Non-standard Limit Distributions Research Articles

Testing for cointegration in nonlinear asymmetric smooth transition error correction models

The present paper proposes an alternative procedure to test for cointegration in asymmetric exponential smooth transition autoregressive (AESTAR) model. The proposed new F-statistics test the null hypothesis of no cointegration against the alternative hypothesis of cointegration with asymmetric ESTAR adjustment. We derive the nonstandard limit distributions analytically via Monte Carlo simulations. We also study the finite sample properties and the power of proposed tests with Monte Carlo simulations and find that the empirical sizes of two tests are quite close to the nominal one and the proposed tests do not lead to over-rejections of the null hypothesis of no cointegration.

A residual-based multivariate constant correlation test

We propose a new multivariate constant correlation test based on residuals. This test takes into account the whole correlation matrix instead of the considering merely marginal correlations between bivariate data series. In financial markets, it is unrealistic to assume that the marginal variances are constant. This motivates us to develop a constant correlation test which allows for non-constant marginal variances in multivariate time series. However, when the assumption of constant marginal variances is relaxed, it can be shown that the residual effect leads to nonstandard limit distributions of the test statistics based on residual terms. The critical values of the test statistics are not directly available and we use a bootstrap approximation to obtain the corresponding critical values for the test. We also derive the limit distribution of the test statistics based on residuals under the null hypothesis. Monte Carlo simulations show that the test has appealing size and power properties in finite samples. We also apply our test to the stock returns in Euro Stoxx 50 and integrate the test into a binary segmentation algorithm to detect multiple break points.

General Limit Distributions for Sums of Random Variables with a Matrix Product Representation

The general limit distributions of the sum of random variables described by a finite matrix product ansatz are characterized. Using a mapping to a Hidden Markov Chain formalism, non-standard limit distributions are obtained, and related to a form of ergodicity breaking in the underlying non-homogeneous Hidden Markov Chain. The link between ergodicity and limit distributions is detailed and used to provide a full algorithmic characterization of the general limit distributions.

Statistics of sums of correlated variables described by a matrix product ansatz

We determine the asymptotic distribution of the sum of correlated variables described by a matrix product ansatz with finite matrices, considering variables with finite variances. In cases in which the correlation length is finite, the law of large numbers is obeyed, and the rescaled sum converges to a Gaussian distribution. In contrast, when the correlation extends over the system size, we observe either a breaking of the law of large numbers, with the onset of giant fluctuations, or a generalization of the central limit theorem with a family of non-standard limit distributions. The corresponding distributions are found as mixtures of delta functions for the generalized law of large numbers, and as mixtures of Gaussian distributions for the generalized central limit theorem. Connections with statistical physics models are emphasized.

Renormalization flow for extreme value statistics of random variables raised to a varying power

Using a renormalization approach, we study the asymptotic limit distribution of the maximum value in a set of independent and identically distributed random variables raised to a power qn that varies monotonically with the sample size n. Under these conditions, a non-standard class of max-stable limit distributions, which mirror the classical ones, emerges. Furthermore, a transition mechanism between the classical and the non-standard limit distributions is brought to light. If qn grows slower than a characteristic function q*n, the standard limit distributions are recovered, while if qn behaves asymptotically as λq*n, non-standard limit distributions emerge.

UNIFORM ASYMPTOTIC NORMALITY IN STATIONARY AND UNIT ROOT AUTOREGRESSION

While differencing transformations can eliminate nonstationarity, they typically reduce signal strength and correspondingly reduce rates of convergence in unit root autoregressions. The present paper shows that aggregating moment conditions that are formulated in differences provides an orderly mechanism for preserving information and signal strength in autoregressions with some very desirable properties. In first order autoregression, a partially aggregated estimator based on moment conditions in differences is shown to have a limiting normal distribution that holds uniformly in the autoregressive coefficient ρ, including stationary and unit root cases. The rate of convergence is $\root \of n $ when $\left| \rho \right| < 1$ and the limit distribution is the same as the Gaussian maximum likelihood estimator (MLE), but when ρ = 1 the rate of convergence to the normal distribution is within a slowly varying factor of n. A fully aggregated estimator (FAE) is shown to have the same limit behavior in the stationary case and to have nonstandard limit distributions in unit root and near integrated cases, which reduce both the bias and the variance of the MLE. This result shows that it is possible to improve on the asymptotic behavior of the MLE without using an artificial shrinkage technique or otherwise accelerating convergence at unity at the cost of performance in the neighborhood of unity. Confidence intervals constructed from the FAE using local asymptotic theory around unity also lead to improvements over the MLE.

Tests of integration in circular autoregressive models

Two likelihood-ratio tests of the hypothesis of integration of order 1 in Gaussian circular vector autoregressive models (CAR) are derived. The tests have non-standard limit distributions which can be expressed as functionals of Brownian motion. Inference on the unit roots can be decomposed and addressed in independent subsystems with 1 or 2 components.

Large sample estimation in nonstationary autoregressive processes with multiple observations

The asymptotic distributions of the least-squares estimators of the parameters in autoregressive processes with multiple observations are derived for the two nonstationary cases, viz., (a) the explosive case and (b) the unstable case. It is shown that nonstandard limit distributions are obtained.