Given observation-pairs (xi ,yi ), i = 1,...,n , taken to be independent observations of the random pair (X ,Y), we sometimes want to form a nonparametric estimate of m(x) = E(Y/ X = x). Let YE have the empirical distribution of the yi , and let (XS ,YS ) have the kernel-smoothed distribution of the (xi ,yi ). Then the standard estimator, the Nadaraya-Watson form mNW(x) can be interpreted as E(YE?XS = x). The smoothed-distribution estimator ms (x)=E(YS/XS = x) is a more general form than mNW (x) and often has better properties. Similar considerations apply to estimating Var(Y/X = x), and to local polynomial estimation. The discussion generalizes to vector (xi ,yi ).

Taking as our point of departure a model proposed by David Card (2001), we suggest new methods for analyzing wage dispersion in a partially unionized labor market. Card's method disaggregates the labor population into skill categories, which procedure entails some loss of information. Accordingly, we develop a model in which each worker individually is assigned a union-membership probability and predicted union and nonunion wages. The model yields a natural three-way decomposition of variance. The decomposition permits counterfactual analysis, using concepts and techniques from the theory of factorial experimental design. We examine causes of the increase in U.K. wage dispersion between 1983 and 1995. Of the factors initially considered, the most influential was a change in the structure of remuneration inside both the union and nonunion sectors. Next in importance was the decrease in union membership. Finally, exogenous changes in labor force characteristics had, for most groups considered, only a small negative effect. We supplement this preliminary three-factorial analysis with a five-factorial analysis that allows us to examine effects from the wage-equation parameters in greater detail.

This paper examines the effects of union decline in Britain on changes in earnings dispersion between 1983 and 1995. As part and parcel of the exercise, the effects of changes in the wage gap and the variance gap are also calculated. Detailed findings are provided by gender and broad sector, allowing for worker characteristics and the skill gradient. Deunionisation is shown to account for surprisingly little of the increase in earnings dispersion in the private sector for either males or females. Although union decline has been more muted in the public sector, union effects are actually stronger here. In the public sector, unions no longer reduce earnings variation as much they once did by virtue of their growing tendency to organise more skilled groups.

In what follows all processes referred to are weakly stationary. Let us call the real part of a complex ARMA(p,q) process a Re CARMA(p,q) process. Every real ARMA(p,q) process can trivially be written as a Re CARMA(p,q) process. Provided the moment properties of complex linear processes are appropriately specified, the following inverse result is available: every Re CARMA(p,q) process is spectrally equivalent to a real ARMA(2p,p + q) process or some simpler process. Thus the ARMA and Re CARMA classes are spectrally equivalent. The question of whether an ARMA or a Re CARMA parametrization is better in a given context then arises. If cyclicality is present, and especially if we wish to treat cycles, growth, and decay together, in a model whose parameters are easy to interpret, then a Re CARMA approach may be helpful.The author thanks Paolo Paruolo, A.M. Robert Taylor, and an anonymous referee for helpful suggestions.

Suppose that the Ordinary Least Squares regressors X follow a vector Ornstein–Uhlenbeck process, with growth matrix bA. The limiting sample variance matrix V is of interest. If A=kI, k⩾0, then ∂V/∂b⩾0. Remarkably, this inequality can fail for any other A.

In this paper we consider the effect of ambiguity on the private provision of public goods. Equilibrium is shown to exist and be unique. We examine how provision of the public good changes as the size of the population increases. We show that when there is uncertainty there may be less free-riding in large societies.

In this paper we consider the problem of testing the null hypothesis that a series has a constant level (possibly as part of a more general deterministic mean) against the alternative that the level follows a random walk. This problem has previously been studied by, inter alia, Nyblom and Makelainen (1983) in the context of the orthogonal Gaussian random walk plus noise model. This model postulates that the noise component and the innovations to the random walk are uncorrelated. We generalize their work by deriving the locally best invariant test of a fixed level against a random walk level in the non-orthogonal case. Here the noise and random walk components are contemporaneously correlated with correlation coefficient p. We demonstrate that the form of the optimal test in this setting is independent of p; i.e. the test statistic previously derived for the case of p = 0 remains the locally optimal test for all p. This is a very useful result: it states that the locally optimal test may be achieved without prior knowledge of p. Moreover, we show that the limiting distribution of the resulting statistic under both the null and local alternatives does not depend on p, behaving exactly as if p = 0. Finite sample simulations of these effects are provided to illustrate and generalizations to models with dependent errors are considered.

We obtain an inequality for the sample variance of a vector Brownian motion on [0,1] and an associated Ornstein–Uhlenbeck process. The result is applied to a regression involving near-integrated regressors, and establishes that in the limit the dispersion of the least squares estimator is greater in the near-integrated than in the integrated case. Our proof uses a quite general integral inequality, which appears to be new.

The number of Arrovian constitutions, when N agents are to rank n alternatives, is p(n) p(n) N , where p(n) is the number of weak orderings of n alternatives. For n≤15, p(n) is the nearest integer to n!/2(log2) n +1, the dominant term of a series derived by contour integration of the generating function. For large n, about n/17 additional terms in the series suffice to compute p(n) exactly.