Ridge regression estimation of the Rotterdam model

The best guesses of unknown coefficients specified in Theil's model of introspection are like predictions and not like de Finetti's prevision and therefore not the values taken by random variables. Constrained least squares procedures can be formulated which are free of these difficulties. The ridge estimator is a simple version of a constrained least squares estimator which can be made operational even when little prior information is available. Our operational ridge estimators are nearly minimax and are not less stable than least squares in the presence of high multicollinearity. Finally, we have presented the ridge estimates for the Rotterdam demand model.     

Consistent estimation of a regression with errors in the variables

Summary As is well known, least squares estimates of regression coefficients are inconsistent if the variables are measured with random errors. In the classical case of known variances and covariances for these error variables, consistent estimates can be derived. It is shown that these estimators generally have a joint asymptotic normal distribution, the covariance matrix of which is derived. No use is made of normality assumptions, but knowledge of the third and fourth moments of error variables is utilized.     

A panel model for political efficacy and responsiveness: an application of LISREL 7 with weighted least squares

Political efficacy is considered to be one of the most important attitudes in theories of political participation and democratic politics. It has been assumed that political efficacy is a stable, persistent orientation rather than a transient attitude. Several studies have examined the stability of political efficacy over time. In most of these studies, based on the analysis of the traditional SRC items, the stability assumption has been questioned. In this paper, we reconsider the stability issue but we adopt a different approach. We distinguish between two components of political efficacy: internal efficacy, a personal attribute and responsiveness, a system attribute, and we study their stability over time. To study the stability of political efficacy and responsiveness over time, we analyse the data with PRELIS and we develop a panel model using LISREL 7. As the observed variables are only ordinal, the estimation of the parameters of the model is based on polychoric correlations and on the weighted least squares method. Our analysis makes use of the Political Action Survey panel data for the USA. This data contains the six SRC efficacy items measured at two occasions. We find that the stability coefficients are higher than those reported in previous research. The difference in the values of the stability coefficients for each component seems to indicate that the personal component is more stable than the system component.     

Multicollinearity in regression: an efficiency comparison between L<Subscript>p</Subscript>-norm and least squares estimators

Multicollinearity is one of the most important issues in regression analysis, as it produces unstable coefficients’ estimates and makes the standard errors severely inflated. The regression theory is based on specific assumptions concerning the set of error random variables. In particular, when errors are uncorrelated and have a constant variance, the ordinary least squares estimator produces the best estimates among all linear estimators. If, as often happens in reality, these assumptions are not met, other methods might give more efficient estimates and their use is therefore recommendable. In this paper, after reviewing and briefly describing the salient features of the methods, proposed in the literature, to determine and address the multicollinearity problem, we introduce the L_pmin method, based on L_p-norm estimation, an adaptive robust procedure that is used when the residual distribution has deviated from normality. The major advantage of this approach is that it produces more efficient estimates of the model parameters, for different degrees of multicollinearity, than those generated by the ordinary least squares method. A simulation study and a real-data application are also presented, in order to show the better results provided by the L_pmin method in the presence of multicollinearity.     

Optimality of least squares in the seemingly unrelated regression equation model

In this article the authors have investigated the situations in which the single-equation least squares estimator is identical with the generalized least squares estimator in the seemingly unrelated regression model. The condition obtained turned out to be advantageous from an empirical point of view as it permits one to decide whether to go for a single-equation least squares method or Zellner's method with estimated disturbance variance covariance matrix for estimating the coefficients in the model.     

Recursions for the two-stage least-squares estimators

This paper presents recursion formulae for the two-stage least-squares estimators of the structural coefficients in a simultaneous equation model and for the residual sum of squares used in estimating the asymptotic covariance matrix. Included are formulae for updating estimates when a new set of observations is obtained and for revising estimates when a set of observations is discarded. The recursion formulae should prove to be of both practical and theoretical interest to econometricians.     

On finite sample properties of alternative estimators of coefficients in a structural equation with many instruments

We compare four different estimation methods for the coefficients of a linear structural equation with instrumental variables. As the classical methods we consider the limited information maximum likelihood (LIML) estimator and the two-stage least squares (TSLS) estimator, and as the semi-parametric estimation methods we consider the maximum empirical likelihood (MEL) estimator and the generalized method of moments (GMM) (or the estimating equation) estimator. Tables and figures of the distribution functions of four estimators are given for enough values of the parameters to cover most linear models of interest and we include some heteroscedastic cases and nonlinear cases. We have found that the LIML estimator has good performance in terms of the bounded loss functions and probabilities when the number of instruments is large, that is, the micro-econometric models with “many instruments” in the terminology of recent econometric literature.     

Homogeneity and endogeneity in systems of demand equations

In estimating systems of demand equations one of the right-hand-side explanatory variables, expenditure, may be endogenous in the sense that it is correlated with the equation error. If the assumption of homogeneity of degree zero in prices and nominal income is imposed on the system, it turns out it is still possible to estimate the parameters of the system even when expenditure is endogenous. The estimation procedure is simple requiring just one additional ordinary least squares regression.The paper also demostrates that a model in which homogeneity is tested with expenditure assumed exogenous is exactly equivalent to a model in which the exogeneity of expenditure is tested with homogeneity imposed. Previous tests of demand systems which have rejected the homogeneity postulate might therefore be reinterpreted instead as rejecting the hypothesis of exogeneity of expenditure with homogeneity of degree zero in prices and nominal income taken as given.     

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.     

Macroeconomic forecasting using approximate factor models with outliers

In this paper we consider estimating an approximate factor model in which candidate predictors are subject to sharp spikes such as outliers or jumps. Given that these sharp spikes are assumed to be rare, we formulate the estimation problem as a penalized least squares problem by imposing a norm penalty function on those sharp spikes. Such a formulation allows us to disentangle the sharp spikes from the common factors and estimate them simultaneously. Numerical values of the estimates can be obtained by solving a principal component analysis (PCA) problem and a one-dimensional shrinkage estimation problem iteratively. In addition, it is easy to incorporate methods for selecting the number of common factors in the iterations. We compare our method with PCA by conducting simulation experiments in order to examine their finite-sample performances. We also apply our method to the prediction of important macroeconomic indicators in the U.S., and find that it can deliver performances that are comparable to those of the PCA method.     

Least squares learning and the US Treasury bill rate

Understanding how agents formulate their expectations about Fed behavior is important for market participants because they can potentially use this information to make more accurate estimates of stock and bond prices. Although it is commonly assumed that agents learn over time, there is scant empirical evidence in support of this assumption. Thus, in this paper we test if the forecast of the three month T-bill rate in the Survey of Professional Forecasters (SPF) is consistent with least squares learning when there are discrete shifts in monetary policy. We first derive the mean, variance and autocovariances of the forecast errors from a recursive least squares learning algorithm when there are breaks in the structure of the model. We then apply the Bai and Perron (1998) test for structural change to a forecasting model for the three month T-bill rate in order to identify changes in monetary policy. Having identified the policy regimes, we then estimate the implied biases in the interest rate forecasts within each regime. We find that when the forecast errors from the SPF are corrected for the biases due to shifts in policy, the forecasts are consistent with least squares learning.     

Market Values of Environmental Amenities: A Latent Variable Approach

This study presents a latent variable framework to provide consistent and efficient estimates of market values of amenities. A model for property values of residential housing using different indicators for neighborhood quality and property value is estimated using data from the U.S. American Housing Survey. The estimated effect of neighborhood quality on property values is positive and more significant compared to the estimates obtained by ordinary least squares and instrumental variable methods. Variances of errors of measurement and variances of the latent structures are shown to be positive and significant without imposing nonnegativity restrictions.     

Interpretation of Cointegrating Coefficients in the Cointegrated Vector Autoregressive Model*

Regression coefficients are interpreted by a counterfactual experiment. For simultaneous equations this experiment can be implemented if the coefficients are identified, and throws some light on the role of instruments and the method of indirect least squares. This paper discusses another counterfactual experiment in the vector autoregressive model in order to interpret the coefficients of an identified cointegrating relation. The dynamics of the model is used to implement a long‐run change by changing the current values. The counterfactual experiment can be conducted precisely when the cointegrating relation is identified.     

Residential preferences, neighborhood filtering and neighborhood change : Comment and corrections

The purpose of this Comment is to correct the estimating technique used by Little in “Residential Preferences, Neighborhood Filtering and Neighborhood Change.” Little uses the factor load matrix rather than the factor score matrix in his computations of the implicit regression coefficients. We correct Little's estimates and also present additional results to compare his corrected results with principal component estimates and ordinary least squares.     

De theoretische zijde van de methode der kleinste kwadraten

The theoretical aspect of least squares.
This article contains a slightly modified presentation of the Markoff theory of least squares as developed along different lines by Aitken and by David and Neyman. The modifications aim at a more complete treatment and a geometrical illustration of the connection between best linear estimates and generalized least squares. The unbiasedness of ordinary least squares estimates in the case of heteroscedastic and correlated errors is stressed and the loss of efficiency is shown to be generally small. Topics like orthogonalization, partial correlation and what is called "over-correlation" are treated in passing.
Matrices are constantly used, being the adequate tools in this matter. In the appendix a special relevant matrix theorem is derived, viz. a generalization of the well known Cauchy inequality.     

Censored regression quantiles with endogenous regressors

This paper develops a semiparametric method for estimation of the censored regression model when some of the regressors are endogenous (and continuously distributed) and instrumental variables are available for them. A “distributional exclusion” restriction is imposed on the unobservable errors, whose conditional distribution is assumed to depend on the regressors and instruments only through a lower-dimensional “control variable,” here assumed to be the difference between the endogenous regressors and their conditional expectations given the instruments. This assumption, which implies a similar exclusion restriction for the conditional quantiles of the censored dependent variable, is used to motivate a two-stage estimator of the censored regression coefficients. In the first stage, the conditional quantile of the dependent variable given the instruments and the regressors is nonparametrically estimated, as are the first-stage reduced-form residuals to be used as control variables. The second-stage estimator is a weighted least squares regression of pairwise differences in the estimated quantiles on the corresponding differences in regressors, using only pairs of observations for which both estimated quantiles are positive (i.e., in the uncensored region) and the corresponding difference in estimated control variables is small. The paper gives the form of the asymptotic distribution for the proposed estimator, and discusses how it compares to similar estimators for alternative models.     

Optimal nested row-column designs for blocks of size 2×2 under dependence

This paper examines nested 2 ×2 row-column designs when within-block observations are assumed to be dependent. The model considered has fixed block effects, which may also include row and/or column effects. Optimal binary and non-binary designs, constructed from semi-balanced arrays, are given under both generalised and ordinary least squares estimation. It is shown that binary designs are optimal when dependence is low. In general, however, the optimal designs are highly specific to the correlation values. Received: October 1999     

The effect of environmental regulation on environmental innovation

Environmental expenditure estimates resulting from US environmental policy are based on current technology which may overstate policy's true costs. Existing evidence shows that ex ante cost estimates are greater than realized costs due to unexpected technological progress. This research programme asks whether innovation is a response to environmental regulation or whether the true regulatory compliance costs are overestimated ex ante when technological advancement is ignored? The author conducts an empirical study of the US manufacturing industry's environmental patent activities and environmental regulation as measured by pollution abatement and control expenditure (PACE) data. She finds a statistically significant positive relationship between environmental regulation and innovation when estimated by ordinary least squares (OLS). However, the OLS coefficient of pollution abatement costs is inconsistent because of a correlation between the explanatory variable and unobservable variables. Two-staged least squares addresses the inconsistency problem, resulting in positive and significant PACE coefficients. Thus, there is evidence that innovation is a response to environmental regulation. © 1998 John Wiley & Sons, Ltd and ERP Environment.     

Improved forecasting of autoregressive series by weighted least squares approximate REML estimation

Restricted maximum likelihood (REML) estimation has recently been shown to provide less biased estimates in autoregressive series. A simple weighted least squares approximate REML procedure has been developed that is particularly useful for vector autoregressive processes. Here, we compare the forecasts of such processes using both the standard ordinary least squares (OLS) estimates and the new approximate REML estimates. Forecasts based on the approximate REML estimates are found to provide a significant improvement over those obtained using the standard OLS estimates.     

Full-information estimates of a nonlinear macroeconometric model

Results of estimating a large-scale, nonlinear macroeconometric model by full-information maximum-likelihood, nonlinear three-stage least squares, and nonlinear two-stage least squares are reported in this paper. The computation of the estimates is first discussed, and then the differences among the estimates are examined.