Timo Teräsvirta

Properties of three well-known and frequently applied first-order models for modelling and forecasting volatility in daily or weekly financial series such as stock and exchange rate returns are considered. These are the standard Generalized Autoregressive Conditional Heteroskedasticity (GARCH), the Exponential GARCH and the Autoregressive Stochastic Volatility model. The focus is on finding out how well these models are able to reproduce characteristic features of such series, also called stylized facts. These include high kurtosis and a rather low-starting and slowly decaying autocorrelation function of the squared or absolute-valued observations. Another stylized fact is that the autocorrelations of absolute-valued returns raised to a positive power are maximized when this power equals unity. Not unexpectedly, a conclusion that emerges from these considerations, largely based on results on the moment structure of these models, is that none of the models dominates the others when it comes to reproducing stylized facts in typical financial time series.

In this article we derive a parameter constancy test of a stationary vector autoregressive model against the hypothesis that the parameters of the model change smoothly over time. A single structural break is contained in this alternative hypothesis as a special case. The test is a generalization of a single-equation test of a similar hypothesis proposed in the literature. An advantage here is that the asymptotic distribution theory is standard. The performance of the tests is compared to that of generalized Chow-tests and found satisfactory in terms of both size and power.

 In this paper we propose a Lagrange multiplier test for volatility interactions among markets or assets. The null hypothesis is the Constant Conditional Correlation GARCH model in which volatility of an asset is described only through lagged squared innovations and volatility of its own. The alternative hypothesis is an extension of that model in which volatility is modelled as a linear combination not only of its own lagged squared innovations and volatility but also of those in the other equations while keeping the conditional correlation structure constant. This configuration enables us to test for volatility transmissions among variables in the model. Monte Carlo experiments show that the proposed test has satisfactory finite sample properties. The size distortions become negligible when the sample size reaches 2500. The test is applied to pairs of foreign exchange returns and individual stock returns. Results indicate that there seem to be volatility interactions in the pairs considered, and that significant interaction effects typically result from the lagged squared innovations of the other variables.

In this article, we derive a set of necessary and sufficient conditions for positivity of the vector conditional variance equation in multivariate GARCH models with explicit modelling of conditional correlation. These models include the constant conditional correlation GARCH model of Bollerslev (1990) and its extensions. Under the new conditions, it is possible to introduce negative volatility spillovers in the model. An empirical example illustrates usefulness of having such conditions in practice.

In this paper a Lagrange multiplier test of the hypothesis that the covariance matrix of a multivariate time series model is constant over time is considered. It is assumed that under the alternative, the error variances are time-varying, whereas the correlations remain constant over time. Under the parameterized alternative hypothesis the variances may change continuously as a function of time or some observable stochastic variables. Small-sample properties of the test statistic are investigated by simulation. The assumption of constant correlations does not appear overly restrictive.

In this paper, we use Monte Carlo (MC) testing techniques for testing linearity against smooth transition models. The MC approach allows us to introduce a new test that differs in two respects from the tests existing in the literature. First, the test is exact in the sense that the probability of rejecting the null when it is true is always less than or equal to the nominal size of the test. Secondly, the test is not based on an auxiliary regression obtained by replacing the model under the alternative by approximations based on a Taylor expansion. We also apply MC testing methods for size correcting the test proposed by Luukkonen, Saikkonen and Terasvirta (Biometrika, Vol. 75, 1988, p. 491). The results show that the power loss implied by the auxiliary regression-based test is non-existent compared with a supremum-based test but is more substantial when compared with the three other tests under consideration.

In this paper, we propose a sequential method for determining the number of regimes in threshold autoregressive models. The proposed method relies on the superconsistency of sequential threshold estimates and uses general linearity tests to determine the number of thresholds. A simulation study is performed in order to find out the finite-sample properties of our procedure and to compare it with two other procedures available in the literature. We find that our method works reasonably well for both single and multiple threshold models.

A definition for a common factor for bivariate time series is suggested by considering the decomposition of the conditional density into the product of the marginals and the copula, with the conditioning variable being a common factor if it does not directly enter the copula. We show the links between this definition and the idea of a common factor as a dominant feature in standard linear representations. An application using a business cycle indicator as the common factor in the relationship between U.S. income and consumption found that both series held the factor in their marginals but not in the copula.

In this paper we introduce a flexible target zone model that is capable of characterizing the dynamic behaviour of an exchange rate implied by the original target zone model of Krugman (Quart. J. Econom. 106 (1991) 669) and its modifications. Our framework also enables the modeller to estimate an implicit target zone if it exists. A modelling cycle consisting of specification, estimation, and evaluation stages is constructed. The model is fitted to series of daily observations of the Swedish and the Norwegian currency indices and the estimated models are evaluated.

This paper contains two novelties. First, a unified framework for testing and evaluating the adequacy of an estimated autoregressive conditional duration (ACD) model is presented. Second, two new classes of ACD models, the smooth transition ACD model and the time-varying ACD model, are introduced and their properties discussed. A number of new misspecification tests for the ACD class of models are introduced. They are Lagrange multiplier and Lagrange multiplier type tests against general forms of additive and multiplicative misspecification of the conditional mean function. These forms include tests against higher-order models, tests of no remaining ACD in the standardized durations, as well as tests of linearity and parameter constancy. In addition to its generality, the advantage of this testing approach is its ease of application, since all the resulting asymptotic null distributions are standard. The finite sample properties of the tests are investigated by simulation. A general observation is that the tests are well-sized and have good power. Versions of the test statistics robust to deviations from distributional assumptions other than those being explicitly tested are also given. The smooth transition and time-varying ACD models are introduced, their main properties are examined, and they serve as alternatives in the tests of linearity and parameter constancy. Finally, the tests are applied to ACD models of the IBM stock traded at the New York Stock Exchange.