Box-Cox Model Research Articles

On the Use of Spline Smoothing in Estimating Hedonic Housing Price Models: Empirical Evidence Using Hong Kong Data

Traditionally in estimating hedonic housing price functions, investigators use parametric models involving specific functional forms and a finite number of unknown parameters. Some investigators have questioned whether the underlying theory is capable of conveying sufficient information to enable a correct and successful specification of parametric models and have instead proposed the less restrictive semiparametric approach to the problem. In this paper, we illustrate how the technique of smoothing splines can be used to estimate hedonic housing price models. Smoothing splines are a powerful approach to the analysis of housing data as they are exceptionally flexible in their functional forms and provide a computationally tractable method even with a large number of explanatory variables. Our illustration takes the form of a rather limited, but very promising, application with Hong Kong data. In the forecasting comparison, the spline smoothing procedure outperforms the traditional parametric Box–Cox model in mean square error terms for out‐of‐sample predictions. Our results also suggest that the distortion caused by underfitting the model is smaller for spline smoothing than for the kernel and k‐nearest‐neighbor semiparametric procedures.

Modeling freight distribution using artificial neural networks

Studies about freight distribution modeling are limited due to the limitations in data availability. Existing studies in this subject, generally either use the conventional gravity models or the regression based models as modeling techniques. The present study, using the 1993 US Commodity Flow Survey Data, models inter-regional commodity flows for 48 continental states of the US with three different artificial neural networks (ANN). The results are compared with those of Celik and Guldmann's (2002) Box–Cox Regression Model. The ANN using conventional gravity model variables provides a slight improvement with respect to this Box–Cox model. However, the ANNs using theoretically relevant variables provide surprising improvements in comparison to the Box–Cox model. It is concluded that ANN architecture is a very promising technique for predicting short-term inter-regional commodity flows.

Valorisation économique des effets de la pollution atmosphérique

Cet article presente les resultats d’une enquete d’evaluation contingente specifique aux risques lies a la pollution de l’air (contextuelle) et estime les consentements a payer individuels pour les effets sanitaires (morbidite et mortalite) et non sanitaires. L’utilisation d’un scenario original aupres de 1 273 habitants des Bouches-du-Rhone et un traitement econometrique adapte (modele Box-Cox avec censure) conduisent a une valeur estimee de 68,60 euros par mois et par menage pour une reduction de moitie du nombre de jours pollues, tous effets confondus. Sur la base d’un modele d’utilite avec duree de vie aleatoire, nous evaluons pour la premiere fois une valeur d’evitement d’un deces dans le contexte de la pollution atmospherique. Elle s’etablit a 0,8 million d’euros.

What is a statistical model?

This paper addresses two closely related questions, "What is a statistical model?" and "What is a parameter?" The notions that a model must "make sense," and that a parameter must "have a well-defined meaning" are deeply ingrained in applied statistical work, reasonably well understood at an instinctive level, but absent from most formal theories of modelling and inference. In this paper, these concepts are defined in algebraic terms, using morphisms, functors and natural transformations. It is argued that inference on the basis of a model is not possible unless the model admits a natural extension that includes the domain for which inference is required. For example, prediction requires that the domain include all future units, subjects or time points. Although it is usually not made explicit, every sensible statistical model admits such an extension. Examples are given to show why such an extension is necessary and why a formal theory is required. In the definition of a subparameter, it is shown that certain parameter functions are natural and others are not. Inference is meaningful only for natural parameters. This distinction has important consequences for the construction of prior distributions and also helps to resolve a controversy concerning the Box-Cox model.

La prévention du tabagisme des adolescents français

Résumé Les pouvoirs publics, représentés par la Mission Interministérielle à la Lutte contre la Drogue et les Toxicomanies (MILDT), considèrent que la prévention par l’information est un moyen efficace de lutte contre le tabagisme juvénile. Devrait-on alors privilégier quelques relais précis pour les campagnes de prévention ? Dans cet article, nous mesurons l’impact de différentes sources d’information sur la consommation de tabac des adolescents scolarisés en France, à l’aide de données de l’INSERM. Nous montrons que les relais traditionnels de l’action sanitaire publique, comme les médecins et l’école, n’assurent pas de prévention primaire du risque tabagique. Ce dernier est lié à des hétérogénéités inobservables fortement corrélées au moral du jeune. La prévention du tabagisme passe donc avant tout par une détection des jeunes en situation de détresse psychologique.

Testing the Semiparametric Box-cox Model with the Bootstrap

This paper considers tests of the transformation parameter of the Box-Cox model when the distribution of the error is unknown. Monte Carlo experiments are carried out to investigate the rejection probabilities of the GMM-based Wald and Lagrange Multiplier (LM) tests when the null hypothesis is true. The results show that the differences between empirical and nominal levels can be large when asymptotic critical values are used. In most cases, the bootstrap reduces the differences between the empirical and nominal levels, and, in many cases, essentially removes the distortions in levels that occur with asymptotic critical values. Experiments are also carried out to investigate the ability of the bootstrap to provide improved finite-sample critical values with Wald tests based on the semiparametric estimation procedure recently developed by Foster, Tian and Wei (2001).

LM Tests for Functional Form and Spatial Error Correlation

This paper derives Lagrangian multiplier (LM) tests to jointly test for functional form and spatial error correlation. In particular, this paper tests for linear and log-linear models with no spatial error dependence against a more general Box-Cox model with spatial error correlation. Conditional LM tests and modified Rao score tests that guard against local misspecification are also derived. These tests are easy to implement and are illustrated using Anselin’s crime data. The performance of these tests is also compared using Monte Carlo experiments.

Double-length regressions for the Box–Cox difference model with heteroskedasticity or autocorrelation

This paper derives Lagrange multiplier tests based on artificial double length regressions (DLR) to jointly test for differenced linear or loglinear models with no heteroskedasticity or autocorrelation against a more general differenced Box–Cox model with heteroskedasticity or autocorrelation. These tests are easy to implement and are illustrated using an empirical example.

Leapfrog estimation of a fixed-effects model with unknown transformation of the dependent variable

This paper considers a fixed-effects panel version of the linear transformation model, in which the dependent variable is h( y t ) for an unspecified, strictly monotonic h. Examples of the model include the multiple-spell proportional hazards model and dependent-variable transformation models (e.g., the Box–Cox model) with fixed effects. A semiparametric estimator, called the leapfrog estimator, is introduced and shown to be n -consistent and asymptotically normal. The leapfrog estimator allows for h to vary over time and for heteroskedasticity across observational units. Related semiparametric estimators are considered, and a general covariance result for estimators based on second-order U-processes is presented.

Testing for Unit Roots and Non‐linear Transformations

It is well known that the augmented Dickey–Fuller (ADF) test of unit roots in univariate time series is sensitive to non‐linear transformations: a common example is when variables expressed in logarithms are found to be stationary, whereas the same variables in levels are found to be non‐stationary. In this paper, the effects of non‐linear transformations on the ADF auxiliary regression are investigated within the class of the Box–Cox model, and a test of non‐linear transformation is developed to assess the adequacy of the ADF regression. The proposed test is computationally simple and is calculated as thet ratio of an added variable in the ADF regression. Cointegration among series which are subject to non‐linear transformations is also analysed, and a simple procedure is developed to test the non‐linear transformation used in cointegration analysis. Several empirical examples taken from the Nelson–Plosser data set illustrate the practical relevance of the proposed test for univariate series, and a second empirical example is used to illustrate the test of non‐linear transformation for cointegrated series

A simple algorithm for estimating Box-Cox models

SUMMARY In this paper we propose a simple algorithm for computing maximum likelihood estimates of an extended Box-Cox model where the same transformation is applied to the dependent variable as well as to the independent variable(s). Illustrative examples presented show that our algorithm performs very well in practical applications. The algorithm can easily be modified to accommodate more general models of the Box-Cox type including the classic Box-Cox model.

Semiparametric estimation of a hedonic price function

Previous work on the preferred specification of hedonic price models usually recommended a Box-Cox model. In this paper we note that any parametric model involves implicit restrictions and they can be reduced by using a semiparametric model. We estimate a benchmark parametric model which passes several common specification tests, before showing that a semiparametric model outperforms it significantly. In addition to estimating the model, we compare the predictions of the models by deriving the distribution of the predicted log(price) and then calculating the associated prediction intervals. Our data show that the semiparametric model provides more accurate mean predictions than the benchmark parametric model.

A forecast comparison of residential housing prices by parametric versus semiparametric conditional mean estimators

The hedonic price model involves examining how the price of a commodity varies with the set of characteristics it possesses and has been used extensively in the housing market literature. The majority of the previous literature has concentrated on the parametric specifications of the hedonic price model by estimating econometric specifications such as ordinary least squares or Box-Cox models; all restrict the functional form and interaction between regressors. Recent evidence suggests that non-parametric and semiparametric techniques fir the data substantially better than the parametric specifications. Here, the parsimony of the parametric and semiparametric hedonic price models are examined by their out-of-sample forecast comparisons. The evidence presented here reveals that the semiparametric model provides the smallest out-of-sample mean square prediction error in comparison with the parametric specifications such as the ordinary least squares regression, the Box-Cox and the Wooldridge transformations. The results of this paper suggest that semiparametric regression can be successfully used for prediction and assessment of residential housing prices.

Forecast Comparisons of Residential Housing Prices by Parametric and Semiparametric Regression

The hedonic price theory involves examining how the price of a commodity varies with respect to its characteristics. Rosen (1974) presented an integrated treatment of the hedonic price theory and outlined the theory for estimation of the demand and supply functions which determine a hedonic price model. Unfortunately, economic theory has provided little guidance concerning the functional form of the dependence of price on quality. Thus, researchers have used econometric estimation techniques which allow flexible parametric functional forms such as the Box-Cox (1964) models. The objective of this paper is to examine the parsimony of the parametric and semiparametric conditional mean estimators by their out-of-sample forecast comparisons within the context of residential housing prices. In the construction of the parametric models, there are three choices: ordinary least squares (OLS hereafter) regression, Box-Cox (1964), and the Wooldridge (1992) transformations. The Box-Cox (1964) transformation is popular because of its flexibility in the model specification. The advantage of the Wooldridge transformation over the Box-Cox is that no second moment or other distributional assumptions are relied upon to obtain consistent estimates or to perform asymptotically valid inferences. We estimate all of the three parametric models and compare their out-of-sample forecasts as measured by the mean square prediction error. For all forecast horizons, the semiparametric model provides a much smaller mean square prediction error in comparison to the Box-Cox and Wooldridge transformations and the OLS regression.

Generalized functional forms and demand for money in an open economy: the case of the United Kingdom

The purpose of this paper are twofold:first, to apply a Box–Cox model to the UK money demand relationship within an open economy framework in order to empirically investigate the proper functional form supported by the data in this general setting. Secondly, to test for the porper scale variable in the UK money demand function within the open economy Box–Cox specification. These improvements enhance the accuracy of our measures of monetary and fiscal policy effects and our understanding of the interdependence between different economies. The empirical results derived here reject the restrictive linear and log specifications in favour of the general Box–Cox model under both income and consumption-based money demand specifications. The traditional income-based model, however,escaps unharmed from the challenge put forward by Mankiw and Summers (1986).

Alternative transformations in a class of limited dependent variable models: alcohol consumption by US women

This study investigates alcohol consumption by women in the United States,using alternative specifications of the double-hurdle models which accommodate non-normal errors. The IHS double-hurdle model performs marginally better than the Box-Cox double-hurdle model but both models suggest very similar elasticities. The profile of a drinker includes white, black, smoker, homeowner, and residing in urban areas. Women who are older, pregnant or lactating, on special diet, and those with higher weight-height ratio are less likely to consume alcohol than others. Seasonal and regional variations are apparent. Smokers are more likely to consume alcohol and also consume more, while older women are less likely to drink and also drink less than others.

Shrinkage estimation in nonlinear regression The Box-Cox transformation

The risk superiority of the Stein-rule estimator over the maximum likelihood (ML) estimator is known in the context of the normal linear regression model. We present a positive-part Stein-like estimator that dominates the ML and pretest estimators under quadratic loss, weighted by the information matrix, in nonlinear regression. The risk properties of ML, constrained ML, pretest, and shrinkage estimators for the Box-Cox model are discussed. In a Monte Carlo experiment, the shrinkage estimator dominates the ML estimator under the unweighted quadratic loss. However, the ML estimator may have lower risk than the shrinkage estimator under other weighted quadratic loss functions in small samples.

A Monte Carlo Investigation of the Box-Cox Model and a Nonlinear Least Squares Alternative

This paper reports a Monte Carlo study of the Box-Cox model and a nonlinear least squares alternative. Key results include the following: the transformation parameter in the Box-Cox model appears to be inconsistently estimated in the presence of conditional heteroskedasticity; the constant term in both the Box-Cox and the nonlinear least squares models is poorly estimated in small samples; conditional mean forecasts tend to underestimate their true value in the Box-Cox model when the transformation parameter is not equal to one; and conditional heteroskedasticity tends to worsen the bias in the Box-Cox predicted values. Copyright 1994 by MIT Press.

Pitfalls of Rescaling Regression Models with Box-Cox Transformations

To facilitate maximum likelihood estimation for Box-Cox models, several authors have suggested dividing the dependent variable by its sample geometric mean. This paper points out previously unmentioned drawbacks of this 'recalling.' First, the 'resealed' model is not actually equivalent to the untransformed one, so that the procedure involves more than a unit change. Second, there is no clear interpretation of the parameters after such resealing. The authors suggest an interpretation but find that the usual formulas for standard errors and confidence intervals are not asymptotically valid. Only tests for zero coefficients are valid. Thirdly, the authors discuss the appropriate way of measuring elasticities in such models. Copyright 1994 by MIT Press.

The relationship between ATP and R-values in postmortem bovine Longissimus dorsi muscle

To study the relationship between ATP and each of three R-values (R248, R250 and R258) in beef longissimus dorsi muscle data, appropriate mathematical functional forms were estimated by the Box-Cox transformation and then tested for linearity. Two separate tests for linearity indicated that the relationships between ATP and the three R-values were nonlinear. A shifting parameter restriction on the Box-Cox model was used to produce a curve that was more parallel to the y axis in the lower range of x axis, but the restriction lowered the R 2 compared with the Box-Cox model. ATP (1 μmol) could be estimated by R248 of 0·925, R250 of 0·967 and R258 of 1·038 with standard errors of 0·033, 0·034 and 0·032, respectively. There were no significant differences among the three R-values for the prediction of ATP, but there was a suggestion that R258 was the preferred R-value for predicting ATP in beef muscle.