Tomohiro Ando

A Model-Averaging Approach for High-Dimensional Regression

This article considers high-dimensional regression problems in which the number of predictors p exceeds the sample size n. We develop a model-averaging procedure for high-dimensional regression problems. Unlike most variable selection studies featuring the identification of true predictors, our focus here is on the prediction accuracy for the true conditional mean of y given the p predictors. Our method consists of two steps. The first step is to construct a class of regression models, each with a smaller number of regressors, to avoid the degeneracy of the information matrix. The second step is to find suitable model weights for averaging. To minimize the prediction error, we estimate the model weights using a delete-one cross-validation procedure. Departing from the literature of model averaging that requires the weights always sum to one, an important improvement we introduce is to remove this constraint. We derive some theoretical results to justify our procedure. A theorem is proved, showing that delete-one cross-validation achieves the lowest possible prediction loss asymptotically. This optimality result requires a condition that unravels an important feature of high-dimensional regression. The prediction error of any individual model in the class for averaging is required to be higher than the classic root n rate under the traditional parametric regression. This condition reflects the difficulty of high-dimensional regression and it depicts a situation especially meaningful for p > n. We also conduct a simulation study to illustrate the merits of the proposed approach over several existing methods, including lasso, group lasso, forward regression, Phase Coupled (PC)-simple algorithm, Akaike information criterion (AIC) model-averaging, Bayesian information criterion (BIC) model-averaging methods, and SCAD (smoothly clipped absolute deviation). This approach uses quadratic programming to overcome the computing time issue commonly encountered in the cross-validation literature. Supplementary materials for this article are available online.

Quantile regression models with factor‐augmented predictors and information criterion

For situations with a large number of series, N, each with T observations and each containing a certain amount of information for prediction of the variable of interest, we propose a new statistical modelling methodology that first estimates the common factors from a panel of data using principal component analysis and then employs the estimated factors in a standard quantile regression. A crucial step in the model‐building process is the selection of a good model among many possible candidates. Taking into account the effect of estimated regressors, we develop an information‐theoretic criterion. We also investigate the criterion when there is no estimated regressors. Results of Monte Carlo simulations demonstrate that the proposed criterion performs well in a wide range of situations.

Bayesian panel data analysis for exploring the impact of subprime financial crisis on the US stock market

The effects of recent subprime financial crisis on the US stock market are analyzed. To investigate this problem, a Bayesian panel data analysis to identify common factors that explain the movement of stock returns when the dimension is high is developed. For high-dimensional panel data, it is known that previously proposed approaches cannot estimate accurately the variance-covariance matrix. An advantage of the proposed method is that it considers parameter uncertainty in variance-covariance estimation and factor selection. Two new criteria for determining the number of factors in the data are developed and the consistency of the selection criteria as both the number of observations and the cross-section dimension tend to infinity is established. An empirical analysis indicates that the US stock market was subject to 8 common factors before the outbreak of the subprime crisis, but the number of factors reduced substantially after the outbreak. In particular, a small number of common factors govern the fluctuations of the stock market after the collapse of Lehman Brothers. In other words, empirical evidence that the structure of US stock market has changed drastically after the subprime crisis is obtained. It is also shown that the factor models selected by the proposed criteria work well in out-of-sample forecasting of asset returns.

Clustering Huge Number of Financial Time Series: A Panel Data Approach with High-Dimensional Predictors and Factor Structures

This article introduces a new procedure for clustering a large number of financial time series based on high-dimensional panel data with grouped factor structures. The proposed method attempts to capture the level of similarity of each of the time series based on sensitivity to observable factors as well as to the unobservable factor structure. The proposed method allows for correlations between observable and unobservable factors and also allows for cross-sectional and serial dependence and heteroscedasticities in the error structure, which are common in financial markets. In addition, theoretical properties are established for the procedure. We apply the method to analyze the returns for over 6000 international stocks from over 100 financial markets. The empirical analysis quantifies the extent to which the U.S. subprime crisis spilled over to the global financial markets. Furthermore, we find that nominal classifications based on either listed market, industry, country or region are insufficient to characterize the heterogeneity of the global financial markets. Supplementary materials for this article are available online.

Bayesian corporate bond pricing and credit default swap premium models for deriving default probabilities and recovery rates

This paper develops a Bayesian method by jointly formulating a corporate bond (CB) pricing model and credit default swap (CDS) premium pricing models to estimate the term structure of default probabilities and the recovery rate. These parameters are formulated by incorporating firm characteristics such as industry, credit rating and Balance Sheet/Profit and Loss information. A cross-sectional model valuing all given CB prices and CDS premiums is considered. The quantities derived are regarded as what market participants infer in forming CB prices and CDS premiums. We also develop a statistical significance test procedure without any distributional assumptions for the specified model. An empirical analysis is conducted using Japanese CB and CDS market data.

A Predictive Approach for Selection of Diffusion Index Models

In this article, we propose a predictive mean squared error criterion for selecting diffusion index models, which are useful in forecasting when many predictors are available. A special feature of the proposed criterion is that it takes into account the uncertainty in estimated common factors. The new criterion is based on estimating the predictive mean squared error in forecasting with correction for asymptotic bias. The resulting estimate of bias-corrected forecast-error is shown to be consistent. The proposed criterion is a natural extension of the traditional Akaike information criterion (AIC), but it does not require the distributional assumptions for the likelihood. Results of real data analysis and Monte Carlo simulations demonstrate that the proposed criterion works well in comparison with the commonly used AIC and Bayesian information criteria.

Generalized predictive information criteria for the analysis of feature events

This paper develops two weighted measures for model selection by generalizing the Kullback-Leibler divergence measure. The concept of a model selection process that takes into account the special features of the underlying model is introduced using weighted measures. New information criteria are defined using the bias correction of an expected weighted loglikelihood estimator. Using weight functions that match the features of interest in the underlying statistical models, the new information criteria are applied to simulated studies of spline regression and copula model selection. Real data applications are also given for predicting the incidence of disease and for quantile modeling of environmental data.

Merchant selection and pricing strategy for a platform firm in the online group buying market

The online group-buying market is characterized by intense competition between brokers, called platform firms, which function as intermediaries between merchants and consumers. In an environment of intense competition, merchant selection and pricing strategies are critical for platform firms. This paper employs business analytics to support strategy formulation for these firms by forecasting market demand and analyzing competitive environments. We apply the proposed decision framework, which relies on business analytics, to a study of the online group-buying market in Japan.

Selecting the regularization parameters in high-dimensional panel data models: Consistency and efficiency

ABSTRACT This article considers panel data models in the presence of a large number of potential predictors and unobservable common factors. The model is estimated by the regularization method together with the principal components procedure. We propose a panel information criterion for selecting the regularization parameter and the number of common factors under a diverging number of predictors. Under the correct model specification, we show that the proposed criterion consistently identifies the true model. If the model is instead misspecified, the proposed criterion achieves asymptotically efficient model selection. Simulation results confirm these theoretical arguments.

Quantile co-movement in financial markets: A panel quantile model with unobserved heterogeneity

This paper introduces a new procedure for analyzing the quantile co-movement of a large number of financial time series based on a large-scale panel data model with factor structures. The proposed method attempts to capture the unobservable heterogeneity of each of the financial time series based on sensitivity to explanatory variables and to the unobservable factor structure. In our model, the dimension of the common factor structure varies across quantiles, and the factor structure is allowed to be correlated with the explanatory variables. The proposed method allows for both cross-sectional and serial dependence, and heteroskedasticity, which are common in financial markets. We propose new estimation procedures for both frequentist and Bayesian frameworks. Consistency and asymptotic normality of the proposed estimator are established. We also propose a new model selection criterion for determining the number of common factors together with theoretical support. We apply the method to analyze the returns for over 6,000 international stocks from over 60 countries during the subprime crisis, European sovereign debt crisis, and subsequent period. The empirical analysis indicates that the common factor structure varies across quantiles. We find that the common factors for the quantiles and the common factors for the mean are different.