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

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

LM-Tests for Linearity Against Smooth Transition Alternatives: A Bootstrap Simulation Study

The universal method for testing linearity against smooth transition autoregressive (STAR) alternatives is the linearization of the STAR model around the null nuisance parameter value, and performing F-tests on polynomial regressions in the spirit of the RESET test. Polynomial regressors, however, are poor proxies for the nonlinearity associated with STAR processes, and are not consistent (asymptotic power of one) against STAR alternatives, let alone general deviations from the null. Moreover, the most popularly used STAR forms of nonlinearity, exponential and logistic, are known to be exploitable for consistent conditional moment tests of functional form, cf. Bierens and Ploberger (1997). In this paper, pushing asymptotic theory aside, we compare the small sample performance of the standard polynomial test with an essentially ignored consistent conditional moment test of linear autoregression against smooth transition alternatives. In particular, we compute an LM sup-statistic and characterize the asymptotic p-value by Hansen's (1996) bootstrap method. In our simulations, we randomly select all STAR parameters in order not to bias experimental results based on the use of "safe", "interior" parameter values that exaggerate the smooth transition nonlinearity. Contrary to past studies, we find that the traditional polynomial regression method performs only moderately well, and that the LM sup-test out-performs the traditional test method, in particular for small samples and for LSTAR processes.Smooth transition AR, consistent conditional moment test, Lagrange Multiplier, bootstrap

On Tail Index Estimation for Dependent, Heterogenous Data

In this paper we analyze the asymptotic properties of the popular distribution tail index estimator by B. Hill (1975) for possibly heavy- tailed, heterogenous, dependent processes. We prove the Hill estimator is weakly consistent for processes with extremes that form mixingale sequences, and asymptotically normal for processes with extremes that are near-epoch-dependent on the extremes of a mixing process. Our limit theory covers infinitely many ARFIMA and FIGARCH processes, stochastic recurrence equations, and simple bilinear processes. Moreover, we develop a simple non-parametric kernel estimator of the asymptotic variance of the Hill estimator, and prove consistency for extremal-NED processes.Hill estimator; regular variation; infinite variance; near epoch dependence; mixingale; kernel estimator; tail array sum.

Causation Delays and Causal Neutralization: The Money-Output Relationship Revisited

In this paper, we develop a parametric test procedure for multiple horizon ”Granger” causality and apply the procedure to the well established problem of determining causal patterns in aggregate monthly U.S. money and output. As opposed to most papers in the parametric causality literature, we are interested in whether money ever "causes" (can ever be used to forecast) output, when causation occurs, and how (through which causal chains). For brevity, we consider only causal patterns up to horizon h = 3. Our tests are based on new recursive parametric characterizations of causality chains which help to distinguish between mere noncausation (the total absence of indirect causal routes) and causal neutralization, in which several causal routes exists that cancel each other out such that noncausation occurs. In many cases the recursive characterizations imply greatly simplified linear compound hypotheses for multi-step ahead causation, and permit Wald tests with the usual asymptotic chi-square distribution. A simulation study demonstrates that a sequential test method does not generate the type of size distortions typically reported in the literature, and null rejection frequencies depend entirely on how we define the "null hypothesis" of non-causality (at which horizon, if any). Using monthly data employed in Stock and Watson (1989), and others, we demonstrate that while Friedman and Kuttner’s (1993) result that detrended money growth fails to cause output one month ahead continues into the third quarter of 2003, a significant causal lag may exist through a variety of short-term interest rates: money appears to cause output after at least one month passes, although in some cases using recent data conflicting evidence suggests money may never cause output and be truly irrelevant in matters of real decisions.multiple horizon causation, multivariate time series, sequential tests

Strong Orthogonal Decompositions and Nonlinear Impulse Response Functions for Infinite-Variance Processes

In this paper we prove Wold-type decompositions with strongorthogonal prediction innovations exist in smooth, re‡exive Banach spaces of discrete time processes if and only if the projection operator generating the innovations satisfies the property of iterations. Our theory includes as special cases all previous Wold-type decompositions of discrete time processes; completely characterizes when nonlinear heavy-tailed processes obtain a strong-orthogonal moving average representation; and easily promotes a theory of nonlinear impulse response functions for infinite variance processes. We exemplify our theory by developing a nonlinear impulse response func tion for smooth transition threshold processes, we discuss how to test de composition innovations for strong orthogonality and whether the proposed model represents the best predictor, and we apply the methodology to currency exchange rates.Orthogonal decompositions, Banach spaces, projection iterations, infinite variance, moving average, nonlinear impulse response function, smooth transition autoregression, Lp-metric projection, Lp-GMM.

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 non-causation 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 test statistics to the 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