Partial Consistency with Sparse Incidental Parameters

Penalized estimation principle is fundamental to high-dimensional problems. In the literature, it has been extensively and successfully applied to various models with only structural parameters. As a contrast, in this paper, we apply this penalization principle to a linear regression model with a finite-dimensional vector of structural parameters and a high-dimensional vector of sparse incidental parameters. For the estimators of the structural parameters, we derive their consistency and asymptotic normality, which reveals an oracle property. However, the penalized estimators for the incidental parameters possess only partial selection consistency but not consistency. This is an interesting partial consistency phenomenon: the structural parameters are consistently estimated while the incidental ones cannot. For the structural parameters, also considered is an alternative two-step penalized estimator, which has fewer possible asymptotic distributions and thus is more suitable for statistical inferences. We further extend the methods and results to the case where the dimension of the structural parameter vector diverges with but slower than the sample size. A data-driven approach for selecting a penalty regularization parameter is provided. The finite-sample performance of the penalized estimators for the structural parameters is evaluated by simulations and a real data set is analyzed

Risks of Large Portfolios

Estimating and assessing the risk of a large portfolio is an important topic in financial econometrics and risk management. The risk is often estimated by a substitution of a good estimator of the volatility matrix. However, the accuracy of such a risk estimator for large portfolios is largely unknown, and a simple inequality in the previous literature gives an infeasible upper bound for the estimation error. In addition, numerical studies illustrate that this upper bound is very crude. In this paper, we propose factor-based risk estimators under a large amount of assets, and introduce a high-confidence level upper bound (H-CLUB) to assess the accuracy of the risk estimation. The H-CLUB is constructed based on three different estimates of the volatility matrix: sample covariance, approximate factor model with known factors, and unknown factors (POET, Fan, Liao and Mincheva, 2013). For the first time in the literature, we derive the limiting distribution of the estimated risks in high dimensionality. Our numerical results demonstrate that the proposed upper bounds significantly outperform the traditional crude bounds, and provide insightful assessment of the estimation of the portfolio risks. In addition, our simulated results quantify the relative error in the risk estimation, which is usually negligible using 3-month daily data. Finally, the proposed methods are applied to an empirical study

Exact solution to the steady-state dynamics of a periodically-modulated resonator

We provide an analytic solution to the coupled-mode equations describing the steady-state of a single periodically-modulated optical resonator driven by a monochromatic input. The phenomenology of this system was qualitatively understood only in the adiabatic limit, i.e. for low modulation speed. However, both in and out of this regime, we find highly non-trivial effects for specific parameters of the modulation. For example, we show complete suppression of the transmission even with zero detuning between the input and the static resonator frequency. We also demonstrate the possibility for complete, lossless frequency conversion of the input into the side-band frequencies, as well as for optimizing the transmitted signal towards a given target temporal waveform. The analytic results are validated by first-principle simulations