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The generalized Additive Nonparametric GARCH Model --With application to the Chinese stock market

Ai Jun Hou Department of Economics School of Economics and Management Lund University. Lund Fudan Economic Forum , Nov 2007. The generalized Additive Nonparametric GARCH Model --With application to the Chinese stock market. 1. Introduction. 2. The basic Model. 3. Simulation experiments.

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The generalized Additive Nonparametric GARCH Model --With application to the Chinese stock market

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  1. Ai Jun Hou Department of Economics School of Economics and Management Lund University Lund Fudan Economic Forum, Nov 2007 The generalized Additive Nonparametric GARCH Model--With application to the Chinese stock market

  2. 1. Introduction 2. The basic Model 3. Simulation experiments 4. Real Examples 5. Conclusion

  3. 1. Introduction • Stationary Times series showing Volatility asymmetries and clustering • Provides the impetus grounded under GARCH family models ARCH (Engle,1982), standard GARCH (Bollerslev, 1986), EGARCH and Threshold GARCH models • Volatility parametrically depends on Lagged volatility and innovations • Nonparametric GARCH model (BÜlman and McNeil, 2002) (NP Model) • Local polynomial smoothing • iteration (procedures )algorithm

  4. Introduction (con.)‏ • Attractive Iterative method: no requirement of specification of the functional form of volatility and the innovation distributions • Problem of -the curse of dimensionality (Härdle, 2004)‏ • To apply an generalized additive model (Hastie and Tibshirani, 1986) • lower dimensional smoothing • the backfitting algorithm

  5. Introduction (Con.) • We apply the iterative algorithm of NP model to the Generalized Additive GARCH Model (NP GAM model), use it to adjust the volatility estimated from the GARCH model • Results from the simulation and real data: • NP and NP GAM model outperform parametric GARCH models if market got asymmetric information • NP GAM model dominates NP model in most cases • there is a moderate improvement in In-sample forecast, and a more clear improvement in Out-of-sample forecast • Why Chinese stock market? • The Chinese market is still young and yet develops quickly. • SHSE and SZSE were established in 1990 and 1991 • Mid October, 856 companies (SHSE), floated market value RMB 6 trillion, 644 companies (SZSE), RMB 2.7 trillion

  6. Introduction (con.) • However, Studies on the Chinese stock market are still limited (Tang and Chen, 2002), to our knowledge, no one has applied the NP GAM model to the Chinese stock market, besides: -- Zou and Wang (2007) examine currency market, , Lu ( 2004) examines the Chinese stock market with NP model , but not NP GAM model • A distinct asymmetric effect exists in the Chinese stock market (Wang, et al.,2005) --it is attractive to fit the Chinese data with the new iterative method • Our contributions: --apply a newly proposed method to examine a new market --fit the models with residuals under both t and normal distributions --show that the nonparametric model could be an effective auxiliary tool to test if the parametric model is an appropriate one which fits the volatility well

  7. 2. The basic model (2.1)

  8. The Basic Model (con.)

  9. 2.1 Estimation algorithm and lagged

  10. Estimation algorithm (con.) • and For computational convenience, we perform the NP GAM (1,1)‏ and compare our results with GARCH(1,1) , EGARCH(1,1), and TGARCH (1,1) models, and also with the NP (1,1) model The parametric models are simulated and estimated in Matlab with a maximumlikelihood method, while the nonparametric procedures and backfitting algorithm are performed in S-PLUS student version

  11. 3. Simulation experiments • We consider three designed process: • Process A: Process B: Process C: For both processes, we work with n=1000 observations, generate 50 realizations, and the maximum iterating is M=8, and a final smooth is performed by averaging over the last four (K=5) iterations. We fit the processed with both T and normal distributed innovations. The performance of each models are evaluated by MSE and MAE, which are the average of the volatility estimation errors of each realization, the first 20 points are omitted from the calculation

  12. Simulation experiments (con.) The above figure shows t´the volatility surface of process A and B, under the asymmetric information effects, there is a significant broken segment on the volatility surface, the results from our simulations show that the nonparametric models smooth surface quite well and outperform the parametric GARCH models

  13. Simulation results : process A When there is no asymmetric effect, the standard GARCH model dominates EGARCH, TGARCH, and nonparametric models. The estimations with t distributed errors perform slightly better than the normal fitting. However, nonparametric models provide the nearly identical results, which disregarding the innovation distribution.

  14. Simulation results: process B

  15. iteration1 iteration 2 iteration 3 iteration 4 iteration 5 iteration 6 iteration 7 iteration 8 final smooth Simulation results : process B

  16. Simulation results: process C

  17. 4. Application to China Stock market • Widely accepted SHCI and SZCI • the daily price of SHCI and SZCI from 2nd January, 1997 to 31st August,  2007. • Are Converted to daily log return and multiply 100 • In-sample group (from 2nd January, 1997 to 31 August, 2006) and an out-of-sample group (from 1st September 2006 to 31 August 2007)‏ • 2379 observations for in-sample and 243 observations for out-of -sample forecast • Realized volatility, extracted from high frequency data (5 minutes) for true volatility proxy for out-of-sample forecast

  18. For out-of-sample forecast, we use two criteria: MSE MAE We use both Performanceevaluation criteria • For in-sample forecast we use three indicators: • Mean Squared Error between squared innovation and squared volatility (LL2) • MSE • MAE • we use : and realized volatility As the true volatility proxy as true volatility

  19. SHCI SZCI Mean 0,025 0,011 Std. 1,514 1,652 Skewness 0,008 0,061 Kurtosis 8,747 8,072 JB Test 0,001 0,001 DF Test -48,491 -46,644 Data description

  20. In-Sample Result Fit with AR(0)-GARCH (1,1)

  21. In Sample forecast all models appear to be adequate in describing the linear dependence in the return and volatility series. The leverage parameters from EGARCH and GJR indicates moderate asymmetric characteristics.

  22. In Sample forecast (con.)

  23. In-sample forecast results • GARCH model with student t distributed errors performs better than the one fitted with normal distributed innovations • Nonparametric models outperform parametric ones (5% for SHCI, 3% for SZCI)‏ • Nonparmetric models fit data better than EGARCH and TGARCH although EGARCH with T distribution got very good result • Nonparametric models disregarding the innovation distributions • No need to see other models for the dynamic changes in the market

  24. Out-the-Sample Predictability Shanghai Composite Index Shenzhen Component Index Model Distribution Realized Volatility Realized Volatility MSE MAE MSE MAE MSE MAE MSE MAE GARCH Normal 2,130 1,129 0,596 0,581 2,602 1,257 0,731 0,653 T 2,088 1,114 0,587 0,576 2,559 1,241 0,709 0,642 EGARCH Normal 2,026 1,087 0,573 0,554 2,531 1,228 0,696 0,644 T 1,983 1,064 0,577 0,552 2,490 1,218 0,683 0,620 JGR Normal 2,138 1,123 0,639 0,593 2,607 1,256 0,724 0,643 T 2,120 1,109 0,625 0,590 2,563 1,238 0,744 0,655 GAM Normal 1,905 1,047 0,515 0,530 2,411 1,195 0,626 0,604 T 1,903 1,045 0,517 0,530 2,403 1,192 0,628 0,604 NP Normal 1,930 1,056 0,526 0,533 2,468 1,205 0,690 0,620 T 1,928 1,055 0,528 0,533 2,472 1,207 0,692 0,619 Out-of-sample forecast results Parametric Nonparametric The improvement of the nonparametric models are more significant in the out-of-sample forecast. (e.x. For MSE, 10% for SHCI and 5% for SZCI)

  25. Out-of-sample forecast results

  26. Conclusion • apply the iterative algorithm of the nonparametric GARCH model (NP model), which is first proposed by the BÜhlman and McNeil (2002), to the Generalized Additive Model (NP GAM model), and use it to adjust the volatility estimated by the parametric GARCH model. • nonparametric iterative technique can provide an improvement for the estimation of the hidden volatility process when the market is complicated e.g. exists asymmetric effects, and this improvement is more clear for an out-of-sample forecast. • The NP GAM model appears to be a more stable method with the computational convenience, and in the most cases outperforms the NP model. • An attractive method: no specification of the functional form of the volatility process nor that of the innovation distributions is required for such an additive algorithm. • It could be also used to test if the parametric model is an appropriate one which fits the volatility process well

  27. Conclusion (con.) • limitations of this method: • e.g. Several assumptions of the models have not been able to be proved (BÜhlman and McNeil, 2002). • This additive iteration can not fit the stochastic volatility model, where the volatility process is fully hidden. • Furthermore, it is well known that the volatility jumps.

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