Comment letters to the National Commission on Commission on Fraudulent Financial Reporting, 1987 (Treadway Commission) Vol. 1

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Bagging Binary and Quantile Predictors for Time Series: Further Issues

Bagging (bootstrap aggregating) is a smoothing method to improve predictive ability under the presence of parameter estimation uncertainty and model uncertainty. In Lee and Yang (2006), we examined how (equal-weighted and BMA-weighted) bagging works for onestep ahead binary prediction with an asymmetric cost function for time series, where we considered simple cases with particular choices of a linlin tick loss function and an algorithm to estimate a linear quantile regression model. In the present paper, we examine how bagging predictors work with different aggregating (averaging) schemes, for multi-step forecast horizons, with a general class of tick loss functions, with different estimation algorithms, for nonlinear quantile regression models, and in different data frequencies. Bagging quantile predictors are constructed via (weighted) averaging over predictors trained on bootstrapped training samples, and bagging binary predictors are conducted via (majority) voting on predictors trained on the bootstrapped training samples. We find that median bagging and trimmed-mean bagging can alleviate the problem of extreme predictors from bootstrap samples and have better performance than equally-weighted bagging predictors; that bagging works more with longer forecast horizons; that bagging works well with highly nonlinear quantile regression models (e.g., artificial neural network), and with general tick loss functions. We also find that the performance of bagging may be affected by using different quantile estimation algorithms (in small sample, even if the estimation is consistent) and by using the different frequency of the time series data