Asymptotics for LS,GLS, and feasible GLS statistics in an AR(1) model with conditional heteroskedasticity

We consider a first-order autoregressive model with conditionally heteroskedastic innovations. The asymptotic distributions of least squares (LS), infeasible generalized least squares (GLS), and feasible GLS estimators and $t$ statistics are determined. The GLS procedures allow for misspecification of the form of the conditional heteroskedasticity and, hence, are referred to as quasi-GLS procedures. The asymptotic results are established for drifting sequences of the autoregressive parameter ${\rho }_n$ and the distribution of the time series of innovations. In particular, we consider the full range of cases in which ${\rho }_n$ satisfies $n(1?{\rho }_n)\to \infty$ and $n(1?{\rho }_n)\to h_1\in [0,\infty )$ as $n\to \infty$, where $n$ is the sample size. Results of this type are needed to establish the uniform asymptotic properties of the LS and quasi-GLS statistics.     

Characterizing lexicographic preferences

We characterize lexicographic preferences on product sets of finitely many coordinates. The main new axiom is a robustness property. It roughly requires this: Suppose $x$ is preferred to $y$; many of its coordinates indicate that the former is better and only a few indicate the opposite. Then the decision maker is allowed a change of mind turning one coordinate in favor of $x$ to an indifference: even if one less argument supports the preference, the fact that we started with many arguments in favor of $x$ suggests that such a small change is not enough to give rise to the opposite preference.     

When is the probabilistic serial assignment uniquely efficient and envy-free?

We study the problem of allocating objects using lotteries. For each economy, the serial assignment, the assignment selected by the (probabilistic) serial rule, is $sd$-efficient and $sd$-envy-free (“$sd$” stands for stochastic dominance) but in general, it is not the only such assignment. Our question is when the uniqueness also holds. First, we provide a necessary condition for uniqueness, termed top-objects divisibility. Exploiting the structure revealed by top-objects divisibility, we then provide two sufficient conditions: preference richness and recursive decomposability. Existing sufficient conditions are restrictive in that they are satisfied only if there are sufficiently many agents relative to the number of objects; and that they only focus on preferences, ignoring other aspects of the problem that are also relevant to uniqueness. Our conditions overcome these limitations and can explain uniqueness for a wide range of economies.     

Statistical tests for multiple forecast comparison

We consider a multivariate version of the Diebold–Mariano test for equal predictive ability of three or more forecasting models. The Wald-type test, $S$, which has a null distribution that is asymptotically chi-squared, is shown to be generally invariant with respect to the ordering of the models being compared. Finite-sample corrections for the test are also developed. Monte Carlo simulations indicate that $S$ has reasonable size properties in large samples but tends to be oversized in moderate samples. The finite-sample correction succeeds in correcting for size, but only partially. For the size-adjusted tests, power increases with sample size, as expected. It is speculated that further finite-sample improvements can be achieved using Hotelling’s $T^2$ or bootstrap critical values.