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.     

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.     

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.     

An index formula for production economies with externalities

We prove that the degree of the equilibrium correspondence of an economy with increasing returns and external effects is equal to (−1)^L−1${(-1)}^{L-1}$ where L is the dimension of the space of goods. This allows us to infer existence, finiteness and uniqueness results.     

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.     

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.     

Indivisible commodities and the nonemptiness of the weak core

We consider a sufficient condition for the nonemptiness of the weak core in a finite exchange economy where every commodity is available only in integer quantities. We show that if the aggregate upper contour set is discretely convex, then the weak core is nonempty. In addition, we give two sufficient conditions for the aggregate upper contour set to be discretely convex. One is that every upper contour set of every agent is M^?${\text{M}}^{?}$-convex. The other is that the number of commodities is two and every agent’s preference relation is weakly monotone and discretely convex.     

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.     

Options and efficiency in spaces of bounded claims

In a seminal contribution, Ross (1976) showed that a static finite state-space market can be completed by supplementing the primitive securities with ordinary call and put options. Galvani (2009) extends this result to norm separable _Lp$L_p$-spaces, with 1≤p<∞$1\le p<\infty$. This study concludes that options maintain the same spanning power in the space of bounded payoffs topologized by the duality with the space of the state price densities. In particular, under mild assumptions on the probability space, options written on a claim that is a.s. equal to an injective function complete the market.     

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.     

Asymptotic distribution of factor augmented estimators for panel regression

In this paper we derive an asymptotic theory for linear panel regression augmented with estimated common factors. We give conditions under which the estimated factors can be used in place of the latent factors in the regression equation. For the principal components estimate of the factor space it is shown that these conditions are satisfied when T/N→0$T/N\to 0$ and N/T³→0$N/T^3\to 0$ under regularity. Monte Carlo studies verify the asymptotic theory.     

Regularization of nonparametric frontier estimators

In production theory and efficiency analysis, we estimate the production frontier, the locus of the maximal attainable level of an output (the production), given a set of inputs (the production factors). In other setups, we estimate rather an input (or cost) frontier, the minimal level of the input (cost) attainable for a given set of outputs (goods or services produced). In both cases the problem can be viewed as estimating a surface under shape constraints (monotonicity, …). In this paper we derive the theory of an estimator of the frontier having an asymptotic normal distribution. It is based on the order-m$m$ partial frontier where we let the order m$m$ to converge to infinity when n→∞$n\to \infty$ but at a slow rate. The final estimator is then corrected for its inherent bias. We thus can view our estimator as a regularized frontier. In addition, the estimator is more robust to extreme values and outliers than the usual nonparametric frontier estimators, like FDH and than the unregularized order-_mn$m_n$ estimator of Cazals et al. (2002) converging to the frontier with a Weibull distribution if _mn→∞$m_n\to \infty$ fast enough when n→∞$n\to \infty$. The performances of our estimators are evaluated in finite samples and compared to other estimators through some Monte-Carlo experiments, showing a better behavior (in terms of robustness, bias, MSE and achieved coverage of the resulting confidence intervals). The practical implementation and the robustness properties are illustrated through simulated data sets but also with a real data set.     

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.     

Cointegrating rank selection in models with time-varying variance

Reduced rank regression (RRR) models with time varying heterogeneity are considered. Standard information criteria for selecting cointegrating rank are shown to be weakly consistent in semiparametric RRR models in which the errors have general nonparametric short memory components and shifting volatility provided the penalty coefficient _Cn→∞$C_n\to \infty$ and _Cn/n→0$C_n/n\to 0$ as n→∞$n\to \infty$. The AIC criterion is inconsistent and its limit distribution is given. The results extend those in Cheng and Phillips (2009a) and are useful in empirical work where structural breaks or time evolution in the error variances is present. An empirical application to exchange rate data is provided.     

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.