The Jevons double coincidence condition and local uniqueness of money: An example

Jevons’s double coincidence of wants condition is derived as the result of household level transaction costs in general equilibrium where N   commodities are traded at (1/2)N(N−1)$(1/2)N(N-1)$ commodity-pairwise trading posts. Each household experiences a set-up cost on entering an additional trading post. Budget constraints are enforced at each trading post separately implying demand for a carrier of value between trading posts, commodity money. General equilibrium consists of prices so that each trading post clears. Existence and local uniqueness of commodity money in equilibrium can follow from the scale economy implied by the household set-up cost.     

A complete asymptotic series for the autocovariance function of a long memory process

An infinite-order asymptotic expansion is given for the autocovariance function of a general stationary long-memory process with memory parameter d∈(−1/2,1/2)$d\in (-1/2,1/2)$. The class of spectral densities considered includes as a special case the stationary and invertible ARFIMA(p,d,q$p,d,q$) model. The leading term of the expansion is of the order O(1/k^1−2d)$O(1/k^{1-2d})$, where k$k$ is the autocovariance order, consistent with the well known power law decay for such processes, and is shown to be accurate to an error of O(1/k^3−2d)$O(1/k^{3-2d})$. The derivation uses Erdélyi’s [Erdélyi, A., 1956. Asymptotic Expansions. Dover Publications, Inc, New York] expansion for Fourier-type integrals when there are critical points at the boundaries of the range of integration - here the frequencies {0,2π}$\{0,2\pi \}$. Numerical evaluations show that the expansion is accurate even for small k$k$ in cases where the autocovariance sequence decays monotonically, and in other cases for moderate to large k$k$. The approximations are easy to compute across a variety of parameter values and models.     

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.     

Edgeworth expansions and normalizing transforms for inequality measures

Finite sample distributions of studentized inequality measures differ 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 first and third cumulant of the studentized inequality measure. We state distribution-free expressions for the bias and skewness coefficients. In the second part we improve over first-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)$O\left(n^{-1}\right)$. This leads to improved confidence intervals and applying a subsequent bootstrap leads to a further improvement to order O(n^−3/2)$O\left(n^{-3/2}\right)$. We illustrate our procedure with an application to regional inequality measurement in Côte d’Ivoire.     

Commodity money equilibrium in a convex trading post economy with transaction costs

Existence and efficiency of general equilibrium with commodity money is investigated in an economy where N   commodities are traded at N(N−1)/2$N(N-1)/2$ commodity-pairwise trading posts. Trade is a resource-using activity recovering transaction costs through the spread between bid (wholesale) and ask (retail) prices. Budget constraints, enforced at each trading post separately, imply demand for a carrier of value between trading posts. Existence of general equilibrium is established under conventional convexity and continuity conditions while structuring the price space to account for distinct bid and ask price ratios. Commodity money flows are identified as the difference between gross and net inter-post trades.     

Estimating quadratic variation consistently in the presence of endogenous and diurnal measurement error

We propose an econometric model that captures the effects of market microstructure on a latent price process. In particular, we allow for correlation between the measurement error and the return process and we allow the measurement error process to have a diurnal heteroskedasticity. We propose a modification of the TSRV estimator of quadratic variation. We show that this estimator is consistent, with a rate of convergence that depends on the size of the measurement error, but is no worse than n^−1/6$n^{-1/6}$. We investigate in simulation experiments the finite sample performance of various proposed implementations.     

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.     

Estimating deterministic trends with an integrated or stationary noise component

We propose a test for the slope of a trend function when it is a priori unknown whether the series is trend-stationary or contains an autoregressive unit root. The procedure is based on a Feasible Quasi Generalized Least Squares method from an AR(1) specification with parameter α$\alpha$, the sum of the autoregressive coefficients. The estimate of α$\alpha$ is the OLS estimate obtained from an autoregression applied to detrended data and is truncated to take a value 1 whenever the estimate is in a T^−δ$T^{-\delta }$ neighborhood of 1. This makes the estimate “super-efficient” when α=1$\alpha =1$ and implies that inference on the slope parameter can be performed using the standard Normal distribution whether α=1$\alpha =1$ or |α|<1$\left|\alpha \right|<1$. Theoretical arguments and simulation evidence show that δ=1/2$\delta =1/2$ is the appropriate choice. Simulations show that our procedure has better size and power properties than the tests proposed by [Bunzel, H., Vogelsang, T.J., 2005. Powerful trend function tests that are robust to strong serial correlation with an application to the Prebish–Singer hypothesis. Journal of Business and Economic Statistics 23, 381–394] and [Harvey, D.I., Leybourne, S.J., Taylor, A.M.R., 2007. A simple, robust and powerful test of the trend hypothesis. Journal of Econometrics 141, 1302–1330].     

Distribution-free tests of stochastic monotonicity

This article proposes a nonparametric test of monotonicity for conditional distributions and its moments. Unlike previous proposals, our method does not require smooth estimation of the derivatives of nonparametric curves. Distinguishing features of our approach are that critical values are pivotal under the null in finite samples and that the test is invariant to any monotonic continuous transformation of the explanatory variable. The test statistic is the sup-norm of the difference between the empirical copula function and its least concave majorant with respect to the explanatory variable coordinate. The resulting test is able to detect local alternatives converging to the null at the parametric rate n^−1/2$n^{-1/2}$, with n$n$ the sample size. The finite sample performance of the test is examined by means of a Monte Carlo experiment and an application to testing intergenerational income mobility.     

Beyond panel unit root tests: Using multiple testing to determine the nonstationarity properties of individual series in a panel

Most panel unit root tests are designed to test the joint null hypothesis of a unit root for each individual series in a panel. After a rejection, it will often be of interest to identify which series can be deemed to be stationary and which series can be deemed nonstationary. Researchers will sometimes carry out this classification on the basis of n$n$ individual (univariate) unit root tests based on some ad hoc significance level. In this paper, we suggest and demonstrate how to use the false discovery rate (FDR)$\left(FDR\right)$ in evaluating I(1)/I(0)$I\left(1\right)/I\left(0\right)$ classifications.     

Multi-agent bilateral bargaining and the Nash bargaining solution

This paper studies a bargaining model where n   players negotiate how to share a pie through (n−1)$(n-1)$ bilateral bargaining sessions. In each session, two players bargain for a partial agreement that specifies who exits and who moves on to the next session (if there is any) via the alternating-proposal framework of Rubinstein [Rubinstein, A., 1982. Perfect equilibrium in a bargaining model. Econometrica 50, 97–109]. We consider two bargaining procedures under which the subgame perfect equilibrium outcomes converge to the Nash [Nash, J., 1950. The bargaining problem. Econometrica 18, 155–162] bargaining solution for the corresponding bargaining problem as the players’ discount factor goes to one. Hence, the model studied here provides a non-cooperative foundation for the Nash cooperative bargaining solution in the multilateral case.     

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.     

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.     

Exact local Whittle estimation of fractionally cointegrated systems

Semiparametric estimation of a bivariate fractionally cointegrated system is considered. We propose a two-step procedure that accommodates both (asymptotically) stationary (δ<1/2)$(\delta <1/2)$ and nonstationary (δ≥1/2)$(\delta \ge 1/2)$ stochastic trend and/or equilibrium error. A tapered version of the local Whittle estimator of Robinson (2008) is used as the first-stage estimator, and the second-stage estimator employs the exact local Whittle approach of Shimotsu and Phillips (2005). The consistency and asymptotic distribution of the two-step estimator are derived. The estimator of the memory parameters has the same Gaussian asymptotic distribution in both the stationary and the nonstationary case. The convergence rate and the asymptotic distribution of the estimator of the cointegrating vector are affected by the difference between the memory parameters. Further, the estimator has a Gaussian asymptotic distribution when the difference between the memory parameters is less than 1/2.     

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.     

Adaptive estimation of the dynamics of a discrete time stochastic volatility model

This paper is concerned with the discrete time stochastic volatility model _Yi=exp(_Xi/2)_ηi$Y_i=exp(X_i/2){\eta }_i$, _Xi+1=b(_Xi)+σ(_Xi)_ξi+1$X_{i+1}=b\left(X_i\right)+\sigma \left(X_i\right){\xi }_{i+1}$, where only (_Yi)$\left(Y_i\right)$ is observed. The model is rewritten as a particular hidden model: _Zi=_Xi+_εi$Z_i=X_i+{\epsilon }_i$, _Xi+1=b(_Xi)+σ(_Xi)_ξi+1$X_{i+1}=b\left(X_i\right)+\sigma \left(X_i\right){\xi }_{i+1}$, where (_ξi)$\left({\xi }_i\right)$ and (_εi)$\left({\epsilon }_i\right)$ are independent sequences of i.i.d. noise. Moreover, the sequences (_Xi)$\left(X_i\right)$ and (_εi)$\left({\epsilon }_i\right)$ are independent and the distribution of ε$\epsilon$ is known. Then, our aim is to estimate the functions b$b$ and σ²${\sigma }^2$ when only observations _Z1,…,_Zn$Z_1,\dots ,Z_n$ are available. We propose to estimate bf$bf$ and (b²+σ²)f$(b^2+{\sigma }^2)f$ and study the integrated mean square error of projection estimators of these functions on automatically selected projection spaces. By ratio strategy, estimators of b$b$ and σ²${\sigma }^2$ are then deduced. The mean square risk of the resulting estimators are studied and their rates are discussed. Lastly, simulation experiments are provided: constants in the penalty functions defining the estimators are calibrated and the quality of the estimators is checked on several examples.