Strictly monotonic preferences on continuum of goods commodity spaces

We consider a set K of differentiated commodities. A preference relation on the set of consumption plans is strictly monotonic whenever to consume more of at least one commodity is more preferred. It is an easy task to find examples of strictly monotonic preference relations when K   is finite or countable. However, it is not easy for spaces like ?^∞([0,1])${?}^{\infty }([0\text{,}1])$, the space of bounded functions on the unit interval.     

Core and competitive equilibria of a coalitional exchange economy with infinite time horizon

The core and competitive equilibria of a large exchange economy on the commodity space ?^∞${?}^{\infty }$ will be discussed. We define the economy as a measure on the space of consumers’ characteristics following Hart and Kohlberg (1974), and prove the existence of competitive equilibria and their equivalence with the core without assuming the convexity of preferences.     

On the properties of the coefficient of determination in regression models with infinite variance variables

We examine the asymptotic properties of the coefficient of determination, R²$R^2$, in models with α-stable$\alpha \text{-stable}$   random variables. If the regressor and error term share the same index of stability α<2$\alpha <2$, we show that the R²$R^2$  statistic does not converge to a constant but has a nondegenerate distribution on the entire [0,1]$[0,1]$ interval. We provide closed-form expressions for the cumulative distribution function and probability density function of this limit random variable, and we show that the density function is unbounded at 0 and 1. If the indices of stability of the regressor and error term are unequal, we show that the coefficient of determination converges in probability to either 0 or 1, depending on which variable has the smaller index of stability, irrespective of the value of the slope coefficient. In an empirical application, we revisit the Fama and MacBeth (1973) two-stage regression and demonstrate that in the infinite-variance case the R²$R^2$  statistic of the second-stage regression converges to 0 in probability even if the slope coefficient is nonzero. We deduce that a small value of the R²$R^2$  statistic should not, in itself, be used to reject the usefulness of a regression model.     

Instrumental variable quantile regression: A robust inference approach

In this paper, we develop robust inference procedures for an instrumental variables model defined by Y=D^′α(U)$Y=D^{\prime }\alpha (U)$ where D^′α(U)$D^{\prime }\alpha (U)$ is strictly increasing in U and U is a uniform variable that may depend on D but is independent of a set of instrumental variables Z. The proposed inferential procedures are computationally convenient in typical applications and can be carried out using software available for ordinary quantile regression. Our inferential procedure arises naturally from an estimation algorithm and has the important feature of being robust to weak and partial identification and remains valid even in cases where identification fails completely. The use of the proposed procedures is illustrated through two empirical examples.     

Efficiency bounds for estimating linear functionals of nonparametric regression models with endogenous regressors

Let Y=μ^∗(X)+ε$Y={\mu }^{\ast }\left(X\right)+\epsilon$, where μ^∗${\mu }^{\ast }$ is unknown and E[ε|X]≠0$\mathbb{E}\left[\epsilon \right|X]\ne 0$ with positive probability but there exist instrumental variables W$W$ such that E[ε|W]=0$\mathbb{E}\left[\epsilon \right|W]=0$ w.p.1. It is well known that such nonparametric regression models are generally “ill-posed” in the sense that the map from the data to μ^∗${\mu }^{\ast }$ is not continuous. In this paper, we derive the efficiency bounds for estimating certain linear functionals of μ^∗${\mu }^{\ast }$ without assuming μ^∗${\mu }^{\ast }$ itself to be identified.     

Semiparametric binary regression models under shape constraints with an application to Indian schooling data

We consider estimation of the regression function in a semiparametric binary regression model defined through an appropriate link function (with emphasis on the logistic link) using likelihood-ratio based inversion. The dichotomous response variable Δ$\Delta$ is influenced by a set of covariates that can be partitioned as (X,Z)$(X,Z)$ where Z$Z$ (real valued) is the covariate of primary interest and X$X$ (vector valued) denotes a set of control variables. For any fixed X$X$, the conditional probability of the event of interest (Δ=1$\Delta =1$) is assumed to be a non-decreasing function of Z$Z$. The effect of the control variables is captured by a regression parameter β$\beta$. We show that the baseline conditional probability function (corresponding to X=0$X=0$) can be estimated by isotonic regression procedures and develop a likelihood ratio based method for constructing asymptotic confidence intervals for the conditional probability function (the regression function) that avoids the need to estimate nuisance parameters. Interestingly enough, the calibration of the likelihood ratio based confidence sets for the regression function no longer involves the usual χ²${\chi }^2$ quantiles, but those of the distribution of a new random variable that can be characterized as a functional of convex minorants of Brownian motion with quadratic drift. Confidence sets for the regression parameter β$\beta$ can however be constructed using asymptotically χ²${\chi }^2$ likelihood ratio statistics. The finite sample performance of the methods are assessed via a simulation study. The techniques of the paper are applied to data sets on primary school attendance among children belonging to different socio-economic groups in rural India.     

Estimating first-price auctions with an unknown number of bidders: A misclassification approach

In this paper, we consider nonparametric identification and estimation of first-price auction models when N^∗$N^{\ast }$, the number of potential bidders, is unknown to the researcher, but observed by bidders. Exploiting results from the recent econometric literature on models with misclassification error, we develop a nonparametric procedure for recovering the distribution of bids conditional on the unknown N^∗$N^{\ast }$. Monte Carlo results illustrate that the procedure works well in practice. We present illustrative evidence from a dataset of procurement auctions, which shows that accounting for the unobservability of N^∗$N^{\ast }$ can lead to economically meaningful differences in the estimates of bidders’ profit margins.     

Euclidean preferences

This note is devoted to the question: how restrictive is the assumption that preferences be Euclidean in d dimensions. In particular it is proven that any preference profile with I individuals and A alternatives can be represented by Euclidean utilities with d   dimensions if and only if d≥min?(I,A−1)$d\ge \min ?(I\text{,}A-1)$. The paper also describes the systems of A points which allow for the representation of any profile over A alternatives, and provides similar results when only strict preferences are considered. These findings contrast with the observation that if preferences are only required to be convex then two dimensions are always sufficient.     

A bootstrap theory for weakly integrated processes

This paper develops a bootstrap theory for models including autoregressive time series with roots approaching to unity as the sample size increases. In particular, we consider the processes with roots converging to unity with rates slower than n^-1$n^{-1}$. We call such processes weakly   integrated processes. It is established that the bootstrap relying on the estimated autoregressive model is generally consistent for the weakly integrated processes. Both the sample and bootstrap statistics of the weakly integrated processes are shown to yield the same normal asymptotics. Moreover, for the asymptotically pivotal statistics of the weakly integrated processes, the bootstrap is expected to provide an asymptotic refinement and give better approximations for the finite sample distributions than the first order asymptotic theory. For the weakly integrated processes, the magnitudes of potential refinements by the bootstrap are shown to be proportional to the rate at which the root of the underlying process converges to unity. The order of boostrap refinement can be as large as o(n^-1/2+?)$o(n^{-1/2+?})$ for any ?>0$?>0$. Our theory helps to explain the actual improvements observed by many practitioners, which are made by the use of the bootstrap in analyzing the models with roots close to unity.     

Treatment effects in sample selection models and their nonparametric estimation

In a sample-selection model with the ‘selection’ variable Q$Q$ and the ‘outcome’ variable Y^∗$Y^{\ast }$, Y^∗$Y^{\ast }$ is observed only when Q=1$Q=1$. For a treatment D$D$ affecting both Q$Q$ and Y^∗$Y^{\ast }$, three effects are of interest: ‘participation  ’ (i.e., the selection) effect of D$D$ on Q$Q$, ‘visible performance  ’ (i.e., the observed outcome) effect of D$D$ on Y≡QY^∗$Y\equiv Q{Y}^{\ast }$, and ‘invisible performance  ’ (i.e., the latent outcome) effect of D$D$ on Y^∗$Y^{\ast }$. This paper shows the conditions under which the three effects are identified, respectively, by the three corresponding mean differences of Q$Q$, Y$Y$, and Y|Q=1$Y|Q=1$ (i.e., Y^∗|Q=1$Y^{\ast }|Q=1$) across the control (D=0$D=0$) and treatment (D=1$D=1$) groups. Our nonparametric estimators for those effects adopt a two-sample framework and have several advantages over the usual matching methods. First, there is no need to select the number of matched observations. Second, the asymptotic distribution is easily obtained. Third, over-sampling the control/treatment group is allowed. Fourth, there is a built-in mechanism that takes into account the ‘non-overlapping support problem’, which the usual matching deals with by choosing a ‘caliper’. Fifth, a sensitivity analysis to gauge the presence of unobserved confounders is available. A simulation study is conducted to compare the proposed methods with matching methods, and a real data illustration is provided.     

Robust adaptive rate-optimal testing for the white noise hypothesis

A new test is proposed for the weak white noise null hypothesis. The test is based on a new automatic selection of the order for a Box–Pierce (1970) test statistic or the test statistic of Hong (1996). The heteroskedasticity and autocorrelation-consistent (HAC) critical values from Lee (2007) are used, allowing for estimation of the error term. The data-driven order selection is tailored to detect a new class of alternatives with autocorrelation coefficients which can be o(n^−1/2)$o\left(n^{-1/2}\right)$ provided there are sufficiently many of such coefficients. A simulation experiment illustrates the good statistical properties of the test both under the weak white noise null and the alternative.     

Quasi ML estimation of the panel AR(1) model with arbitrary initial conditions

In this paper we show that the Quasi ML estimation method yields consistent Random and Fixed Effects estimators for the autoregression parameter ρ$\rho$ in the panel AR(1) model with arbitrary initial conditions and possibly time-series heteroskedasticity even when the error components are drawn from heterogeneous distributions. We investigate both analytically and by means of Monte Carlo simulations the properties of the QML estimators for ρ$\rho$. The RE(Q)MLE for ρ$\rho$ is asymptotically at least as robust to individual heterogeneity and, when the data are i.i.d. and normal, at least as efficient as the FE(Q)MLE for ρ$\rho$. Furthermore, the QML estimators for ρ$\rho$ only suffer from a ‘weak moment conditions’ problem when ρ$\rho$ is close to one if the cross-sectional average of the variances of the errors is (almost) constant over time, e.g. under time-series homoskedasticity. However, in this case the QML estimators for ρ$\rho$ are still consistent when ρ$\rho$ is local to or equal to one although they converge to a non-normal possibly asymmetric distribution at a rate that is lower than N^1/2$N^{1/2}$ but at least N^1/4$N^{1/4}$. Finally, we study the finite sample properties of two types of estimators for the standard errors of the QML estimators for ρ$\rho$, and the bounds of QML based confidence intervals for ρ$\rho$.     

A likelihood ratio test for stationarity of rating transitions

We study the time-stationarity of rating transitions, modelled by a time-continuous discrete-state Markov process and derive a likelihood ratio test. For multiple Markov processes from a multiplicative intensity model, maximum likelihood parameter estimates can be written as martingale transform of the processes, counting transitions between the rating states, so that the profile partial likelihood ratio is asymptotically χ²${\chi }^2$-distributed. An application to an internal rating data set reveals highly significant instationarity.     

Predictable returns and asset allocation: Should a skeptical investor time the market?

We investigate optimal portfolio choice for an investor who is skeptical about the degree to which excess returns are predictable. Skepticism is modeled as an informative prior over the R²$R^2$ of the predictive regression. We find that the evidence is sufficient to convince even an investor with a highly skeptical prior to vary his portfolio on the basis of the dividend-price ratio and the yield spread. The resulting weights are less volatile and deliver superior out-of-sample performance as compared to the weights implied by an entirely model-based or data-based view.     

Nonstationary discrete choice: A corrigendum and addendum

We correct the limit theory presented in an earlier paper by Hu and Phillips [2004a. Nonstationary discrete choice. Journal of Econometrics 120, 103–138] for nonstationary time series discrete choice models with multiple choices and thresholds. The new limit theory shows that, in contrast to the binary choice model with nonstationary regressors and a zero threshold where there are dual rates of convergence (n^1/4$n^{1/4}$ and n^3/4$n^{3/4}$), all parameters including the thresholds converge at the rate n^3/4$n^{3/4}$. The presence of nonzero thresholds therefore materially affects rates of convergence. Dual rates of convergence reappear when stationary variables are present in the system. Some simulation evidence is provided, showing how the magnitude of the thresholds affects finite sample performance. A new finding is that predicted probabilities and marginal effect estimates have finite sample distributions that manifest a pile-up, or increasing density, towards the limits of the domain of definition.     

Essential supremum and essential maximum with respect to random preference relations

In the first part of the paper, we study concepts of supremum and maximum as subsets of a topological space X$X$ endowed by preference relations. Several rather general existence theorems are obtained for the case where the preferences are defined by countable semicontinuous multi-utility representations. In the second part of the paper, we consider partial orders and preference relations “lifted” from a metric separable space X$X$ endowed by a random preference relation to the space L⁰(X)$L^0\left(X\right)$ of X$X$-valued random variables. We provide an example of application of the notion of essential maximum to the problem of the minimal portfolio super-replicating an American-type contingent claim under transaction costs.