This paper investigates the limit properties of mean-variance (mv) and arbitrage pricing (ap) trading strategies using a general dynamic factor model, as the number of assets diverge to infinity. It extends the results obtained in the literature for the exact pricing case to two other cases of asymptotic no-arbitrage and the unconstrained pricing scenarios. The paper characterizes the asymptotic behaviour of the portfolio weights and establishes that in the non-exact pricing cases the ap and mv portfolio weights are asymptotically equivalent and, moreover, functionally independent of the factors conditional moments. By implication, the paper sheds light on a number of issues of interest such as the prevalence of short-selling, the number of dominant factors and the granularity property of the portfolio weights.

In prediction problems with more predictors than observations, it can sometimes be helpful to use a joint probability model, π(Y, X), rather than a purely conditional model, π(Y | X), where Y is a scalar response variable and X is a vector of predictors. This approach is motivated by the fact that in many situations the marginal predictor distribution π(X) can provide useful information about the parameter values governing the conditional regression. However, under very mild misspecification, this marginal distribution can also lead conditional inferences astray. Here, we explore these ideas in the context of linear factor models, to understand how they play out in a familiar setting. The resulting Bayesian model performs well across a wide range of covariance structures, on real and simulated data. 1

This paper investigates the limit properties of mean-variance (mv) and arbitrage pricing (ap) trading strategies using a general dynamic factor model, as the number of assets diverge to in…nity. It extends the results obtained in the literature for the exact pricing case to two other cases of asymptotic no-arbitrage and the unconstrained pricing scenarios. The paper characterizes the asymptotic behaviour of the portfolio weights and establishes that in the non-exact pricing cases the ap and mv portfolio weights are asymptotically equivalent and, moreover, functionally independent of the factors conditional moments. By implication, the paper sheds light on a number of issues of interest such as the prevalence of short-selling, the number of dominant factors and the granularity property of the portfolio weights.

This paper adapts sparse factor models for exploring covariation in multivariate binary data, with an application to measuring latent factors in U.S. Congressional roll-call voting patterns. We focus on the advantages of using formal probability models for inference in this context, drawing parallels with the seminal findings of Poole and Rosenthal (1991). Our methodological innovation is to introduce a sparsity prior on a latent covariance matrix that descibes common factors in binary and ordinal outcomes. We apply the method to analyze sixty years of roll-call votes from the United States Senate, focusing primarily on the interpretation of posterior summaries that arise from the model. We also explore two advantages of our approach over traditional factor analysis. First, patterns of sparsity in the factor-loadings matrix often have natural subject-matter interpretations. For the roll-call vote data, the sparsity prior enables one to conduct a formal hypothesis test about whether a given vote can be explained exclusively by partisanship. Moreover, the factor scores provide a novel way of ranking Senators by the partisanship of their voting patterns. Second, by introducing sparsity into existing factor-analytic probit models, we effect a favorable bias–variance tradeoff in estimating the latent covariance matrix. Our model can thus be used in situations where the number of variables is very large relative to the number of observations. Key words: covariance estimation; factor models; multivariate probit models; voting patterns 1 Corresponding author.

No. DMS-0112069. Any opinions, findings, and conclusions or recommendations expressed in this

Finding an unconstrained and statistically interpretable reparameterization of a covariance matrix is still an open problem in statistics. Its solution is of central importance in covariance estimation, particularly in the recent high-dimensional data environment where enforcing the positive-definiteness constraint could be computationally expensive. We provide a survey of the progress made in modeling covariance matrices from the perspectives of generalized linear models (GLM) or parsimony and use of covariates in low dimensions, regularization (shrinkage, sparsity) for high-dimensional data, and the role of various matrix factorizations. A viable and emerging regressionbased setup which is suitable for both the GLM and the regularization approaches is to link a covariance matrix, its inverse or their factors to certain regression models and then solve the relevant (penalized) least squares problems. We point out several instances of this regression-based setup in the literature. A notable case is in the Gaussian graphical models where linear regressions with LASSO penalty are used to estimate the neighborhood of one node at a time (Meinshausen and Bühlmann, 2006). Some advantages

This paper is concerned with testing the time series implications of the capital asset pricing model (CAPM) due to Sharpe (1964) and Lintner (1965), when the number of securities, N, is large relative to the time dimension, T, of the return series. Two new tests of CAPM are proposed that exploit recent advances on the analysis of large panel data models, and are valid even if N> T. When the errors are Gaussian and cross sectionally independent, a test, denoted by ^ J;1, is proposed which is N(0; 1) as N! 1, with T …xed. Even when the errors are non-Gaussian we are still able to show that ^ J;1 tends to N(0; 1) so long as the errors are cross-sectionally independent and N=T 3! 0, with N and T! 1, jointly. In the case of cross sectionally correlated errors, using a threshold estimator of the average squares of pair-wise error correlations, a modi…ed version of ^ J;1, denoted by ^ J;2, is proposed. Small sample properties of the tests are compared using Monte Carlo experiments designed speci…cally to match the correlations, volatilities, and other distributional features of the residuals of Fama-French three factor regressions of individual securities in the Standard & Poor 500 index. Overall, the proposed tests perform best in terms of power, with empirical sizes very close to the chosen nominal value even in cases where N is much larger than T. The ^J;2 test (which allows for non-Gaussian and weakly cross correlated errors) is applied to all securities in the S&P 500 index with 60 months of return data at the end of each month over the period September 1989-September 2011. Statistically signi…cant evidence against Sharpe-Lintner CAPM is found mainly during the recent …nancial crisis. Furthermore, a strong negative correlation is found between a twelve-month moving average p-values of the ^ J;2 test and the returns of long/short equity strategies relative to the return on S&P 500 over the period December 2006 to September 2011, suggesting that abnormal pro…ts are earned during episodes of market ine ¢ ciencies.

estimation

Abstract: The paper proposes a method for constructing a sparse estimator for the inverse covariance (concentration) matrix in high-dimensional settings. The estimator uses a penalized normal likelihood approach and forces sparsity by using a lasso-type penalty. We establish a rate of convergence in the Frobenius norm as both data dimension p and sample size n are allowed to grow, and show that the rate depends explicitly on how sparse the true concentration matrix is. We also show that a correlationbased version of the method exhibits better rates in the operator norm. The estimator is required to be positive definite, but we avoid having to use semi-definite programming by re-parameterizing the objective function

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