Evaluating Specification Tests for Markov‐Switching Time‐Series Models

Abstract. We evaluate the performance of several specification tests for Markov regime‐switching time‐series models. We consider the Lagrange multiplier (LM) and dynamic specification tests of Hamilton (1996) and Ljung–Box tests based on both the generalized residual and a standard‐normal residual constructed using the Rosenblatt transformation. The size and power of the tests are studied using Monte Carlo experiments. We find that the LM tests have the best size and power properties. The Ljung–Box tests exhibit slight size distortions, though tests based on the Rosenblatt transformation perform better than the generalized residual‐based tests. The tests exhibit impressive power to detect both autocorrelation and autoregressive conditional heteroscedasticity (ARCH). The tests are illustrated with a Markov‐switching generalized ARCH (GARCH) model fitted to the US dollar–British pound exchange rate, with the finding that both autocorrelation and GARCH effects are needed to adequately fit the data.     

Semiparametric Estimation in Time‐Series Regression with Long‐Range Dependence

Abstract. We consider semiparametric estimation in time‐series regression in the presence of long‐range dependence in both the errors and the stochastic regressors. A central limit theorem is established for a class of semiparametric frequency domain‐weighted least squares estimates, which includes both narrow‐band ordinary least squares and narrow‐band generalized least squares as special cases. The estimates are semiparametric in the sense that focus is on the neighbourhood of the origin, and only periodogram ordinates in a degenerating band around the origin are used. This setting differs from earlier studies on time‐series regression with long‐range dependence, where a fully parametric approach has been employed. The generalized least squares estimate is infeasible when the degree of long‐range dependence is unknown and must be estimated in an initial step. In that case, we show that a feasible estimate which has the same asymptotic properties as the infeasible estimate, exists. By Monte Carlo simulation, we evaluate the finite‐sample performance of the generalized least squares estimate and the feasible estimate.     

On a Mixture GARCH Time‐Series Model

Abstract. Recently, there has been a lot of interest in modelling real data with a heavy‐tailed distribution. A popular candidate is the so‐called generalized autoregressive conditional heteroscedastic (GARCH) model. Unfortunately, the tails of GARCH models are not thick enough in some applications. In this paper, we propose a mixture generalized autoregressive conditional heteroscedastic (MGARCH) model. The stationarity conditions and the tail behaviour of the MGARCH model are studied. It is shown that MGARCH models have tails thicker than those of the associated GARCH models. Therefore, the MGARCH models are more capable of capturing the heavy‐tailed features in real data. Some real examples illustrate the results.     

Time‐Transformed Unit Root Tests for Models with Non‐Stationary Volatility

Abstract. Conventional unit root tests are known to be unreliable in the presence of permanent volatility shifts. In this paper, we propose a new approach to unit root testing which is valid in the presence of a quite general class of permanent variance changes which includes single and multiple (abrupt and smooth transition) volatility change processes as special cases. The new tests are based on a time transformation of the series of interest which automatically corrects their form for the presence of non‐stationary volatility without the need to specify any parametric model for the volatility process. Despite their generality, the new tests perform well even in small samples. We also propose a class of tests for the null hypothesis of stationary volatility in (near‐) integrated time‐series processes.     

CUSUM of Squares‐Based Tests for a Change in Persistence

Abstract. Using standardized cumulative sums of squared sub‐sample residuals, we propose a new ratio‐based test of the null hypothesis that a time series exhibits no change in its persistence structure [specifically that it displays constant I(1) behaviour] against the alternative of a change in persistence from trend stationarity to difference stationarity, or vice versa. Neither the direction nor location of any possible change under the alternative hypothesis need be assumed known. A key feature of our proposed test which distinguishes it from extant tests for persistence change [certain of which test the null hypothesis of constant I(0) behaviour while others, like our proposed test, test the null hypothesis of constant I(1) behaviour] is that it displays no tendency to spuriously over‐reject when applied to series which, although not constant I(1) series, do not display a change in persistence [specifically are constant I(0) processes]. Moreover, where our ratio test correctly rejects the null of no persistence change, the tail in which the rejection occurs can also be used to identify the direction of change since, even in relatively small samples, the test almost never rejects in the right [left] tail when there is a change from I(0) to I(1) [I(1) to I(0)]. Again this useful property is not shared by existing tests. As a by‐product of our analysis, we also propose breakpoint estimators which are consistent where the timing of the change in persistence is unknown.     

Bootstrap Unit‐Root Tests: Comparison and Extensions

Abstract. In this article, we study and compare the properties of several bootstrap unit‐root tests recently proposed in the literature. The tests are Dickey–Fuller (DF) or Augmented DF, based either on residuals from an autoregression and the use of the block bootstrap or on first‐differenced data and the use of the stationary bootstrap or sieve bootstrap. We extend the analysis by interchanging the data transformations (differences vs. residuals), the types of bootstrap and the presence or absence of a correction for autocorrelation in the tests. We show that two sieve bootstrap tests based on residuals remain asymptotically valid. In contrast to the literature which focuses on a comparison of the bootstrap tests with an asymptotic test, we compare the bootstrap tests among themselves using response surfaces for their size and power in a simulation study. This study leads to the following conclusions: (i) augmented DF tests are always preferred to standard DF tests; (ii) the sieve bootstrap performs better than the block bootstrap; (iii) difference‐based tests appear to have slightly better size properties, but residual‐based tests appear more powerful.     

Asymptotics for the Conditional‐Sum‐of‐Squares Estimator in Multivariate Fractional Time‐Series Models

This article proves consistency and asymptotic normality for the conditional‐sum‐of‐squares estimator, which is equivalent to the conditional maximum likelihood estimator, in multivariate fractional time‐series models. The model is parametric and quite general and, in particular, encompasses the multivariate non‐cointegrated fractional autoregressive integrated moving average (ARIMA) model. The novelty of the consistency result, in particular, is that it applies to a multivariate model and to an arbitrarily large set of admissible parameter values, for which the objective function does not converge uniformly in probability, thus making the proof much more challenging than usual. The neighbourhood around the critical point where uniform convergence fails is handled using a truncation argument.     

On the Evaluation of the Information Matrix for Multiplicative Seasonal Time‐Series Models

Abstract. This paper gives a procedure for evaluating the Fisher information matrix for a general multiplicative seasonal autoregressive moving average time‐series model. The method is based on the well‐known integral specification of Whittle [Ark. Mat. Fys. Astr. (1953) vol. 2. pp. 423–434] and leads to a system of linear equations, which is independent of the seasonal period and has a closed solution. It is shown to be much simpler, in general, than the method of Klein and Mélard [Journal of Time Series Analysis (1990) vol. 11, pp. 231–237], which depends on the seasonal period. It is also shown that the nonseasonal method of McLeod [Biometrika (1984) vol. 71, pp. 207–211] has the same basic features as that of Klein and Mélard. Explicit solutions are obtained for the simpler nonseasonal and seasonal models in common use, a feature which has not been attempted with the Klein–Mélard or the McLeod approaches. Several illustrations of these results are discussed in detail.     

Testing the Null of Co‐integration in the Presence of Variance Breaks

Abstract. We show that changes in the innovation covariance matrix of a vector of series can generate spurious rejections of the null hypothesis of co‐integration when applying standard residual‐based co‐integration tests. A bootstrap solution to the inference problem is suggested which is shown to perform well in practice, redressing the size problems associated with the standard test but not losing power relative to the standard test under the alternative.