indices are trend stationary with significant observation of coefficient of determination. As achieving the stationarity in modeling the time series is an important issue therefore inclusion of non-linear time trend is important for modeling National Stock Indices.

Acknowledgement

The author gratefully acknowledges the financial support from CST to carry out the present work.

References

Andrews, D. W. K. (1993), Tests for Parametric Instability and Structural Change with Unknown Change Point, Econometrica, 61, 821-856.

Bai, J. (1996), Testing for Parametric Constancy in Linear Regression, An Empirical Distribution Approach, Econometrica, 64, 597-622.

Box. G. E. P and G. M. Jenkins (1970), Time Series Analysis Forecasting and Control, Holden- Day, San Francisco.

Chaturvedi, A. and Jitendra Kumar (2005), Bayesian Unit Root Test for Time Series Model with Maintained Trend, Statistics & Probability, Letters, 74, 109-115.

Chaturvedi, A. and Jitendra Kumar (2007), Bayesian Unit Root Test for Time Series Models with Structural Breaks, American Journal of Mathematical and Management Sciences, Vol. 1& 2, 243-268.

Christiano, L. I. (1988), Searching for a Break in G.N.D, working paper No. 2695, National Bureau of Economic Research, Cambridge, M.A.

Dickey, D. A. and W. A. Fuller (1979), Distribution of the Estimators for Autoregressive Time

Series with a Unit Root, Journal of the American Statistical Association, 74, 427-431.

Dickey, D. A. and W. A. Fuller (1981), Likelihood Ratio Tests for Autoregressive Time Series with a Unit Root, Econometrica, 49, 1057-1072.

Hasegawa, H., A. Chaturvedi and Tran Van Hoa (2000), Bayesian Unit root Test in Non-normal AR (1) Model, Journal of Time Series Analysis, 31, 261-280.

Jitendra Kumar and A. Shukla (2009), Unit Root Test of Autoregressive Time Series Model with Partially Linear Time Trend: A Bayesian Approach, Journal of International Academy of Physical Sciences, Vol. 13 No. 4, 1-11.

Jitendra Kumar, A. Shukla and A. Chaturvedi (2010), Bayesian Analysis of Exchange Rates with Partially Linear Time Trend, International Journal of Statistics and Systems, Vol. 5, No. 3,439-453.

Kwaitkowski, D., P.C.B. Phillips, P. J. Schmidt and Y. Shin (1992), Testing the Null Hypothesis of Stationarity against the Alternative of a Unit Root: How Sure are we that Economic Time Series have a Unit Root, Journal of Econometrics, 54, 159-178.

Lin, S. and J. Yang (1999), Testing Shifts in Financial Market with Conditional Heteroskedasticity: An Empirical Distribution Financial Approach, Manuscript, Department of Economics, University of Western Ontario.

Maddala, G. S. and, I. M. Kim (1999), Unit Roots, Co-integration, and Structural Change, Cambridge University Press.

Perron, P. (1989), The Great Crash, the Oil Price Shock and the unit Root Hypothesis, Econometrica, 57, 1361-1401.

Perron, P. (1990), Testing for a Unit Root in a Time Series with a Changing Mean, Journal of Business and Economic Statistics, 8(2), 153-162.

Perron P. (1992), Trend Unit Root Structural Change: A Multi-country Study with Historical Data, Proceedings of the American Statistical Association, Business and Economic Statistics Section, 144-149.

Rishi Kumar, Kumar Jitendra and A. Chaturvedi (2009), "Bayesian Unit Root Test for Time Series Models with Structural Break in Variance" IMST-FIM XVIII, Jaypee University of Information Technology, Waknaghat, Distt. Solan, Himanchal Pradesh.

Sims, C. A. (1988), Bayesian Skepticism on Unit Root Econometrics, Journal of Economic Dynamics and Control, 12, 463-474.

Sims, C. A. and H. Uhling (1991), Understanding unit Rooters: A Helicopter Tour, Econometrica, 59, 1591-1599.

Silvapulle, P. and R. Maddock (1992), Unit root and Structural Break, Australian Economic Society Meeting, Monash University.

Silvapulle, P. (1995), Unit Root Tests are Structural Breaks, Discussion paper, La Trode University Australia.

Zivot, E. and D. W. K. Andrews (1992), Further Evidence the Great Crash, the Oil Price Shock and the Unit Root Hypothesis, Journal of Business and Economics Statistics, 10, 251-270.

Zivot, E. and P.C.B. Phillips (1994), A Bayesian Analysis of Trend Determination in Economic Time Series, Economica, 13, 291-336.

JITENDRA KUMAR, Institute for Development and Research in Banking Technology Castle Hills Road #1, Masab Tank, Hyderabad-500057, AP, India

Table 1
Summary Statistics

                          Mean       SE    Kurtosis   Skewness

S&P CNX NIFTY           3676.55   150.82      -0.61       0.38
S&P CNX DEITY           2950.99   145.74      -0.51       0.64
CNX NIFTY JUNIOR        6687.42   293.69       0.18       0.80
CNX IT                  3978.75   140.33      -0.79      -0.29
S&P CNX 500             3052.43   124.60      -0.32       0.55
BANK NIFTY              5547.74   237.65       0.59       1.05
CNX MIDCAP              4938.99   207.22       0.06       0.75
CNX 100                 3563.57   146.71      -0.52       0.46
CNX INFRASTRUCTURE      3016.51   167.24      -0.08       0.74
NIFTY MIDCAP 50         2038.40    84.38       0.40       0.62
S&P ESG INDIA INDEX     1749.67    68.59      -0.53       0.48
CNX REALTY               835.12    76.25      -0.63      -0.11
S&P CNX 500 SHARIAH     1057.85    48.80      -0.54      -0.17
S&P CNX NIFTY SHARIAH   1056.94    45.18      -0.57      -0.27

                        Minimum   Maximum

S&P CNX NIFTY           1987.10    5963.57
S&P CNX DEITY           1573.89    5246.61
CNX NIFTY JUNIOR        3962.72   12001.50
CNX IT                  2141.84    5625.53
S&P CNX 500             1751.74    5123.76
BANK NIFTY              3389.29    9906.63
CNX MIDCAP              2951.33    8671.24
CNX 100                 1971.33    5865.67
CNX INFRASTRUCTURE      1346.16    5819.42
NIFTY MIDCAP 50         1076.11    3619.35
S&P ESG INDIA INDEX      960.58    2839.49
CNX REALTY               175.74    1596.74
S&P CNX 500 SHARIAH      638.46    1528.40
S&P CNX NIFTY SHARIAH    664.43    1470.34

Table 2
Model with Intercept

                        [[rho].sub.Estimate]   Stationarity

S&P CNX NIFTY                         0.9372         2.2306
S&P CNX DEFTY                         0.9520         2.2631
CNX NIFTY JUNIOR                      0.9423         1.5608
CNX IT                                0.9726         1.6806
S&P CNX 500                           0.9379         1.9513
BANK NIFTY                            0.9330         1.4964
CNX MIDCAP                            0.9348         1.5958
CNX 100                               0.9382         2.1409
CNX INFRASTRUCTURE                    0.9462         2.7163
NIFTY MIDCAP 50                       0.9338         1.3673
S&P ESG INDIA INDEX                   0.9248         2.0620
CNX REALTY                            0.9881         1.9908
S&P CNX 500 SHARIAH                   0.9805         1.2055
S&P CNX NIFTY SHARIAH                 0.9811         1.1871

                        Break in mean   Break in variance

S&P CNX NIFTY                  2.9571            6.80E-09
S&P CNX DEFTY                  2.7348            9.25E-06
CNX NIFTY JUNIOR               1.8304            4.30E-01
CNX IT                         1.6061            1.27E-28
S&P CNX 500                    2.4765            1.62E-02
BANK NIFTY                     1.8089            1.94E+02
CNX MIDCAP                     1.9536            6.34E-08
CNX 100                        2.8275            6.01E-09
CNX INFRASTRUCTURE             3.4836            2.03E-05
NIFTY MIDCAP 50                1.5379            1.18E-05
S&P ESG INDIA INDEX            2.8482            3.65E-20
CNX REALTY                     0.4813            5.38E-11
S&P CNX 500 SHARIAH            0.4486            8.77E-12
S&P CNX NIFTY SHARIAH          0.5108            8.87E-11

Table 3
Model with Linear Time Trend

NSE Indices             [POR.sub.Linear]   [R.sup.2]

S&P CNX NIFTY                     1.1582      0.9871
S&P CNX DEFTY                     5.7770      0.9995
CNX NIFTY JUNIOR                  1.0402      0.9742
CNX IT                            1.7774      0.9700
S&P CNX 500                       1.0920      0.9820
BANK NIFTY                        0.8497      0.9729
CNX MIDCAP                        1.0225      0.9767
CNX 100                           1.1645      0.9862
CNX INFRASTRUCTURE                1.3947      0.9921
NIFTY MIDCAP 50                   0.9093      0.9606
S&P ESG INDIA INDEX               0.6940      0.9688
CNX REALTY                        2.4007      0.9120
S&P CNX 500 SHARIAH               1.8871      0.9161
S&P CNX NIFTY SHARIAH             1.8665      0.9137

NSE Indices             [[rho].sub.Linear]     SE(r)

S&P CNX NIFTY                       0.9875   1.74E-04
S&P CNX DEFTY                       0.9999   1.71E-04
CNX NIFTY JUNIOR                    0.9744   7.84E-05
CNX IT                              0.9703   1.56E-04
S&P CNX 500                         0.9824   1.97E-04
BANK NIFTY                          0.9731   1.06E-04
CNX MIDCAP                          0.9770   1.17E-04
CNX 100                             0.9866   1.74E-04
CNX INFRASTRUCTURE                  0.9925   1.60E-04
NIFTY MIDCAP 50                     0.9612   2.67E-04
S&P ESG INDIA INDEX                 0.9697   3.91E-04
CNX REALTY                          0.9134   6.06E-04
S&P CNX 500 SHARIAH                 0.9181   8.52E-04
S&P CNX NIFTY SHARIAH               0.9158   9.25E-04

Table 4
Model with Quadratic Time Trend

                        [POR.sub.Partial]   [R.sup.2]

S&P CNX NIFTY                      0.8759      0.7785
S&P CNX DEFTY                      0.8977      0.7552
CNX NIFTY JUNIOR                   0.6711      0.6629
CNX IT                             0.6311      0.8913
S&P CNX 500                        0.8333      0.7277
BANK NIFTY                         0.6012      0.6328
CNX MIDCAP                         0.7133      0.6514
CNX 100                            0.8502      0.7632
CNX INFRASTRUCTURE                 0.8774      0.7565
NIFTY MIDCAP 50                    0.7115      0.5883
S&P ESG INDIA INDEX                0.8513      0.7030
CNX REALTY                         0.6197      0.7823
S&P CNX 500 SHARIAH                0.8981      0.8236
S&P CNX NIFTY SHARIAH              0.9129      0.8338

                        [r.sub.Estimate]      SE(r)

S&P CNX NIFTY                     0.8665   2.59E-04
S&P CNX DEFTY                     0.9044   2.35E-04
CNX NIFTY JUNIOR                  0.8730   1.08E-04
CNX IT                            0.6706   4.32E-04
S&P CNX 500                       0.8699   2.86E-04
BANK NIFTY                        0.8900   1.29E-04
CNX MIDCAP                        0.8790   1.55E-04
CNX 100                           0.8690   2.57E-04
CNX INFRASTRUCTURE                0.9030   2.12E-04
NIFTY MIDCAP 50                   0.8556   3.66E-04
S&P ESG INDIA INDEX               0.8686   5.25E-04
CNX REALTY                        0.7341   1.02E-03
S&P CNX 500 SHARIAH               0.6868   1.74E-03
S&P CNX NIFTY SHARIAH             0.6702   1.94E-03

                        [C.sub.Quadratic]   SE([C.sub.Quadratic])

S&P CNX NIFTY                     -0.8322              1.32E-03
S&P CNX DEFTY                     -0.7219              1.22E-03
CNX NIFTY JUNIOR                  -1.6761              1.22E-03
CNX IT                            -1.8277              2.46E-03
S&P CNX 500                       -0.6978              1.29E-03
BANK NIFTY                        -1.2219              1.08E-03
CNX MIDCAP                        -1.1329              1.17E-03
CNX 100                           -0.8145              1.31E-03
CNX INFRASTRUCTURE                -0.7584              1.17E-03
NIFTY MIDCAP 50                   -0.5129              1.22E-03
S&P ESG INDIA INDEX               -0.3439              1.19E-03
CNX REALTY                        -1.3040              6.01E-03
S&P CNX 500 SHARIAH               -1.1058              7.26E-03
S&P CNX NIFTY SHARIAH             -1.0727              7.46E-03

Table 5
Model with Partial Time Trend g(t )= t*log(t)

                        [POR.sub.Partial]   [R.sup.2]

S&P CNX NIFTY                      1.1929      0.7346
S&P CNX DEFTY                      0.8877      0.7086
CNX NIFTY JUNIOR                   1.0299      0.5987
CNX IT                             3.3902      0.8675
S&P CNX 500                        1.1613      0.6751
BANK NIFTY                         0.8400      0.5654
CNX MIDCAP                         1.0168      0.5847
CNX 100                            1.1507      0.7166
CNX INFRASTRUCTURE                 0.8833      0.7153
NIFTY MIDCAP 50                    1.2336      0.5133
S&P ESG INDIA INDEX                1.2363      0.6446
CNX REALTY                         1.0364      0.8309
S&P CNX 500 SHARIAH                1.7105      0.8476
S&P CNX NIFTY SHARIAH              1.8509      0.8535

                        [[rho].sub.Estimate]     SE(r)    Partial

S&P CNX NIFTY                         0.9132   2.24E-04   -22.8820
S&P CNX DEFTY                         0.9419   2.07E-04   -20.4600
CNX NIFTY JUNIOR                      0.9127   9.52E-05   -46.8160
CNX IT                                0.7948   3.16E-04   -43.7060
S&P CNX 500                           0.9131   2.50E-04   -19.3130
BANK NIFTY                            0.9239   1.19E-04   -34.9410
CNX MIDCAP                            0.9170   1.40E-04   -31.7870
CNX 100                               0.9145   2.22E-04   -22.4380
CNX INFRASTRUCTURE                    0.9371   1.89E-04   -22.0470
NIFTY MIDCAP 50                       0.8958   3.29E-04   -14.1320
S&P ESG INDIA INDEX                   0.9082   4.72E-04    -9.4759
CNX REALTY                            0.7302   9.02E-04   -32.7150
S&P CNX 500 SHARIAH                   0.7146   1.44E-03   -23.7580
S&P CNX NIFTY SHARIAH                 0.7054   1.59E-03   -22.4240

                              SE
                        (Partial)

S&P CNX NIFTY            4.35E-02
S&P CNX DEFTY            4.07E-02
CNX NIFTY JUNIOR         4.10E-02
CNX IT                   6.84E-02
S&P CNX 500              4.28E-02
BANK NIFTY               3.78E-02
CNX MIDCAP               4.02E-02
CNX 100                  4.32E-02
CNX INFRASTRUCTURE       3.98E-02
NIFTY MIDCAP 50          4.17E-02
S&P ESG INDIA INDEX      4.07E-02
CNX REALTY               1.19E-01
S&P CNX 500 SHARIAH      1.35E-01
S&P CNX NIFTY SHARIAH    1.37E-01

Table 6
Model with Partial Time Trend g(t)= log(t)

                        [POR.sub.Partial]   [R.sup.2]

S&P CNX NIFTY                      0.3778      0.6794
S&P CNX DEFTY                      0.2632      0.6349
CNX NIFTY JUNIOR                   0.3812      0.5045
CNX IT                             0.4631      0.8399
S&P CNX 500                        0.3741      0.6062
BANK NIFTY                         0.3838      0.4555
CNX MIDCAP                         0.3804      0.4926
CNX 100                            0.3670      0.6561
CNX INFRASTRUCTURE                 0.3272      0.6514
NIFTY MIDCAP 50                    0.4288      0.4199
S&P ESG INDIA INDEX                0.4390      0.5717
CNX REALTY                         0.7082      0.8326
S&P CNX 500 SHARIAH                0.6457      0.8343
S&P CNX NIFTY SHARIAH              0.6477      0.8384

                        [[rho].sub.Estimate]      SE(r)    Partial

S&P CNX NIFTY                         0.9519   1.96E-04   178.4800
S&P CNX DEFTY                         0.9734   1.86E-04   161.7500
CNX NIFTY JUNIOR                      0.9460   8.57E-05   361.4100
CNX IT                                0.8982   2.19E-04   265.0300
S&P CNX 500                           0.9493   2.21E-04   150.1500
BANK NIFTY                            0.9512   1.11E-04   277.6700
CNX MIDCAP                            0.9484   1.27E-04   249.9800
CNX 100                               0.9523   1.95E-04   174.7600
CNX INFRASTRUCTURE                    0.9668   1.71E-04   178.8700
NIFTY MIDCAP 50                       0.9299   2.98E-04   106.4900
S&P ESG INDIA INDEX                   0.9406   4.30E-04    72.3280
CNX REALTY                            0.8051   7.38E-04   189.8700
S&P CNX 500 SHARIAH                   0.8102   1.11E-03   119.3200
S&P CNX NIFTY SHARIAH                 0.8059   1.22E-03   110.5600

                               SE
                        (Partial)

S&P CNX NIFTY              0.4533
S&P CNX DEFTY              0.4358
CNX NIFTY JUNIOR           0.4395
CNX IT                     0.5650
S&P CNX 500                0.4502
BANK NIFTY                 0.4215
CNX MIDCAP                 0.4371
CNX 100                    0.4513
CNX INFRASTRUCTURE         0.4314
NIFTY MIDCAP 50            0.4487
S&P ESG INDIA INDEX        0.4416
CNX REALTY                 0.7395
S&P CNX 500 SHARIAH        0.7931
S&P CNX NIFTY SHARIAH      0.8007

Table 7
Model with Partial Time Trend g(t)= t exp(t)

                        [POR.sub.Partial]   [R.sup.2]

S&P CNX NIFTY                      0.1617      0.5680
S&P CNX DEFTY                     14.8500      0.4934
CNX NIFTY JUNIOR                   0.2058      0.3049
CNX IT                             0.1236      0.7898
S&P CNX 500                        0.1560      0.4575
BANK NIFTY                         0.3820      0.3515
CNX MIDCAP                         0.2254      0.3103
CNX 100                            0.1573      0.5277
CNX INFRASTRUCTURE                 0.1661      0.5018
NIFTY MIDCAP 50                    0.2788      0.2733
S&P ESG INDIA INDEX                0.2868      0.4433
CNX REALTY                         0.3121      0.6365
S&P CNX 500 SHARIAH                0.2379      0.6934
S&P CNX NIFTY SHARIAH              0.2324      0.7080

                        [[rho].sub.Estimate]     SE(r)     Partial

S&P CNX NIFTY                         1.0051   2.03E-04    1.66E-20
S&P CNX DEFTY                         1.0129   1.98E-04    1.28E-20
CNX NIFTY JUNIOR                      0.9803   8.87E-05    1.35E-20
CNX IT                                0.9908   1.74E-04    2.47E-20
S&P CNX 500                           0.9959   2.28E-04    1.15E-20
BANK NIFTY                            0.9523   1.19E-04   -3.70E-20
CNX MIDCAP                            0.9792   1.36E-04    3.30E-21
CNX 100                               1.0024   2.02E-04    1.51E-20
CNX INFRASTRUCTURE                    0.9999   1.82E-04    8.25E-21
NIFTY MIDCAP 50                       0.9612   3.05E-04   -1.97E-23
S&P ESG INDIA INDEX                   0.9775   4.56E-04    3.29E-21
CNX REALTY                            0.9279   6.41E-04    5.60E-12
S&P CNX 500 SHARIAH                   0.9432   9.30E-04    5.64E-12
S&P CNX NIFTY SHARIAH                 0.9433   1.01E-03    5.56E-12

                              SE
                        (Partial)

S&P CNX NIFTY            9.84E-23
S&P CNX DEFTY            9.74E-23
CNX NIFTY JUNIOR         9.54E-23
CNX IT                   9.44E-23
S&P CNX 500              9.74E-23
BANK NIFTY               9.43E-23
CNX MIDCAP               9.75E-23
CNX 100                  9.79E-23
CNX INFRASTRUCTURE       9.59E-23
NIFTY MIDCAP 50          9.63E-23
S&P ESG INDIA INDEX      9.82E-23
CNX REALTY               8.10E-14
S&P CNX 500 SHARIAH      8.35E-14
S&P CNX NIFTY SHARIAH    8.38E-14

Table 8
Model with Partial Time Trend g(t)= exp(t)

                        [POR.sub.Partial]   [R.sup.2]

S&P CNX NIFTY                      0.1619      0.5683
S&P CNX DEFTY                     17.6140      0.4945
CNX NIFTY JUNIOR                   0.2062      0.3047
CNX IT                             0.1236      0.7901
S&P CNX 500                        0.1555      0.4575
BANK NIFTY                         0.3861      0.3538
CNX MIDCAP                         0.2265      0.3103
CNX 100                            0.1574      0.5279
CNX INFRASTRUCTURE                 0.1663      0.5016
NIFTY MIDCAP 50                    0.2781      0.2734
S&P ESG INDIA INDEX                0.2829      0.4428
CNX REALTY                         0.3107      0.6367
S&P CNX 500 SHARIAH                0.2364      0.6938
S&P CNX NIFTY SHARIAH              0.2315      0.7084

                        [[rho].sub.Estimate]      SE(r)     Partial

S&P CNX NIFTY                         1.0054   2.03E-04    7.85E-19
S&P CNX DEFTY                         1.0132   1.99E-04    6.08E-19
CNX NIFTY JUNIOR                      0.9804   8.89E-05    6.40E-19
CNX IT                                0.9911   1.75E-04    1.16E-18
S&P CNX 500                           0.9961   2.29E-04    5.45E-19
BANK NIFTY                            0.9519   1.19E-04   -1.75E-18
CNX MIDCAP                            0.9792   1.36E-04    1.53E-19
CNX 100                               1.0027   2.03E-04    7.14E-19
CNX INFRASTRUCTURE                    1.0001   1.82E-04    3.90E-19
NIFTY MIDCAP 50                       0.9611   3.05E-04   -2.67E-21
S&P ESG INDIA INDEX                   0.9777   4.57E-04    1.56E-19
CNX REALTY                            0.9283   6.43E-04    1.52E-10
S&P CNX 500 SHARIAH                   0.9441   9.34E-04    1.54E-10
S&P CNX NIFTY SHARIAH                 0.9443   1.02E-03    1.52E-10

                               SE
                        (Partial)

S&P CNX NIFTY            4.63E-21
S&P CNX DEFTY            4.58E-21
CNX NIFTY JUNIOR         4.49E-21
CNX IT                   4.44E-21
S&P CNX 500              4.58E-21
BANK NIFTY               4.44E-21
CNX MIDCAP               4.59E-21
CNX 100                  4.61E-21
CNX INFRASTRUCTURE       4.51E-21
NIFTY MIDCAP 50          4.53E-21
S&P ESG INDIA INDEX      4.63E-21
CNX REALTY               2.19E-12
S&P CNX 500 SHARIAH      2.26E-12
S&P CNX NIFTY SHARIAH    2.27E-12

Table 9
Model with Partial Time Trend g(t)= 1/t

                        [POR.sub.Partial]   [R.sup.2]

S&P CNX NIFTY                      0.2925      0.6240
S&P CNX DEFTY                      0.1664      0.5499
CNX NIFTY JUNIOR                   0.2708      0.4055
CNX IT                             0.2847      0.8137
S&P CNX 500                        0.2798      0.5364
BANK NIFTY                         0.2944      0.3422
CNX MIDCAP                         0.2808      0.4012
CNX 100                            0.2820      0.5940
CNX INFRASTRUCTURE                 0.2383      0.5792
NIFTY MIDCAP 50                    0.3159      0.3393
S&P ESG INDIA INDEX                0.3933      0.5084
CNX REALTY                         0.4412      0.7825
S&P CNX 500 SHARIAH                0.3875      0.7890
S&P CNX NIFTY SHARIAH              0.3848      0.7947

                        [[rho].sub.Estimate]     SE(r)     Partial

S&P CNX NIFTY                         0.9734   1.81E-04   -3.23E+02
S&P CNX DEFTY                         0.9904   1.76E-04   -2.82E+02
CNX NIFTY JUNIOR                      0.9637   8.08E-05   -6.28E+02
CNX IT                                0.9427   1.76E-04   -4.29E+02
S&P CNX 500                           0.9694   2.05E-04   -2.68E+02
BANK NIFTY                            0.9648   1.08E-04   -5.14E+02
CNX MIDCAP                            0.9656   1.21E-04   -4.48E+02
CNX 100                               0.9731   1.81E-04   -3.14E+02
CNX INFRASTRUCTURE                    0.9833   1.63E-04   -3.17E+02
NIFTY MIDCAP 50                       0.9487   2.79E-04   -1.81E+02
S&P ESG INDIA INDEX                   0.9581   4.06E-04   -1.26E+02
CNX REALTY                            0.8689   6.49E-04   -2.67E+02
S&P CNX 500 SHARIAH                   0.8737   9.49E-04   -1.54E+02
S&P CNX NIFTY SHARIAH                 0.8704   1.04E-03   -1.43E+02

                              SE
                        (Partial)

S&P CNX NIFTY              1.1684
S&P CNX DEFTY              1.1496
CNX NIFTY JUNIOR           1.1558
CNX IT                     1.2661
S&P CNX 500                1.1668
BANK NIFTY                 1.1382
CNX MIDCAP                 1.1555
CNX 100                    1.1667
CNX INFRASTRUCTURE         1.1451
NIFTY MIDCAP 50            1.1713
S&P ESG INDIA INDEX        1.1623
CNX REALTY                 1.3991
S&P CNX 500 SHARIAH        1.4538
S&P CNX NIFTY SHARIAH      1.4613