Central quantile subspace

et al. 2020

Stat

In this work, we construct a lack‐of‐fit test for testing parametric single‐index quantile regression models. We apply the kernel smoothing technique for the multivariate nonparametric estimation involved in this task. To avoid the “curse of dimensionality” in multivariate nonparametric estimation and to fully utilize the information contained in the model, we employ a sufficient dimension reduction technique to identify the corresponding dimensionally reduced subspace and then construct our test statistic in this subspace. At different quantile levels, the test statistics given in this paper can quickly detect local alternative hypotheses, which are different from the null hypothesis for small and moderate sample sizes. A new wild bootstrap method is applied to approximate the critical values of the quantile regression model test. The effectiveness of the method is verified by simulation experiments and a real data application.

“…Christou (2020) proved that when q is given,

\overset{B}{true} false(q false)

Section: Asymptotic Propertiesmentioning

confidence: 97%

“…Similarly, the τ th central quantile subspace (CQS) can be defined as the intersection of all subspaces span{ B τ } that satisfy the condition

normalT ⫫ {normalQ}_{scriptT} ((), normalY false| normalX) false| {normalB}_{τ}^{⊤} normalX

, which can be denoted as

{scriptS}_{Q_{τ} false(Y false| X false)}

(refer to Christou, 2020). For simplicity, in the remainder of this paper, we write the structural dimension as

q = q_{Q_{τ} false(Y false| X false)}

and the basis matrix as B = B τ .…”

Section: Model Checking For Quantile Regressionmentioning

confidence: 99%

Section: Model Checking For Quantile Regressionmentioning

confidence: 99%

“…A typical choice for C n is √n . In fact,

P false(\overset{q}{true} = q false) \to 1

, under the assumption that

\overset{B}{true}

is a consistent estimate of B (refer to Christou, 2020).…”

Section: Model Checking For Quantile Regressionmentioning

confidence: 99%

See 2 more Smart Citations

Model checking for parametric single‐index quantile models

Yuan

et al. 2020

Stat

“…Along the lines of Ding and Wang (2011) and Guo et al (2014), we choose

C_{n} = \sqrt{n}

to avoid overestimating or underestimating

q

. In fact,

P false(\overset{q}{true} = q false) \to 1

, under the assumption that

\overset{B}{true}

is a consistent estimate of

B

(refer to Christou, 2020). Notably, in Algorithm 1, the m‐BIC criterion is used twice in steps 1 and 6.…”

Section: Model Checking For Quantile Regression Under Marmentioning

confidence: 99%

Powerful nonparametric checks for parametric single‐index quantile models with missing responses

Yuan

et al. 2022

Stat

This paper considers the lack‐of‐fit test of parametric single‐index quantile models when the response variable is missing at random. The model's coefficients are estimated by an estimation method suitable for the quantile regression coefficients of the missing data. Simultaneously, an algorithm for solving the central subspace of the multidimensional quantile regression model with missing responses is proposed. Based on the central quantile regression subspace, we construct two‐dimensional reduction adaptive‐to‐model test statistics suitable for randomly missing response variables to avoid the curse of dimensionality. Under the null hypothesis and local alternative hypothesis, the asymptotic properties of the test statistics are obtained. The proposed testing methods are shown to be consistent and able to detect local alternative hypothetical models converging to the null model at the rate of order Ofalse(n−1false/2h−1false/4false). A consistent bootstrap method is proposed to determine the critical values, and its asymptotic properties are established. The simulation results show that the proposed method is superior to existing methods in terms of both empirical size and power in the case of multidimensional and even high‐dimensional covariables. The ACTG Protocol 175 data set is analyzed to demonstrate the application of the testing procedures. Supplementary materials for this article are available online.

Semiparametric estimation of expected shortfall and its application in finance

Yan

et al. 2022

Journal of Forecasting

Measuring risk effectively is crucial for managing risk in financial markets. The expected shortfall has become an increasingly popular metric for risk in recent years. How to estimate it is important in statistics and financial econometrics. Based on the single index quantile regression, we introduce a new semiparametric approach, namely, weighted single index quantile regression. We assess the performance of the proposed expected shortfall estimator with backtesting. Our simulation results indicate that the estimator has a good finite sample performance and often outperforms existing methods. By applying the new method to both a market index and individual stocks, we show that it not only exhibits the best performance but also reveals an insight about the effect of the COVID pandemic, that is, the pandemic increases the market risk.