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

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

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 (B Ü lman and McNeil, 2002) (NP Model)  Local polynomial smoothing  iteration (procedures )algorithm

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

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

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

2. The basic model (2.1)

The Basic Model (con.)

2.1 Estimation algorithm and lagged

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 maximum likelihood method, while the nonparametric procedures and backfitting algorithm are performed in S-PLUS student version

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

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

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.

Simulation results: process B

iteration1 iteration 2iteration 3 iteration 4 iteration 5iteration 6 iteration 7iteration 8final smooth

Simulation results: process C

4. Application to China Stock market Widely accepted SHCI and SZCI the daily price of SHCI and SZCI from 2 nd January, 1997 to 31 st August, Are Converted to daily log return and multiply 100 In-sample group (from 2 nd January, 1997 to 31 August, 2006) and an out-of-sample group (from 1 st 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

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

Data description 8,0728,747 Kurtosis 0,0610,008 Skewness 0,001 JB Test -46,644-48,491 DF Test 1,6521,514 Std. 0,0110,025 Mean SZCISHCI

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

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. In Sample forecast

In Sample forecast (con.)

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

Out-of-sample forecast results 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) Parametric Nonparametric

Out-of-sample forecast results

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

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.