Restricted Covariance Research Articles

Hidden Variable Models in Text Classification and Sentiment Analysis

In this paper, we are proposing extensions to the multinomial principal component analysis (MPCA) framework, which is a Dirichlet (Dir)-based model widely used in text document analysis. The MPCA is a discrete analogue to the standard PCA (it operates on continuous data using Gaussian distributions). With the extensive use of count data in modeling nowadays, the current limitations of the Dir prior (independent assumption within its components and very restricted covariance structure) tend to prevent efficient processing. As a result, we are proposing some alternatives with flexible priors such as generalized Dirichlet (GD) and Beta-Liouville (BL), leading to GDMPCA and BLMPCA models, respectively. Besides using these priors as they generalize the Dir, importantly, we also implement a deterministic method that uses variational Bayesian inference for the fast convergence of the proposed algorithms. Additionally, we use collapsed Gibbs sampling to estimate the model parameters, providing a computationally efficient method for inference. These two variational models offer higher flexibility while assigning each observation to a distinct cluster. We create several multitopic models and evaluate their strengths and weaknesses using real-world applications such as text classification and sentiment analysis.

PML versus minimum {chi }^{2}: the comeback

Arellano (J Econ 42:247–265, 1989a) showed that valid equality restrictions on covariance matrices could result in efficiency losses for Gaussian PMLEs in simultaneous equations models. We revisit his two-equation example using finite normal mixtures PMLEs instead, which are also consistent for mean and variance parameters regardless of the true distribution of the shocks. Because such mixtures provide good approximations to many distributions, we relate the asymptotic variance of our estimators to the relevant semiparametric efficiency bound. Our Monte Carlo results indicate that they systematically dominate MD and that the version that imposes the valid covariance restriction is more efficient than the unrestricted one.

Separation-based parameterization strategies for estimation of restricted covariance matrices in multivariate model systems

Many multivariate model systems involve the estimation of a covariance matrix that must be positive-definite. A common strategy to ensure positive definiteness of the covariance matrix is through the use of a Cholesky parameterization of the covariance matrix. However, several model systems require imposing restrictions on the elements of the covariance elements. For instance, modelling systems may require fixing some (or all) of the diagonal elements in the covariance matrix to unity due to identification considerations. However, imposing such restrictions using the traditional Cholesky decomposition approach is not feasible and requires the additional parameterization of the Cholesky elements.In this paper, we explore a separation-based strategy with spherical parameterization of the Cholesky matrix to impose restrictions on the covariance matrix. Importantly, using this separation-based parameterization strategy, we also explore the possibility of restricting some covariance (or correlation) terms to zero. The effectiveness of the proposed strategy is assessed through extensive simulation experiments. The results from the simulation experiments highlight better performance of the separation-based strategy in terms of recovery of model parameters – particularly those in the covariance matrix, than the traditional Cholesky parameterization approach. Finally, the proposed strategy is implemented in a joint multivariate binary probit ordered probit model system to analyze the usage (and the extent of use) of non-private modes of transportation in Bengaluru, India. In doing so, the proposed strategy is implemented to restrict several correlations to zero, thus avoiding the estimation of a profligate correlation matrix and substantially easing the estimation process.

Inference on incomplete information games with multi-dimensional actions

By extending de Paula and Tang (2012) and Aradillas-López and Gandhi (2016), we derive testable restrictions for uniqueness of equilibrium in games with multi-dimensional actions. We discuss two models of payoff functions which imply certain covariance restrictions for players’ actions. These restrictions can be used to construct an identified set of strategic parameters under multiple equilibria.

Control variables approach to estimate semiparametric models of mismeasured endogenous regressors with an application to U.K. twin data

We study the identification and estimation of semiparametric models with mismeasured endogenous regressors using control variables that ensure the conditional covariance restriction on endogenous regressors and unobserved causes. We provide a set of sufficient conditions for identification, which control for both endogeneity and measurement error. We propose a sieve-based estimator and derive its asymptotic properties. Given the sieve approximation, our proposed estimator is easy to implement as weighted least squares. Monte Carlo simulations illustrate that our proposed estimator performs well in the finite samples. In an empirical application, we estimate the return to education on earnings using U.K. twin data, in which self-reported education is potentially measured with error and is also correlated with unobserved factors. Our approach utilizes the twin’s reported education as a control variable to obtain consistent estimates. We find that a one-year increase in education leads to an 11% increase in hourly wage. The estimate is significantly higher than those from OLS and IV approaches which are potentially biased. The application underscores that our proposed estimator is useful to correct for both endogeneity and measurement error in estimating returns to education.

Positive geometries and differential forms with non-logarithmic singularities. Part I

Positive geometries encode the physics of scattering amplitudes in flat space- time and the wavefunction of the universe in cosmology for a large class of models. Their unique canonical forms, providing such quantum mechanical observables, are characterised by having only logarithmic singularities along all the boundaries of the positive geometry. However, physical observables have logarithmic singularities just for a subset of theories. Thus, it becomes crucial to understand whether a similar paradigm can underlie their structure in more general cases. In this paper we start a systematic investigation of a geometric-combinatorial characterisation of differential forms with non-logarithmic singularities, focusing on projective polytopes and related meromorphic forms with multiple poles. We introduce the notions of covariant forms and covariant pairings. Covariant forms have poles only along the boundaries of the given polytope; moreover, their leading Laurent coefficients along any of the boundaries are still covariant forms on the specific boundary. Whereas meromorphic forms in covariant pairing with a polytope are associated to a specific (signed) triangulation, in which poles on spurious boundaries do not cancel completely, but their order is lowered. These meromorphic forms can be fully characterised if the poly- tope they are associated to is viewed as the restriction of a higher dimensional one onto a hyperplane. The canonical form of the latter can be mapped into a covariant form or a form in covariant pairing via a covariant restriction. We show how the geometry of the higher di- mensional polytope determines the structure of these differential forms. Finally, we discuss how these notions are related to Jeffrey-Kirwan residues and cosmological polytopes.

Split Property for Free Massless Finite Helicity Fields

We prove the split property for any finite helicity free quantum fields. Finite helicity Poincar\'e representations extend to the conformal group and the conformal covariance plays an essential role in the argument. The split property is ensured by the trace class condition: Tr (exp(-s L_0)) is finite for all s>0 where L_0 is the conformal Hamiltonian of the M\"obius covariant restriction of the net on the time axis. We extend the argument for the scalar case presented in [7]. We provide the direct sum decomposition into irreducible representations of the conformal extension of any helicity-h representation to the subgroup of transformations fixing the time axis. Our analysis provides new relations among finite helicity representations and suggests a new construction for representations and free quantum fields with non-zero helicity.

The role of intergenerational mobility in internal migration

This paper investigates the role of intergenerational mobility in the internal migration decisions of families. The geographic variation in intergenerational mobility suggests that if parents value their children's human capital accumulation and future outcomes, they would have an incentive to move to areas with a higher upward mobility. To identify the effect of intergenerational mobility on family migration, we first use an instrumental variable approach, based on a heteroskedastic covariance restriction, which addresses measurement-error and omitted-variable biases. Then, we apply the semiparametric maximum score estimation method to our empirical model, which yields a consistent estimator when families' choice sets are partially observed. We find that highly educated families with school-aged children choose areas that favor upward mobility. Our welfare analysis indicates that a unit increase in the absolute upward mobility of a commuting zone is equivalent to approximately a $722 higher mean wage in the local labor market.

Parametric Models for Mutual Kernel Matrix Completion

Recent studies utilize multiple kernel learning to deal with incomplete-data problem. In this study, we introduce new methods that do not only complete multiple incomplete kernel matrices simultaneously, but also allow control of the flexibility of the model by parameterizing the model matrix. By imposing restrictions on the model covariance, overfitting of the data is avoided. A limitation of kernel matrix estimations done via optimization of an objective function is that the positive definiteness of the result is not guaranteed. In view of this limitation, our proposed methods employ the LogDet divergence, which ensures the positive definiteness of the resulting inferred kernel matrix. We empirically show that our proposed restricted covariance models, employed with LogDet divergence, yield significant improvements in the generalization performance of previous completion methods.

Instrument-Free Demand Estimation

We consider the identification of empirical models of supply and demand with imperfect competition. As is well known, a supply-side instrument can resolve price endogeneity in demand estimation. We show that, under common assumptions, two other approaches also yield consistent estimates of the joint model: (i) a demand-side instrument, or (ii) a covariance restriction between unobserved demand and cost shocks. Covariance restrictions can obtain identification even the absence of instruments. Further, supply and demand assumptions alone may bound the structural parameters without additional restrictions. We illustrate the covariance restriction approach with applications to ready-to-eat cereal, cement, and airlines.

Focused Shrinkage with an Application to Portfolio Choice

We propose a shrinkage estimator for parameters θ which improves the mean squared error of functions x (θ) over standard choices. When the restricted model estimator is in the class of minimum distance estimators, we project onto the restricted parameter space using a matrix based on the derivatives of x (θ). The proposed matrix is shown to minimize the bias among estimators in this class. This choice is then incorporated into a shrinkage procedure. We derive a risk bound for shrinkage estimators which allows for arbitrary projection matrices. Using this result, it is shown the proposed projection matrix can lead to substantially lower risk. The improvement is largest when the restricted model has nontrivial bias. This is shown with our motivating example: implementing a mean-variance portfolio choice rule. Our method is applied by shrinking toward a model with restricted covariance matrix. Extensive simulations demonstrate improved risk in this case. The estimator is also implemented in a portfolio choice application to futures data. Our approach outperforms standard procedures here as well.

Does temporary interruption in postsecondary education induce a wage penalty? Evidence from Canada

Almost 40% of Canadian youth who left postsecondary education in 1999 had returned two years later. This paper investigates the extent to which schooling discontinuities affect post-graduation starting wages and whether the latter are influenced by the reasons behind these discontinuities. We use data from the 2007 National Graduate Survey. We apply Lewbel’s (2012) generated instruments approach. The source of identification is a heteroscedastic covariance restriction of the error terms that is a feature of many models of endogeneity. We also perform two-stage quantile regressions. We find a positive effect on wages of temporary interruption for men who held a full-time job during their out-of-school spell(s). Both men and women witness a wage decrease if their interruption depends on health issues. Women bear a wage penalty if their interruption is due to a part-time job, to lack of money, or is caused by reasons other than health, work, and money.

Estimation of games with ordered actions: An application to chain-store entry

We study the estimation of static games where players are allowed to have ordered actions, such as the number of stores to enter into a market. Assuming that payoff functions satisfy general shape restrictions, we show that equilibrium of the game implies a covariance restriction between each player's action and a component of the player's payoff function that we call the strategic index. The strategic index captures the direction of strategic interaction (i.e., patterns of substitutability or complementarity) as well as the relative effects of opponents' decisions on players' payoffs. The covariance restriction we derive is robust to the presence of multiple equilibria, and provides a basis for identification and estimation of the strategic index. We introduce an econometric method for inference in our model that exploits the information in moment inequalities in a computationally simple way. We analyze its properties through Monte Carlo experiments and then apply our approach to study entry behavior by chain stores where there is both an intensive margin of entry (how many stores to open in a market) as well as the usual extensive margin of entry (whether to enter a market or not). Using data from retail pharmacies we find evidence of asymmetries in strategic effects among firms in the industry that has implications for merger policy. We also find that business stealing effects are less pronounced in larger markets, which helps explain the large positive correlation in entry behavior observed in the data. Static games multiple equilibria partial identification conditional moment inequalities entry decisions C01 C14 C57

Statistical and computational trade-offs in estimation of sparse principal components

In recent years, sparse principal component analysis has emerged as an extremely popular dimension reduction technique for high-dimensional data. The theoretical challenge, in the simplest case, is to estimate the leading eigenvector of a population covariance matrix under the assumption that this eigenvector is sparse. An impressive range of estimators have been proposed; some of these are fast to compute, while others are known to achieve the minimax optimal rate over certain Gaussian or sub-Gaussian classes. In this paper, we show that, under a widely-believed assumption from computational complexity theory, there is a fundamental trade-off between statistical and computational performance in this problem. More precisely, working with new, larger classes satisfying a restricted covariance concentration condition, we show that there is an effective sample size regime in which no randomised polynomial time algorithm can achieve the minimax optimal rate. We also study the theoretical performance of a (polynomial time) variant of the well-known semidefinite relaxation estimator, revealing a subtle interplay between statistical and computational efficiency.

Estimation of games with ordered actions: An application to chain‐store entry

We study the estimation of static games where players are allowed to have ordered actions, such as the number of stores to enter into a market. Assuming that payoff functions satisfy general shape restrictions, we show that equilibrium of the game implies a covariance restriction between each player's action and a component of the player's payoff function that we call the strategic index. The strategic index captures the direction of strategic interaction (i.e., patterns of substitutability or complementarity) as well as the relative effects of opponents' decisions on players' payoffs. The covariance restriction we derive is robust to the presence of multiple equilibria, and provides a basis for identification and estimation of the strategic index. We introduce an econometric method for inference in our model that exploits the information in moment inequalities in a computationally simple way. We analyze its properties through Monte Carlo experiments and then apply our approach to study entry behavior by chain stores where there is both an intensive margin of entry (how many stores to open in a market) as well as the usual extensive margin of entry (whether to enter a market or not). Using data from retail pharmacies we find evidence of asymmetries in strategic effects among firms in the industry that has implications for merger policy. We also find that business stealing effects are less pronounced in larger markets, which helps explain the large positive correlation in entry behavior observed in the data. Static games multiple equilibria partial identification conditional moment inequalities entry decisions C01 C14 C57

Does Temporary Interruption in Postsecondary Education Induce a Wage Penalty? Evidence from Canada

Data from the Youth in Transition Survey reveal that almost 40% of Canadian youth who left post-secondary education in 1999 had returned two years later. This paper investigates the extent to which schooling discontinuities affect post-graduation starting real wages and whether the latter are differently influenced by the reasons behind these discontinuities. We analyse this issue using data from the 2007 National Graduate Survey. We take covariates endogeneity into account using Lewbel's (2012) generated instrument approach. The source of identification is a heteroscedastic covariance restriction of the error terms that is a feature of many models of endogeneity. To allow for individual heterogeneity in the causal effect of various reasons for schooling interruption, we also provide results from two-stage quantile regressions using Lewbel's generated instruments. Conditional on the levels of schooling and experience, we find a positive effect on wages of temporary schooling interruption for men who had held a full-time job during their out-of-school spell(s). Both men and women witness a wage decrease if their interruption is associated with health issues. Women also bear a wage penalty if their interruption is due to a part-time job, to lack of money, or is caused by reasons other than health, work, and money.

Does Temporary Interruption in Postsecondary Education Induce a Wage Penalty? Evidence from Canada

Data from the Youth in Transition Survey reveal that almost 40% of Canadian youth who left post-secondary education in 1999 had returned two years later. This paper investigates the extent to which schooling discontinuities affect post-graduation starting real wages and whether the latter are differently influenced by the reasons behind these discontinuities. We analyse this issue using data from the 2007 National Graduate Survey. We take covariates endogeneity into account using Lewbel's (2012) generated instrument approach. The source of identification is a heteroscedastic covariance restriction of the error terms that is a feature of many models of endogeneity.To allow for individual heterogeneity in the causal effect of various reasons for schooling interruption, we also provide results from two-stage quantile regressions using Lewbel's generated instruments. Conditional on the levels of schooling and experience, we find a positive effect on wages of temporary schooling interruption for men who had held a full-time job during their out-of-school spell(s). Both men and women witness a wage decrease if their interruption is associated with health issues. Women also bear a wage penalty if their interruption is due to a part-time job, to lack of money, or is caused by reasons other than health, work, and money.

Consistent N = 8 $$ \mathcal{N}=8 $$ truncation of massive IIA on S 6

Massive type IIA supergravity is shown to admit a consistent truncation on the six-sphere to maximal supergravity in four dimensions with a dyonic ISO(7) gauging. We obtain the complete, non-linear embedding of all the $D=4$ fields into the IIA metric and form potentials, and show its consistency. We first rewrite the IIA theory in an $\textrm{SO}(1,3) \times \textrm{SL}(7)$--covariant way. Then, we employ an ${\cal N}=8$ SL(7)--covariant restriction of the $D=4$ tensor hierarchy in order to find the full embedding. The redundant $D=4$ degrees of freedom introduced by the tensor hierarchy can be eliminated by writing the embedding in terms of the field strengths and exploiting the restricted duality hierarchy. In particular, closed expressions for the Freund-Rubin term are found using this technique which reveal a pattern valid for other truncations. Finally, we show that the present ${\cal N}=8$ truncation of massive IIA on $S^6$ and the ${\cal N}=2$ truncation obtained when $S^6$ is equipped with its nearly-K\"ahler structure, overlap in the ${\cal N}=1$, G$_2$--invariant sector of the former.

Effects of Psychiatric Disorders on Labor Market Outcomes: A Latent Variable Approach Using Multiple Clinical Indicators.

In this paper, we estimate the effect of psychiatric disorders on labor market outcomes using a structural equation model with a latent index for mental illness, an approach that acknowledges the continuous nature of psychiatric disability. We also address the potential endogeneity of mental illness using an approach proposed by Lewbel (2012) that relies on heteroscedastic covariance restrictions rather than questionable exclusion restrictions for identification. Data come from the US National Comorbidity Survey - Replication and the National Latino and Asian American Study. We find that mental illness adversely affects employment and labor force participation and also reduces the number of weeks worked and increases work absenteeism. To assist in the interpretation of findings, we simulate the labor market outcomes of individuals meeting diagnostic criteria for mental disorder if they had the same mental health symptom profile as individuals not meeting diagnostic criteria. We estimate potential gains in employment for 3.5 million individuals, and reduction in workplace costs of absenteeism of $21.6 billion due to the resultant improvement in mental health. Copyright © 2015 John Wiley & Sons, Ltd.

Assessing Identifying Restrictions in SVAR Models

Structural vector autoregressive models are usually identified by combining covariance restrictions implied by the data and identifying restrictions suggested, for example, by economic theory. This paper proposes an approach to assess, in a Bayesian sense, whether or not the data support a candidate set of identifying restrictions. I analyse sign restrictions. The researcher represents her uncertainty regarding the validity of the restrictions using a uniform prior that covers the parameter space both where the restrictions are satisfied and where they are not satisfied. The correlation pattern in the data then guides whether the probability mass in favour of the candidate restrictions increases or not from prior to posterior. To outline the key mechanism I use two examples, a two-equation model of demand and supply and a simplified New Keynesian model.