Solving Multicollinearity in the Process of Fitting Regression Model Using the Nested Estimate Procedure

In the practical cases, we are usually faced with the more difficult problem of multicollinearity in our fitted regression model. Multicollinearity will arise when there are approximate linear relationships between two or more independent variables. It may cause some serious problems in validation, interpretation, and analysis of the model, such as unstable estimates, unreasonable sing, high-standard errors, and so on. Although there are some methods to solve or avoid this problem, we will propose another alternative from the practical view in this paper, called nested estimate procedure. The first half of the paper explains the concept and process of this procedure, and the second half provides two examples to illustrate this procedure’s suitability and reliability.     

On an inappropriate means of reducing multicollinearity

In a recent issue of this journal, Buck and Hakim suggest a means of alleviating the impact of multicollinearity in the context of grouped data and show how hypothesis testing can be conducted using this estimation procedure. This note shows that (a) their suggested procedure is analytically equivalent to the traditional estimation procedure, and (b) their hypothesis-testing procedure is incorrect, invalidating their conclusions concerning the interjurisdictional mobility of urban criminals.     

The Deviance Information Criterion in Comparison of Normal Mixing Models

Model selection from several non‐nested models by using the deviance information criterion within Bayesian inference Using Gibbs Sampling (BUGS) software needs to be treated with caution. This is particularly important if one can specify a model in various mixing representations, as for the normal variance‐mean mixing distribution occurring in financial contexts. We propose a procedure to compare goodness of fit of several non‐nested models, which uses BUGS software in part.     

Markov-switching models with endogenous explanatory variables II: A two-step MLE procedure

This paper proposes a two-step maximum likelihood estimation (MLE) procedure to deal with the problem of endogeneity in Markov-switching regression models. A joint estimation procedure provides us with an asymptotically most efficient estimator, but it is not always feasible, due to the ‘curse of dimensionality’ in the matrix of transition probabilities. A two-step estimation procedure, which ignores potential correlation between the latent state variables, suffers less from the ‘curse of dimensionality’, and it provides a reasonable alternative to the joint estimation procedure. In addition, our Monte Carlo experiments show that the two-step estimation procedure can be more efficient than the joint estimation procedure in finite samples, when there is zero or low correlation between the latent state variables.     

Pseudo-likelihood estimation and bootstrap inference for structural discrete Markov decision models

This paper analyzes the higher-order properties of the estimators based on the nested pseudo-likelihood (NPL) algorithm and the practical implementation of such estimators for parametric discrete Markov decision models. We derive the rate at which the NPL algorithm converges to the MLE and provide a theoretical explanation for the simulation results in Aguirregabiria and Mira [Aguirregabiria, V., Mira, P., 2002. Swapping the nested fixed point algorithm: A class of estimators for discrete Markov decision models. Econometrica 70, 1519–1543], in which iterating the NPL algorithm improves the accuracy of the estimator. We then propose a new NPL algorithm that can achieve quadratic convergence without fully solving the fixed point problem in every iteration and apply our estimation procedure to a finite mixture model. We also develop one-step NPL bootstrap procedures for discrete Markov decision models. The Monte Carlo simulation evidence based on a machine replacement model of Rust [Rust, J., 1987. Optimal replacement of GMC bus engines: An empirical model of Harold Zurcher. Econometrica 55, 999–1033] shows that the proposed one-step bootstrap test statistics and confidence intervals improve upon the first order asymptotics even with a relatively small number of iterations.     

A Note Regarding the Condition Number: The Case of Spurious and Latent Multicollinearity

The detection of multicollinearity in econometric models is usualy based on the so-called condition number (CN) of the data matrix X. However, the computation of the CN, which is the greater condition index, gives misleading results in particular cases and many commercial computer packages produce an inflated CN, even in cases of spurious multicollinearity, i.e. even if no collinearity exists when the explanatory variables are considered. And this is due to the very low total variation of some columns of the transformed data matrix, which is used to compute CN. On the other hand, we may have the problem of latent multocollinearity which can be revealed by additionally computing a revised CN. With all these in mind, we figure out the ill-conditioned situations, suggesting some practical rules of thumb to face such problems using a single diagnostic in a fairly simple procedure. It is noted that this procedure is not mentioned in the relevant literature.     

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.     

Identifying Structural Breaks in Cointegrated Vector Autoregressive Models*

This article suggests an alternative formulation of the cointegrated vector autoregressive (VAR) model such that the coefficients for the deterministic terms have straightforward interpretations. These coefficients can be interpreted as growth rates and cointegration mean level coefficients and express long‐run properties of the model. For example, the growth rate coefficients tell us how much to expect (unconditionally) the variables in the system to grow from one period to the next, representing the underlying (steady state) growth in the variables. The estimation of the proposed formulation is made operationally in GRaM, which is a program for Ox Professional. GRaM can be used for analysing structural breaks when the deterministic terms include shift dummies and broken trends. By applying a formulation with interpretable deterministic components, different types of structural breaks can be identified. Shifts in both intercepts and growth rates, or combinations of these, can be tested for. The ability to distinguish between different types of structural breaks makes the procedure superior compared with alternative procedures. Furthermore, the procedure utilizes the information more efficiently than alternative procedures. Finally, interpretable coefficients of different types of structural breaks can be identified.     

Soil texture prediction via reduced K-means Principal Component Multinomial Regression

Texture is one of the most important physical property of the soils for its influence on other fundamental properties. It is defined according to particle size distribution, that can be accurately measured in laboratory. However, these measurements are costly and very time consuming, therefore valid alternatives are necessary. In last years some statistical techniques have been used to predict textural classification using values of reflectance spectrometry as explicative variables. The estimation of the model parameters can be not too accurate, affecting prediction when there is multicollinearity among predictors. Another issue can be the great number of explicative variables usually necessary to explain the response. In order to improve the accuracy of the prediction in classification problems under multicollinearity and to reduce the dimension of the problem with continuous covariates, in this paper we introduce a new technique, based on classification and dimension reduction methods. We show how the new proposal can improve the accuracy of prediction, considering a problem concerning the textural classification of soils of Campania region.     

A model for non-negative and non-positive distributed lag functions

A new class of distributed lag models which avoid sign changes in the estimated lag weights and can approximate any non-negative or non-positive lag structure to any desired degree of accuracy is investigated. An estimation procedure is proposed and applied to capital appropriations and expenditures data.     

Forecasting from others’ experience: Bayesian estimation of the generalized Bass model

We propose a Bayesian estimation procedure for the generalized Bass model that is used in product diffusion models. Our method forecasts product sales early based on previous similar markets; that is, we obtain pre-launch forecasts by analogy. We compare our forecasting proposal to traditional estimation approaches, and alternative new product diffusion specifications. We perform several simulation exercises, and use our method to forecast the sales of room air conditioners, BlackBerry handheld devices, and compressed natural gas. The results show that our Bayesian proposal provides better predictive performances than competing alternatives when little or no historical data are available, which is when sales projections are the most useful.     

Autocorrelation and dynamic methodology with an application to wage determination models

A demonstration is provided of rigorous, statistical methodology whereby both the type and order of an error process can be identified in dynamic, single equation econometric models. The paper relies heavily upon maximum likelihood estimation, nested likelihood ratio tests and the overfitting or exponentially weighted procedure for model selection. An application of the methodology to a class of quarterly wage determination models is included.     

超越对数函数要素替代弹性公式修正与估计方法比较

对超越对数函数要素替代弹性估计方法进行专门研究,修正了超越对数生产函数要素替代弹性公式错误,梳理超越对数成本函数各类替代弹性定义,综合考虑数据易得性、估计可靠性、定义合意性以判定各类估计方法的相对优劣。研究发现,受困于共线性难题,基于超越对数生产函数计算替代弹性效果不佳;基于超越对数成本函数计算替代弹性更具优势,影子替代弹性凭借出色的理论性质与强稳健性成为首选。     

Improved instrumental variables estimation of simultaneous equations under conditionally heteroskedastic disturbances

In this paper we develop estimation techniques and a specification test for the validity of instrumental variables allowing for conditionally heteroskedastic disturbances. We propose modified two‐stage least squares (2SLS) and modified 3SLS procedures where the conditional heteroskedasticity is taken into account, which are natural extensions of the traditional 2SLS and 3SLS estimators and which achieve a lower variance. We recommend the use of these modified 2SLS and 3SLS procedures in practice instead of alternative estimators like limited‐information maximum likelihood/full‐information maximum likelihood, where the non‐existence of moments leads to extreme values, and also for ease of computation. It is shown theoretically and with simulation that in some cases 2SLS, 3SLS and our modified 2SLS and 3SLS procedures can have very severe biases (including the weak instruments case), and we present bias correction procedures to apply in practice along the lines of Flores‐Lagunes ( 2007 ). Our new estimation procedures can also be used to extend the test for weak instruments of Stock and Yogo ( 2005 ) and to allow for conditional heteroskedasticity. Finally, we show the usefulness of our estimation procedures with an application to the demand and supply of fish. Copyright © 2010 John Wiley & Sons, Ltd.     

A Wald test for the cointegration rank in nonstationary fractional systems

This paper develops new methods for determining the cointegration rank in a nonstationary fractionally integrated system, extending univariate optimal methods for testing the degree of integration. We propose a simple Wald test based on the singular value decomposition of the unrestricted estimate of the long run multiplier matrix. When the “strength” of the cointegrating relationship is less than 1/2, the test statistic has a standard asymptotic distribution, like Lagrange Multiplier tests exploiting local properties. We consider the behavior of our test under estimation of short run parameters and local alternatives. We compare our procedure with other cointegration tests based on different principles and find that the new method has better properties in a range of situations by using information on the alternative obtained through a preliminary estimate of the cointegration strength.     

Generalised estimates of diffusion rates for the oxygen blown steel process: An international comparison

The purpose of this study is to build on the results recently obtained by Poznanski (1990). First, the estimation technique is generalised by removing both the symmetric restrictions and the effects of autocorrelated errors that are present in simple logistic functions. Secondly, attention is drawn to the existence and consequences of multicollinearity when non linear techniques are applied to a general exponential. Thirdly, an attempt is made to quantify the importance of market structure and time lags in adoption as determinants of diffusion speed. Finally, a comparison between these generalised results and Poznanski's is made.The study concludes that, by generalising the estimation technique, some significant changes take place. Changes occur in the ordering and magnitude of diffusion speeds, especially amongst the developing and centrally planned economies. Changes also occur in the nature of the diffusion process, and in the importance attributed to market structure and diffusion lags in determining the rate of diffusion. It is also shown that, for some countries, multicollinearity imposes constraints on the type of generality that can be obtained.     

Ridge estimation of house value determinants

This study considers structural and lot characteristics which are determinants of equilibrium prices of homes. Previous studies which have attempted this estimation have been plagued by problems of multicollinearity. Micro data on detailed house characteristics are used with a biased estimation technique called ridge regression to obtain estimated characteristic prices with more correct signs and increased stability. The ridge prices give a more precise indication of how characteristics influence house prices. As a method of finding house value determinants ridge regression is shown to be superior to ordinary least-squares regression.     

OPTION PRICING AND IMPLICIT VOLATILITIES1

Abstract. This paper demonstrates that Black-Scholes implied volatilities can be used to value options in many situations where the assumptions of the Black-Scholes model are violated, including (i) alternative stock processes, (ii) stochastic interest rates, and (iii) market frictions. Given its computational simplicity, this procedure provides an attractive alternative to the more complex models with a direct estimation procedure.     

Asymptotic least-squares estimation efficiency considerations and applications

This paper is concerned with the large sample efficiency of the asymptotic least-squares (ALS) estimators introduced by Gouriéroux, Monfort, and Trognon (1982, 1985) and Chamberlain (1982, 1984). We show how the efficiency of these estimators is affected when additional information is incorporated into the estimation procedure. The relationship between ALS and maximum likelihood is discussed. It is shown that ALS can be used to obtain asymptotically efficient estimates for a large range of econometric models. Many results from the literature on estimation are special cases of the framework adopted in this paper. An application of ALS to a dynamic rational expections factor demand model in the manufacturing sector in The Netherlands demonstrates the potential of the method in the estimation of the parameters in models which are subject to nonlinear cross-equation restrictions.     

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.