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This paper proposes a robust, shocks-adaptive portfolio in a large-dimensional assets universe where the number of assets could be comparable to or even larger than the sample size. It is well documented that portfolios based on optimizations are sensitive to outliers in return data. We deal with outliers by proposing a robust factor model, contributing methodologically through the development of a robust principal component analysis (PCA) for factor model estimation and a shrinkage estimation for the random error covariance matrix. This approach extends the well-regarded Principal Orthogonal Complement Thresholding (POET) method (Fan et al., 2013), enabling it to effectively handle heavy tails and sudden shocks in data. The novelty of the proposed robust method is its adaptiveness to both global and idiosyncratic shocks, without the need to distinguish them, which is useful in forming portfolio weights when facing outliers. We develop the theoretical results of the robust factor model and the robust minimum variance portfolio. Numerical and empirical results show the superior performance of the new portfolio.

Robust estimation for modern portfolio selection on a large set of assets becomes more important due to large deviation of empirical inference on big data. We propose a distributionally robust methodology for high-dimensional mean-variance portfolio problem, aiming to select an optimal conservative portfolio allocation by taking distribution uncertainty into account. With the help of factor structure, we extend the distributionally robust mean-variance problem investigated by Blanchet et al. (2022, Management Science) to the high-dimensional scenario and transform it to a new penalized risk minimization problem. Furthermore, we propose a data-adaptive method to estimate the quantified uncertainty size, which is the radius around the empirical probability measured by the Wasserstein distance. Asymptotic consistency is derived for the estimation of the population parameters involved in selecting the uncertainty size and the selected portfolio return. Our Monte-Carlo simulation results show that the chosen uncertainty size and target return from the proposed procedure are very close to the corresponding oracle version, and the new portfolio strategy is of low risk. Finally, we conduct empirical studies based on S&P index components to show the robust performance of our proposal in terms of risk controlling and return-risk balancing., Comment: 41 pages, 3 figures, 6 tables

In this paper, we investigate a semiparametric regression model under the context of treatment effects via a localized neural network (LNN) approach. Due to a vast number of parameters involved, we reduce the number of effective parameters by (i) exploring the use of identification restrictions; and (ii) adopting a variable selection method based on the group-LASSO technique. Subsequently, we derive the corresponding estimation theory and propose a dependent wild bootstrap procedure to construct valid inferences accounting for the dependence of data. Finally, we validate our theoretical findings through extensive numerical studies. In an empirical study, we revisit the impacts of a tightening monetary policy action on a variety of economic variables, including short-/long-term interest rate, inflation, unemployment rate, industrial price and equity return via the newly proposed framework using a monthly dataset of the US.

In this paper, we propose a novel robust Principal Component Analysis (PCA) for high-dimensional data in the presence of various heterogeneities, especially the heavy-tailedness and outliers. A transformation motivated by the characteristic function is constructed to improve the robustness of the classical PCA. Besides the typical outliers, the proposed method has the unique advantage of dealing with heavy-tail-distributed data, whose covariances could be nonexistent (positively infinite, for instance). The proposed approach is also a case of kernel principal component analysis (KPCA) method and adopts the robust and non-linear properties via a bounded and non-linear kernel function. The merits of the new method are illustrated by some statistical properties including the upper bound of the excess error and the behaviors of the large eigenvalues under a spiked covariance model. In addition, we show the advantages of our method over the classical PCA by a variety of simulations. At last, we apply the new robust PCA to classify mice with different genotypes in a biological study based on their protein expression data and find that our method is more accurately on identifying abnormal mice comparing to the classical PCA.

Principal component analysis (PCA), the most popular dimension-reduction technique, has been used to analyze high-dimensional data in many areas. It discovers the homogeneity within the data and creates a reduced feature space to capture as much information as possible from the original data. However, in the presence of a group structure of the data, PCA often fails to identify the group-specific pattern, which is known as sub-homogeneity in this study. Group-specific information that is missed can result in an unsatisfactory representation of the data from a particular group. It is important to capture both homogeneity and sub-homogeneity in high-dimensional data analysis, but this poses a great challenge. In this study, we propose a novel iterative complement-clustering principal component analysis (CPCA) to iteratively estimate the homogeneity and sub-homogeneity. A principal component regression based clustering method is also introduced to provide reliable information about clusters. Theoretically, this study shows that our proposed clustering approach can correctly identify the cluster membership under certain conditions. The simulation study and real analysis of the stock return data confirm the superior performance of our proposed methods.

We propose modeling raw functional data as a mixture of a smooth function and a high-dimensional factor component. The conventional approach to retrieving the smooth function from the raw data is through various smoothing techniques. However, the smoothing model is inadequate to recover the smooth curve or capture the data variation in some situations. These include cases where there is a large amount of measurement error, the smoothing basis functions are incorrectly identified, or the step jumps in the functional mean levels are neglected. A factor-augmented smoothing model is proposed to address these challenges, and an iterative numerical estimation approach is implemented in practice. Including the factor model component in the proposed method solves the aforementioned problems since a few common factors often drive the variation that cannot be captured by the smoothing model. Asymptotic theorems are also established to demonstrate the effects of including factor structures on the smoothing results. Specifically, we show that the smoothing coefficients projected on the complement space of the factor loading matrix are asymptotically normal. As a byproduct of independent interest, an estimator for the population covariance matrix of the raw data is presented based on the proposed model. Extensive simulation studies illustrate that these factor adjustments are essential in improving estimation accuracy and avoiding the curse of dimensionality. The superiority of our model is also shown in modeling Australian temperature data., Comment: 88 pages, 11 figures. arXiv admin note: substantial text overlap with arXiv:2102.02580

High-dimensional autocovariance matrices play an important role in dimension reduction for high-dimensional time series. In this article, we establish the central limit theorem (CLT) for spiked eigenvalues of high-dimensional sample autocovariance matrices, which are developed under general conditions. The spiked eigenvalues are allowed to go to infinity in a flexible way without restrictions in divergence order. Moreover, the number of spiked eigenvalues and the time lag of the autocovariance matrix under this study could be either fixed or tending to infinity when the dimension p and the time length T go to infinity together. As a further statistical application, a novel autocovariance test is proposed to detect the equivalence of spiked eigenvalues for two high-dimensional time series. Various simulation studies are illustrated to justify the theoretical findings. Furthermore, a hierarchical clustering approach based on the autocovariance test is constructed and applied to clustering mortality data from multiple countries.

Modelling and forecasting homogeneous age-specific mortality rates of multiple countries could lead to improvements in long-term forecasting. Data fed into joint models are often grouped according to nominal attributes, such as geographic regions, ethnic groups, and socioeconomic status, which may still contain heterogeneity and deteriorate the forecast results. Our paper proposes a novel clustering technique to pursue homogeneity among multiple functional time series based on functional panel data modelling to address this issue. Using a functional panel data model with fixed effects, we can extract common functional time series features. These common features could be decomposed into two components: the functional time trend and the mode of variations of functions (functional pattern). The functional time trend reflects the dynamics across time, while the functional pattern captures the fluctuations within curves. The proposed clustering method searches for homogeneous age-specific mortality rates of multiple countries by accounting for both the modes of variations and the temporal dynamics among curves. We demonstrate that the proposed clustering technique outperforms other existing methods through a Monte Carlo simulation and could handle complicated cases with slow decaying eigenvalues. In empirical data analysis, we find that the clustering results of age-specific mortality rates can be explained by the combination of geographic region, ethnic groups, and socioeconomic status. We further show that our model produces more accurate forecasts than several benchmark methods in forecasting age-specific mortality rates.

This paper proposes a new AR-sieve bootstrap approach on high-dimensional time series. The major challenge of classical bootstrap methods on high-dimensional time series is two-fold: the curse dimensionality and temporal dependence. To tackle such difficulty, we utilise factor modelling to reduce dimension and capture temporal dependence simultaneously. A factor-based bootstrap procedure is constructed, which conducts AR-sieve bootstrap on the extracted low-dimensional common factor time series and then recovers the bootstrap samples for original data from the factor model. Asymptotic properties for bootstrap mean statistics and extreme eigenvalues are established. Various simulations further demonstrate the advantages of the new AR-sieve bootstrap under high-dimensional scenarios. Finally, an empirical application on particulate matter (PM) concentration data is studied, where bootstrap confidence intervals for mean vectors and autocovariance matrices are provided.

This paper considers a model with general regressors and unobservable factors. An estimator based on iterated principal components is proposed, which is shown to be not only asymptotically normal and oracle efficient, but under certain conditions also free of the otherwise so common asymptotic incidental parameters bias. Interestingly, the conditions required to achieve unbiasedness become weaker the stronger the trends in the factors, and if the trending is strong enough unbiasedness comes at no cost at all. In particular, the approach does not require any knowledge of how many factors there are, or whether they are deterministic or stochastic. The order of integration of the factors is also treated as unknown, as is the order of integration of the regressors, which means that there is no need to pre-test for unit roots, or to decide on which deterministic terms to include in the model.

We propose a dual-factor model for high-dimensional functional time series (HDFTS) that considers multiple populations. The HDFTS is first decomposed into a collection of functional time series (FTS) in a lower dimension and a group of population-specific basis functions. The system of basis functions describes cross-sectional heterogeneity, while the reduced-dimension FTS retains most of the information common to multiple populations. The low-dimensional FTS is further decomposed into a product of common functional loadings and a matrix-valued time series that contains the most temporal dynamics embedded in the original HDFTS. The proposed general-form dual-factor structure is connected to several commonly used functional factor models. We demonstrate the finite-sample performances of the proposed method in recovering cross-sectional basis functions and extracting common features using simulated HDFTS. An empirical study shows that the proposed model produces more accurate point and interval forecasts for subnational age-specific mortality rates in Japan. The financial benefits associated with the improved mortality forecasts are translated into a life annuity pricing scheme.

We propose a novel method to extract global and local features of functional time series. The global features concerning the dominant modes of variation over the entire function domain, and local features of function variations over particular short intervals within function domain, are both important in functional data analysis. Functional principal component analysis (FPCA), though a key feature extraction tool, only focus on capturing the dominant global features, neglecting highly localized features. We introduce a FPCA-BTW method that initially extracts global features of functional data via FPCA, and then extracts local features by block thresholding of wavelet (BTW) coefficients. Using Monte Carlo simulations, along with an empirical application on near-infrared spectroscopy data of wood panels, we illustrate that the proposed method outperforms competing methods including FPCA and sparse FPCA in the estimation functional processes. Moreover, extracted local features inheriting serial dependence of the original functional time series contribute to more accurate forecasts. Finally, we develop asymptotic properties of FPCA-BTW estimators, discovering the interaction between convergence rates of global and local features.

Mortality forecasting plays a pivotal role in insurance and financial risk management of life insurers, pension funds, and social securities. Mortality data is usually high-dimensional in nature and favors factor model approaches to modelling and forecasting. This paper introduces a new forecast-driven hierarchical factor model (FHFM) customized for mortality forecasting. Compared to existing models, which only capture the cross-sectional variation or time-serial dependence in the dimension reduction step, the new model captures both features efficiently under a hierarchical structure, and provides insights into the understanding of dynamic variation of mortality patterns over time. By comparing with static PCA utilized in Lee and Carter 1992, dynamic PCA introduced in Lam et al. 2011, as well as other existing mortality modelling methods, we find that this approach provides both better estimation results and superior out-of-sample forecasting performance. Simulation studies further illustrate the advantages of the proposed model based on different data structures. Finally, empirical studies using the US mortality data demonstrate the implications and significance of this new model in life expectancy forecasting and life annuities pricing.

We propose modeling raw functional data as a mixture of a smooth function and a highdimensional factor component. The conventional approach to retrieving the smooth function from the raw data is through various smoothing techniques. However, the smoothing model is not adequate to recover the smooth curve or capture the data variation in some situations. These include cases where there is a large amount of measurement error, the smoothing basis functions are incorrectly identified, or the step jumps in the functional mean levels are neglected. To address these challenges, a factor-augmented smoothing model is proposed, and an iterative numerical estimation approach is implemented in practice. Including the factor model component in the proposed method solves the aforementioned problems since a few common factors often drive the variation that cannot be captured by the smoothing model. Asymptotic theorems are also established to demonstrate the effects of including factor structures on the smoothing results. Specifically, we show that the smoothing coefficients projected on the complement space of the factor loading matrix is asymptotically normal. As a byproduct of independent interest, an estimator for the population covariance matrix of the raw data is presented based on the proposed model. Extensive simulation studies illustrate that these factor adjustments are essential in improving estimation accuracy and avoiding the curse of dimensionality. The superiority of our model is also shown in modeling Canadian weather data and Australian temperature data.

Many existing mortality models follow the framework of classical factor models, such as the Lee-Carter model and its variants. Latent common factors in factor models are defined as time-related mortality indices (such as $\kappa_t$ in the Lee-Carter model). Factor loadings, which capture the linear relationship between age variables and latent common factors (such as $\beta_x$ in the Lee-Carter model), are assumed to be time-invariant in the classical framework. This assumption is usually too restrictive in reality as mortality datasets typically span a long period of time. Driving forces such as medical improvement of certain diseases, environmental changes and technological progress may significantly influence the relationship of different variables. In this paper, we first develop a factor model with time-varying factor loadings (time-varying factor model) as an extension of the classical factor model for mortality modelling. Two forecasting methods to extrapolate the factor loadings, the local regression method and the naive method, are proposed for the time-varying factor model. From the empirical data analysis, we find that the new model can capture the empirical feature of time-varying factor loadings and improve mortality forecasting over different horizons and countries. Further, we propose a novel approach based on change point analysis to estimate the optimal `boundary' between short-term and long-term forecasting, which is favoured by the local linear regression and naive method, respectively. Additionally, simulation studies are provided to show the performance of the time-varying factor model under various scenarios.

This study decomposes the bilateral trade flows using a three-dimensional panel data model. Under the scenario that all three dimensions diverge to infinity, we propose an estimation approach to identify the number of global shocks and country-specific shocks sequentially, and establish the asymptotic theories accordingly. From the practical point of view, being able to separate the pervasive and nonpervasive shocks in a multi-dimensional panel data is crucial for a range of applications, such as, international financial linkages, migration flows, etc. In the numerical studies, we first conduct intensive simulations to examine the theoretical findings, and then use the proposed approach to investigate the international trade flows from two major trading groups (APEC and EU) over 1982-2019, and quantify the network of bilateral trade.

Accurate estimation for extent of cross{sectional dependence in large panel data analysis is paramount to further statistical analysis on the data under study. Grouping more data with weak relations (cross{sectional dependence) together often results in less efficient dimension reduction and worse forecasting. This paper describes cross-sectional dependence among a large number of objects (time series) via a factor model and parameterizes its extent in terms of strength of factor loadings. A new joint estimation method, benefiting from unique feature of dimension reduction for high dimensional time series, is proposed for the parameter representing the extent and some other parameters involved in the estimation procedure. Moreover, a joint asymptotic distribution for a pair of estimators is established. Simulations illustrate the effectiveness of the proposed estimation method in the finite sample performance. Applications in cross-country macro-variables and stock returns from S&P 500 are studied., Comment: 47 pages, 10 figures

We address the problem of forecasting high-dimensional functional time series through a two-fold dimension reduction procedure. The difficulty of forecasting high-dimensional functional time series lies in the curse of dimensionality. In this paper, we propose a novel method to solve this problem. Dynamic functional principal component analysis is first applied to reduce each functional time series to a vector. We then use the factor model as a further dimension reduction technique so that only a small number of latent factors are preserved. Classic time series models can be used to forecast the factors and conditional forecasts of the functions can be constructed. Asymptotic properties of the approximated functions are established, including both estimation error and forecast error. The proposed method is easy to implement especially when the dimension of the functional time series is large. We show the superiority of our approach by both simulation studies and an application to Japanese age-specific mortality rates., Comment: 17 pages, 2 figures, To appear in Journal of Multivariate Analysis

This paper proposes a new statistic to test independence between two high dimensional random vectors ${\mathbf{X}}:p_1\times1$ and ${\mathbf{Y}}:p_2\times1$. The proposed statistic is based on the sum of regularized sample canonical correlation coefficients of ${\mathbf{X}}$ and ${\mathbf{Y}}$. The asymptotic distribution of the statistic under the null hypothesis is established as a corollary of general central limit theorems (CLT) for the linear statistics of classical and regularized sample canonical correlation coefficients when $p_1$ and $p_2$ are both comparable to the sample size $n$. As applications of the developed independence test, various types of dependent structures, such as factor models, ARCH models and a general uncorrelated but dependent case, etc., are investigated by simulations. As an empirical application, cross-sectional dependence of daily stock returns of companies between different sections in the New York Stock Exchange (NYSE) is detected by the proposed test., Comment: Published in at http://dx.doi.org/10.1214/14-AOS1284 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Statistical inferences for sample correlation matrices are important in high dimensional data analysis. Motivated by this, this paper establishes a new central limit theorem (CLT) for a linear spectral statistic (LSS) of high dimensional sample correlation matrices for the case where the dimension p and the sample size $n$ are comparable. This result is of independent interest in large dimensional random matrix theory. Meanwhile, we apply the linear spectral statistic to an independence test for $p$ random variables, and then an equivalence test for p factor loadings and $n$ factors in a factor model. The finite sample performance of the proposed test shows its applicability and effectiveness in practice. An empirical application to test the independence of household incomes from different cities in China is also conducted., Comment: 78 pages