Modelling the dynamics of EU economic sentiment indicators: an interaction-based approach

Abstract

This article estimates a simple univariate model of expectation or opinion formation in continuous time adapting a ‘canonical’ stochastic model of collective opinion dynamics (Weidlich and Haag, 1983; Lux, 1995, 2009a). This framework is applied to a selected data set on survey-based expectations from the rich EU business and consumer survey database for 12 European countries. The model parameters are estimated through Maximum Likelihood (ML) and numerical solution of the transient probability density functions for the resulting stochastic process. The model's success is assessed with respect to its out-of-sample forecasting performance relative to univariate Time Series (TS) models of the Autoregressive Moving Average model, ARMA(p, q) and Autoregressive Fractionally Integrated Moving Average, ARFIMA(p, d, q) varieties. These tests speak for a slight superiority of the canonical opinion dynamics model over the alternatives in the majority of cases.

Keywords:

• expectation formation
• survey-based expectations
• opinion dynamics
• Fokker–Planck equation
• forecasting

JEL Classification::

• E32
• C83
• C53

Acknowledgements

We thank Xiaokang Wang for excellent research assistance. We are also grateful to an anonymous referee and to participants at the Eurostat conference on ‘New Techniques and Technologies for Statistics’, Brussels, 18–20 February 2009 for helpful comments. We gratefully acknowledge financial support by the Volkswagen Foundation through their grant on ‘Complex Networks As Interdisciplinary Phenomena’. Jaba Ghonghadze expresses his gratitude to DAAD/OSI for financial support through their doctoral scholarship programme. A portion of this article was written while the first author visited the Chair of Systems Design at ETH Zurich, whose hospitality is gratefully acknowledged.

Notes

¹ Cf., Acemoglu and Scott, 1994; Delorme et al., 2001, and the survey by Nardo (2003).

² Other instances of empirical research utilizing EU survey-based expectations data include, e.g. Madsen (1996), who explores how producers form their inflation expectations, Cotsomitis and Kwan (2006), who examine the ability of consumer confidence indices to forecast household spending within a multicountry framework, and Drakos (2008), who investigates the predictive content of survey-based expectations on investment.

³ On monthly and quarterly bases.

⁴ The programme was set up in 1961 and is currently managed by the Commission.

⁵About 125 000 firms and almost 40 000 consumers are currently surveyed every month across the EU. Source: http://ec.europa.eu/economy_finance/indicators/business_consumer_surveys/userguide_en.pdf.

⁶ Note that N* includes the number of ‘neutral’ agents, N^∼, i.e. N* = N⁺ + N⁻ + N^∼.

⁷ Balance series are usually referred to as ‘opinion index’, ‘climate index’, or ‘diffusion index’ in the literature.

⁸ In the monthly surveys, the total number of questions referring to future expectations is 18. The sectoral breakdown of these is as follows: three in industry, three in services, six in consumer, four in retail trade, and two in construction surveys. See Section IV on the selection criterion for the countries. Since the survey in the service sector had only been launched during the 1990s, we did not include it in our selection.

⁹ Note that in the case of the last question the balance is calculated aswith the intuitive notation of N⁺⁺ (N⁻⁻) being the number of ‘very optimistic’ (‘very pessimistic’) respondents.

¹⁰ The Master equation represents the general and exact system of equations tracking the flow of probabilities between states, see Weidlich and Haag (1983) or van Kampen (2007).

¹¹Note the notational change: p(n, t) → P(x, t) and ω(n) → w(x).

¹² It might, however, be mentioned that we could also design a slightly modified framework allowing for a neutral disposition along with the ‘+’ and ‘−’ choices. We leave this for future research.

¹³ See Weidlich and Haag (1983), Lux (1997, 2009a), Gardiner (2004), and van Kampen (2007) for more details.

¹⁴ Our function U(·) resembles the utility function within a discrete choice framework (cf. Brock and Durlauf, 2001). However, there is no clear utility component to survey responses so that we prefer the notion of a ‘forcing function’. The major difference of our framework to studies of Discrete Choice problems with Social Interaction (DSCI) is that we investigate a dynamic model of aggregate opinion formation while DSCI models are typically applied to cross-sections of micro data.

¹⁵ Using the hyperbolic trigonometric functions, the drift and diffusion function can also be written as

¹⁶ Note that this is only an approximation to our population dynamics in that the microscopic sources of randomness have been proxied by a macroscopic noise factor W_t. See Gardiner (2004, Ch. 3) and the appendix of Lux (2009a) for technical details on the diffusion approximation to Markov jump processes.

¹⁷ Cf. Weidlich and Haag (1983), Lux (2009a).

¹⁸ This information can be obtained from the EU Business and Consumer Surveys database. It actually varies widely across countries and sections.

¹⁹ The idea is that despite the inclusion of a social interaction term our model might not capture all correlation between respondents. For example, there might be groups that always switch simultaneously which would, indeed, reduce the number of effectively independent agents. Of course, the officially reported number should be an upper boundary to the ‘effective’ number. However, in quite a few cases we obtained higher estimates for N than this boundary. In the tables we report these estimates without applying the corresponding upper boundaries.

²⁰ ARMA models have been estimated via standard ML, while for ARFIMA models first the parameter of fractional differentiation, d, has been estimated from the frequency spectrum and the remaining parameters have been estimated via ML. This approach proved to be more robust than full ML in previous studies with small samples (cf. Lux and Kaizoji, 2007).

²¹ Note also that often large SEs are associated with the estimates of N. This happens mostly if the estimated values of α₁ in Models 3 and 4 are much lower than 1. As has been demonstrated in Lux (2009a), in this case the relationship between parameters ν and N is close to collinear so that one of both could be fixed without much deterioration of the likelihood.

²² See Diebold and Mariano (1995).

²³ We considered only 1-month-ahead forecasts.

²⁴ Note that using the large official numbers of respondents would lead to very low predicted volatility due to the law of large numbers. This can to a certain degree be overcome by high sensitivity of the system to changes. This is what characterizes the neighbourhood of α₁ while moving away from this benchmark in both directions leads to more persistent macroscopic dynamics.

²⁵ As an additional test for predictive accuracy we have used the Kullback–Leibler information criterion (following the methodology of Mitchell and Hall, 2005) to compare out-of-sample predictive densities of all models considered in this article. The overall conclusion is that, while ARMA density forecasts are quite poor, in more than 50% of the cases, the hypothesis of equal predictive accuracy between opinion and ARFIMA models can be rejected at the 5% level. The tests again point towards the superiority of the OM density forecasts.

²⁶ Note that the value of outside of the interval (−0.5, 0.5) indicates that the series might be nonstationary (cf. Brockwell and Davis, 1991).

²⁷ Alfarano and Lux (2007) demonstrate that a closely related model mimics the long-term dependency that is the defining feature of ARFIMA models. Lux (2009b) shows that both a behavioural OM and a parsimonious diffusion process provide nearly equivalent fits to a financial sentiment index.

²⁸ Lux (2009a) considered various macroeconomic factors in the analysis of a German business climate index but found surprisingly little value added compared to the ‘canonical’ model.