Combination schemes for turning point predictions☆
Highlights
► This paper proposes to use Bayesian inference to combining turning point forecasts from linear and non-linear models. ► The first methodology combines the forecasts from the models and then detects the turning points. ► The second methodology detects the turning points from the models and then combines them. ► We find that the forecast abilities of the two strategies are cycle-specific and need to be evaluated in the problem at hand.
Introduction
In recent years, interest has increased in the ability of business cycle models to forecast economic growth rates and turning points or structural breaks in economic activity. The early contributions in this stream of literature consider nonlinear models such as the Markov-switching (MS) models (see for example Goldfeld & Quandt, 1973 and Hamilton, 1989) and the threshold autoregressive models (see Tong, 1983 and Potter, 1995), both of which are able to capture the asymmetry and the turning points in business cycle dynamics. See also Clements and Krolzig (1998), Kim and Murray (2002), Kim and Piger (2000) and Krolzig (2000) for further extensions. In our paper we consider MS models and apply them to US and EU industrial production data, for a period of time including the 2009 recession and find that four regimes (strong-recession, contraction, normal-growth, and high-growth) are necessary to capture some important features of the US and EU cycle in the strong-recession phases. As most of the forecast errors are due to shifts in the deterministic factors (see Krolzig, 2000), we consider a model with shifts in the intercept and in the volatility. Evidence of more than two regimes, even in forecasting applications, is rather common in finance and has suggestive economic meanings. See Guidolin (2011) for an up-to-date literature review with a deep discussion of this aspect.
The first contribution of this paper is to exploit the time-varying forecast ability of linear and nonlinear models to produce potentially better forecasts. More specifically, in some empirical investigations and simulation studies, there is evidence that MS models are superior in in-sample fit, but not always in forecasting and that the relative forecast performance of the MS models depends on the regime present at the time the forecast is made (see for example Clements & Krolzig, 1998). It seems thus possible to obtain better forecasts by dynamically combining in a suitable way various model forecasts.
The second main contribution of this paper is to study the relationship between forecast combination and turning point identification when many forecasts are available from different models for the same variable of interest. When many models are used for forecasting turning points, one can then alternatively combine the forecasts from the models and detect the turning points on the combined forecasts, or detect the turning points on the model forecasts and then combine the turning point indicators. We tackle this problem and show that the turning point forecasts are not invariant with respect to the order of the forecast combination and turning point identification, and that the best combination should be evaluated in the specific case at hand. For this aspect, our paper is related to Stock and Watson (2010), who consider the issue of dating the turning point for a reference cycle when many series are available. In this context, it is possible to detect clusters of turning points that are cycle-specific, and the problem of their aggregation becomes crucial to determine a reference cycle.
Another relevant contribution is the proposal of a new model selection scheme which relies upon non-parametric measures, i.e. concordance statistics, that counts the proportion of time during which the predicted and the reference turning point series are in the same state. The proposed scheme extends the literature on Bayesian model averaging (BMA) procedure (see Grunwald, Raftery, & Guttorp, 1993, for a review) for turning point forecasts. In the proposed scheme, we follow a Bayesian inference approach and account thus for both model and parameter uncertainty. The use of a Bayesian approach to forecast combination in business cycle analysis has been discussed in Min and Zellner (1993). They consider both autoregressive (AR) models and AR models with time-varying parameters for predicting international output growth rates. Canova and Ciccarelli (2004) propose a Bayesian inference approach to the estimation of a multi-country panel model with time-varying parameters, lagged interdependencies and country specific effects. They follow Zellner, Hong, and Min (1991) and predict turning points by using the predictive densities from their model. In this paper, we extend the previous literature and propose AR models with discontinuous (Markov-switching) dynamics in the parameters. The Bayesian approach proposed in this paper is based on a numerical approximation algorithm, Gibbs sampler, which is general enough to account not only for parameter uncertainty, but also for possible non-normality of the prediction error, as well as for nonlinearities of the data generating process. Another advantage of the Gibbs sampling procedures is that they naturally provide approximation of predictive density and forecast intervals for the variable of interest. Then, following Canova and Ciccarelli (2004), we use the approximated marginal predictive density for turning point detection, and extend the existing literature (see for example Krolzig, 2004) which instead applies the Markov-switching smoothing probabilities. The advantage in using the marginal predictive is that the forecast will include all the information that is contained in both the observable variable and the hidden state predictive densities.
Finally, we study different strategies to specify combination weights. More specifically, we compare in terms of forecast performances two weighting schemes. The first one computes model weights based on recursive updating of the prediction errors for the level of the variable of interest. The second one is based on the prediction of the turning points. We apply them to predict the level and the turning points of the EU and the US business cycles, measured in terms of industrial production. We find that the performances of the different combination strategies rank differently in predicting industrial production growth and turning points and are country-specific. This suggests that both combination strategies should be considered in applications and the best one selected for the problem at hand.
The paper is structured as follows. Section 2 introduces the Markov-switching model used in the analysis of the business cycle. Sections 3 and 4 present a Bayesian approach to inference and to forecast combination, respectively. Section 5 provides a comparison between the performance of different methods for the EU area and the US business cycles. Section 6 concludes the paper.
Section snippets
Predicting with Markov-switching models
Let yt, with t = 1, …, T, be a set of observations for a variable of interest. We consider two alternative autoregressive models for yt. First, we assume that yt follows the Gaussian AR process of order p, denoted with AR(p),t = 1, …, T, where ν is the intercept; ϕl, l = 1, …, p, are the autoregressive coefficients and σ the volatility. In the following we will assume that the initial values, (y−p+1, …, y0), of the process are known. More generally, it is
Data augmentation
In this paper we follow a Bayesian inference approach. One of the reasons of this choice, is that inference for latent variable models calls for simulation based methods, which can be naturally included in a Bayesian framework. Moreover, model selection and averaging can be easily performed in an elegant and efficient way within a Bayesian framework, overcoming difficulties of the frequentist approach in dealing with model selection for non-nested models.
In this paper we propose a Bayesian
Combining linear and non-linear models
In this section we describe the rules used for combining the forecasts from linear (AR) and non-linear (MS-AR) models and for predicting the turning points of the business cycle. We propose to combine the models through the use of two alternative schemes. The first one is a Bayesian Model Averaging (BMA) procedure based on the forecasting performance for the variable of interest. The second one is based on the performance of the models in terms of turning point forecasts.
The BMA procedure gives
Data and reference cycle
In our study we consider the Industrial Production Index (IPI) from OECD at a monthly frequency for the United States (US), from February 1949 to January 2011, and for the Euro Area (EU), from January 1971 to January 2011. Data for both US and EU economies are seasonally and working day adjusted. We employ revised data from the April 2011 vintage (see Hamilton, 2011 and Nalewaik, 2012, for business cycle analysis using real-time data). To obtain the IPI at the Euro zone level a
Conclusion
We focus on the analysis of turning points of the business cycle and follow a Bayesian model averaging approach to combine their forecasts obtained from different prediction models. The new combination scheme relies upon non-parametric measures, i.e. concordance statistics, that counts the proportion of time during which the predicted and the reference chronologies are in the same phase. We compare empirically our combination approach with a combination strategy based on the predictive
References (64)
- et al.
Forecasting and turning point prediction in a bayesian panel var model
Journal of Econometrics
(2004) - et al.
Combining probability forecasts
International Journal of Forecasting
(2011) - et al.
Stationarity of multivariate Markov-switching ARMA models
Journal of Econometrics
(2001) Calling recessions in real time
International Journal of Forecasting
(2011)- et al.
Dissecting the Cycle: A methodological investigation
Journal of Monetary Economics
(2002) - et al.
Bayesian forecasting using stochastic search variable selection in a VAR subject to breaks
International Journal of Forecasting
(2010) - et al.
Bayesian long-run prediction in time series models
Journal of Econometrics
(1995) - et al.
Bayesian and non-Bayesian methods for combining models and forecasts with applications to forecasting international growth rates
Journal of Econometrics
(1993) Bayesian Skepticism on unit root econometrics
Journal of Economic Dynamics and Control
(1988)- et al.
Combined forecasts from linear and nonlinear time series models
International Journal of Forecasting
(2002)
Forecasting turning points in international output growth rates using Bayesian exponentially weighted autoregression, time varying parameter, and, pooling techniques
Journal of Econometrics
Bayes inference via Gibbs sampling of autoregressive time series subject to Markov mean and variance shifts
Journal of Business and Economic Statistics
Business Cycle Analysis with Multivariate Markov Switching Models
A turning point chronology for the Euro-zone classical and growth cycle
A System for dating and detecting turning points in the Euro area
The Manchester School
Regime switches in interest rates
Journal of Business and Economic Statistics
Identifying business cycle turning points with sequential Monte Carlo methods: an online and real-time application to the Euro area
Journal of Forecasting
Beta autoregressive transition Markov-switching models for business cycle analysis
Studies in Nonlinear Dynamics and Econometrics
Does forecast combination improve Norges bank inflation forecasts?
Oxford Bulletin of Economics and Statistics
Comparison of the real-time performance of business cycle dating methods
Journal of Business and Economic Statistics
A comparison of the forecast performances of Markov-switching and threshold autoregressive models of US GNP
Econometrics Journal
The simulation smoother for time series models
Biometrika
Bayesian near-boundary analysis in basic macroeconomic time series models
Advances in Econometrics
Measuring business cycles: A modern perspective
The Review of Economics and Statistics
Estimation of finite mixture distributions through Bayesian sampling
Journal of the Royal Statistical Society B
Duration-dependent transition in Markov model of U.S. GNP growth
Journal of Business and Economics Statistics
Business cycle phases and their transitional dynamics
Journal of Business and Economics Statistics
Mixture and Markov-switching models
Cited by (0)
- ☆
We thank the co-editor, Massimo Guidolin, two anonymous referees and Anne Sofie Jore for their very useful comments on an earlier version of our paper. We also thank the participants at: the 31st Annual International Symposium on Forecasting, 2011, Prague and the 5th CSDA International Conference on Computational and Financial Econometrics, 2011, London. The views expressed in this paper are our own and do not necessarily reflect those of Norges Bank.