This paper considers the percentage impact of a dummy variable regressor on the level of the dependent variable in a semilogarithmic regression equation with normal disturbances. We derive an exact unbiased estimator, its variance, and an exact unbiased estimator of the variance. The main practical contribution lies in a convenient approximation for the unbiased estimator of the variance, which can be reported together with Kennedy's approximate unbiased estimator of the percentage change. The two approximations are very simple, yet highly reliable. The results are applied to teacher earnings and further illustrated by examples from the literature.

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Finite sample distributions of studentized inequality measures di¤er substantially from their asymptotic normal distribution in terms of location and skewness. We study these aspects formally by deriving the second order expansion of the …rst and third cumulant of the studentized inequality measure. We state distribution-free expressions for the bias and skewness coe¢ cients. In the second part we improve over …rst-order theory by deriving Edgeworth expansions and normalizing transforms. These normalizing transforms are designed to eliminate the second order term in the distributional expansion of the studentized transform and converge to the Gaussian limit at rate O(n 1 ). This leads to improved con…dence intervals and applying a subsequent bootstrap leads to a further improvement to order O(n 3=2 ). We illustrate our procedure with an application to regional inequality measurement in Côte d'Ivoire.

Curved exponential models have the property that the dimension of the minimal sufficient statistic is larger than the number of parameters in the model. Many econometric models share this feature. The first part of the paper shows that, in fact, econometric models with this property are necessarily curved exponential. A method for constructing an explicit set of minimal sufficient statistics, based on partial scores and likelihood ratios, is given. The difference in dimension between parameterand statistic and the curvature of these models have important consequences for inference. It is not the purpose of this paper to contribute significantly to the theory of curved exponential models, other than to show that the theory applies to many econometric models and to highlight some multivariate aspects. Using the methods developed in the first part, we show that demand systems, the single structural equation model, the seemingly unrelated regressions, and autoregressive models are all curved exponential models.

Exact expressions for the statistical curvature and related geometric quantities in ®rst-order autoregressive models are derived. We present a method for calculating moments that is applicable in general autoregressive models. It combines the algebra of differential and difference operators to simplify the problem, and to obtain results valid for all sample sizes. The exact covariance matrix for the minimal suf®cient statistic is also derived.

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Let g~l! be the spectral density of a stationary process and let f u~l !, u ʦ Q, be a fitted spectral model for g~l!+ A minimum contrast estimator Z u n of u is defined that minimizes a distance D~f u , [ g n ! between f u and [ g n , where [ g n is a nonparametric spectral density estimator based on n observations+ It is known that Z u n is asymptotically Gaussian efficient if g~l! ϭ f u~l !+ Because there are infinitely many candidates for the distance function D~f u , [ g n !, this paper discusses higher order asymptotic theory for Z u n in relation to the choice of D+ First, the second-order Edgeworth expansion for Z u n is derived+ Then it is shown that the bias-adjusted version of Z u n is not second-order asymptotically efficient in general+ This is in sharp contrast with regular parametric estimation, where it is known that if an estimator is first-order asymptotically efficient, then it is automatically secondorder asymptotically efficient after a suitable bias adjustment~e+g+, Ghosh, 1994, Higher Order Asymptotics, p+ 57!+ The paper establishes therefore that for semiparametric estimation it does not hold in general that "first-order efficiency implies second-order efficiency+" The paper develops verifiable conditions on D that imply second-order efficiency+ This paper was written while the first author was visiting the University of Bristol as a Benjamin Meaker Professor+ The second author was previously at Bristol and is now supported by a fellowship of the Royal Netherlands Academy of Arts and Sciences+ We are grateful to the co-editor Pentti Saikkonen and two anonymous referees for their valuable comments, which significantly

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