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Marked and Weighted Empirical Processes of Residuals with Applications to Robust Regressions

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  • Vanessa Berenguer Rico
  • Bent Nielsen

Abstract

A new class of marked and weighted empirical processes of residuals is introduced. The framework is general enough to accommodate both stationary and non-stationary regressions as well as a wide class of estimation procedures with applications in misspecification testing and robust statistics. Two applications are presented. First, we analyze the relationship between truncated moments and linear statistical functionals of residuals. In particular, we show that the asymptotic behaviour of these functionals, expressed as integrals with respect to their empirical distribution functions, can be easily analyzed given the main theorems of the paper. In our context the integrands can be unbounded provided that the underlying distribution meets certain moment conditions. A general first order asymptotic approximation of the statistical functionals is derived and then applied to some cases of interest. Second, the consequences of using the standard cumulant based normality test for robust regressions are analyzed. We show that the rescaling of the moment based statistic is case dependent, i.e., it depends on the truncation and the estimation method being used. Hence, using the standard least squares normalizing constants in robust regressions will lead to incorrect inferences. However, if appropriate normalizations, which we derive, are used then the test statistic is asymptotically chi-square.

Suggested Citation

  • Vanessa Berenguer Rico & Bent Nielsen, 2017. "Marked and Weighted Empirical Processes of Residuals with Applications to Robust Regressions," Economics Series Working Papers 841, University of Oxford, Department of Economics.
    • Handle: RePEc:oxf:wpaper:841
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    References listed on IDEAS

    as
    1. Søren Johansen & Bent Nielsen, 2016. "Asymptotic Theory of Outlier Detection Algorithms for Linear Time Series Regression Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(2), pages 321-348, June.
    2. Søren Johansen & Bent Nielsen, 2016. "Rejoinder: Asymptotic Theory of Outlier Detection Algorithms for Linear Time Series Regression Models," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 43(2), pages 374-381, June.
    3. Juan Carlos Escanciano, 2004. "Model Checks Using Residual Marked Empirical Processes," Faculty Working Papers 13/04, School of Economics and Business Administration, University of Navarra.
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    Cited by:

    1. Vanessa Berenguer-Rico & Søren Johansen & Bent Nielsen, 2019. "Models where the Least Trimmed Squares and Least Median of Squares estimators are maximum likelihood," CREATES Research Papers 2019-15, Department of Economics and Business Economics, Aarhus University.
    2. Takamitsu Kurita & Bent Nielsen, 2019. "Partial Cointegrated Vector Autoregressive Models with Structural Breaks in Deterministic Terms," Econometrics, MDPI, vol. 7(4), pages 1-35, October.
    3. David H. Bernstein & Bent Nielsen, 2019. "Asymptotic Theory for Cointegration Analysis When the Cointegration Rank Is Deficient," Econometrics, MDPI, vol. 7(1), pages 1-24, January.
    4. Xiyu Jiao & Felix Pretis, 2022. "Testing the Presence of Outliers in Regression Models," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 84(6), pages 1452-1484, December.
    5. Takamitsu Kurita & B. Nielsen, 2018. "Partial cointegrated vector autoregressive models with structural breaks in deterministic terms," Economics Papers 2018-W03, Economics Group, Nuffield College, University of Oxford.

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