Forecasting economic variables with nonlinear models

This article is concerned with forecasting from nonlinear conditional mean models. First, a number of often applied nonlinear conditional mean models are introduced and their main properties discussed. The next section is devoted to techniques of building nonlinear models. Ways of computing multi-step ahead forecasts from nonlinear models are surveyed. Tests of forecast accuracy in the case where the models generating the forecasts are nested are discussed. There is a numerical example, showing that even when a stationary nonlinear process generates the observations, future obervations may in some situations be better forecast by a linear model with a unit root. Finally, some empirical studies that compare forecasts from linear and nonlinear models are discussed.Forecast accuracy; forecast comparison; hidden Markov model; neural network; nonlinear modelling; recursive forecast; smooth transition regression; switching regression;

An introduction to univariate GARCH models

This paper contains a survey of univariate models of conditional heteroskedasticity. The classical ARCH model is mentioned, and various extensions of the standard GARCH model are highlighted. This includes the Exponential GARCH model. Stochastic volatility models remain outside this review.ARCH; conditional heteroskedasticity; GARCH; nonlinear GARCH; volatility modelling

Conditional Correlation Models of Autoregressive Conditional Heteroskedasticity with Nonstationary GARCH Equations

In this paper we investigate the effects of careful modelling the long-run dynamics of the volatilities of stock market returns on the conditional correlation structure. To this end we allow the individual unconditional variances in Conditional Correlation GARCH models to change smoothly over time by incorporating a nonstationary component in the variance equations. The modelling technique to determine the parametric structure of this time-varying component is based on a sequence of specification Lagrange multiplier-type tests derived in Amado and Teräsvirta (2011). The variance equations combine the long-run and the short-run dynamic behaviour of the volatilities. The structure of the conditional correlation matrix is assumed to be either time independent or to vary over time. We apply our model to pairs of seven daily stock returns belonging to the S&P 500 composite index and traded at the New York Stock Exchange. The results suggest that accounting for deterministic changes in the unconditional variances considerably improves the fit of the multivariate Conditional Correlation GARCH models to the data. The effect of careful specification of the variance equations on the estimated correlations is variable: in some cases rather small, in others more discernible. As a by-product, we generalize news impact surfaces to the situation in which both the GARCH equations and the conditional correlations contain a deterministic component that is a function of time.Multivariate GARCH model; Time-varying unconditional variance; Lagrange multiplier test; Modelling cycle; Nonlinear time series.

Determining the Number of Regimes in a Threshold Autoregressive Model Using Smooth Transition Autoregressions

In this paper we propose a method for determining the number of regimes in threshold autoregressive models using smooth transition autoregression as a tool. As the smooth transition model is just an approximation to the threshold autoregressive one, no asymptotic properties are claimed for the proposed method. Tests available for testing the adequacy of a smooth transition autoregressive model are applied sequentially to determine the number of regimes. A simulation study is performed in order to find out the finite-sample properties of the 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.Model specification; model selection criterion; nonlinear modelling; sequential testing; switching regression

Modelling Volatility by Variance Decomposition

In this paper, we propose two parametric alternatives to the standard GARCH model. They allow the variance of the model to have a smooth time-varying structure of either additive or multiplicative type. The suggested parameterisations describe both nonlinearity and structural change in the conditional and unconditional variances where the transition between regimes over time is smooth. The main focus is on the multiplicative decom- position that decomposes the variance into an unconditional and conditional component. A modelling strategy for the time-varying GARCH model based on the multiplicative decomposition of the variance is developed. It is heavily dependent on Lagrange multiplier type misspeci.cation tests. Finite-sample properties of the strategy and tests are examined by simulation. An empirical application to daily stock returns and another one to daily exchange rate returns illustrate the functioning and properties of our modelling strategy in practice. The results show that the long memory type behaviour of the sample autocorrelation functions of the absolute returns can also be explained by deterministic changes in the unconditional variance.Conditional heteroskedasticity; Structural change; Lagrange multiplier test; Misspecification test; Nonlinear time series; Time-varying parameter model.

An application of the analogy between vector ARCH and vector random coefficient autoregressive models

In this paper we derive conditions for the conditional covariance matrix to be positive definite in a general vector ARCH model. The conditions can be easily extended to the diagonal vector GARCH model. For the general vector GARCH model, analytical expressions for the conditions in terms of the parameters become complicated, but their validity can in principle be checked numerically once the values of the parameters are given.conditional covariance matrix; multivariate GARCH; multivariate volatility model; random coefficient model; volatility forecasting

Modelling Conditional and Unconditional Heteroskedasticity with Smoothly Time-Varying Structure

In this paper, we propose two parametric alternatives to the standard GARCH model. They allow the conditional variance to have a smooth time-varying structure of either additive or multiplicative type. The suggested parameterizations describe both nonlinearity and structural change in the conditional and unconditional variances where the transition between regimes over time is smooth. A modelling strategy for these new time-varying parameter GARCH models is developed. It relies on a sequence of Lagrange multiplier tests, and the adequacy of the estimated models is investigated by Lagrange multiplier type misspecification tests. Finite-sample properties of these procedures and tests are examined by simulation. An empirical application to daily stock returns and another one to daily exchange rate returns illustrate the functioning and properties of our modelling strategy in practice. The results show that the long memory type behaviour of the sample autocorrelation functions of the absolute returns can also be explained by deterministic changes in the unconditional variance.Conditional heteroskedasticity; Structural change; Lagrange multiplier test; Misspecification test; Nonlinear time series; Time-varying parameter model.

Modelling economic high-frequency time series with STAR-STGARCH models

In this paper we introduce the STAR-STGARCH model that can characterize nonlinear behaviour both in the conditional mean and the conditional variance. A modelling cycle for this family of models, consisting of specification, estimation, and evaluation stages is constructed. Misspecification tests for the estimated model are obtained using standard asymptotic distribution theory. We illustrate the actual modelling by applying the STAR-STGARCH model family to two series of daily observations, the Swedish OMX index and the exchange rate JPY-USD.Financial time series; model misspecification test; nonlinear time series; smooth transition autoregressive model; smooth transition GARCH; time series model specification.

Error correction in DHSY

In this note, we consider the contradiction between the fact that the best fit for the UK consumption data in Davidson et al. (1978) is obtained using an equation with an intercept but without an error correction term, whereas the equation with error correction and without the intercept has better post-sample forecasting properties than the former equation. This contradiction is explained and the two equations reconciled in a nonlinear framework by applying a smooth transition regression model to the data.consumption equation; model misspecification testing; nonlinearity; smooth transition regression

Forecasting inflation with gradual regime shifts and exogenous information

We propose a new method for medium-term forecasting using exogenous information. We first show how a shifting-mean autoregressive model can be used to describe characteristic features in inflation series. This implies that we decompose the inflation process into a slowly moving nonstationary component and dynamic short-run fluctuations around it. An important feature of our model is that it provides a way of combining the information in the sample and exogenous information about the quantity to be forecast. This makes it possible to form a single model-based inflation forecast that also incorporates the exogenous information. We demonstrate, both theoretically and by simulations, how this is done by using the penalised likelihood for estimating the model parameters. In forecasting inflation, the central bank inflation target, if it exists, is a natural example of such exogenous information. We illustrate the application of our method by an out-of-sample forecasting experiment for euro area and UK inflation. We find that for euro area inflation taking the exogenous information into account improves the forecasting accuracy compared to that of a number of relevant benchmark models but this is not so for the UK. Explanations to these outcomes are discussed. JEL Classification: C22, C52, C53, E31, E47Nonlinear forecast, nonlinear model, nonlinear trend, penalised likelihood, structural shift, time-varying parameter