Identifying, Estimating and Testing Restricted Cointegrated Systems: An Overview

The notion of cointegration has lead to a renewed interest in the identification and estimation of structural relations among economic time series, a field to which Henri Theil has made many pioneering contributions. This paper reviews the different approaches that have been put forward in the literature for identifying cointegrating relationships and imposing (possibly over-identifying) restrictions on them. Next, various algorithms to obtain (approximate) maximum likelihood estimates and likelihood ratio statistics are reviewed, with an emphasis on so-called switching algorithms. The implementation of these algorithms is discussed and illustrated using an empirical example.

Behavioral Heterogeneity in Stock Prices

We estimate a dynamic asset pricing model characterized by heterogeneous boundedly rational agents. The fundamental value of the risky asset is publicly available to all agents, but they have different beliefs about the persistence of deviations of stock prices from the fundamental benchmark. An evolutionary selection mechanism based on relative past profits governs the dynamics of the fractions and switching of agents between different beliefs or forecasting strategies. A strategy attracts more agents if it performed relatively well in the recent past compared to other strategies. We estimate the model to annual US stock price data from 1871 until 2003. The estimation results support the existence of two expectation regimes, and a bootstrap F-test rejects linearity in favor of our nonlinear two-type heterogeneous agent model. One regime can be characterized as a fundamentalists regime, because agents believe in mean reversion of stock prices toward the benchmark fundamental value. The second regime can be characterized as a chartist, trend following regime because agents expect the deviations from the fundamental to trend. The fractions of agents using the fundamentalists and trend following forecasting rules show substantial time variation and switching between predictors. The model offers an explanation for the recent stock prices run-up. Before the 90s the trend following regime was active only occasionally. However, in the late 90s the trend following regime persisted and created an extraordinary deviation of stock prices from the fundamentals. Recently, the activation of the mean reversion regime has contributed to drive stock prices back closer to their fundamental valuation.

Testing for Cointegration with Nonstationary Volatility

The paper generalises recent unit root tests for nonstationary volatility to a multivariate context. Persistent changes in the innovation variance matrix lead to size distortions in conventional cointegration tests, and possibilities of increased power by taking the time-varying volatilities and correlations into account. The testing procedures are based on a likelihood analysis of the vector autoregressive model with a conditional covariance matrix that may be estimated nonparametrically. We find that under suitable conditions, adaptation with respect to the volatility matrix process is possible, in the sense that nonparametric volatility estimation does not lead to a loss of asymptotic local power

Testing for a Unit Root with Near-Integrated Volatility

This paper considers tests for a unit root when the innovations follow a near-integrated GARCH process. We compare the asymptotic properties of the likelihood ratio statistic with that of the least-squares based Dickey-Fuller statistic. We first use asymptotics where the GARCH variance process is stationary with fixed parameters, and then consider parameter sequences such that the GARCH process converges to a diffusion process. In the fixed-parameter case, the asymptotic local power gain of the likelihood ratio test is only marginal for realistic parameter values. However, under near-integrated parameter sequences the difference in power is more pronounced.

Temporal aggregation in a periodically integrated autoregressive process

Time Series;Aggregation;statistics

Wake me up before you GO-GARCH

In this paper we present a new three-step approach to the estimation of Generalized Orthogonal GARCH (GO-GARCH) models, as proposed by van der Weide (2002). The approach only requires (non-linear) least-squares methods in combination with univariate GARCH estimation, and as such is computationally attractive, especially in largerdimensional systems, where a full likelihood optimization is often infeasible. The eï¬~@ectiveness of the method is investigated using Monte Carlo simulations as well as a number of empirical applications.

Why Frequency Matters for Unit Root Testing

It is generally believed that for the power of unit root tests, only the time span and not the observation frequency matters. In this paper we show that the observation frequency does matter when the high-frequency data display fat tails and volatility clustering, as is typically the case for financial time series such as exchange rate returns. Our claim builds on recent work on unit root and cointegration testing based non-Gaussian likelihood functions. The essential idea is that such methods will yield power gains in the presence of fat tails and persistent volatility clustering, and the strength of these features (and hence the power gains) increases with the observation frequency. This is illustrated using both Monte Carlo simulations and empirical applications to real exchange rates

Revised calendar date for the Taupo eruption derived by ¹⁴C wiggle-matching using a New Zealand kauri ¹⁴C calibration data set

Taupo volcano in central North Island, New Zealand, is the most frequently active and productive rhyolite volcano on Earth. Its latest explosive activity about 1800 years ago generated the spectacular Taupo eruption, the most violent eruption known in the world in the last 5000 years. We present here a new accurate and precise eruption date of AD 232 ± 5 (1718 ± 5 cal. BP) for the Taupo event. This date was derived by wiggle-matching 25 high-precision ¹⁴C dates from decadal samples of Phyllocladus trichomanoides from the Pureora buried forest near Lake Taupo against the high-precision, first-millennium AD subfossil Agathis australis (kauri) calibration data set constructed by the Waikato Radiocarbon Laboratory. It shows that postulated dates for the eruption estimated previously from Greenland ice-core records (AD 181 ± 2) and putative historical records of unusual atmospheric phenomena in ancient Rome and China (c. AD 186) are both untenable. However, although their conclusion of a zero north–south ¹⁴C offset is erroneous, and their data exhibit a laboratory bias of about 38 years (too young), Sparks et al. (Sparks RJ, Melhuish WH, McKee JWA, Ogden J, Palmer JG and Molloy BPJ (1995) ¹⁴C calibration in the Southern Hemisphere and the date of the last Taupo eruption: Evidence from tree-ring sequences. Radiocarbon 37: 155–163) correctly utilized the Northern Hemisphere calibration curve of Stuiver and Becker (Stuiver M and Becker B (1993) High-precision decadal calibration of the radiocarbon timescale, AD 1950–6000 BC. Radiocarbon 35: 35–65) to obtain an accurate wiggle-match date for the eruption identical to ours but less precise (AD 232 ± 15). Our results demonstrate that high-agreement levels, indicated by either agreement indices or χ² data, obtained from a ¹⁴C wiggle-match do not necessarily mean that age models are accurate. We also show that laboratory bias, if suspected, can be mitigated by applying the reservoir offset function with an appropriate error value (e.g. 0 ± 40 years). Ages for eruptives such as Taupo tephra that are based upon individual ¹⁴C dates should be considered as approximate only, and confined ideally to short-lived material (e.g. seeds, leaves, small branches or the outer rings of larger trees)

Testing for periodic integration

A periodic autoregressive time-series model assumes that the autoregressive parameters vary with the season. This model can also be represented by a multivariate model for the annual vector containing the seasonal observations. When this multivariate model contains one unit root, a time-series is said to be periodically integrated of order 1. In this paper we propose tests for such a single unit root. These tests for periodic integration are applied to a periodic model for the quarterly German consumption series