Investigating the effect of different input sample size with Nested Conditional Mean and Variance Models over market returns forecast in volatile market

conditions of 2008

*ABV-Indian Institute of Information Technology and Management, Gwalior-474010,India

Abstract---In highly volatile market conditions it’s always difficult to predict returns using heteroscedastic Garch models. This paper tries to investigate the impact of sample data inputs over forecast using nested conditional mean ARMAX(2,2,0) and conditional variance Garch(1,1), Gjr-garch(1,1) and Egarch(1,1) models. Research also tries to indentify relationship between outcome of formal hypothesis tests, the Ljung-Box-Pierce Q-test and Engle's ARCH test, sample input and forecast results . Study is conducted over two diversified stock markets index of America (NASDAQ Composite) and Asia (Nikkei 225). Returns are forecasted for the volatile month of aug-08 using sample inputs of last one month, two months up to seven months before aug-08 and a complete year before input i.e from 1-aug-07 to 31-jul-08 of 246 trading days has also been taken. Graphical and correlative comparison of forecasted and observed returns is also done to identify any trend followed by Garch models in relation to size of sample inputs and forecast. Results show that forecast of mean of returns is far more accurate with Nikkei as compared to Nasdaq.

Keywords: - ARMAX; GARCH ; Arch test; Ljung-Box-Pierce Q-test ; Heteroscedasticity.

1. INTRODUCTION

Forecast of any time series data depends upon the size of the sample input of observed data taken to train the model. Highly fluctuating market conditions which has arisen due to credit crunch in US [1] has affected world economy and trends have become more difficult to predict, atleast accurate average returns forecast would also help in pricing options [2] Moreover the heteroscedastic properties of sample under consideration do greatly affect the forecast.

Two samples under consideration here are of Nasdaq composite and Nikkei 225 index from the period 1-aug-08 to 31-jul-08. Data has been taken from [3], [4] respectively. Small samples of months are selected from this large chunk of data for forecasting returns for month of aug-08.

1.1 Analysis of Sample data

Table 1.1 below shows analysis of one year sample data

Parameter Nasdaq Nikkei Mean (returns) -0.0003651 -0.0009511 Std (returns) 0.01458 0.01713

Range(returns) 0.07945 0.09998 Max ( index) 2859 17460 Min (index) 2169 11790

Figure 1.1a ACF of Returns of Nasdaq

Figure 1.1b ACF of Squared Returns of Nasdaq

Figures (1.1a) and (1.1b) are plots of Auto Correlation Function of returns and squared returns

Joydeep Dhar *ABV-IIITM

jdhar@iiitm.ac.in

Utkarsh Shrivastava ABV-IIITM

utkarsh@students.iiitm.ac.in

2009 International Conference on Information and Financial Engineering

978-0-7695-3606-4/09 $25.00 © 2009 IEEE

DOI 10.1109/ICIFE.2009.38

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of Nasdaq respectively. Its shows no significant correlation in the data.

Figure 1.1c ACF of Returns of Nikkei

Figure 1.1d ACF of Squared Returns of Nikkei

Figure (1.1c) and (1.1d) are plots of ACF of means and squared means of Nikkei 225 respectively. It shows no significant correlation in raw returns but for lags of 4, 11 and 16 squared returns are highly auto correlated, which is the good indication of Arch effect in the return series.

2. METHODOLOGY AND MODELS

2.1 Nested Conditional Mean and Variance Models

To capture the property of time-varying volatility, Engle (1982) introduced the ARCH [5] [6] (Autoregressive Conditional Heteroskedasticity model) . Bollerslev's (1986) extension of this model, the GARCH [7] (Generalized ARCH) model, has gained widespread acceptance in the literature and is often used for modeling stochastic volatility in financial time series.". Although GARCH models give adequate fits for exchange-rate dynamics, these models often fail to perform well in modeling the volatility of stock returns because GARCH models assume that there is asymmetric response between volatility and returns. Therefore, GARCH models are not able to capture the "leverage effect" of stock returns. Nelson (1991) introduced the EGARCH [8] (Exponential GARCH) model in order to model asymmetric variance effects.

2.2 GJR (P,Q) Conditional Variance Model

The general GJR (P,Q) [9] model for the conditional variance of the innovations with leverage terms is:-

σ κ GP σ AQ ε LQ S ε

where , S 1, if ε 0 and S 0 otherwise

and ∑ GP ∑ AQ ∑ LQ 1 κ 0, G ≥ 0 i = 1,2..P ), A 0 j 1,2. . Q ) A L 0 j 1,2. . Q

2.3 EGARCH (P,Q) Conditional Variance Model

The general EGARCH(P,Q) model for the conditional variance of the innovations, with leverage terms and an explicit probability distribution assumption, is logσ κ ∑ G logP σ ∑ AQ | |E | | ∑ LQ

where, E | | for gaussian distribution

E | | = ) for Student's t

distribution, with degrees of freedom > 2.

In this case distribution has been assumed to be student’s t as forecast sample size is less than 30. EGARCH models are fundamentally different from GARCH and GJR models, in that the standardized innovation, ε serves as the forcing variable for both the conditional variance and the error. GARCH and GJR models allow for volatility clustering (persistence) via a combination of the G and A terms. The G terms capture persistence in EGARCH models.

2.4 Conditional Mean Models

The general ARMAX (R,M,Nx) model for the conditional mean is :-

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,

applies to all variance models with autoregressive coefficients { }, moving average coefficients { }, innovations{ }, and returns { }.X is an explanatory regression matrix in which each column is a time series. x(t, k) denotes the row and column of this matrix. In this paper the model chosen for obtaining conditional mean is ARMAX (2, 2, 0) [10]. Hence nested ARMAX (2,2,0) and Garch(1,1), Gjr-garch(1,1), Egarch(1,1) have been applied for computing forecast.

2.5 Heteroscedastic Analysis Tests

Formal hypothesis tests, such as the Ljung-Box-Pierce Q-test and Engle's ARCH test are applied over Nasdaq and Nikkei observed returns series one year sample. Ljung-Box-Pierce Q-test for a departure from randomness has been implemented based on the ACF of the data [11] (page 186). Under the null hypothesis of no serial correlation, the Q-test statistic is asymptotically Chi-Square distributed [12] (page 314). In ARCH test under the null hypothesis that a time series is a random sequence of Gaussian disturbances (that is, no ARCH effects exist), this test statistic is also asymptotically Chi-Square distributed [13]. The first output, H of the above tests is a Boolean decision flag. H = 0 implies that no significant correlation exists (that is, do not reject the null hypothesis). H = 1 means that significant correlation exists (that is, reject the null hypothesis). The remaining outputs are the p-value, the test statistic (Stat), and the critical value of the Chi-Square distribution (Critical).

2.6 Forecast Procedure

Computation of minimum mean square error (MMSE) forecasts of the conditional mean of returns {y has been done. Conditional mean and variance models have been viewed from a linear filtering perspective, then application of iterated conditional expectations to the recursive equations have been done , one forecast period at a time. For example for the month of July sample data has been taken from 1-31 July and forecast using above defined method has been done , for calculation of observed returns

and its comparison with forecasted results market data from 31jul - 31aug has been taken as return for 1st august has also to be calculated.

3. RESULTS AND CONCLUSION

For Nasdaq 4-5 months sample input has given best possible results which too are very far away from (table 3.1d) actual while for Nikkei first 1-2 months input then 6-7 month sample input (table 3.1e) produced approximately same mean of returns as observed (table 3.1a). Egarch(1,1) model has produced most number of favorable forecasts and best also, with least error of 0.000006 for Nikkei taking one month sample input. Correlation analysis (table 3.1f) of forecast and actual result shows that Nasdaq’s returns forecast has been more correlated for aug-08 while Nikkei has not been too correlated . Figures 3.1a-f shows that Nasdaq forecast has been consistently flat and highly deviated which can be attributed to unfavorable results of Arch test and LBQ test for squared residuals while favorable results for Nikkei has lead to more accurate forecast. Hence, analysis shows that in volatile market scenario sample input size do play important role in forecasts of market returns and Arch and Lbq tests statistics can be used to justify variations in forecasts.

Table 3.1a Test Statistics for 1yr sample of Nasdaq

Test Lag H p-value Stat Critical

LBQ test for returns

10 0 0.4561 0.2599 18.3070

15 0 0.6798 11.9903 24.995820 0 0.6468 17.0946 31.4104

LBQ(squared Residuals)

10 0 0.3788 10.7322 18.3070

15 0 0.1865 19.6327 24.995820 0 0.1894 25.3212 31.4104

Arch test 10 0 0.4063 10.3975 18.3070

15 0 0.2883 17.5285 24.995820 0 0.2599 23.6081 31.4104

Table 3.1b Test Statistics for 1yr sample Nikkei 225

Test Lag H p-value Stat CriticalLBQ test (returns)

10 0 0.1333 14.9670 18.307015 0 0.2995 17.3304 24.995820 0 0.2898 22.9798 31.4104

LBQ test (squared returns)

10 1 0.0000 26.6114 18.307015 1 0.0000 53.7472 24.995820 1 0.0000 64.6136 31.4104

ARCH test 10 0 0.1037 10.3975 18.307015 1 0.0001 17.5285 24.995820 1 0.0002 23.6081 31.4104

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Table 3.1c Observed data statistics for august 08

Nasdaq Nikkei 225

Mean(returns) Std(returns) Range(returns) Mean(returns) Std(returns) Range(returns)

0.0008517 0.01341 0.04828 -0.001094 0.01417 0.04909

Table 3.1d Sample wise analysis of forecasted returns mean for Nasdaq

Garch (1,1) Gjr-garch(1,1) Egarch(1,1)D Mean Std Range Mean Std Range Mean Std Range 1 -0.0006699 0.001284 0.00527 1.087e-006 0.001556 0.006374 -0.002033 0.0006631 0.003641 2 -0.001935 0.001788 0.00698 -0.00196 0.002345 0.007634 -0.002026 0.002601 0.008964 3 -0.002042 0.001741 0.006631 -0.002041 0.001741 0.00663 -0.001326 0.001521 0.006295 4 -0.0002984 0.001122 0.004793 -0.0002819 0.001078 0.004652 0.001333 0.0007247 0.004111 5 0.0002523 0.0004802 0.002341 0.000424 0.000245 0.001134 8.421e-005 0.0003214 0.00136 6 -0.0002723 0.0006956 0.003183 -0.0002399 0.0004907 0.002223 -0.001084 2.79e-005 0.0001245 7 -0.0008124 0.0003824 0.001732 -0.0006578 0.0003853 0.001713 -0.001397 0.0001329 0.0005874 8 -0.000462 0.0002005 0.0008732 -0.0002376 0.0001847 0.0007749 -0.0004518 0.0002341 0.000958

Table 3.1e Sample wise analysis of forecasted returns mean for Nikkei

Garch (1,1) Gjr-garch(1,1) Egarch (1,1)

*D Mean Std Range Mean Std Range Mean Std Range

1 -0.0008442 0.007388 0.01482 -0.001246 0.005679 0.01286 -0.001088 0.002803 0.00589 2 -0.003698 0.00183 0.006403 -0.003838 .0009215 0.003386 -0.002965 0.002356 0.01262 3 -0.0006537 0.003173 0.009589 -0.0007013 0.002522 0.008235 0.0003506 .0008909 0.003793 4 0.0002487 0.0023 0.008212 0.0001787 0.005284 0.01603 0.001444 0.00162 0.006654 5 0.0008136 0.001963 0.006929 -0.0001204 0.002146 0.007556 0.001634 0.003569 0.00961 6 -0.001044 0.003105 0.008107 -0.0004563 0.003044 .0095555 -0.001274 0.003084 0.008025 7 -0.0007553 0.002224 0.008155 -0.002088 0.00281 0.007067 -0.00118 0.00346 0.01629 1yr -0.001652 0.001083 0.00435 -0.001899 0.001143 0.004519 -0.001036 0.00215 0.01139 *Tables 3.1c and 3.1e show mean ,std deviation and range of forecasted returns for various sizes of sample input in first column ( D ). Duration of sample i.e D = 1 implies sample of one month before august-08 i.e july-08 and D = 4 implies sample containing past 4 months data i.e from april-july 08 .

Table 3.1f Correlation Analysis (correlation coeff of forecast result with actual data during aug-08)

Input duration

Nassdaq Nikkei Garch(1,1) Gjr (1,1) Egarch(1,1) Garch (1,1) Gjr (1,1) Egarch (1,1)

1 month -0.05851 0.028009 -0.0003 -0.05883 -0.06899 -0.04781 2 month 0.103147 0.147346 0.114163 -0.07344 -0.07344 0.266593 3 month -0.00506 -0.00506 0.054521 -0.30988 -0.17941 -0.23704 4 month 0.041638 0.039011 0.15358 -0.20602 -0.25124 -0.23879 5 month -0.01178 0.361605 0.05418 -0.18873 -0.19395 -0.08471 6 month -0.0143 -0.02838 0.122135 -0.09775 -0.18416 -0.09895 7 month -0.02783 -0.00424 0.192549 -0.21248 -0.17336 -0.25125 1 year 0.033704 -0.16544 -0.02211 -0.2097 -0.20284 -0.24055

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Figure 3.1 (a-f) Plots of best forecast results

a) Nasdaq garch (1,1) forecast using last 5month input

b) Nasdaq gjr (1,1) forecast for aug-08 using 5 month input

c) Nikkei egarch(1,1) forecast using last 1month input

d) Nikkei garch(1,1) forecast using last 6 month input

f) Nikkei egarch(1,1) forecast using last 7 month input

where, ------------ Observed Returns, +++++++ Forecasted returns

REFERENCES

[1] Wall Street Journal. "TED Spread spikes in July 2007". Wall Street Journal.

[2] Hull, John (2006). Options, Futures and Other Derivatives, New Delhi: Pearson Education. [3] http://finance.yahoo.com/q?s=^N225

[4] http://finance.yahoo.com/q?s=^IXIC

[5] Engle, R.F.,“ Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation”, Econometrica, vol. 50, pp. 987-1007, 1982. [6] Bollerslev, T., Engle, R.F., and Nelson, D.B., "ARCH Models, Handbook of GARCH” , Volume IV, Chapter 49, pp. 2959–3038, Elsevier Science B.V., Amsterdam, The Netherlands, 1994. [7] Bollerslev, T., “Generalized Autoregressive Conditional Heteroskedasticity”, Journal of Economtric, 31, pp.307-327, 1986.

[8] Campbell, J.Y., A.W. Lo, and MacKinlay, A.C.,"The GARCH of Financial Markets," Nonlinearities in Financial Data, Chapter 12, Princeton University Press, Princeton, NJ, 1997.

[9] Greene, W.H., Econometric Analysis, Fifth edition, Prentice Hall, Upper Saddle River, NJ, 2003. [10] Greene, W.H., Econometric Analysis, Fifth edition, Prentice Hall, Upper Saddle River, NJ, 2003.

[11] Box, G.E.P. , Jenkins, G.M. , and Reinsel,G.C. , Time Series Analysis: Forecasting and Control, Third edition, Prentice Hall, Upper Saddle River, NJ, 1994.

[12] Shreve, S.E., Stochastic Calculus for Finance II:Continuous-Time Models, Springer-Verlag, 2004. [13] Engle, R.F., "Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation," Econometrica, Vol 50, 1982, pp.98.

e)Nikkei egarch(1,1) forecast using last 1 year input

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