A Test of the Conditional Independence Assumption in Sample Selection Models

Identification in most sample selection models depends on the independence of the regressors and the error terms conditional on the selection probability. All quantile and mean functions are parallel in these models; this implies that quantile estimators cannot reveal any—per assumption non‐existing—heterogeneity. Quantile estimators are nevertheless useful for testing the conditional independence assumption because they are consistent under the null hypothesis. We propose tests of the Kolmogorov–Smirnov type based on the conditional quantile regression process. Monte Carlo simulations show that their size is satisfactory and their power sufficient to detect deviations under plausible data‐generating processes. We apply our procedures to female wage data from the 2011 Current Population Survey and show that homogeneity is clearly rejected. Copyright © 2015 John Wiley & Sons, Ltd.     

Real‐time forecast combinations for the oil price

Baumeister and Kilian (Journal of Business and Economic Statistics, 2015, 33(3), 338–351) combine forecasts from six empirical models to predict real oil prices. In this paper, we broadly reproduce their main economic findings, employing their preferred measures of the real oil price and other real‐time variables. Mindful of the importance of Brent crude oil as a global price benchmark, we extend consideration to the North Sea‐based measure and update the evaluation sample to 2017:12. We model the oil price futures curve using a factor‐based Nelson–Siegel specification estimated in real time to fill in missing values for oil price futures in the raw data. We find that the combined forecasts for Brent are as effective as for other oil price measures. The extended sample using the oil price measures adopted by Baumeister and Kilian yields similar results to those reported in their paper. Also, the futures‐based model improves forecast accuracy at longer horizons.     

Modelling Group Heterogeneity for Small Area Estimation Using M‐Quantiles

Small area estimation typically requires model‐based methods that depend on isolating the contribution to overall population heterogeneity associated with group (i.e. small area) membership. One way of doing this is via random effects models with latent group effects. Alternatively, one can use an M‐quantile ensemble model that assigns indices to sampled individuals characterising their contribution to overall sample heterogeneity. These indices are then aggregated to form group effects. The aim of this article is to contrast these two approaches to characterising group effects and to illustrate them in the context of small area estimation. In doing so, we consider a range of different data types, including continuous data, count data and binary response data.     

Finite sample inference for quantile regression models

Under minimal assumptions, finite sample confidence bands for quantile regression models can be constructed. These confidence bands are based on the “conditional pivotal property” of estimating equations that quantile regression methods solve and provide valid finite sample inference for linear and nonlinear quantile models with endogenous or exogenous covariates. The confidence regions can be computed using Markov Chain Monte Carlo (MCMC) methods. We illustrate the finite sample procedure through two empirical examples: estimating a heterogeneous demand elasticity and estimating heterogeneous returns to schooling. We find pronounced differences between asymptotic and finite sample confidence regions in cases where the usual asymptotics are suspect.     

Forecasting value at risk with intra-day return curves

Methods for incorporating high resolution intra-day asset price data into risk forecasts are being developed at an increasing pace. Existing methods such as those based on realized volatility depend primarily on reducing the observed intra-day price fluctuations to simple scalar summaries. In this study, we propose several methods that incorporate full intra-day price information as functional data objects in order to forecast value at risk (VaR). Our methods are based on the recently proposed functional generalized autoregressive conditionally heteroscedastic (GARCH) models and a new functional linear quantile regression model. In addition to providing daily VaR forecasts, these methods can be used to forecast intra-day VaR curves, which we considered and studied with companion backtests to evaluate the quality of these intra-day risk measures. Using high-frequency trading data from equity and foreign exchange markets, we forecast the one-day-ahead daily and intra-day VaR with the proposed methods and various benchmark models. The empirical results suggested that the functional GARCH models estimated based on the overnight cumulative intra-day return curves exhibited competitive performance with benchmark models for daily risk management, and they produced valid intra-day VaR curves.     

An approximation method to derive confidence intervals for quantiles with some applications

Standard jackknife confidence intervals for a quantile Q _y (β) are usually preferred to confidence intervals based on analytical variance estimators due to their operational simplicity. However, the standard jackknife confidence intervals can give undesirable coverage probabilities for small samples sizes and large or small values of β. In this paper confidence intervals for a population quantile based on several existing estimators of a quantile are derived. These intervals are based on an approximation for the cumulative distribution function of a studentized quantile estimator. Confidence intervals are empirically evaluated by using real data and some applications are illustrated. Results derived from simulation studies show that proposed confidence intervals are narrower than confidence intervals based on the standard jackknife technique, which assumes normal approximation. Proposed confidence intervals also achieve coverage probabilities above to their nominal level. This study indicates that the proposed method can be an alternative to the asymptotic confidence intervals, which can be unknown in practice, and the standard jackknife confidence intervals, which can have poor coverage probabilities and give wider intervals.     

Realized extreme quantile: A joint model for conditional quantiles and measures of volatility with EVT refinements

We propose a new framework exploiting realized measures of volatility to estimate and forecast extreme quantiles. Our realized extreme quantile (REQ) combines quantile regression with extreme value theory and uses a measurement equation that relates the realized measure to the latent conditional quantile. Model estimation is performed by quasi maximum likelihood, and a simulation experiment validates this estimator in finite samples. An extensive empirical analysis shows that high‐frequency measures are particularly informative of the dynamic quantiles. Finally, an out‐of‐sample forecast analysis of quantile‐based risk measures confirms the merit of the REQ.     

Comparison of intraday probabilistic forecasting of solar irradiance using only endogenous data

Accurate solar forecasts are necessary to improve the integration of solar renewables into the energy grid. In recent years, numerous methods have been developed for predicting the solar irradiance or the output of solar renewables. By definition, a forecast is uncertain. Thus, the models developed predict the mean and the associated uncertainty. Comparisons are therefore necessary and useful for assessing the skill and accuracy of these new methods in the field of solar energy.The aim of this paper is to present a comparison of various models that provide probabilistic forecasts of the solar irradiance within a very strict framework. Indeed, we consider focusing on intraday forecasts, with lead times ranging from 1 to 6 h. The models selected use only endogenous inputs for generating the forecasts. In other words, the only inputs of the models are the past solar irradiance data. In this context, the most common way of generating the forecasts is to combine point forecasting methods with probabilistic approaches in order to provide prediction intervals for the solar irradiance forecasts. For this task, we selected from the literature three point forecasting models (recursive autoregressive and moving average (ARMA), coupled autoregressive and dynamical system (CARDS), and neural network (NN)), and seven methods for assessing the distribution of their error (linear model in quantile regression (LMQR), weighted quantile regression (WQR), quantile regression neural network (QRNN), recursive generalized autoregressive conditional heteroskedasticity (GARCHrls), sieve bootstrap (SB), quantile regression forest (QRF), and gradient boosting decision trees (GBDT)), leading to a comparison of 20 combinations of models.None of the model combinations clearly outperform the others; nevertheless, some trends emerge from the comparison. First, the use of the clear sky index ensures the accuracy of the forecasts. This derived parameter permits time series to be deseasonalized with missing data, and is also a good explanatory variable of the distribution of the forecasting errors. Second, regardless of the point forecasting method used, linear models in quantile regression, weighted quantile regression and gradient boosting decision trees are able to forecast the prediction intervals accurately.     

Confidence intervals for the quantile of treatment effects in randomized experiments

In this paper, we explore partial identification and inference for the quantile of treatment effects for randomized experiments. First, we propose nonparametric estimators of sharp bounds on the quantile of treatment effects and establish their asymptotic properties under general conditions. Second, we construct confidence intervals for the bounds and the true quantile by using the approach in Chernozhukov et al. (2009). Third, under additional conditions, we develop a new approach to construct confidence intervals for the bounds and the true quantile and refer to it as the order statistic approach. A simulation study is conducted to investigate the finite sample performance of both approaches.     

Conformal prediction interval estimation and applications to day-ahead and intraday power markets

In this study, we investigated the application of the conformal prediction (CP) concept in the context of short-term electricity price forecasting. In particular, we determined the most important aspects related to the utility of CP, as well as explaining why this simple but highly effective idea has proved useful in other application areas and why its characteristics make it promising for short-term power applications. We compared the performance of CP with various state-of-the-art electricity price forecasting models, such as quantile regression averaging, in an empirical out-of-sample study of three short-term electricity time series. We combined CP with various underlying point forecast models to demonstrate its versatility and behavior under changing conditions. Our findings suggest that CP yields sharp and reliable prediction intervals in short-term power markets. We also inspected the effects of each of the model components to provide path-based guideline regarding how to find the best CP model for each market.     

A Comparison of Methods for Poverty Estimation in Developing Countries

Small area estimation is a widely used indirect estimation technique for micro‐level geographic profiling. Three unit level small area estimation techniques—the ELL or World Bank method, empirical best prediction (EBP) and M‐quantile (MQ) — can estimate micro‐level Foster, Greer, & Thorbecke (FGT) indicators: poverty incidence, gap and severity using both unit level survey and census data. However, they use different assumptions. The effects of using model‐based unit level census data reconstructed from cross‐tabulations and having no cluster level contextual variables for models are discussed, as are effects of small area and cluster level heterogeneity. A simulation‐based comparison of ELL, EBP and MQ uses a model‐based reconstruction of 2000/2001 data from Bangladesh and compares bias and mean square error. A three‐level ELL method is applied for comparison with the standard two‐level ELL that lacks a small area level component. An important finding is that the larger number of small areas for which ELL has been able to produce sufficiently accurate estimates in comparison with EBP and MQ has been driven more by the type of census data available or utilised than by the model per se.     

Valuation ratios and long‐horizon stock price predictability

Using annual data for 1872–1997, this paper re‐examines the predictability of real stock prices based on price–dividend and price–earnings ratios. In line with the extant literature, we find significant evidence of increased long‐horizon predictability; that is, the hypothesis that the current value of a valuation ratio is uncorrelated with future stock price changes cannot be rejected at short horizons but can be rejected at longer horizons based on bootstrapped critical values constructed from linear representations of the data. While increased statistical power at long horizons in finite samples provides a possible explanation for the pattern of predictability in the data, we find via Monte Carlo simulations that the power to detect predictability in finite samples does not increase at long horizons in a linear framework. An alternative explanation for the pattern of predictability in the data is nonlinearities in the underlying data‐generating process. We consider exponential smooth‐transition autoregressive models of the price–dividend and price–earnings ratios and their ability to explain the pattern of stock price predictability in the data. Copyright © 2005 John Wiley & Sons, Ltd.     

SCAD‐penalized quantile regression for high‐dimensional data analysis and variable selection

The present penalized quantile variable selection methods are only applicable to finite number of predictors or do not have oracle property associated with estimator. This technique is considered as an alternative to ordinary least squares regression in case of the outliers and the heavy‐tailed errors existing in linear models. The variable selection through quantile regression with diverging number of parameters is investigated in this paper. The convergence rate of estimator with smoothly clipped absolute deviation penalty function is also studied. Moreover, the oracle property with proper selection of tuning parameter for quantile regression under certain regularity conditions is also established. In addition, the rank correlation screening method is used to accommodate ultra‐high dimensional data settings. Monte Carlo simulations demonstrate finite performance of the proposed estimator. The results of real data reveal that this approach provides substantially more information as compared with ordinary least squares, conventional quantile regression, and quantile lasso.     

Intergenerational Transmission of Fertility Patterns*

Recent studies by economists have focused on cultural transmission from the origin country rather than the origin family. Our paper extends this research by investigating how family‐specific‘cultural transmission’ can affect fertility rates. Following Machado and Santos Silva [Journal of the American Statistical Association (2005) Vol. 100, p. 1226] and Miranda [Journal of Population Economics (2008) Vol. 21, p. 67], we estimate count data quantile regression models using the British Household Panel Survey. We find that a woman's origin‐family size is positively associated with completed fertility in her destination family. A woman's country of birth also matters for her fertility. For a sub‐sample of continuously partnered men and women, both partners’ origin‐family sizes significantly affect destination‐family fertility.     

Quantile regression for dynamic panel data with fixed effects

This paper studies a quantile regression dynamic panel model with fixed effects. Panel data fixed effects estimators are typically biased in the presence of lagged dependent variables as regressors. To reduce the dynamic bias, we suggest the use of the instrumental variables quantile regression method of Chernozhukov and Hansen (2006) along with lagged regressors as instruments. In addition, we describe how to employ the estimated models for prediction. Monte Carlo simulations show evidence that the instrumental variables approach sharply reduces the dynamic bias, and the empirical levels for prediction intervals are very close to nominal levels. Finally, we illustrate the procedures with an application to forecasting output growth rates for 18 OECD countries.     

The effect of household characteristics on living standards in South Africa 1993–1998: a quantile regression analysis with sample attrition

This paper examines whether the dismantling of apartheid has resulted in an improvement in the standard of living for the vast majority of South Africans. The study is based on a panel data set from the Kwazulu‐Natal province. We use weighted quantile regressions to examine the distribution of standards of living, which corrects for the potential bias arising from non‐random sample attrition. Our results show that there has been a significant increase in the spread of the distribution of household expenditure of the non‐white households residing in Kwazulu‐Natal province. Copyright © 2006 John Wiley & Sons, Ltd.     

Smoothed binary regression quantiles

This paper extends results regarding smoothed median binary regression to general smoothed binary quantile regression, discusses the interpretation of the resulting estimators under alternative assumptions, and shows how they may be used to obtain semiparametric estimates of counterfactual probabilities. The estimators are applied to a model of labour force participation of married women in the USA. We find that the elasticity with respect to non‐labour income is significantly negative only for women that belong to the middle of the conditional willingness‐to‐participate (WTP) distribution. In comparing the quantile models with parametric logit and semiparametric single‐index specifications, we find that the models agree closely for women around the centre of the WTP distribution, but there are considerable disagreements as we move towards the tails of the distribution. Copyright © 2006 John Wiley & Sons, Ltd.     

CEO Stock‐Based Incentive Compensation and Firm Performance: A Quantile Regression Approach

This study employs the quantile regression model to examine the non‐monotonic impact of CEO stock‐based compensation on firm performance, using the data for U.S. non‐financial firms from 1993 to 2005. The results indicate that while the impact of CEO stock‐based pay on firm performance is positive for firms in the higher earnings quantile levels, the impact is negative for firms in the lower levels. In addition, the “V‐shaped” relationship between CEO stock‐based pay and firm performance satisfactorily explains the longstanding disagreement among earlier studies with regard to whether CEO stock‐based pay can enhance firm performance. Furthermore, the quantile‐varying pattern of the impact of stock‐based compensation on firm performance is robust after controlling for the industrial and yearly effects. It is also robust to the use of the pay‐for‐performance sensitivity as an alternative explanatory variable or the market‐based measure of performance as the dependent variable, or the consideration of the suspected endogenous problem between firm performance and stock‐based compensation.     

Modelling firm‐size distribution using Box–Cox heteroscedastic regression

Using the Box–Cox regression model with heteroscedasticity (BCHR), we re‐examine the size distribution of the Portuguese manufacturing firms studied by Machado and Mata ( 2000 ) using the Box–Cox quantile regression (BCQR) method. We show that the BCHR model compares favourably against the BCQR method. In particular, the BCHR model can answer the key questions addressed by the BCQR method, with the advantage that the estimated quantile functions are monotonic. Furthermore, estimation of the BCHR model is straightforward and the confidence intervals of the BCHR regression quantiles are easy to compute. Copyright © 2006 John Wiley & Sons, Ltd.