Abstract

For a symmetric matrix B, we determine the class of Q such that Q t BQ is non-negative definite and apply it to panel data estimation and forecasting: the Hausman test for testing the endogeneity of the random effects in panel data models. We show that the test can be performed if the estimated error variances in the fixed and random effects models satisfy a specific inequality. If it fails, we discuss the restrictions under which the test can be performed. We show that estimators satisfying the inequality exist. Furthermore, we discuss an application to a constrained quadratic minimization problem with an indefinite objective function.

Highlights

  • Optimization of quadratic structures and corrections to the construction of covariance matrices has a long history in econometrics and financial economics

  • The importance of the above in forecasting cannot be overstated as they all relate to decision-making at some future time period: a good panel data model can be used in generating out-of-sample forecasts, a well-constructed covariance matrix can be used in optimizing the weights of a portfolio or for building a model for volatility and correlation forecasting

  • In this paper we present a novel mathematical approach for solving a particular class of quadratic optimization problems with applications in econometrics, statistics and portfolio construction

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Summary

Introduction

Optimization of quadratic structures and corrections to the construction of covariance matrices has a long history in econometrics and financial economics. Stats 2020, 3 ground to all is rank deficiency or rank indeterminacy based on redundant information in the variables from which we compute that said matrix: a large number of variables involved or some deficiency in the structure of the underlying problem The problem of this indeterminacy leads to other problems in different set-ups: in the context of a matrix-version of the well-known Hausman test in econometrics a difference in covariance matrices required for the application of the test might not be positive definite; in the context of a large portfolio optimization the covariance matrix of the financial returns might not be positive definite. Such a model can be used in forecasting

Notations
Preliminaries
Non-Negative Definiteness of Qt BQ
Remarks
Hausman Test
NT is invariant under the choices of generalized inverses of
A Quadratic Optimization Problem
A Quadratic Optimization Problem with Non-Homogeneous Linear Constraints
Full Text
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