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This article proposes a test for the martingale difference hypothesis (MDH) using dependence measures related to the characteristic function. The MDH typically has been tested using the sample autocorrelations or in the spectral domain using the periodogram. Tests based on these statistics are inconsistent against uncorrelated non-martingales processes. Here, we generalize the spectral test of Durlauf (1991) for testing the MDH taking into account linear and nonlinear dependence. Our test considers dependence at all lags and is consistent against general pairwise nonparametric Pitman's local alternatives converging at the parametric rate n 1=2 ; with n the sample size. Furthermore, with our methodology there is no need to choose a lag order, to smooth the data or to formulate a parametric alternative. Our approach could be extended to specification testing of the conditional mean of possibly nonlinear models. The asymptotic null distribution of our test depends on the data generating process, so a bootstrap procedure is proposed and theoretically justified. Our bootstrap test is robust to higher order dependence, in particular to conditional heteroskedasticity. A Monte Carlo study examines the finite sample performance of our test and shows that it is more powerful than some competing tests. Finally, an application to the S&P 500 stock index and exchange rates highlights the merits of our approach. r 2005 Elsevier B.V. All rights reserved.JEL classification: C12

This paper proposes a consistent test for the goodness-of-fit of parametric regression models which overcomes two important problems of the existing tests, namely, the poor empirical power and size performance of the tests due to the curse of dimensionality and the choice of subjective parameters like bandwidths, kernels or integrating measures. We overcome these problems by using a residual marked empirical process based on projections (RMPP). We study the asymptotic null distribution of the test statistic and we show that our test is able to detect local alternatives converging to the null at the parametric rate. It turns out that the asymptotic null distribution of the test statistic depends on the data generating process, so a bootstrap procedure is considered. Our bootstrap test is robust to higher order dependence, in particular to conditional heteroskedasticity. For completeness, we propose a new minimum distance estimator constructed through the same RMPP as in the testing procedure. Therefore, the new estimator inherits all the good properties of the new test. We establish the consistency and asymptotic normality of the new minimum distance estimator. Finally, we present some Monte Carlo evidence that our testing procedure can play a valuable role in econometric regression modeling.Juan Carlos Escanciano Reyero Universidad de Navarra, Departamento de Métodos Cuantitativos Campus Universitario, 31080 Pamplona jescanci@unav.es AcknowledgmentsThe author thanks Carlos Velasco and Miguel A. Delgado for useful comments. The paper has also benefited from the comments of two referees and the Co-editor. Research funded by the Spanish Ministry of Education and Science reference number SEJ2004-04583/ECON and by the Universidad de Navarra reference number 16037001. A CONSISTENT DIAGNOSTIC TEST FOR REGRESSION MODELS USING PROJECTIONS J. Carlos EscancianoUniversidad de Navarra June 1, 2005Abstract This paper proposes a consistent test for the goodness-of-…t of parametric regression models which overcomes two important problems of the existing tests, namely, the poor empirical power and size performance of the tests due to the curse of dimensionality and the choice of subjective parameters like bandwidths, kernels or integrating measures. We overcome these problems by using a residual marked empirical process based on projections (RMPP). We study the asymptotic null distribution of the test statistic and we show that our test is able to detect local alternatives converging to the null at the parametric rate. It turns out that the asymptotic null distribution of the test statistic depends on the data generating process, so a bootstrap procedure is considered. Our bootstrap test is robust to higher order dependence, in particular to conditional heteroskedasticity. For completeness, we propose a new minimum distance estimator constructed through the same RMPP as in the testing procedure. Therefore, the new estimator inherits all the good properties of the new test. We establish the consistency and asymptotic nor...

The Basel Committee on Banking Supervision (BIS) has recently sanctioned Expected Shortfall (ES) as the market risk measure to be used for banking regulatory purposes, replacing the well-known Value-at-Risk (V aR). This change is motivated by the appealing theoretical properties of ES as a measure of risk and the poor ones of V aR. In particular, V aR fails to control for "tail risk". In this transition, the major challenge faced by …nancial institutions is the unavailability of simple tools for evaluation of ES forecasts (i.e. backtesting ES). The main purpose of this article is to propose such tools. Speci…cally, we propose a conditional backtest for ES based on cumulative violations, which is the natural analogue of the commonly used conditional backtest for V aR. We establish the asymptotic properties of the test, and investigate its …nite sample performance through some Monte Carlo simulations. An empirical application to three major stock indexes shows that V aR is generally unresponsive to extreme events such as those experienced during the recent …nancial crisis, while ES provides a more accurate description of the risk involved.

We give a general construction of debiased/locally robust/orthogonal (LR) moment functions for GMM, where the derivative with respect to first step nonparametric estimation is zero and equivalently first step estimation has no effect on the influence function. This construction consists of adding an estimator of the influence function adjustment term for first step nonparametric estimation to identifying or original moment conditions. We also give numerical methods for estimating LR moment functions that do not require an explicit formula for the adjustment term.LR moment conditions have reduced bias and so are important when the first step is machine learning. We derive LR moment conditions for dynamic discrete choice based on first step machine learning estimators of conditional choice probabilities.We provide simple and general asymptotic theory for LR estimators based on sample splitting. This theory uses the additive decomposition of LR moment conditions into an identifying condition and a first step influence adjustment. Our conditions require only mean square consistency and a few (generally either one or two) readily interpretable rate conditions.LR moment functions have the advantage of being less sensitive to first step estimation. Some LR moment functions are also doubly robust meaning they hold if one first step is incorrect. We give novel classes of doubly robust moment functions and characterize double robustness. For doubly robust estimators our asymptotic theory only requires one rate condition.