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.     

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.     

Core equivalence with differentiated commodities

This paper presents improved core equivalence results for atomless economies with differentiated commodities in the framework of Ostroy and Zame (1994). Commodity bundles are elements of the space $M\left(K\right)$ of signed Borel measures on a compact space $K$ of commodity characteristics. Ostroy and Zame provide two sufficient conditions for core equivalence: It is sufficient that markets are “physically thick”, so that there are many suppliers of every commodity, or that markets are “economically thick”, so that consumers are sufficiently willing to substitute commodities with a similar composition for each other. The sufficient conditions in Ostroy and Zame (1994) all imply that there are “many more agents than commodities”, an idea of Aumann that was formalized and discussed in Tourky and Yannelis (2001) and Greinecker and Podczeck (2016). We generalize the framework in Ostroy and Zame (1994) and weaken their sufficient conditions to not imply the presence of “many more agents than commodities”. In particular, we drop the requirement that $K$ is metrizable from the basic model, the requirement of an uniform bound on endowments from the condition of “physically thick markets”, and the requirement that preferences are weak${}^1$-continuous from the condition of “economically thick markets”. Core equivalence still holds, showing that “many more agents than commodities” are not needed for core equivalence in models of commodity differentiation.     

A unified approach to validating univariate and multivariate conditional distribution models in time series

Modeling conditional distributions in time series has attracted increasing attention in economics and finance. We develop a new class of generalized Cramer–von Mises (GCM) specification tests for time series conditional distribution models using a novel approach, which embeds the empirical distribution function in a spectral framework. Our tests check a large number of lags and are therefore expected to be powerful against neglected dynamics at higher order lags, which is particularly useful for non-Markovian processes. Despite using a large number of lags, our tests do not suffer much from loss of a large number of degrees of freedom, because our approach naturally downweights higher order lags, which is consistent with the stylized fact that economic or financial markets are more affected by recent past events than by remote past events. Unlike the existing methods in the literature, the proposed GCM tests cover both univariate and multivariate conditional distribution models in a unified framework. They exploit the information in the joint conditional distribution of underlying economic processes. Moreover, a class of easy-to-interpret diagnostic procedures are supplemented to gauge possible sources of model misspecifications. Distinct from conventional CM and Kolmogorov–Smirnov (KS) tests, which are also based on the empirical distribution function, our GCM test statistics follow a convenient asymptotic $N(0,1)$ distribution and enjoy the appealing “nuisance parameter free” property that parameter estimation uncertainty has no impact on the asymptotic distribution of the test statistics. Simulation studies show that the tests provide reliable inference for sample sizes often encountered in economics and finance.     

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).