Distribution Theory of the Least Squares Averaging Estimator

This paper derives the limiting distributions of least squares averaging estimators for linear regression models in a local asymptotic framework. We show that the averaging estimators with fixed weights are asymptotically normal and then develop a plug-in averaging estimator that minimizes the sample analog of the asymptotic mean squared error. We investigate the focused information criterion (Claeskens and Hjort, 2003), the plug-in averaging estimator, the Mallows model averaging estimator (Hansen, 2007), and the jackknife model averaging estimator (Hansen and Racine, 2012). We find that the asymptotic distributions of averaging estimators with data-dependent weights are nonstandard and cannot be approximated by simulation. To address this issue, we propose a simple procedure to construct valid confidence intervals with improved coverage probability. Monte Carlo simulations show that the plug-in averaging estimator generally has smaller expected squared error than other existing model averaging methods, and the coverage probability of proposed confidence intervals achieves the nominal level. As an empirical illustration, the proposed methodology is applied to cross-country growth regressions

Distribution Theory of the Least Squares Averaging Estimator

This paper derives the limiting distributions of least squares averaging estimators for linear regression models in a local asymptotic framework. We show that the averaging estimators with fixed weights are asymptotically normal and then develop a plug-in averaging estimator that minimizes the sample analog of the asymptotic mean squared error. We investigate the focused information criterion (Claeskens and Hjort, 2003), the plug-in averaging estimator, the Mallows model averaging estimator (Hansen, 2007), and the jackknife model averaging estimator (Hansen and Racine, 2012). We find that the asymptotic distributions of averaging estimators with data-dependent weights are nonstandard and cannot be approximated by simulation. To address this issue, we propose a simple procedure to construct valid confidence intervals with improved coverage probability. Monte Carlo simulations show that the plug-in averaging estimator generally has smaller expected squared error than other existing model averaging methods, and the coverage probability of proposed confidence intervals achieves the nominal level. As an empirical illustration, the proposed methodology is applied to cross-country growth regressions

Model selection and model averaging in nonparametric instrumental variables models

This paper considers the problem of choosing the regularization parameter and the smoothing parameter in nonparametric instrumental variables estimation. We propose a simple Mallows’ Cp-type criterion to select these two parameters simultaneously. We show that the proposed selection criterion is optimal in the sense that the selected estimate asymptotically achieves the lowest possible mean squared error among all candidates. To account for model uncertainty, we introduce a new model averaging estimator for nonparametric instrumental variables regressions. We propose a Mallows criterion for the weight selection and demonstrate its asymptotic optimality. Monte Carlo simulations show that both selection and averaging methods generally achieve lower root mean squared error than other existing methods. The proposed methods are applied to two empirical examples, the effect of class size question and Engel curve

Model selection and model averaging in nonparametric instrumental variables models

This paper considers the problem of choosing the regularization parameter and the smoothing parameter in nonparametric instrumental variables estimation. We propose a simple Mallows’ Cp-type criterion to select these two parameters simultaneously. We show that the proposed selection criterion is optimal in the sense that the selected estimate asymptotically achieves the lowest possible mean squared error among all candidates. To account for model uncertainty, we introduce a new model averaging estimator for nonparametric instrumental variables regressions. We propose a Mallows criterion for the weight selection and demonstrate its asymptotic optimality. Monte Carlo simulations show that both selection and averaging methods generally achieve lower root mean squared error than other existing methods. The proposed methods are applied to two empirical examples, the effect of class size question and Engel curve

Testing Monotonicity of Mean Potential Outcomes in a Continuous Treatment

While most treatment evaluations focus on binary interventions, a growing literature also considers continuously distributed treatments, e.g. hours spent in a training program to assess its effect on labor market outcomes. In this paper, we propose a Cram\'er-von Mises-type test for testing whether the mean potential outcome given a specific treatment has a weakly monotonic relationship with the treatment dose under a weak unconfoundedness assumption. This appears interesting for testing shape restrictions, e.g. whether increasing the treatment dose always has a non-negative effect, no matter what the baseline level of treatment is. We formally show that the proposed test controls asymptotic size and is consistent against any fixed alternative. These theoretical findings are supported by the method's finite sample behavior in our Monte-Carlo simulations. As an empirical illustration, we apply our test to the Job Corps study and reject a weakly monotonic relationship between the treatment (hours in academic and vocational training) and labor market outcomes like earnings or employment

Model Averaging in Predictive Regressions

This paper considers forecast combination in a predictive regression. We construct the point forecast by combining predictions from all possible linear regression models given a set of potentially relevant predictors. We derive the asymptotic risk of least squares averaging estimators in a local asymptotic framework. We then develop a frequentist model averaging criterion, an asymptotically unbiased estimator of the asymptotic risk, to select forecast weights. Monte Carlo simulations show that our averaging estimator compares favorably with alternative methods such as weighted AIC, weighted BIC, Mallows model averaging, and jackknife model averaging. The proposed method is applied to stock return predictions

Model Averaging in Predictive Regressions

This paper considers forecast combination in a predictive regression. We construct the point forecast by combining predictions from all possible linear regression models given a set of potentially relevant predictors. We derive the asymptotic risk of least squares averaging estimators in a local asymptotic framework. We then develop a frequentist model averaging criterion, an asymptotically unbiased estimator of the asymptotic risk, to select forecast weights. Monte Carlo simulations show that our averaging estimator compares favorably with alternative methods such as weighted AIC, weighted BIC, Mallows model averaging, and jackknife model averaging. The proposed method is applied to stock return predictions

Model Averaging in Predictive Regressions

This paper considers forecast combination in a predictive regression. We construct the point forecast by combining predictions from all possible linear regression models given a set of potentially relevant predictors. We propose a frequentist model averaging criterion, an asymptotically unbiased estimator of the mean squared forecast error (MSFE), to select forecast weights. In contrast to the existing literature, we derive the MSFE in a local asymptotic framework without the i.i.d. normal assumption. This result allows us to decompose the MSFE into the bias and variance components and also to account for the correlations between candidate models. Monte Carlo simulations show that our averaging estimator has much lower MSFE than alternative methods such as weighted AIC, weighted BIC, Mallows model averaging, and jackknife model averaging. We apply the proposed method to stock return predictions

New Insights on 30 Dor B Revealed by High-Quality Multi-wavelength Observations

The supernova remnant (SNR) 30 Dor B is associated with the \ion{H}{2} region ionized by the OB association LH99. The complex interstellar environment has made it difficult to study the physical structure of this SNR. We have used Hubble Space Telescope H$\alpha$ images to identify SNR shocks and deep Chandra X-ray observations to detect faint diffuse emission. We find that 30 Dor B hosts three zones with very different X-ray surface brightnesses and nebular kinematics that are characteristic of SNRs in different interstellar environments and/or evolutionary stages. The ASKAP 888 MHz map of 30 Dor B shows counterparts to all X-ray emission features except the faint halo. The ASKAP 888 MHz and 1420 MHz observations are used to produce a spectral index map, but its interpretation is complicated by the background thermal emission and the pulsar PSR J0537$-$6910's flat spectral index. The stellar population in the vicinity of 30 Dor B indicates a continuous star formation in the past 8--10 Myr. The observed very massive stars in LH99 cannot be coeval with the progenitor of 30 Dor B's pulsar. Adopting the pulsar's spin-down timescale, 5000 yr, as the age of the SNR, the X-ray shell would be expanding at $\sim$4000 km\,s$^{-1}$ and the post-shock temperature would be 1--2 orders of magnitude higher than that indicated by the X-ray spectra. Thus, the bright central region of 30 Dor B and the X-ray shell requires two separate SN events, and the faint diffuse X-ray halo perhaps other older SN events