Unconstrained Estimation Research Articles

Implications of multiple structural changes in event studies

Detection of unbiased abnormal returns in the classic event study depends on the validity of the assumption that the parameters of the return generating process remain constant throughout the sample period. However, given the substantial amount of evidence to support the fact that the market model parameters are not stable over time, previous research has suggested that the classic methodology may result in mismeasurement of the magnitude and timing of the market response and lead to incorrect conclusions regarding the impact of the event. This study addresses the issue of parameter nonstationarity in event studies by examining its implication for measuring the market response. Using an unconstrained estimation model that allows multiple structural changes in the security pricing model, our results indicate that the usual finding of statistically significant positive abnormal returns on and around the split announcement is robust to the more general specification of the underlying return generating process.

Tensor methods for full-information maximum likelihood estimation: Unconstrained estimation

In this study, we present a new method, called a tensor method, for the computation of unconstrained Full-Information Maximum Likelihood (FIML) estimates. The new techniqus is based upon a fourth order approximation to the log-likelihood function, rather than the second order approximation used in standard methods. The higher order terms are low rank third and fourth order tensors that are computed, at very little storage or computation cost, using information from previous iterations. We form and solve the tensor model, then present test results showing that the tensor method is far more efficient than the standard Newton's method for a wide range of unconstrained FIML estimation problems.

The use of constraints to eliminate artifacts in maximum-likelihood image estimation for emission tomography

Images produced in emission tomography with unconstrained maximum-likelihood estimation techniques exhibit two artifacts as the likelihood hill is climbed and the images converge toward one with maximum likelihood. The first artifact is a speckled appearance on the image, a noise artifact. The second is an estimation error near edges of the underlying radioactivity concentration. However, if mathematical constraints are used that blur the estimated image and the image being estimated, these artifacts are eliminated. The authors review the nature of these constraints, demonstrate them, and measure the performance of the modified algorithm relative to the classical linear image-estimation algorithm.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The role of likelihood and entropy in incomplete-data problems: Applications to estimating point-process intensities and toeplitz constrained covariances

The principle of maximum entropy has played an important role in the solution of problems in which the measurements correspond to moment constraints on some many-to-one mapping h(x). In this paper we explore its role in estimation problems in which the measured data are statistical observations and moment constraints on the observation function h(x) do not exist. We conclude that: 1) For the class of likelihood problems arising in a complete-incomplete data context in which the complete data x are nonuniquely determined by the measured incomplete data y via the many-to-one mapping y = h(x), the density maximizing entropy is identical to the conditional density of the complete data given the incomplete data. This equivalence results by viewing the measurements as specifying the domain over which the density is defined, rather than as a moment constraint on h(x). 2) The identity between the maximum entropy and the conditional density results in the fact that maximum-likelihood estimates may be obtained via a joint maximization (minimization) of the entropy function (Kullback-Liebler divergence). This provides the basis for the iterative algorithm of Dempster, Laird, and Rubin [1] for the maximization of likelihood functions. 3) This iterative method is used for maximum-likelihood estimation of image parameters in emission tomography and gammaray astronomy. We demonstrate that unconstrained likelihood estimation of image intensities from finite data sets yields unstable estimates. We show how Grenander's method of sieves can be used with the iterative algorithm to remove the instability. A bandwidth sieve is introduced resulting in an estimator which is smoothed via exponential splines. 4) We also derive a recursive algorithm for the generation of Toeplitz constrained maximum-likelihood estimators which at each iteration evaluates conditional mean estimates of the lag products based on the previous estimate of the covariance, from which the updated Toeplitz covariance is generated. We prove that the sequence of Toeplitz estimators has the property that they increase in likelihood, remain in the set of positive-definite Toeplitz covariances, and has all of its limit points stable and satisfying the necessary conditions for maximizing the likelihood.

An examination of two-step estimators for models with lagged dependent variables and autocorrelated errors

The possible roles of the Durbin Equation and the first observation correction in improving the efficiency of parameter estimates in the lagged dependent variables-serial correlation model are examined. Unconstrained estimation of the Durbin Equation results in an estimate of ρ which is inefficient and its use in feasible generalized least squares does not provide asymptotically efficient estimates. Evidently, the first observation correction is a very important determinant of small sample properties in the present model. Asymptotically inefficient estimators which use a first observation correction frequently outperform Hatanaka's asymptotically efficient estimator in finite samples, essentially because it does not use the first observation.