The Performance of GARCH Models with Short-memory, Long-memory, or Jump Dynamics: Evidence from Global Financial Markets

Yaw-Huei Wang National Central University

Co-authored with Chih-Chiang Hsu National Central University

Motivation

Volatility is a key input for both risk management and derivative pricing.

ARCH-type models are the most successful and popular framework for describing volatility.

In the past two decades, the model has been developed to be more realistic, but more complicated unfortunately.

Research Questions

Would the more complicated model performs any better for a particular purpose? Conditional distribution: skewed, fat-tailed Memory: short or long

If so, can such improved performance be globally valid for different markets?

Objectives

Develop a nested volatility model based on the EGARCH framework to investigate the performance of

(1)short-memory, (2)long-memory, and (3)jump models

in terms of

(1)model fitting, (2)volatility forecasting, and (3)VaR prediction

for 8 relatively large stock markets.

The Model

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The Model

Estimation: MLE Maximize the log likelihood function

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The Model

Short-memory: EGARCH (Nelson, 1991) λt = d = 0.

Long-memory: FIEGARCH (Bollerslev & Mikkelsen, 1996) λt = 0.

Jump dynamics: EGARCH-jump (Maheu & McCurdy, 2004) d = 0.

EGARCH-skewed-t (Hansen, 1994)

Measures of Performance

Model fitting: Likelihood ratio test: Akaike information criteria (AIC)

Volatility forecasting: Mean squared errors (MSEs)

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Measures of Performance

VaR prediction:

Likelihood ratio test

In practice, a preferred model should have a violation rate which is no greater than the threshold.

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Data

The Datastream stock market indices of US, Japan, the UK, Germany, France, Canada, Italy and Spain .

From July 1990 to June 2005. Excluding holiday, there are, on average, abo

ut 3,785 observations. Preliminary tests for the absolute values of re

turns support the existence of long memory in volatility, particularly clear for US and Canada.

Empirical Results

Model fitting: The EGARCH-jump model has the smallest AIC

and the highest LR statistic globally. The EGARCH-skewed-t model, with two more

parameters, can also provide substantial improvements.

The FIEGARCH model does not necessarily result in any significant improvement.

Model Type

Tests for Model Fit b

EGARCH FIEGARCH EGARCH

-Jump EGARCH -Skewed t

Panel A: US

AIC -6.4614 -6.4767 -6.4909 -6.4827

LR test based on EGARCH 44.32 95.48 62.84

Panel B: Japan

AIC -6.1490 -6.1499 -6.1826 -6.1755

LR test based on EGARCH 4.58 105.82 76.38

Panel C: UK

AIC -6.7300 -6.7305 -6.7629 -6.7438

LR test based on EGARCH 3.34 105.86 42.58

Panel D: France

AIC -6.2116 -6.2124 -6.2385 -6.2238

LR test based on EGARCH 4.08 89.32 38.20

Panel E: Germany

AIC -6.3047 -6.3053 -6. 3414 -6.3231

LR test based on EGARCH 3.56 116.78 55.58

Panel F: Canada

AIC -6.8889 -6.8937 -6.9497 -6.9402

LR test based on EGARCH 15.50 183.32 146.78

Panel G: Italy

AIC -6.1072 -6.1121 -6.1505 -6.1301

LR test based on EGARCH 15.68 134.94 68.04

Panel H: Spain

AIC -6.2555 -6.2563 -6.2982 -6.2852

LR test based on EGARCH 4.26 134.18 87.54

Empirical Results

Volatility forecasting The EGARCH-jump performs best for some

countries, particularly good for German and Canadian markets.

FIEGARCH has fairly satisfactory improvement as well (but with higher variation), although the performance in model fitting is bad.

By contrast, the EGARCH-skewed-t model does not provide any improvement, although the performance in model fitting is good.

Model Type Volatility Forecast

Period EGARCH FIEGARCH EGARCH-Jump EGARCH-Skewed t

Panel A: US (1E-6)

0.0038 0.0031 (18.42) 0.0032 (16.05) 0.0037 (2.63)

Panel B: Japan (1E-6)

0.0209 0.0206 (1.44) 0.0213 (-1.91) 0.0208 (0.48)

Panel C: UK (1E-7)

0.0141 0.0136 (3.55) 0.0152 (-0.78) 0.0142 (-0.71)

Panel D: France (1E-7)

0.0466 0.0470 (-0.86) 0.0514 (-10.30) 0.0462 (0.86)

Panel E: Germany (1E-7)

0.0349 0.0341 (2.29) 0.0320 (8.31) 0.0345 (1.15)

Panel F: Canada (1E-7)

0.0222 0.0213 (4.05) 0.0208 (6.31) 0.0215 (3.15)

Panel G: Italy (1E-6)

0.0096 0.0098 (-2.08) 0.0095 (1.04) 0.0099 (-3.13)

Panel H: Spain (1E-7)

0.0533 0.0520 (2.44) 0.0576 (-8.07) 0.0529 (0.75)

Empirical Results

VaR prediction Almost of all models pass the LR tests for all coun

tries at the 5% significance level. Both the EGARCH-jump model and the EGARCH-

skewed-t model pass the LR tests for all countries and at all significance levels.

However, for the EGARCH-jump model, 87.5% of all violation rates are lower than the corresponding significance levels, while 62.5% for the EGARCH-skewed-t model.

Model Type and Statistics

EGARCH FIEGARCH EGARCH

-Jump EGARCH -Skewed t

Volatility Forecast Period

1% 5% 1% 5% 1% 5% 1% 5%

Panel A: US

Share of Total (%) 1.48 5.46 1.37 5.50 0.94 4.88 1.05 5.25

LRb 0.02 0.27 0.06 0.23 0.75 0.77 0.79 0.55

Panel B: Japan

Share of Total (%) 1.32 4.69 1.32 4.58 0.84 5.05 0.88 4.84

LR 0.11 0.45 0.11 0.30 0.58 0.87 0.51 0.69

Panel C: UK

Share of Total (%) 1.36 5.55 1.32 5.73 0.86 4.62 0.75 5.16

LR 0.07 0.19 0.10 0.08 0.44 0.34 0.16 0.70

Panel D: France

Share of Total (%) 1.61 5.11 1.46 5.03 0.75 4.43 1.07 4.71

LR 0.00 0.79 0.02 0.93 0.16 0.15 0.70 0.48

Panel E: Germany

Share of Total (%) 1.36 5.42 1.36 5.42 0.96 4.93 0.89 4.89

LR 0.07 0.30 0.07 0.30 0.84 0.85 0.55 0.78

Panel F: Canada

Share of Total (%) 1.62 4.96 1.55 5.03 0.93 4.49 1.01 4.81

LR 0.00 0.92 0.00 0.94 0.72 0.21 0.97 0.65

Panel G: Italy

Share of Total (%) 1.43 5.51 1.18 5.37 1.00 5.37 0.93 5.33

LR 0.03 0.22 0.34 0.37 0.99 0.37 0.70 0.42

Panel H: Spain

Share of Total (%) 1.49 4.79 1.31 4.79 0.99 4.54 0.99 4.76

LR 0.01 0.60 0.11 0.60 0.97 0.25 0.97 0.54

Conclusions

The EGARCH-jump model outperforms all other models in all aspects, with the single exception of volatility forecasting for some indices.

However, the computation load is substantially increased.

Conclusions

If less-expensive volatility are preferred, alternative models include the use of the EGARCH-skew-t model for model fi

tting and VaR prediction. the FIEGARCH model for volatility forecasting,

since these models also demonstrate fairly good performance for these particular purposes.

Interestingly, the FIEGARCH model performs relatively satisfactory for the US market only.