On existence and uniqueness of maximum likelihood estimates in quantal and ordinal response models

Summary For quantal and ordinal response models, conditions on existence and uniqueness of maximum likelhood estimates are presented. Results are derived from general results on direction sets and spaces associated with a proper concave function. If each summand of the log likelihood is in any direction either strictly concave or affine, necessary and sufficient conditions are obtained. If all cell counts are strictly positive, then it is shown that estimates always exist, and that they are unique if all parameters are identifiable. If estimates exist without being unique, results on uniquely estimable linear functions are given, paralleling corresponding results in linear regression. An extension of the maximum likelihood principle is outlined yielding similar results even if the likelihood does not attain its supremum. The logit model, the linear probability model, cumulative and sequential models and binomial response models are considered in detail.     

Minimum kolmogorov distance estimates of parameters and parametrized distributions

Minimum Kolmogorov distance estimates of arbitrary parameters are considered. They are shown to be strongly consistent if the parameter space metric is topologically weaker than the metric induced by the Kolmogorov distance of distributions from the statistical model. If the parameter space metric can be locally uniformly upper-bounded by the induced metric then these estimates are shown to be consistent of ordern ^−1/2. Similar results are proved for minimum Kolmogorov distance estimates of densities from parametrized families where the consistency is considered in theL ₁-norm. The presented conditions for the existence, consistency, and consistency of ordern ^−1/2 are much weaker than those established in the literature for estimates with similar properties. It is shown that these assumptions are satisfied e.g. by all location and scale models with parent distributions different from Dirac, and by all standard exponential models. Supported by the scientific exchange program between the Hungarian Academy of Sciences and the Royal Belgian Academy of Sciences, and by GACR grant 201/93/0232.     

Ecological Fallacy and Covariates: New Insights based on Multilevel Modelling of Individual Data

The paper provides a new and more explicit formulation of the assumptions needed by the ordinary ecological regression to provide unbiased estimates and clarifies why violations of these assumptions will affect any method of ecological inference. Empirical evidence is obtained by showing that estimates provided by three main ecological inference methods are heavily biased when compared with multilevel logistic regression applied to a unique set of individual data on voting behaviour. The main findings of our paper have two important implications that can be extended to all situations where the assumptions needed to apply ecological inference are violated in the data: (i) only ecological inference methods that allow one to model the effect of covariates have a chance to produce unbiased estimates, and (ii) there are certain data generating mechanisms producing a kind of bias in ecological estimates that cannot be corrected by modelling the effect of covariates.     

Median unbiased estimates for M.L.R.-families

Summary Lehmann [p. 83] has shown that some families of probability measures with monotone likelihood ratios (m.l.r.) admit median unbiased estimates which are optimum in the sense that among all median unbiased estimates they minimize the expected loss for any loss function which assumes its minimal value zero for the “true” parameter value and is nondecreasing as the parameter moves away from the true value in either direction. This very strong optimum property was proved under the assumption that all probability measures of the m.l.r.-family have continuous distribution functions, that they are mutually absolutely continuous and that each element of the support is the median of somep-measure of the family. This result does therefore not cover important cases such as the binomial families or thePoisson family. The purpose of the present paper is to show the existence ofrandomized median unbiased estimates with the same optimum property for m.l.r.-families which are closed and connected with respect to the strong topology. Such families are always dominated. We do, however, neither assume that thep-measures are mutually absolutely continuous nor that the distribution functions are continuous. We remark that the use of randomized estimates is indispensable here because nonrandomized median unbiased estimates do not always exist in the general case.     

Capital-land substitution in urban housing: A survey of empirical estimates

Recent empirical studies of capital-land substitution in urban housing are examined to determine the best estimate of the elasticity of substitution parameter σ. Studies based upon a cross section of metropolitan areas produce a rather narrow range for σ. Studies of individual metropolitan areas produce a wide range of estimates for σ, suggesting that a may vary across metropolitan areas. However, all estimates of σ are probably biased toward zero by errors in the measurement of land values.     

How far can reduced childcare prices push female labour supply?

Exploiting exogenous variation in childcare prices stemming from a childcare price reform, this paper estimates effects of reductions in childcare costs on female labour supply. The reform introduced a cap on childcare prices, and lead to considerable reductions in prices depending on family type and region of residence. Since the price is determined by a handful of observed characteristics, we are able to match households that are similar in all relevant aspects, but experienced quite different price changes. Our difference-in-differences regression matching estimates are very precise, and close to zero.     

Growth and convergence in a multi-country empirical stochastic Solow model

The paper considers international per capita output and its growth using a panel of data for 102 countries between 1960 and 1989. It sets out an explicitly stochastic Solow growth model and shows that this has quite different properties from the standard approach where the output equation is obtained by adding an error term to the linearized solution of a deterministic Solow model. It examines the econometric properties of estimates of beta convergence as traditionally defined in the literature and shows that all these estimates are subject to substantial biases. Our empirical estimates clearly reflect the nature and the magnitude of these biases as predicted by econometric theory. Steady state growth rates differ significantly across countries and once this heterogeneity is allowed for the estimates of beta are substantially higher than the consensus in the literature. But they are very imprecisely estimated and difficult to interpret. The paper also discusses the economic implications of these results for sigma convergence. © 1997 John Wiley & Sons, Ltd.     

Estimating restricted structural change models

This paper considers issues related to multiple structural changes, occurring at unknown dates, in the linear regression model when restrictions are imposed on the parameters. This includes, for example, the important special case where different nonadjacent regimes are the same. The estimates are constructed as global minimizers of the restricted sum of squared residuals and we provide an extension of the algorithm discussed in Bai and Perron [2003b, Computation and analysis of multiple structural change models. Journal of Applied Econometrics 18, 1–22] to efficiently compute them. We show that the estimates of the break dates have the same asymptotic properties with or without the restrictions imposed; that is, in large samples, there is no efficiency gain from imposing valid restrictions as far as the estimates of the break dates are concerned. Of course, efficiency gains occur for the other parameters of the model. Simulations show that in small samples, all parameters are more efficiently estimated using the restrictions. We also consider tests of the null hypothesis of no structural change. These are also more powerful when the restrictions are imposed. A Gauss code for all the procedures discussed in this paper is available from the authors.     

Technical efficiency and region The U.S. manufacturing sector 1972–1977

In this paper we estimate the extent of technological bias in an interregional context for U.S. manufacturing during the period 1972–1977 using a factor augmenting production function approach. We present estimates of the elasticity of factor substitution for each of the 48 states in the sample using a variable elasticity of substitution production function. Next, we use these estimates to generate estimates of the rates of change in the efficiencies of capital and labor inputs and compare these estimates across states and census regions. We also examine and compare estimates of total factor productivity across states and regions. We ,find that the average annual rates of growth of capital efficiency during the period are 5.5 percent in the Northeast, 5.3 percent in the Northcentral, 5.6 percent in the West and 3.1 percent in the South. The rates of change of the efficiencies of labor are found to be negative across all regions except the South. The rates of change of total factor productivity are found to be 1.7 percent in the Northeast, 2.3 percent in the Northcentral, 2.4 percent in the West and 1.7 percent in the South. We also find that between 33 percent and 56 percent of the growth of output across regions is due to technical progress.     

Career progression and comparative advantage

This paper constructs and estimates a structural dynamic model of occupational choice in which all occupations are characterized in a skill requirement space using data from the Dictionary of Occupational Titles and the NLSY79. This skill requirement space approach has its merit in computational simplicity as well as ease of interpretation: it allows the model to include hundreds of occupations at the three-digit census classification level without a large number of parameters. Parameter estimates indicate that wages grow with the skill requirements of an occupation and that educated and experienced individuals are better rewarded in a cognitive and interpersonal skill demanding occupation. They also suggest that ignoring self-selection into occupations and individual heterogeneity may result in counter-intuitive and biased estimates of the returns to skill requirements.     

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.     

Estimating an economic model of crime using panel data from North Carolina

This paper replicates the Cornwell and Trumbull ( 1994 ) estimation of a crime model using panel data on 90 counties in North Carolina over the period 1981–1987. While the Between and Within estimates are replicated, the fixed effects 2SLS as well as the 2SLS estimates are not. In fact, the fixed effects 2SLS estimates turn out to be insignificant for all important deterrent variables as well as legal opportunity variables. We argue that the usual Hausman test, based on the difference between fixed effects and random effects, may lead to misleading inference when endogenous variables of the conventional simultaneous equation type are among the regressors. We estimate the model using random effects 2SLS and perform a Hausman test based on the difference between fixed effects 2SLS and random effects 2SLS. We cannot reject the consistency of the random effects 2SLS estimator and this estimator yields plausible and significant estimates of the crime model. This result should be tempered by the legitimacy of the chosen instruments. Copyright © 2006 John Wiley & Sons, Ltd.     

Substitution Bias and Cost of Living Variability for U.S. Demographic Groups:

A bstract . Intertemporal cost of living variability is analyzed for households with differing income levels and family characteristics. These indexes are based upon the parameter estimates of a comprehensive system of expenditure equations, the quadratic expenditure system. Despite considerable differences in the group-specific share parameters as well as nonlinearities in the Engle curves for each group, little variation occurs in these indexes for several U.S. price series over the 1967–1984 time period. As a result, we find little evidence that group-specific fixed weight indexes are better cost of living approximations than a general Consumer Price Index even though all substitution bias estimates, by income and household type, are quite small.     

COMPARISON OF MODEL AVERAGING TECHNIQUES: ASSESSING GROWTH DETERMINANTS

This paper investigates the replicability of three important studies on growth theory uncertainty that employed Bayesian model averaging tools. We compare these results with estimates obtained using alternative, recently developed model averaging techniques. Overall, we successfully replicate all three studies, find that the sign and magnitude of these new estimates are reasonably close to those produced via traditional Bayesian methods and deploy a novel strategy to implement one of the new averaging estimators. Copyright © 2012 John Wiley & Sons, Ltd.     

Estimation of a non-invertible moving average process: The case of overdifferencing

The effect of differencing all of the variables in a properly specified regression equation is examined. Excessive use of the difference transformation induces a non-invertible moving average (MA) process in the disturbances of the transformed regression. Monte Carlo techniques are used to examine the effects of overdifferencing on the efficiency of regression parameter estimates, inferences based on these estimates, and tests for overdifferenccing based on the estimator of the MA parameter for the disturbances of the differences regression. Overall, the problem of overdifferencing is not serious if careful attention is paid to the properties of the disturbances of regression equations.     

A Bayesian Multiple Comparison Procedure for Order-Restricted Mixed Models

A Bayesian hierarchical mixed model is developed for multiple comparisons under a simple order restriction. The model facilitates inferences on the successive differences of the population means, for which we choose independent prior distributions that are mixtures of an exponential distribution and a discrete distribution with its entire mass at zero. We employ Markov Chain Monte Carlo (MCMC) techniques to obtain parameter estimates and estimates of the posterior probabilities that any two of the means are equal. The latter estimates allow one both to determine if any two means are significantly different and to test the homogeneity of all of the means. We investigate the performance of the model-based inferences with simulated data sets, focusing on parameter estimation and successive-mean comparisons using posterior probabilities. We then illustrate the utility of the model in an application based on data from a study designed to reduce lead blood concentrations in children with elevated levels. Our results show that the proposed hierarchical model can effectively unify parameter estimation, tests of hypotheses and multiple comparisons in one setting.     

Least squares and stochastic difference equations

In this paper we study the asymptotic properties of least squares estimates of parameters in a stochastic difference equation. The difference equation is assumed to be linear with constant real coefficients. Moreover, the roots of the associated characteristic polynomial are all assumed to have absolute value different from one. Finally, the difference equation is assumed to be non-homogeneous with fixed (i.e., non-random) initial values, and to have ‘error terms’ that are independently and identically distributed with mean zero and finite variance. We show that under these conditions the least squares estimates of the coefficients of the difference equation converge with probability one to the true values.     

Estimates of relative risk

Summary For the two-sample problem with proportional hazard functions, we consider estimation of the constant of proportionality, known as relative risk, using complete uncensored data. For this very special case of Cox’s (1972) regression model for survival data, we find a two-step estimate which is asymptotically equivalent to Cox’s partial likelihood estimate, and we show that both estimates are asymptotically optimal (in the sense of minimum asymptotic variance) among all regular rank estimates of relative risk. Supported by NSF grant 81-00748.     

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.