1. What is the proposed framework for estimating expected shortfall regression coefficients?
The proposed framework for estimating expected shortfall regression coefficients involves a two-step l1-penalized approach. In the first step, a lasso-penalized estimator is used to estimate the quantile regression coefficients. In the second step, a l1-penalized orthogonal-score least squares estimator is used to estimate the expected shortfall regression coefficients. The framework addresses the challenge of non-tractable limiting distributions by introducing a new score function and proposing a debiased estimator for valid inference. The decorrelated score test is used to test the hypothesis, and a confidence interval is constructed using a debiased estimator and a Wald-type approach.
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2. How to estimate s2s and s2o in high-dimensional linear models?
To estimate s2s and s2o in high-dimensional linear models, a refitted cross-validation method is proposed. The dataset is randomly split into two parts, S1 and S2. The first half, S1, is used to compute bcv, thcv, and gcv by solving equations (3.2), (3.3), and (3.4) respectively, with tuning parameters lq, le, and lm determined by cross-validation. The selected variables with cardinalities sq, se, and sm are denoted as Sq, Se, and Sm. The second half, S2, is used to compute intermediate variance estimators s2s,1 and s2o,1 using equations (3.12). This method helps to address false discoveries and downward bias in the lasso residual sum of squares estimator.
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3. What is the weakest model consistency property for sparse regression?
The weakest model consistency property for sparse regression in the high-dimensional setting is the sure screening condition, as verified in Fan et al. 2012. This condition ensures consistency by requiring a summation of products of two residual squares, along with three sources of bias in b, th, and g. The support of the lasso estimators from the first stage, namely Sq, Se, and Sm, and their corresponding cardinalities sq, se, and sm, are also crucial for verifying consistency. Additionally, the assumption that g*1 is bounded and penalization parameters lq, le log(p)/n, and the sample size satisfying max(s2, s20) log(p) = o(n) contribute to the consistency of the refitted cross-validated variance estimator s2s/s4o as defined in (3.13).
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4. What is the proposed two-step method for estimation?
The proposed two-step method involves computing an l1-penalized quantile regression estimator and a lasso-type estimator with the adjusted response variable defined in (2.5). It uses the R package conquer with default tuning parameters to obtain an l1-penalized smoothed quantile regression estimator, which is statistically equivalent to the l1-penalized quantile estimator in (3.2). The method compares the proposed estimator with the oracle expected shortfall estimator obtained by regressing {Z i (b *)} n i=1 on {t X i, S * e } n i=1. It calculates the relative l2-error, true positive rate, and false positive rate for each method. The simulation results show that the cross-validated two-step estimator performs slightly worse than the oracle estimator, but the estimation errors decrease with increasing sample size. The refitted estimator has a similar estimation error to the oracle estimator for true positive coefficients but a high estimation error for false positives. More extensive numerical studies are conducted in Section C of the online Supplementary Materials.
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