Quality of local equilibria in discrete exchange economies

This paper defines the notion of a local equilibrium of quality $(r,s)$, $0\le r,s$, in a discrete exchange economy: a partial allocation and item prices that guarantee certain stability properties parametrized by the numbers $r$ and $s$. The quality $(r,s)$ measures the fit between the allocation and the prices: the larger $r$ and $s$ the closer the fit. For $r,s\le 1$ this notion provides a graceful degradation for the conditional equilibria of Fu, Kleinberg and Lavi (2012) which are exactly the local equilibria of quality $(1,1)$. For $1<r,s$ the local equilibria of quality $(r,s)$ are more stable than conditional equilibria. Any local equilibrium of quality $(r,s)$ provides, without any assumption on the type of the agents’ valuations, an allocation whose value is at least $\frac{rs}{1+rs}$ the optimal fractional allocation. In any economy in which all agents’ valuations are $a$-submodular, i.e., exhibit complementarity bounded by $a\ge 1$, there is a local equilibrium of quality $(\frac{1}{a},\frac{1}{a})$. In such an economy any greedy allocation provides a local equilibrium of quality $(1,\frac{1}{a})$. Walrasian equilibria are not amenable to such graceful degradation.     

Pythagorean generalization of testing the equality of two symmetric positive definite matrices

We provide a new test for equality of two symmetric positive-definite matrices that leads to a convenient mechanism for testing specification using the information matrix equality or the sandwich asymptotic covariance matrix of the GMM estimator. The test relies on a new characterization of equality between two $k$ dimensional symmetric positive-definite matrices $A$ and $B$: the traces of $A{B}^{?1}$ and $B{A}^{?1}$ are equal to $k$ if and only if $A=B$. Using this simple criterion, we introduce a class of omnibus test statistics for equality and examine their null and local alternative approximations under some mild regularity conditions. A preferred test in the class with good omni-directional power is recommended for practical work. Monte Carlo experiments are conducted to explore performance characteristics under the null and local as well as fixed alternatives. The test is applicable in many settings, including GMM estimation, SVAR models and high dimensional variance matrix settings.     

The cointegrated vector autoregressive model with general deterministic terms

In the cointegrated vector autoregression (CVAR) literature, deterministic terms have until now been analyzed on a case-by-case, or as-needed basis. We give a comprehensive unified treatment of deterministic terms in the additive model $X_t=\gamma Z_t+Y_t$, where $Z_t$ belongs to a large class of deterministic regressors and $Y_t$ is a zero-mean CVAR. We suggest an extended model that can be estimated by reduced rank regression, and give a condition for when the additive and extended models are asymptotically equivalent, as well as an algorithm for deriving the additive model parameters from the extended model parameters. We derive asymptotic properties of the maximum likelihood estimators and discuss tests for rank and tests on the deterministic terms. In particular, we give conditions under which the estimators are asymptotically (mixed) Gaussian, such that associated tests are ${\chi }^2$-distributed.