GARCH Time Series Model (Free)

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收藏 2008-01-11

【书名】 Generalized Autoregressive Conditional Heteroscedastic Time Series Models
【作者】Michael S. Lo
【出版社】SIMON FRASER UNIVERSITY
【出版日期】April 2003
【文件格式】PDF 【页数】61
【资料类别】计量经济文章

【内容简介】

Autoregressive and Moving Average time series models and their combination are
reviewed. Autoregressive Conditional Heteroscedastic (ARCH) and Generalized
Autoregressive Conditional Heteroscedastic (GARCH) models are extensions of these
models. These are de ned and compared to the class of Autoregressive Moving
Average models. Maximum likelihood estimation of parameters is examined.
Conditions for existence and stationarity of GARCH models are discussed and the
moments of the observations and the conditional variance are derived. Characteristics
of low order GARCH models are explored further through simulations with
di erent initial parameter values. As examples, GARCH models with di erent orders
are tted to the Standard & Poor's 500 Stock Price Index.
【目录】Contents
Approval Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Time Series Concepts and Models . . . . . . . . . . . . . . . . . . . . 3
2.1 Stationarity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Standard Time Series Models . . . . . . . . . . . . . . . . . . 5
2.2.1 General Autoregressive Models . . . . . . . . . . . . 5
2.2.2 General Moving Average Models . . . . . . . . . . . 8
2.2.3 General Autoregressive Moving Average Models . . . 10
2.3 Financial Time Series Models . . . . . . . . . . . . . . . . . . 12
2.3.1 Autoregressive Conditional Heteroscedastic (ARCH)
Models . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.3.2 Generalized Autoregressive Conditional Heteroscedastic
(GARCH) Models . . . . . . . . . . . . . . . . . . 14
2.3.3 The ARCH(q) and the GARCH(1; 1) Models . . . . . 14
3 The GARCH(1; 1) Model . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1 Existence of the GARCH(1; 1) Process . . . . . . . . . . . . . 16
3.2 Moments of Xt and ht . . . . . . . . . . . . . . . . . . . . . . 18
3.3 Stationarity of the GARCH(1; 1) Process . . . . . . . . . . . . 20
vi
3.4 Data Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4.1 The Likelihood Function and Estimation of
Parameters . . . . . . . . . . . . . . . . . . . . . . . 23
3.5 A typical GARCH(1; 1) Example . . . . . . . . . . . . . . . . 25
3.5.1 Results from the Monte Carlo Simulations . . . . . . 25
3.5.2 Identi ability of parameters . . . . . . . . . . . . . . 27
3.6 Results from Further Simulations . . . . . . . . . . . . . . . . 31
3.6.1 Characteristics and Behaviour of the Estimates . . . 31
3.6.2 Negative estimates . . . . . . . . . . . . . . . . . . . 32
4 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.1 Data Description . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.2 Fitting ARCH(1) Models and Diagnostics . . . . . . . . . . . 39
4.2.1 Ljung-Box Q-Statistic . . . . . . . . . . . . . . . . . 41
4.3 Fitting GARCH(1; 1) Models . . . . . . . . . . . . . . . . . . 44
4.4 Diagnostics for the GARCH(1; 1) Models . . . . . . . . . . . . 46
4.5 Final Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
vii