Volume 8, Issue 1 -

Right-tailed Dickey–Fuller-type unit root tests against the explosive alternative have become popular in economics and finance for detecting asset price bubbles. This paper studies the size properties of fixed sample and recursive right-tailed Dickey–Fuller tests if the relevant series contains a unit root, but a structural break in the drift parameter occurs. It is shown that positive size distortion and therefore spurious rejections of the unit root null hypothesis in favour of the explosive alternative can be a problem for both types of test. Some possible solutions to this problem are briefly discussed.

In this note we extend the standard results for the limit theory of the popular quasi-maximum likelihood estimator (QMLE) in the context of the non-stationary autoregressive conditional heteroskedastic ARCH(1) model by allowing the innovation process not to possess fourth moments. Depending on the value of the index of stability, we either derive α$\alpha $-stable weak limits with non-standard rates or inconsistency and non-tightness. We obtain the limit theory by the derivation of a limit theorem for multiplicative “martingale” transforms with limit mixtures of α$\alpha $-stable distributions for any α∈0,2$\alpha \in \left({0,2} \right]$.

This paper proposes an improved likelihood-based method to test the hypothesis that the disturbances of a linear regression model are generated by a first-order autoregressive process against the alternative that they follow a first-order moving average scheme. Compared with existing tests which usually rely on the asymptotic properties of the estimators, the proposed method has remarkable accuracy, particularly in small samples. Simulations studies are provided to show the superior accuracy of the method compared to the traditional tests. An empirical example using Canada real interest rate illustrates the implementation of the proposed method in practice.

This paper details particle Markov chain Monte Carlo (PMCMC) techniques for analysis of unobserved component time series models using several economic data sets. The objective of this paper is to explain the basics of the methodology and provide computational applications that justify applying PMCMC in practice. For instance, we use PMCMC to estimate a stochastic volatility model with a leverage effect, Student-t distributed errors or serial dependence. We also model time series characteristics of monthly US inflation rate by considering a heteroskedastic ARFIMA model where heteroskedasticity is specified by means of a Gaussian stochastic volatility process.