The impact of a Hausman pretest on the size of a hypothesis test: The panel data case

The size properties of a two-stage test in a panel data model are investigated where in the first stage a Hausman (1978) specification test is used as a pretest of the random effects specification and in the second stage, a simple hypothesis about a component of the parameter vector is tested, using a t$t$-statistic that is based on either the random effects or the fixed effects estimator depending on the outcome of the Hausman pretest. It is shown that the asymptotic size of the two-stage test equals 1 for empirically relevant specifications of the parameter space. The size distortion is caused mainly by the poor power properties of the pretest. Given these results, we recommend using a t$t$-statistic based on the fixed effects estimator instead of the two-stage procedure.     

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.     

Bayesian Estimation of Willingness‐to‐pay Where Respondents Mis‐report Their Preferences*

We introduce a modified conditional logit model that takes account of uncertainty associated with mis‐reporting in revealed preference experiments estimating willingness‐to‐pay (WTP). Like Hausman et al. [Journal of Econometrics (1988) Vol. 87, pp. 239–269], our model captures the extent and direction of uncertainty by respondents. Using a Bayesian methodology, we apply our model to a choice modelling (CM) data set examining UK consumer preferences for non‐pesticide food. We compare the results of our model with the Hausman model. WTP estimates are produced for different groups of consumers and we find that modified estimates of WTP, that take account of mis‐reporting, are substantially revised downwards. We find a significant proportion of respondents mis‐reporting in favour of the non‐pesticide option. Finally, with this data set, Bayes factors suggest that our model is preferred to the Hausman model.     

PRACTITIONERS CORNER A POSITIVE SEMI-DEFINITE COVARIANCE MATRIX FOR HAUSMAN SPECIFICATION TESTS OF CONDITIONAL AND MARGINAL DENSITIES

ABSTRACT When the joint density of data can be factorized into a conditional and marginal densities, Hausman test can be used for diagnosing misspecifications of these densities. However, since common covariance estimates of the difference of the two estimators used in Hausman test need not be positive semi-definite in finite samples, the test statistic may be negative. This paper presents a simple and consistent covariance matrix which is positive semi-definite in any finite sample.     

The Mundlak Approach in the Spatial Durbin Panel Data Model

Abstract

This paper extends the Mundlak approach to the spatial Durbin panel data model (SDM) to help the applied researcher to determine the adequacy of the random effects specification in this setup. We propose a likelihood ratio (LR) test that assesses the significance of the correlation between regressors and individual effects. By contrast to the Hausman test, the Mundlak approach identifies (to some extent) the regressors correlated with individual effects. The second advantage is that once the correlation with individual effects has been modelled through an auxiliary regression, the random effects specification provides consistent estimators and the effect of time-constant variables can be estimated. Some Monte Carlo simulations study the properties of this proposed LR test in small samples and show that in some cases, it has a better behaviour than the Hausman test. We finally illustrate the usefulness of the extended Mundlak approach by estimating a house price model where some of the price determinants are time-constant. We show that ignoring the endogeneity of regressors with respect to individual effects leads to unreliable estimated parameters while results obtained using the Mundlak approach and the fixed effects specification are similar (concerning time-varying variables), implying that correlation between regressors and individual effects is well captured.

RÉSUMÉ la présente communication applique l'approche de Mundlak au modèle de données spatiales de Durbin pour aider le chercheur appliqué à déterminer dans quelle mesure la spécification des effets aléatoires est adéquate dans cette configuration. Nous proposons un test de ratio de vraisemblance évaluant l'importance de la corrélation entre régresseurs et effets individuels. Contrairement au test de Hausman, l'approche de Mundlak identifie (dans une certaine mesure) les régresseurs corrélés à des effets individuels. Le deuxième avantage est que lorsque la corrélation avec les effets individuels a été modélisée via une régression auxiliaire, la spécification des effets aléatoires fournit des estimateurs convergents, et il est alors possible d’évaluer l'effet de variables constantes dans le temps. Des simulations Monte Carlo étudient les propriétés de ce test de ratio de vraisemblance proposé dans des échantillons de taille finie, et indiquent que, dans certains cas, il présente un meilleur comportement que le test de Hausman. Nous illustrons enfin l'utilité de l'approche étendue de Mundlak en évaluant un modèle de prix des maisons, dans lequel certains déterminants des prix sont constants dans le temps. Nous montrons que si on ne prend pas en compte l'endogénéité des régresseurs par rapport aux effets individuels, on obtient des paramétres estimés non fiables, alors que les résultats obtenus avec l'approche de Mundlak et la spécification des effets fixes sont similaires (sur le plan des variables variant dans le temps), ce qui implique que la corrélation entre régresseurs et effets individuels est bien captée.

EXTRACTO Este estudio extiende el planteamiento Mundlak al modelo espacial de datos de panel (SDM) Durbin para ayudar al investigador aplicado a determinar la idoneidad de la especificación de efectos aleatorios dentro de esta configuración. Proponemos una prueba de relación de la probabilidad (LR) que evalúa la significancia de la correlación entre regresores y efectos individuales. En contraste con la prueba Hausman, el planteamiento Mundlak identifica (hasta cierto punto) los regresores correlacionados con efectos individuales. La segunda ventaja es que, una vez modelada la correlación con efectos individuales a través de una regresión auxiliar, la especificación de efectos aleatorios proporciona estimadores consistentes y puede estimarse el efecto de las variables constantes en el tiempo. Algunas simulaciones de Monte Carlo estudian las propiedades de esta prueba LR propuesta en muestras pequeñas y demuestran que, en algunos casos, se comporta mejor que la prueba Hausman. Finalmente, ilustramos la utilidad del planteamiento Mundlak ampliado estimando el precio de una vivienda donde varios determinantes del precio son constantes en el tiempo. Mostramos que ignorar la endogeneidad de los regresores con respecto a efectos individuales conduce a parámetros estimados no fiables, mientras que los resultados obtenidos mediante el planteamiento Mundlak y la especificación de efectos fijos son similares (en lo concerniente a variables que varan en el tiempo), sugiriendo que la correlación entre regresores y efectos individuales se ha capturado satisfactoriamente

     

On the Problem of Identification in Multiplicative Intensity-Rate Models with Multiple Interactions

In this paper we examine a multiplicative intensity model in which a covariate interacts with two other covariates in the same model. We demonstrate, analytically, that in such situations a log-linear parameterization based on two pairs of baseline levels cannot be transformed, uniquely, to the, otherwise equivalent, multiplicative parameterization. We show that the problem lies in an oversight of the conditional independence between the two covariates interacting with a common third covariate. As a solution, therefore, we propose an approach that takes due account of such dependence. Our proposed approach uses a common baseline level for the three covariates involved in interaction while estimating the corresponding relative intensities. The issues addressed are illustrated with a demographic data set involving the estimation of rates of transition to parenthood.     

Practical Considerations for Choosing Between Tobit and SCLS or CLAD Estimators for Censored Regression Models with an Application to Charitable Giving*

Practical considerations for choosing between Tobit, symmetrically censored least squares (SCLS) and censored least absolute deviations (CLAD) estimators are offered. Practical considerations deal with when a Hausman test is better than a conditional moment test for judging the severity of a misspecification, the need to bootstrap the sampling distributions of the Hausman tests, what to look for in a graphical examination of the residuals and the limited value of SCLS. The practical considerations are applied to a model of the intergenerational transmission of charitable giving using new data from the Panel Study of Income Dynamics (PSID). The paper shows how to use relative distribution methods to calculate CLAD‐based marginal effects on the observable dependent variable.     

A Review of the Use of Conditional Likelihood in Capture‐Recapture Experiments

We present a modern perspective of the conditional likelihood approach to the analysis of capture‐recapture experiments, which shows the conditional likelihood to be a member of generalized linear model (GLM). Hence, there is the potential to apply the full range of GLM methodologies. To put this method in context, we first review some approaches to capture‐recapture experiments with heterogeneous capture probabilities in closed populations, covering parametric and non‐parametric mixture models and the use of covariates. We then review in more detail the analysis of capture‐recapture experiments when the capture probabilities depend on a covariate.     

Is it a Timing or a Probability Effect? Four Simulations and an Application of Transition Rate Models to the Analysis of Unemployment Exit

Among the applications of event history analysis, in the last 10 years the lion's share has been played by proportional transition rate model. This type of models suffers from a major draw-back: it does not allow us to distinguish whether a covariate affects the event timing (the event occurs sooner/later) or the overall probability of the ultimate event occurrence (the chances of occurrence are constantly higher/lower). Thus, a positive/negative effect of a covariate found using a proportional transition rate model might reflect an acceleration/deceleration in the timing of the event and/or a high/low probability of the ultimate event occurrence (Yamaguchi, 1992). This paper shows how this problem can be reformulated in terms of the proportionality/non proportionality of the covariate effects. A twofold solution to disentangle the timing/probability problem is presented: this solution consists of a test of the proportionality of the covariate effects and a computation of the survival function at the end of the time interval studied. Two applications are discussed. The first one is based on four simulated processes. The second is based on an analysis of unemployment exit in Italy, with particular attention being paid to the effects of unemployment benefits. In the conclusion, implications for future applications of event history analysis are discussed.     

Testing for time-invariant unobserved heterogeneity in generalized linear models for panel data

Recent literature on panel data emphasizes the importance of accounting for time-varying unobservable individual effects, which may stem from either omitted individual characteristics or macro-level shocks that affect each individual unit differently. In this paper, we propose a simple specification test of the null hypothesis that the individual effects are time-invariant against the alternative that they are time-varying. Our test is an application of Hausman (1978) testing procedure and can be used for any generalized linear model for panel data that admits a sufficient statistic for the individual effect. This is a wide class of models which includes the Gaussian linear model and a variety of nonlinear models typically employed for discrete or categorical outcomes. The basic idea of the test is to compare two alternative estimators of the model parameters based on two different formulations of the conditional maximum likelihood method. Our approach does not require assumptions on the distribution of unobserved heterogeneity, nor it requires the latter to be independent of the regressors in the model. We investigate the finite sample properties of the test through a set of Monte Carlo experiments. Our results show that the test performs well, with small size distortions and good power properties. We use a health economics example based on data from the Health and Retirement Study to illustrate the proposed test.     

Semiparametric Estimation of a Duration Model

Within the framework of the proportional hazard model proposed in Cox (1972), Han and Hausman (1990) consider the logarithm of the integrated baseline hazard function as constant in each time period. We, however, proposed an alternative semiparametric estimator of the parameters of the covariate part. The estimator is considered as semiparametric since no prespecified functional form for the error terms (or certain convolution) is needed. This estimator, proposed in Lewbel (2000) in another context, shows at least four advantages. The distribution of the latent variable error is unknown and may be related to the regressors. It takes into account censored observations, it allows for heterogeneity of unknown form and it is quite easy to implement since the estimator does not require numerical searches. Using the Spanish Labour Force Survey, we compare empirically the results of estimating several alternative models, basically on the estimator proposed in Han and Hausman (1990) and our semiparametric estimator.     

Semiparametric estimation of logistic regression model with missing covariates and outcome

We consider a semiparametric method to estimate logistic regression models with missing both covariates and an outcome variable, and propose two new estimators. The first, which is based solely on the validation set, is an extension of the validation likelihood estimator of Breslow and Cain (Biometrika 75:11–20, 1988). The second is a joint conditional likelihood estimator based on the validation and non-validation data sets. Both estimators are semiparametric as they do not require any model assumptions regarding the missing data mechanism nor the specification of the conditional distribution of the missing covariates given the observed covariates. The asymptotic distribution theory is developed under the assumption that all covariate variables are categorical. The finite-sample properties of the proposed estimators are investigated through simulation studies showing that the joint conditional likelihood estimator is the most efficient. A cable TV survey data set from Taiwan is used to illustrate the practical use of the proposed methodology.     

Analysis of Survival Data with Multiple Causes of Failure

The purpose of the paper is to compare results of estimation and inference concerning covariate effects as obtained from two approaches to the analysis of survival data with multiple causes of failure. The first approach involves a dynamic model for the cause-specific hazard rate. The second is based on a static logistic regression model for the conditional probability of having had an event of interest. The influence of sociodemographic characteristics on the rate of family initiation and, more importantly, on the choice between marriage and cohabitation as a first union, is examined. We found that results, generally, are similar across the methods considered. Some issues in relation to censoring mechanisms and independence among causes of failure are discussed.     

Maximum likelihood estimation of a binomial proportion using one‐sample misclassified binary data

In this article, we construct two likelihood‐based confidence intervals (CIs) for a binomial proportion parameter using a double‐sampling scheme with misclassified binary data. We utilize an easy‐to‐implement closed‐form algorithm to obtain maximum likelihood estimators of the model parameters by maximizing the full‐likelihood function. The two CIs are a naïve Wald interval and a modified Wald interval. Using simulations, we assess and compare the coverage probabilities and average widths of our two CIs. Finally, we conclude that the modified Wald interval, unlike the naïve Wald interval, produces close‐to‐nominal CIs under various simulations and, thus, is preferred in practice. Utilizing the expressions derived, we also illustrate our two CIs for a binomial proportion parameter using real‐data example.     

Estimating Transition Probabilities from a Time Series of Independent Cross Sections

This paper considers the implementation of a nonstationary, heterogeneous Markov model for the analysis of a binary dependent variable in a time series of independent cross sections. The model, previously considered by M offitt (1993), offers the opportunity to estimate entry and exit transition probabilities and to examine the effects of time-constant and time-varying covariates on the hazards. We show how ML estimates of the parameters can be obtained by Fisher's method-of-scoring and how to estimate both fixed and time-varying covariate effects. The model is exemplified with an analysis of the labor force participation decision of Dutch women using data from the Socio-economic Panel (SEP) study conducted in the Netherlands between 1986 and 1995. We treat the panel data as independent cross sections and compare the employment status sequences predicted by the model with the observed sequences in the panel. Some open problems concerning the application of the model are also discussed.     

Minimax regret treatment choice with finite samples

This paper applies the minimax regret criterion to choice between two treatments conditional on observation of a finite sample. The analysis is based on exact small sample regret and does not use asymptotic approximations or finite-sample bounds. Core results are: (i) Minimax regret treatment rules are well approximated by empirical success rules in many cases, but differ from them significantly–both in terms of how the rules look and in terms of maximal regret incurred–for small sample sizes and certain sample designs. (ii) Absent prior cross-covariate restrictions on treatment outcomes, they prescribe inference that is completely separate across covariates, leading to no-data rules as the support of a covariate grows. I conclude by offering an assessment of these results.     

Generalized confidence intervals for process capability indices in the one-way random model

The method of generalized confidence intervals is proposed as an alternative method for constructing confidence intervals for process capability indices under the one-way random model for balanced as well as unbalanced data. The generalized lower confidence limits and the coverage probabilities for three commonly used capability indices were studied via simulation, separately for balanced and unbalanced cases. Simulation results showed that the generalized confidence interval procedure is quite satisfactory both in the balanced and unbalanced cases. Examples are provided to illustrate the results.     

Detecting overdispersion in INARCH(1) processes

A statistical test for the degree of overdispersion of count data time series based on the empirical version of the (Poisson) index of dispersion is considered. The test design relies on asymptotic properties of this index of dispersion, which in turn have been analyzed for time series stemming from a compound Poisson (Poisson‐stopped sum) INAR(1) model. This approach is extended to the popular Poisson INARCH(1) model, which exhibits unconditional overdispersion but has an (equidispersed) conditional Poisson distribution. The asymptotic distribution of the index of dispersion if applied to time series stemming from such a model is derived. These results allow us to investigate the ability of the dispersion test to discriminate between Poisson INAR(1) and INARCH(1) models. Furthermore, the question is considered if the index of dispersion could be used to test the null of a Poisson INARCH(1) model against the alternative of an INARCH(1) model with additional conditional overdispersion.     

Time series properties of ARCH processes with persistent covariates

We investigate the time series properties of a volatility model, whose conditional variance is specified as in ARCH with an additional persistent covariate. The included covariate is assumed to be an integrated or nearly integrated process, with its effect on volatility given by a wide class of nonlinear volatility functions. In the paper, such a model is shown to generate many important characteristics that are commonly observed in financial time series. In particular, the model yields persistence in volatility, and also well predicts leptokurtosis. This is true for any type of volatility functions considered in the paper, as long as the covariate is integrated or nearly integrated. Stationary covariates cannot produce important characteristics observed in many financial time series. We present two empirical applications of the model, which show that the default premium (the yield spread between Baa and Aaa corporate bonds) affects stock return volatility and the interest rate differential between two countries accounts for exchange rate return volatility. The forecast evaluation shows that the model generally outperforms GARCH and FIGARCH at relatively lower frequencies.     

Interval estimation of value-at-risk based on GARCH models with heavy-tailed innovations

ARCH and GARCH models are widely used to model financial market volatilities in risk management applications. Considering a GARCH model with heavy-tailed innovations, we characterize the limiting distribution of an estimator of the conditional value-at-risk (VaR), which corresponds to the extremal quantile of the conditional distribution of the GARCH process. We propose two methods, the normal approximation method and the data tilting method, for constructing confidence intervals for the conditional VaR estimator and assess their accuracies by simulation studies. Finally, we apply the proposed approach to an energy market data set.