Robust estimation and inference for heavy tailed GARCH

We develop two new estimators for a general class of stationary GARCH models with possibly heavy tailed asymmetrically distributed errors, covering processes with symmetric and asymmetric feedback like GARCH, Asymmetric GARCH, VGARCH and Quadratic GARCH. The first estimator arises from negligibly trimming QML criterion equations according to error extremes. The second imbeds negligibly transformed errors into QML score equations for a Method of Moments estimator. In this case, we exploit a sub-class of redescending transforms that includes tail-trimming and functions popular in the robust estimation literature, and we re-center the transformed errors to minimize small sample bias. The negligible transforms allow both identification of the true parameter and asymptotic normality. We present a consistent estimator of the covariance matrix that permits classic inference without knowledge of the rate of convergence. A simulation study shows both of our estimators trump existing ones for sharpness and approximate normality including QML, Log-LAD, and two types of non-Gaussian QML (Laplace and Power-Law). Finally, we apply the tail-trimmed QML estimator to financial data.Comment: Published at http://dx.doi.org/10.3150/14-BEJ616 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

A Smoothed P-Value Test When There is a Nuisance Parameter under the Alternative

We present a new test when there is a nuisance parameter under the alternative hypothesis. The test exploits the p-value occupation time [PVOT], the measure of the subset of a nuisance parameter on which a p-value test rejects the null hypothesis. Key contributions are: (i) An asymptotic critical value upper bound for our test is the significance level, making inference easy. Conversely, test statistic functionals need a bootstrap or simulation step which can still lead to size and power distortions, and bootstrapped or simulated critical values are not asymptotically valid under weak or non-identification. (ii) We only require the test statistic to have a known or bootstrappable limit distribution, hence we do not require root(n)-Gaussian asymptotics, and weak or non-identification is allowed. Finally, (iii) a test based on the sup-p-value may be conservative and in some cases have nearly trivial power, while the PVOT naturally controls for this by smoothing over the nuisance parameter space. We give examples and related controlled experiments concerning PVOT tests of: omitted nonlinearity; GARCH effects; and a one time structural break. Across cases, the PVOT test variously matches, dominates or strongly dominates standard tests based on the supremum p-value, or supremum or average test statistic (with wild bootstrapped p-value

Kernel Methods for Small Sample and Asymptotic Tail Inference for Dependent, Heterogeneous Data

This paper considers tail shape inference techniques robust to substantial degrees of serial dependence and heterogeneity. We detail a new kernel estimator of the asymptotic variance and the exact small sample mean-squared-error, and a simple representation of the bias of the B. Hill (1975) tail index estimator for dependent, heterogeneous data. Under mild assumptions regarding the tail fractile sequence, memory and heterogeneity, choosing the sample fractile by non-parametrically minimizing the mean-squared-error leads to a consistent and asymptotically normal estimator. A broad simulation study demonstrates the merits of the resulting minimum MSE estimator for autoregressive and GARCH data. We analyze the distribution of a standardiz ed Hill-estimator in order to asses the accuracy of the kernel e stimator of the asymptotic variance, and the distribution of the minimum MSE estimator. Finally, we apply the estimators to a small study of the tail shape of equity markets returns.Hill estimator; regular variation; extremal near epoch dependence; kernel estimator; mean-square-error.

On Functional Central Limit Theorems for Dependent, Heterogeneous Tail Arrays with Applications to Tail Index and Tail Dependence Estimators

We establish functional central limit theorems for a broad class of dependent, heterogeneous tail arrays encountered in the extreme value literature, including extremal exceedances, tail empirical processes and tail empirical quantile processes. We trim dependence assumptions down to a minimum by constructing extremal versions of mixing and Near-Epoch-Dependence properties, covering mixing, ARFIMA, FIGARCH, bilinear, random coefficient autoregressive, nonlinear distributed lag and Extremal Threshold processes, and stochastic recurrence equations. Of practical importance our theory can be used to characterize the functional limit distributions of sample means and covariances of tail arrays, including popular tail index estimators, the tail quantile function, and multivariate extremal dependence measures under substantially general conditions.Functional central limit theorem; extremal processes; tail empirical process; cadlag space; mixingale; near-epoch-dependence; regular variation; Hill estimator; tail dependence.

Super-Consistent Tests of Lp-Functional Form

This paper develops a consistent test of best Lp-predictor functional form for a time series process. By functionally relating two moment conditions with different nuisance parameters we are able to construct a vector moment condition in which at least one element must be non-zero under the alternative. Specifically, we provide a sufficient condition for moment conditions of the type characterized by Stinchcombe and White (1998) to reveal model mis-specification for any nuisance parameter value. When the sufficient condition fails an alternative moment condition is guaranteed to work. A simulation study clearly demonstrates the superiority of a randomized test: randonly selecting the nuisance parameter leads to more power than average- and supremum-test functionals, and obtains empirical power nearly equivelant to uniformly most powerful tests in most cases.consistent test; conditional moment test; best Lp-predictor; nonlinear model.

Asymptotically Nuisance-Parameter-Free Consistent Tests of Lp-Functional Form

We develop a consistent conditional moment test of Lp-best predictor functional form, 1nonlinear regression models, consistent conditional moment test, nuisance parameter-free test, Lp-best predictor

Efficient Tests of Long-Run Causation in Trivariate VAR Processes with a Rolling Window Study of the Money-Income Relationship

This paper develops a simple sequential multiple horizon noncausation test strategy for trivariate VAR models (with one auxiliary variable). We apply the test strategy to a rolling window study of money supply and real income, with the price of oil, the unemployment rate and the spread between the Treasury bill and commercial paper rates as auxiliary processes. Ours is the first study to control simultaneously for common stochastic trends, sensitivity of causality tests to chosen sample period, null hypothesis over-rejection, sequential test size bounds, and the possibility of causal delays. Evidence suggests highly significant direct or indirect causality from M1 to real income, in particular through the unemployment rate and M2 once we control for cointegration.multiple horizon causality, Wald tests, parametric bootstrap, money-income causality, rolling windows, cointegration