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

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

Abstract

IV estimators for the semi-strong ARCH(1) model that rely on past squared residuals alone as instruments do not extend to the GARCH case. Efficient IV estimators of the semi-strong GARCH(1,1) model require the derivative of the conditional variance as well as both the third and fourth conditional moments to be included within the instrument vector. This paper proposes IV estimators for the semi-strong GARCH(1,1) model that only rely on past residuals and past squared residuals as instruments. These estimators are based on the autocovariances of squared residuals, as in the ARCH(1) case described above, as well as on the covariances between squared residuals and past residuals. These latter covariances are nonzero if the residuals are skewed. Jackknife GMM estimators and jackknife continuous updating estimators (CUE) eliminate the bias caused by many (weak) instruments. The jackknife CUE is new and applies to cases where the optimal weighting matrix is unavailable out of a concern over the existence of higher moments. In these cases, a robust analog to the variance-covariance matrix determines the weighting matrix. A Monte Carlo study shows that a CUE based on the optimal weighting matrix as well as the jackknife CUE outperforms QMLE in finite samples. An empirical application involving Australian Dollar and Japanese Yen spot returns is also included.

• 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.

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.

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.

Hafner, C.M., 2003, Fourth moment structure of multivariate GARCH models, Journal of Financial Econometrics, 1, 26-54.

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, B., 1994, Autoregressive conditional density estimation, International Economic Review, 35, 705-730.

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.

Harvey, C. and A. Siddique, 1999, Autoregressive conditional skewness, Journal of Financial and Quantitative Analysis, 34, 465-487.

Kendall, M., 1938, A new measure of rank correlation, Biometrica, 30, 81-89.

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.

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.

Rich, R.W., J. Raymond and J.S. Butler, 1991, Generalized instrumental variables estimation of autoregressive conditional heteroskedastic models, Economics Letters, 35, 179-185.

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.

Zadrozny, P.A., 2005, Necessary and sufficient restrictions for existence of a unique fourth moment of a univariate GARCH(p,q) process, CESIFO Working Paper No. 1505.