Simple, Skewness-Based GMM Estimation of the Semi-Strong GARCH(1,1) Model

Todd, Prono (2009): Simple, Skewness-Based GMM Estimation of the Semi-Strong GARCH(1,1) Model.

Abstract

IV estimators with an instrument vector composed only of past squared residuals, while applicable to the semi-strong ARCH(1) model, do not extend to the semi-strong GARCH(1,1) case because of underidentification. Augmenting the instrument vector with past residuals, however, renders traditional IV estimation feasible, if the residuals are skewed. The proposed estimators are much simpler to implement than efficient IV estimators, yet they retain improved finite sample performance over QMLE. Jackknife versions of these estimators deal with the issues caused by many (potentially weak) instruments. A Monte Carlo study is included, as is an empirical application involving foreign currency spot returns.

• Item Type: MPRA Paper
• Language: English
• Keywords: GARCH, GMM, instrumental variables, continuous updating, many moments, robust estimation
• Subjects: C - Mathematical and Quantitative Methods > C1 - Econometric and Statistical Methods and Methodology: General > C13 - Estimation: General; C - Mathematical and Quantitative Methods > C5 - Econometric Modeling > C53 - Forecasting and Prediction Methods ; Simulation Methods; C - Mathematical and Quantitative Methods > C2 - Single Equation Models ; Single Variables > C22 - Time-Series Models ; Dynamic Quantile Regressions ; Dynamic Treatment Effect Models ; Diffusion Processes

References

Andrews, D.W.K., 1988, Laws of large numbers for dependent non-identically distributed random variables, Econometric Theory, 4, 458-467.

Angrist, J., G. Imbens and A. Kreuger 1999, Jackknife instrumental variables estimation, Journal of Applied Econometrics, 14, 57-67.

Baillie, R.T., and H. Chung, 2001, Estimation of GARCH models from the autocorrelations of the squares of a process, Journal of Time Series Analysis, 22, 631-650.

Bodurtha, J.N. and N.C. Mark, 1991, Testing the CAPM with time-varying risks and returns, Journal of Finance, 46, 1485-1505.

Bollerslev, T., 1986, Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics, 31, 307--327.

Brown, B.W. and W.K. Newey, 2002, Generalized method of moments, efficient bootstrapping, and improved inference, Journal of Business and Economic Statistics, 20, 507-571.

Carlstein, E., 1986, The use of subseries methods for estimating the variance of a general statistic from a stationary time series, Annals of Statistics, 14, 1171-1179.

Carrasco, M. and X. Chen, 2002, Mixing and moment properties of various GARCH and stochastic volatility models, Econometric Theory, 18, 17-39.

Chamberlain, G., 1982, Multivariate regression models for panel data, Journal of Econometrics, 18, 5-46.

Cragg, J.G., 1983, More efficient estimation in the presence of heteroskedasticity of unknown form, Econometrica, 51, 751-764.

Donald, S.G., G. Imbens and W.K Newey, 2008, Choosing the number of moments in conditional moment restriction models, MIT working paper.

Donald, S.G., and W.K. Newey, 2000, A jackknife interpretation of the continuous updating estimator, Economic Letters, 67, 239 - 243.

Drost, F.C. and T.E. Nijman, 1993, Temporal aggregation of GARCH processes, Econometrica, 61, 909-927.

Engle, R.F., and J. Mezrich, 1996, GARCH for groups, Risk, 9, 36-40.

Escanciano, J.C., 2009, Quasi-maximum likelihood estimation of semi-strong GARCH models, Econometric Theory, 25, 561-570.

Francq, C., L. Horath and J.M. Zakoian, 2009, Merits and drawbacks of variance targeting in GARCH models, NBER-NSF Time Series Conference proceedings.

Guo, B. and P.C.B Phillips, 2001, Efficient estimation of second moment parameters in ARCH models, unpublished manuscript.

Hall, P. and J.L. Horowitz, 1996, Bootstrap critical values for tests based on generalized-method-of-moments estimators, Econometrica, 64, 891-916.

Han, C. and P.C.B. Phillips, 2006, GMM with many moment conditions, Econometrica, 74, 147-192.

Hansen, L.P., 1982, Large sample properties of generalized method of moments estimators, Econometrica, 50, 1029-1054.

Hansen, L.P., J. Heaton and A. Yaron, 1996, Finite-sample properties of some alternative GMM estimators, Journal of Business and Economic Statistics, 14, 262-280.

Hansen, P.R. and A. Lunde, 2005, A forecast comparison of volatility models: does anything beat a GARCH(1,1)?, Journal of Applied Econometrics, 20, 873-889.

He, C. and T. Teräsvirta, 1999, Properties of moments of a family of GARCH processes, Journal of Econometrics, 92, 173-192.

Kristensen, D. and O. Linton, 2006, A closed-form estimator for the GARCH(1,1)-model, Econometric Theory, 22, 323-327.

Lee, S.W, B.E. Hansen, 1994, Asymptotic theory for the GARCH(1,1) qausi-maximum likelihood estimator, Econometric Theory, 10, 29-52.

Lumsdaine, R.L., 1996, Consistency and asymptotic normality of the quasi-maximum likelihood estimator in IGARCH(1,1) and covariance stationary GARCH(1,1) models, Econometrica, 64, 575-596.

Mark, N.C, 1988, Time-varying betas and risk premia in the pricing of forward foreign exchange contracts, Journal of Financial Economics, 22, 335-354.

Meddahi, N. and E. Renault, 1998, Quadratic M-estimators for ARCH-type processes, working paper CIRANO 98s-29.

Newey, W.K. and D. McFadden, 1994, Large sample estimation and hypothesis testing, in R.F. Engle and D. McFadded, eds, Handbook of Econometrics, Vol. 4, Amsterdam North Holland, chapter 36, 2111-2245.

Newey, W.K. and R.J. Smith, 2004, Higher order properties of GMM and generalized empirical likelihood estimators, Econometrica, 72, 219-255.

Newey, W.K. and D.G. Steigerwald, 1997, Asymptotic bias for quasi-maximum-likelihood estimators in conditional heteroskedasticity models, Econometrica, 65, 587-599.

Newey, W.K and F. Windmeijer, 2009, Generalized method of moments with many weak moment conditions, Econometrica, 77, 687-719.

Pakes, A. and D. Pollard, 1989, Simulation and the asymptotics of optimization estimators, Econometrica, 57, 1027-1057.

Schmid, F. and R. Schmidt, 2007, Multivariate extensions of Spearman's rho and related statistics, Statistics and Probability Letters, 77, 407-416.

Skoglund, J., 2001, A simple efficient GMM estimator of GARCH models, unpublished manuscript.

Spearman, C., 1904, The proof and measurement of association between two things, American Journal of Psychology, 15, 72-101.

Stock, J. and J. Wright, 2000, GMM with weak identification, Econometrica, 68, 1055-1096.

Weiss, A.A., 1986, Asymptotic theory for ARCH models: estimation and testing, Econometric Theory, 2, 107-131.

White, H., 1982, Instrumental variables regression with independent observations, Econometrica, 50, 483-499.