A Note on Unwanted Variance in Exploratory Factor Models

One strategy of exploratory factor analysis is to decide on the number of factors to extract by means of the eigenvalues of an initial principal component analysis. The present article proves that there is a non zero covariance of the factors with the components rejected when the number of factors to extract is determined by means of principal components analysis. Thus, some of the variance declared as irrelevant or unwanted in an initial principal component analysis is again part of the final factor model.     

中国上市公司资本结构影响因素的实证分析

文章运用主成分因子分析法和多元线性回归法,对中国287家上市公司资本结构的主要影响因素进行了实证分析。研究结论表明:资本结构与困境成本、收益现金流和资产担保价值正相关;与非负债税盾、规模和盈利能力负相关;与盈余管理有关;与成长性无关。     

Diffusion Indexes With Sparse Loadings

The use of large-dimensional factor models in forecasting has received much attention in the literature with the consensus being that improvements on forecasts can be achieved when comparing with standard models. However, recent contributions in the literature have demonstrated that care needs to be taken when choosing which variables to include in the model. A number of different approaches to determining these variables have been put forward. These are, however, often based on ad hoc procedures or abandon the underlying theoretical factor model. In this article, we will take a different approach to the problem by using the least absolute shrinkage and selection operator (LASSO) as a variable selection method to choose between the possible variables and thus obtain sparse loadings from which factors or diffusion indexes can be formed. This allows us to build a more parsimonious factor model that is better suited for forecasting compared to the traditional principal components (PC) approach. We provide an asymptotic analysis of the estimator and illustrate its merits empirically in a forecasting experiment based on U.S. macroeconomic data. Overall we find that compared to PC we obtain improvements in forecasting accuracy and thus find it to be an important alternative to PC. Supplementary materials for this article are available online.     

中国核心通货膨胀的指数测算、分类特征与冲击传导

为了测算和分析我国核心通货膨胀指数，本文在动态因子模型分析框架下引入了时变因子载荷系数、随机扰动和异常值调整，构建了基于我国城市环比CPI的UCSVO模型和基于八大类城市环比CPI及消费支出权重的MUCSVO模型。研究发现：①UCSVO模型识别出的CPI异常变动时间点符合经济现实，由其测算得出的核心通货膨胀指数适用于我国通货膨胀的实时监测；②MUCSVO模型中共同的趋势成份因子及其载荷系数能体现宏观冲击与价格粘性的现实经济含义，价格粘性的差异是各大类核心通货膨胀指数对宏观冲击产生异质性响应的重要原因；③MUCSVO模型所测算的核心通货膨胀指数的分类权重与消费支出成正比、与波动性成反比，在测算分类以及总体核心通货膨胀指数的同时，还能准确反映各大类CPI的变化特征。     

Factor analysis for ranked data with application to a job selection attitude survey

Summary.  Factor analysis is a powerful tool to identify the common characteristics among a set of variables that are measured on a continuous scale. In the context of factor analysis for non-continuous-type data, most applications are restricted to item response data only. We extend the factor model to accommodate ranked data. The Monte Carlo expectation–maximization algorithm is used for parameter estimation at which the E-step is implemented via the Gibbs sampler. An analysis based on both complete and incomplete ranked data (e.g. rank the top q out of k items) is considered. Estimation of the factor scores is also discussed. The method proposed is applied to analyse a set of incomplete ranked data that were obtained from a survey that was carried out in GuangZhou, a major city in mainland China, to investigate the factors affecting people's attitude towards choosing jobs.     

A skew-normal factor model for the analysis of student satisfaction towards university courses

Classical factor analysis relies on the assumption of normally distributed factors that guarantees the model to be estimated via the maximum likelihood method. Even when the assumption of Gaussian factors is not explicitly formulated and estimation is performed via the iterated principal factors’ method, the interest is actually mainly focussed on the linear structure of the data, since only moments up to the second ones are involved. In many real situations, the factors could not be adequately described by the first two moments only. For example, skewness characterizing most latent variables in social analysis can be properly measured by the third moment: the factors are not normally distributed and covariance is no longer a sufficient statistic. In this work we propose a factor model characterized by skew-normally distributed factors. Skew-normal refers to a parametric class of probability distributions, that extends the normal distribution by an additional shape parameter regulating the skewness. The model estimation can be solved by the generalized EM algorithm, in which the iterative Newthon–Raphson procedure is needed in the M-step to estimate the factor shape parameter. The proposed skew-normal factor analysis is applied to the study of student satisfaction towards university courses, in order to identify the factors representing different aspects of the latent overall satisfaction.     

A Note on Equality and Inequality of Some Factor Score Predictors

Anderson and Rubin (1956), as well as Takeuchi et al. (1982) finally Schneeweiss and Mathes (1995) proposed factor score predictors that were regarded as orthogonal. Anderson and Rubin's and Takeuchi et al.'s factor score predictors are shown to be identical for non zero unique variances, although they have never been claimed being identical. It is shown that the Schneeweiss and Mathe's factor score predictor is not equal to McDonald's factor score predictor, although it has been claimed that these predictors are identical. It is, moreover, shown that the Schneeweiss and Mathe's factor score predictor is orthogonal only for the canonical orthogonal factor model.     

Examples of the application of matrix factors of a design matrix

When the elements of a design matrix are rational numbers and the variances of the observations are rational multiples of a common real constant, the covariances being zero, the design matrix may be factorised into a product of matrices which have usefil statistical interpretations. The main factor matrices have integer elements, while the other factor matrices are diagonal with rational elements. Weights which are rational numbers, and missing observations, are readily accommodated. A computer is usually needed to find the factors. This paper shows how, once the factors have been found, they may be employed for any suitable set of observations without further need for such assistance.     

一类近似因子模型的GMM估计及其统计性质研究

对于一类异质性误差项存在截面相关性的近似因子模型,本文首先提出了估计共同因子向量和因子载荷矩阵的广义矩估计方法（GMM）,该方法推广了Doz等（2012）的极大似然估计方法;其次,分别研究了模型参数广义矩估计的渐近性质和有限样本的统计性质,在适当的条件下,证明了参数的GMM估计是具有渐近正态分布的一致估计;最后,利用近似因子模型对我国各类上市公司增长性的共同驱动因素及其差异性进行了实证分析。     

Factor Analysis Revisited – How Many Factors are There?

We suggest a procedure to improve the overall performances of several existing methods for determining the number of factors in factor analysis by using alternative measures of correlation: Pearson's, Spearman's, Gini's, and a robust estimator of the covariance matrix (MCD). We examine the effect of the choice of the covariance used on the number of factors chosen by the KG rule of one, the 80% rule, the Minimum average partial (MAP), and the Parallel Analysis Methodology (PAM). Extensive simulations show that when the entire (or part) of the data come from heavy-tail (lognormal) distributions, ranking the variables which come from non symmetric distributions improves the performances of the methods. In this case, Gini is slightly better than Spearman. The PAM and MAP procedures are qualitatively superior to the KG and the 80% rules in determining the true number of factors. A real example involving data on document authorship is analyzed.     

Statistical Inferences Based on Extended Weighted Log-Rank Estimating Function for Censored Regression Model

In the regression model with censored data, it is not straightforward to estimate the covariances of the regression estimators, since their asymptotic covariances may involve the unknown error density function and its derivative. In this article, a resampling method for making inferences on the parameter, based on some estimating functions, is discussed for the censored regression model. The inference procedures are associated with a weight function. To find the best weight functions for the proposed procedures, extensive simulations are performed. The validity of the approximation to the distribution of the estimator by a resampling technique is also examined visually. Implementation of the procedures is discussed and illustrated in a real data example.     

The determinacy of the regression factor score predictor based on continuous parameter estimates from categorical variables

The estimation of population parameters of the continuous common factor model from categorical observed variables is meanwhile regularly performed. It is shown that the formula for the calculation of the determinacy of the regression factor score predictor from the estimated model parameters has to be adapted under these conditions. A method for the calculation of this determinacy from the model parameters of the continuous population factor model based on categorical variables is proposed and evaluated by means of simulated population data. It turns out that using the uncorrected formula can lead to serious overestimation of determinacy for categorical variables.     

影响中国上市公司融资结构的因素分析

中国上市公司的融资结构有着明显的经济转轨时期的特点,国有化程度与上市公司的资产负债率有着极强的相关性。文章应用因子分析综合评价方法来分析影响中国上市公司融资结构的因素,进而分析这些因素与公司融资结构的相关性。     

析主成分分析和因子分析——以Frobenius范数下矩阵近似的新视角

本文通过引入数据阵在Frobenius范数下的最优近似等概念来重新探讨主成分和因子分析。我发现，主成分分析中主成分和因子分析中因子得分（通过主成分解因子载荷，然后用最小二乘解因子得分）的估计为数据阵的最优近似（在Frobenius范数下）在不同正交坐标方向矩阵下的坐标。两种方法分别采用了不同的约束条件分解的最优近似（在Frobenius范数下），因为该分解并不唯一。     

Factor Multivariate Stochastic Volatility via Wishart Processes

This paper proposes a high dimensional factor multivariate stochastic volatility (MSV) model in which factor covariance matrices are driven by Wishart random processes. The framework allows for unrestricted specification of intertemporal sensitivities, which can capture the persistence in volatilities, kurtosis in returns, and correlation breakdowns and contagion effects in volatilities. The factor structure allows addressing high dimensional setups used in portfolio analysis and risk management, as well as modeling conditional means and conditional variances within the model framework. Owing to the complexity of the model, we perform inference using Markov chain Monte Carlo simulation from the posterior distribution. A simulation study is carried out to demonstrate the efficiency of the estimation algorithm. We illustrate our model on a data set that includes 88 individual equity returns and the two Fama–French size and value factors. With this application, we demonstrate the ability of the model to address high dimensional applications suitable for asset allocation, risk management, and asset pricing.     

Factor Multivariate Stochastic Volatility via Wishart Processes

This paper proposes a high dimensional factor multivariate stochastic volatility (MSV) model in which factor covariance matrices are driven by Wishart random processes. The framework allows for unrestricted specification of intertemporal sensitivities, which can capture the persistence in volatilities, kurtosis in returns, and correlation breakdowns and contagion effects in volatilities. The factor structure allows addressing high dimensional setups used in portfolio analysis and risk management, as well as modeling conditional means and conditional variances within the model framework. Owing to the complexity of the model, we perform inference using Markov chain Monte Carlo simulation from the posterior distribution. A simulation study is carried out to demonstrate the efficiency of the estimation algorithm. We illustrate our model on a data set that includes 88 individual equity returns and the two Fama-French size and value factors. With this application, we demonstrate the ability of the model to address high dimensional applications suitable for asset allocation, risk management, and asset pricing.     

Factor analysis regression

The paper describes two regression models—principal components and maximum-likelihood factor analysis—which may be used when the stochastic predictor varibles are highly intereorrelated and/or contain measurement error. The two problems can occur jointly, for example in social-survey data where the true (but unobserved) covariance matrix can be singular. Departure from singularity of the sample dispersion matrix is then due to measurement error. We first consider the more elementary principal components regression model, where it is shown that it can be derived as a special case of (i) canonical correlation, and (ii) restricted least squares. The second part consists of the more general maximum-likelihood factor-analysis regression model, which is derived from the generalized inverse of the product of two singular matrices. Also, it is proved that factor-analysis regression can be considered as an instrumental variables estimator and therefore does not depend on whether factors have been “properly” identified in terms of substantive behaviour. Consequently the additional task of rotating factors to “simple structure” does not arise.     

Identification of Unknown Common Factors: Leaders and Followers

This article has the following contributions. First, this article develops a new criterion for identifying whether or not a particular time series variable is a common factor in the conventional approximate factor model. Second, by modeling observed factors as a set of potential factors to be identified, this article reveals how to easily pin down the factor without performing a large number of estimations. This allows the researcher to check whether or not each individual in the panel is the underlying common factor and, from there, identify which individuals best represent the factor space by using a new clustering mechanism. Asymptotically, the developed procedure correctly identifies the factor when N and T jointly approach infinity. The procedure is shown to be quite effective in the finite sample by means of Monte Carlo simulation. The procedure is then applied to an empirical example, demonstrating that the newly developed method identifies the unknown common factors accurately.     

A Likelihood Ratio Test for Idiosyncratic Unit Roots in the Exact Factor Model with Integrated Factors

We consider an exact factor model with unobservable common stochastic trends imposed by non-stationary factors, and study, by simulation, the power of the likelihood ratio test for unit roots in the idiosyncratic components. The power of the test is compared with the analogous Lagrange multiplier test and the Fisher-type test proposed by Bai and Ng. The results suggest that the benefit of the likelihood ratio test is in panels with a small cross-section.     

Asymptotic distributions of functions of a sample covariance matrix under the elliptical distribution

This paper is concerned with asymptotic distributions of functions of a sample covariance matrix under the elliptical model. Simple but useful formulae for calculating asymptotic variances and covariances of the functions are derived. Also, an asymptotic expansion formula for the expectation of a function of a sample covariance matrix is derived; it is given up to the second-order term with respect to the inverse of the sample size. Two examples are given: one of calculating the asymptotic variances and covariances of the stepdown multiple correlation coefficients, and the other of obtaining the asymptotic expansion formula for the moments of sample generalized variance.