Fixed Effects and Random Effects Estimation of Higher-order Spatial Autoregressive Models with Spatial Autoregressive and Heteroscedastic Disturbances

Abstract

This paper develops a unified framework for fixed effects (FE) and random effects (RE) estimation of higher-order spatial autoregressive panel data models with spatial autoregressive disturbances and heteroscedasticity of unknown form in the idiosyncratic error component. We derive the moment conditions and optimal weighting matrix without distributional assumptions for a generalized moments (GM) estimation procedure of the spatial autoregressive parameters of the disturbance process and define both an RE and an FE spatial generalized two-stage least squares estimator for the regression parameters of the model. We prove consistency of the proposed estimators and derive their joint asymptotic distribution, which is robust to heteroscedasticity of unknown form in the idiosyncratic error component. Finally, we derive a robust Hausman test of the spatial random against the spatial FE model.     

The asymptotic behaviour of the estimated generalized least squares method in the linear regression model

Abstract  In the linear regression model the generalized least squares (GLS) method is only applicable if the covariance matrix of the errors is known but for a scalar factor. Otherwise an estimator for this matrix has to be used. Then we speak of the estimated generalized least squares (EGLS) method. In this paper the asymptotic behaviour of both methods is compared. Results are applied to some standard models commonly used in econometrics     

Efficiency of iterated WLS in the linear model with completely unknown heteroskedasticity

Iterated weighted least squares (IWLS) is investigated for estimating the regression coefficients in a linear model with symmetrically distributed errors. The variances of the errors are not specified; it is not assumed that they are unknown functions of the explanatory variables nor that they are given in some parametric way.
IWLS is carried out in a random number of steps, of which the first one is OLS. In each step the error variance at time t is estimated with a weighted sum of m squared residuals in the neighbourhood of t and the coefficients are estimated using WLS. Furthermore an estimate of the co-variance matrix is obtained. If this estimate is minimal in some way the iteration process is stopped.
Asymptotic properties of IWLS are derived for increasing sample size n . Some particular cases show that the asymptotic efficiency can be increased by allowing more than two steps. Even asymptotic efficiency with respect to WLS with the true error variances can be obtained if m is not fixed but tends to infinity with n and if the heteroskedasticity is smooth.     

耦合Sylvester矩阵方程的梯度迭代算法

通过推广求解矩阵方程AX=b或AX+XB=C的递推迭代算法和基于递阶辩识原理的思想,给出了求解广义耦合矩阵方程的梯度迭代算法。并证明了迭代算法的收敛性。分析表明,若矩阵方程有唯一解,则对任意的初始值该算法给出的迭代解都能快速的收敛到其精确解。数值实例验证了该算法的有效性。     

Re-examining 50-year-old OLS estimates of the Klein–Goldberger model

This paper examines the ordinary least squares estimates of the Klein–Goldberger model by Fox ( Journal of Political Economy , 64 , 1956, 128). Because Klein and Goldberger published the data set with the model, it is possible to re-examine Fox's results years later, and investigate the accuracy with which these estimates were calculated. The examination reported in this paper was conducted by making independent estimates using three different modern econometric software packages. This examination reveals that the Fox estimates for a number of the equations of this model are replicable, to the two or three digits reported by Fox. Fox's results for other equations cannot be replicated. Not all the reasons for this lack of replicability can be determined, but in several cases the computational methods used by Fox and his assistants have been found to be faulty by modern computational standards.     

Forecasting mortality with a hyperbolic spatial temporal VAR model

Accurate forecasts of mortality rates are essential to various types of demographic research like population projection, and to the pricing of insurance products such as pensions and annuities. Recent studies have considered a spatial–temporal vector autoregressive (STVAR) model for the mortality surface, where mortality rates of each age depend on the historical values for that age (temporality) and the neighboring cohorts ages (spatiality). This model has sound statistical properties including co-integrated dependent variables, the existence of closed-form solutions and a simple error structure. Despite its improved forecasting performance over the famous Lee–Carter (LC) model, the constraint that only the effects of the same and neighboring cohorts are significant can be too restrictive. In this study, we adopt the concept of hyperbolic memory to the spatial dimension and propose a hyperbolic STVAR (HSTVAR) model. Retaining all desirable features of the STVAR, our model uniformly beats the LC, the weighted functional demographic model, STVAR and sparse VAR counterparties for forecasting accuracy, when French and Spanish mortality data over 1950–2016 are considered. Simulation results also lead to robust conclusions. Long-term forecasting analyses up to 2050 comparing the four models are further performed. To illustrate the extensible feature of HSTVAR to a multi-population case, a two-population illustrative example using the same sample is further presented.     

Optimal design of nonlinear experiments with parameter constraints

The nonlinear regression model with N observations y _i=η(x _i,θ) +ε_i, and with the parameter θ subject to q nonlinear constraints C _j (θ)=0; j=1, …,q, is considered. As an example, the spline regression with unknown nodes is taken. Expressions for the variances (variance matrices) of the LSE are discussed. Because of the complexity of these expressions, and the singularity of the variance matrix of the LSE for θ, the optimality criteria and their properties, in particular the convexity and the equivalence theorem are considered from different aspects. Also the possibility of restriction to designs with limited values of measures of nonlinearity is mentioned. Research supported by the VEGA-grant of the Slovak grant agency No. 1/7295/20.     

An application of approximate finite sample results to parameter estimation in a linear errors-in-variables model

In the simple errors-in-variables model the least squares estimator of the slope coefficient is known to be biased towards zero for finite sample size as well as asymptotically. In this paper we suggest a new corrected least squares estimator, where the bias correction is based on approximating the finite sample bias by a lower bound. This estimator is computationally very simple. It is compared with previously proposed corrected least squares estimators, where the correction aims at removing the asymptotic bias or the exact finite sample bias. For each type of corrected least squares estimators we consider the theoretical form, which depends on an unknown parameter, as well as various feasible forms. An analytical comparison of the theoretical estimators is complemented by a Monte Carlo study evaluating the performance of the feasible estimators. The new estimator proposed in this paper proves to be superior with respect to the mean squared error.     

Simultaneous Parameter Estimation in Exploratory Factor Analysis: An Expository Review

The classical exploratory factor analysis (EFA) finds estimates for the factor loadings matrix and the matrix of unique factor variances which give the best fit to the sample correlation matrix with respect to some goodness-of-fit criterion. Common factor scores can be obtained as a function of these estimates and the data. Alternatively to the classical EFA, the EFA model can be fitted directly to the data which yields factor loadings and common factor scores simultaneously. Recently, new algorithms were introduced for the simultaneous least squares estimation of all EFA model unknowns. The new methods are based on the numerical procedure for singular value decomposition of matrices and work equally well when the number of variables exceeds the number of observations. This paper provides an account that is intended as an expository review of methods for simultaneous parameter estimation in EFA. The methods are illustrated on Harman's five socio-economic variables data and a high-dimensional data set from genome research.     

Issues in Assessing the Effect of Interindustry R&D Spillovers

This paper aims to clarify three issues concerning the weighting methodol ogy generally used to evaluate interindustry R&D spillovers. These issues concern the likely nature of the spillovers estimated through different types of supporting matrices; the similarity between input–output (IO), technology flows and technological proximity matrices; and the relevance of the assumption that a single matrix can be used for different countries. Data analyses of weighting components show that technology flows matrices are in an intermediate position between IO matrices and technological proximity matrices, but closer to the former. The various IO matrices, as well as the three technological proximity matrices, are very similar to each other. The panel data estimates of the effect of different types of interindustry R&D spillovers on industrial productivity growth in the G7 countries reject the hypotheses that a technology flows matrix can be approximated by an IO matrix and that a single IO matrix can be usedfor different countries. By transitivity, the procedure that comprises using a single technology flow for several countries is not reliable. The international comparison shows that each country benefits from different types of R&D externality. In Japan and, to a lesser extent, in the US, the rate of return to direct R&D is very high and is likely to compensate for relatively weak interindustry R&D spillover effects. In the five other industrialized countries, the reverse observation is true: strong social rates of return to R&D counterbal ance the poor performances of direct R&D.     

Analysis of variance on indirectly measured variables

Abstract  When observations from a normal distribution can only be obtained indirectly by counting the number of subjects responding to a previously chosen dose, parameter estimates can be obtained by using probit analysis. Well-known is the maximum likelihood technique of parameter estimation, less known is the approach by weighted least squares. The latter approach is followed to compare the parameters of several normal distributions by testing their equality, in analogy with the analysis of variance. A practical situation gave rise to this study and it is worked out at the end of the paper.     

GMM estimation of a maximum entropy distribution with interval data

We develop a generalized method of moments (GMM) estimator for the distribution of a variable where summary statistics are available only for intervals of the random variable. Without individual data, one cannot calculate the weighting matrix for the GMM estimator. Instead, we propose a simulated weighting matrix based on a first-step consistent estimate. When the functional form of the underlying distribution is unknown, we estimate it using a simple yet flexible maximum entropy density. Our Monte Carlo simulations show that the proposed maximum entropy density is able to approximate various distributions extremely well. The two-step GMM estimator with a simulated weighting matrix improves the efficiency of the one-step GMM considerably. We use this method to estimate the U.S. income distribution and compare these results with those based on the underlying raw income data.     

Minimax robust designs for field experiments

Experimental designs for field experiments are useful in planning agricultural experiments, environmental studies, etc. Optimal designs depend on the spatial correlation structures of field plots. Without knowing the correlation structures exactly in practice, we can study robust designs. Various neighborhoods of covariance matrices are introduced and discussed. Minimax robust design criteria are proposed, and useful results are derived. The generalized least squares estimator is often more efficient than the least squares estimator if the spatial correlation structure belongs to a small neighborhood of a covariance matrix. Examples are given to compare robust designs with optimal designs. The work was partially supported by research grants from the Natural Science and Engineering Research Council of Canada.     

A UNIFYING VIEW ON MULTI‐STEP FORECASTING USING AN AUTOREGRESSION

Abstract This paper unifies two methodologies for multi‐step forecasting from autoregressive time series models. The first is covered in most of the traditional time series literature and it uses short‐horizon forecasts to compute longer‐horizon forecasts, while the estimation method minimizes one‐step‐ahead forecast errors. The second methodology considers direct multi‐step estimation and forecasting. In this paper, we show that both approaches are special (boundary) cases of a technique called partial least squares (PLS) when this technique is applied to an autoregression. We outline this methodology and show how it unifies the other two. We also illustrate the practical relevance of the resultant PLS autoregression for 17 quarterly, seasonally adjusted, industrial production series. Our main findings are that both boundary models can be improved by including factors indicated from the PLS technique.     

Beyond spatial representation

The correspondence between theory and observation is often evaluated by a comparison between a hypothesized constraint matrix and the spatial representation of a pxp similarity matrix. This comparison of constraint and proximity matrices assumes the accurate translation of similarities to proximities. If the translation is not exact (i.e., a stress or alienation coefficient greater than zero), the hypothesized structure is evaluated using a false representation of the observed data. The proposed model eliminates the need for spatial representation by making a direct comparison between the hypothesized constraint matrix and the multivariate structure of the bivariate similarities. Goodness of fit indices are used for three model comparisons; (1) single data set, one hypothesized structure; (2) single data set, two hypothesized structures; and (3) two data sets, one hypothesized structure.     

A Unifying Framework and Comparison of Algorithms for Non-negative Matrix Factorisation

Non-negative matrix factorisation (NMF) is an increasingly popular unsupervised learning method. However, parameter estimation in the NMF model is a difficult high-dimensional optimisation problem. We consider algorithms of the alternating least squares type. Solutions to the least squares problem fall in two categories. The first category is iterative algorithms, which include algorithms such as the majorise–minimise (MM) algorithm, coordinate descent, gradient descent and the Févotte-Cemgil expectation–maximisation (FC-EM) algorithm. We introduce a new family of iterative updates based on a generalisation of the FC-EM algorithm. The coordinate descent, gradient descent and FC-EM algorithms are special cases of this new EM family of iterative procedures. Curiously, we show that the MM algorithm is never a member of our general EM algorithm. The second category is based on cone projection. We describe and prove a cone projection algorithm tailored to the non-negative least square problem. We compare the algorithms on a test case and on the problem of identifying mutational signatures in human cancer. We generally find that cone projection is an attractive choice. Furthermore, in the cancer application, we find that a mix-and-match strategy performs better than running each algorithm in isolation.     

The effect of a Durbin–Watson pretest on confidence intervals in regression

Consider a linear regression model and suppose that our aim is to find a confidence interval for a specified linear combination of the regression parameters. In practice, it is common to perform a Durbin–Watson pretest of the null hypothesis of zero first‐order autocorrelation of the random errors against the alternative hypothesis of positive first‐order autocorrelation. If this null hypothesis is accepted then the confidence interval centered on the ordinary least squares estimator is used; otherwise the confidence interval centered on the feasible generalized least squares estimator is used. For any given design matrix and parameter of interest, we compare the confidence interval resulting from this two‐stage procedure and the confidence interval that is always centered on the feasible generalized least squares estimator, as follows. First, we compare the coverage probability functions of these confidence intervals. Second, we compute the scaled expected length of the confidence interval resulting from the two‐stage procedure, where the scaling is with respect to the expected length of the confidence interval centered on the feasible generalized least squares estimator, with the same minimum coverage probability. These comparisons are used to choose the better confidence interval, prior to any examination of the observed response vector.     

Estimation in monotone single‐index models

Single‐index models are popular regression models that are more flexible than linear models and still maintain more structure than purely nonparametric models. We consider the problem of estimating the regression parameters under a monotonicity constraint on the unknown link function. In contrast to the standard approach of using smoothing techniques, we review different “non‐smooth” estimators that avoid the difficult smoothing parameter selection. For about 30 years, one has had the conjecture that the profile least squares estimator is an ‐consistent estimator of the regression parameter, but the only non‐smooth argmin/argmax estimators that are actually known to achieve this ‐rate are not based on the nonparametric least squares estimator of the link function. However, solving a score equation corresponding to the least squares approach results in ‐consistent estimators. We illustrate the good behavior of the score approach via simulations. The connection with the binary choice and current status linear regression models is also discussed.     

Optimal versus orthogonal and equivalent-estimation design of blocked and split-plot experiments

This article provides an overview of the recent literature on the design of blocked and split-plot experiments with quantitative experimental variables. A detailed literature study introduces the ongoing debate between an optimal design approach to constructing blocked and split-plot designs and approaches where the equivalence of ordinary least squares and generalized least squares estimates are envisaged. Examples where the competing design strategies lead to totally different designs are given, as well as examples in which the optimal experimental designs are orthogonally blocked or equivalent-estimation split-plot designs.     

解加权约束最小二乘问题的行M-不变方法

本文提出了一种新的求解加权约束线性最小二乘问题方法,即利用行M-不变矩阵得到了求解加权约束线性最小二乘的updating问题的递推方法。