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.     

Nonparametric fixed effects model for panel data with locally stationary regressors

We develop methods for inference in nonparametric time-varying fixed effects panel data models that allow for locally stationary regressors and for the time series length $T$ and cross-section size $N$ both being large. We first develop a pooled nonparametric profile least squares dummy variable approach to estimate the nonparametric function, and establish the optimal convergence rate and asymptotic normality of the resultant estimator. We then propose a test statistic to check whether the bivariate nonparametric function is time-varying or the time effect is separable, and derive the asymptotic distribution of the proposed test statistic. We present several simulated examples and two real data analyses to illustrate the finite sample performance of the proposed methods.     

Sparse linear models and l1-regularized 2SLS with high-dimensional endogenous regressors and instruments

We explore the validity of the 2-stage least squares estimator with $l_1$-regularization in both stages, for linear triangular models where the numbers of endogenous regressors in the main equation and instruments in the first-stage equations can exceed the sample size, and the regression coefficients are sufficiently sparse. For this $l_1$-regularized 2-stage least squares estimator, we first establish finite-sample performance bounds and then provide a simple practical method (with asymptotic guarantees) for choosing the regularization parameter. We also sketch an inference strategy built upon this practical method.     

The ZD-GARCH model: A new way to study heteroscedasticity

This paper proposes a first-order zero-drift GARCH (ZD-GARCH(1, 1)) model to study conditional heteroscedasticity and heteroscedasticity together. Unlike the classical GARCH model, the ZD-GARCH(1, 1) model is always non-stationary regardless of the sign of the Lyapunov exponent ${\gamma }_0$, but interestingly it is stable with its sample path oscillating randomly between zero and infinity over time when ${\gamma }_0=0$. Furthermore, this paper studies the generalized quasi-maximum likelihood estimator (GQMLE) of the ZD-GARCH(1, 1) model, and establishes its strong consistency and asymptotic normality. Based on the GQMLE, an estimator for ${\gamma }_0$, a $t$-test for stability, a unit root test for the absence of the drift term, and a portmanteau test for model checking are all constructed. Simulation studies are carried out to assess the finite sample performance of the proposed estimators and tests. Applications demonstrate that a stable ZD-GARCH(1, 1) model is more appropriate than a non-stationary GARCH(1, 1) model in fitting the KV-A stock returns in Francq and Zakoïan (2012).     

Random mechanisms for house allocation with existing tenants

We study the house allocation problem with existing tenants: $n$ houses (stand for “indivisible objects”) are to be allocated to $n$ agents; each agent needs exactly one house and has strict preferences; $k$ houses are initially unowned; $k$ agents initially do not own houses; the remaining $n-k$ agents (the so-called “existing tenants”) initially own the remaining $n-k$ houses (each owns one). In this setting, we consider various randomized allocation rules under which voluntary participation of existing tenants is assured and the randomization procedure either treats agents equally or discriminates against some (or all) of the existing tenants. We obtain two equivalence results, which generalize the equivalence results in Abdulkadiroğlu and Sönmez (1999) and Sönmez and Ünver (2005).     

Robust linear static panel data models using ε-contamination

The paper develops a general Bayesian framework for robust linear static panel data models using$\epsilon$-contamination. A two-step approach is employed to derive the conditional type-II maximum likelihood (ML-II) posterior distribution of the coefficients and individual effects. The ML-II posterior means are weighted averages of the Bayes estimator under a base prior and the data-dependent empirical Bayes estimator. Two-stage and three stage hierarchy estimators are developed and their finite sample performance is investigated through a series of Monte Carlo experiments. These include standard random effects as well as Mundlak-type, Chamberlain-type and Hausman–Taylor-type models. The simulation results underscore the relatively good performance of the three-stage hierarchy estimator. Within a single theoretical framework, our Bayesian approach encompasses a variety of specifications while conventional methods require separate estimators for each case.     

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.