Estimating large-scale general linear and seemingly unrelated regressions models after deleting observations

Abstract

A new numerical method to solve the downdating problem (and variants thereof), namely removing the effect of some observations from the generalized least squares (GLS) estimator of the general linear model (GLM) after it has been estimated, is extensively investigated. It is verified that the solution of the downdated least squares problem can be obtained from the estimation of an equivalent GLM, where the original model is updated with the imaginary deleted observations. This updated GLM has a non positive definite dispersion matrix which comprises complex covariance values and it is proved herein to yield the same normal equations as the downdated model. Additionally, the problem of deleting observations from the seemingly unrelated regressions model is addressed, demonstrating the direct applicability of this method to other multivariate linear models. The algorithms which implement the novel downdating method utilize efficiently the previous computations from the estimation of the original model. As a result, the computational cost is significantly reduced. This shows the great usability potential of the downdating method in computationally intensive problems. The downdating algorithms have been applied to real and synthetic data to illustrate their efficiency.

An alternative numerical method for estimating large-scale time-varying parameter seemingly unrelated regressions models

“…The variance-covariance matrix in (3.22) is such that the effect of the oldest observation is excluded from the current estimate but the new information from the acquired observation will be incorporated. The imaginary unit in (3.21) gives the weight needed to downdate the model, that is, to eliminate the affect of the first datum (Hadjiantoni and Kontoghiorghes, 2017). The rolling window estimation problem is then given by argmin…”

“…Namely, let the fixed size estimation window move forward at one point of time to capture the information from the next data point while excluding the effect of the oldest data point. That is, estimate (3.20) by employing the SR-KF algorithm or by solving the GLLSP (3.23) using an Up-downdating algorithm similar to that in (Hadjiantoni and Kontoghiorghes, 2017). Table 5 reports the total time, in CPU seconds, to estimate the model over a rolling window (of fixed size) which rolls ahead one data point 1000 times.…”

A recursive three-stage least squares method for large-scale systems of simultaneous equations

“…Namely, this is the case where rows of data are excluded after the estimation procedure has been completed and hence a reduced observations model has to be estimated. Observations may have to be deleted when they are considered to be old and misleading, when they have been shown to be outliers or for the identification of influential data [1,7,11,35].…”

“…The problem of re-estimating linear models after adding (updating) or removing (downdating) observations has already been addressed [6,11,13,14,16,17,26,29,33]. Methods had previously been proposed for the effective estimation of the SEM [1,6,8,18,20], however, the sequential derivation of the 3SLS estimator for large-scale SEMs has not, previously, been considered.…”

A numerical method for the estimation of time-varying parameter models in large dimensions

“…gives the weight needed to downdate the model, that is, to eliminate the affect of the first datum (Hadjiantoni and Kontoghiorghes, 2017). The window estimation problem is then given by argmin…”