Nonlinearities in economic convergence. The German-Spanish case in fixed rates and stockexchange markets.

Lorenzo Escot Mangas. Universidad Complutense (lescot@teleline.es)

Ricardo Gimeno Nogués. Universidad Pontificia de Comillas.(rgimeno@cee.upco.es)

Ruth Mateos de Cabo. Universidad San Pablo-CEU (matcab@ceu.es)

Elena Olmedo Fernández. Universidad de Sevilla (olmedo@cica.es)

Jose Ramón Sánchez-Galán. Universidad San Pablo-CEU (sangal@ceu.es)

One of the main targets for the European Union is to level the different economic conditions and opportunities all over the countries of the Union -that is what we call real convergence. It is essential to know if real convergence of economies is being reached, and if nonlinearities are present in this union process in orden to try to explain better the behaviour of that process and the presence of similar patterns among the different countries.

On the other hand, a way to look for real convergence lies on studying the evolution of the stock market indexes and the interest rates series of the long-term bond that the different countries of the European Union may have. Germany has been chosen, for obvious reasons, as a reference country for the Spanish convergence.

We have taken the indexes listed below:

1. IBEX 35, Jan 1^st 1990 = 3.000 (Laspeyres with an aggregate coefficient)

2. 2. DAX 30, Dec 31^st 1987 = 1.000 (Laspeyres with an adjusted coefficient)

3. Spanish National Bond: 10 years. CTD: higher implicit repo or lower negative base (fut x cf – spot). Meff

4. German National Bond: 10 years. Ctd. Liffe.

5. Risk Premium: difference between both bonds. It indicates level of confidence for investors comparing both economies.

The period of time we have considered is July 1^st 1994 to September 23^rd 1999. We need this data to study the possible real convergence. The problem is that this data set is possibly too small.

Usually, before studying the possible presence of nonlinearity or chaos in a time series, we have to adjust an ARIMA model and we work with the residuals. First, because we are going to try to explain the part of the time series which is not explained by usual methods -in other words, by ARIMA methodology. And second, because the Brock’s Residual Test guarantees us that dynamic analysis (correlation dimension, Lyapunov exponents) is not going to change.

So we take logarithms and their respective differences in order to obtain stationary series to be analyse. Then, we determine its linear structure with ARIMA techniques although all the implications included in this methodology. After that, we estimate these models in order to keep the residuals included in the rest of the analysis.

After all, we observe that DAX index does not show clairly any linear structure. It fits better a white noise. And both INDEX and Premium Risk seem to be closer to an ARMA(0,1) structure.

Proposed by Takens, embedding the time series guarantees us that we recover the dynamics of the original (unknown) system. The method works with m-historys. These m-historys are vectors in an m-dimensional space which are windows of the time series. This dimension m is what we call embedding dimension. The principal measures of complexity do not change if we consider the original system or the embedded time series (that is what we call invariants of the systems). One of these invariants is the correlation dimension, and another is the Lyapunov exponent.

We need two parameters to get an embedding of our time series. First, we have to choose an appropriate time delay and after that, we have to choose the embedding dimension. We use the average mutual information to choose the time delay (which is going to be the first minimum of this measure) and the false neighbours method to choose the embedding dimension (the percentage of false neighbours decays as we augment the embedding dimension, before reaching a saturate point. The embedding dimension corresponding to this saturate point is the appropriate embedding dimension). If the system is deterministic, the vale of minimum false neighbours will be cero but, if the system is random or, at least, noisy, we will never reach this value. So the value of saturation could give us an idea of the amount of noise we have in our time series.

As we say before, it is one of the invariants of the system. It is one of the first measures of complexity, and its behaviour is different depending on the system. If the system is random, the correlation dimension does not saturate as the embedding dimension increases but, if the system is deterministic, the correlation dimension reaches a saturate point. The value of this correlation dimension will be the dimension of our system. If our system is low-dimensional chaotic, it will be fractional (if our system is chaotic, it will be probably fractal too).

We can see that nor of the three time series is clearly low-dimensional chaotic, because the correlation dimension does not reach a saturate point. But we can obtain two conclusions:

1. the series corresponding to IBEX and DAX behaves similarly.

2. the correlation dimension of the Premium Risk Residuals behaves more as it reaches a saturate point.

3. we can think these graphics could correspond to a chaotic system, but not low-dimensional or perhaps a deterministic nonlinear system, with noise added.

We can complement the results given by the correlation dimension with those given by the BDS test. This test tries to verify the hypotheses of IID (independence and identically distributed) and proportioned an statistic which follows a normal distribution if the hypotheses of IID is verified. If we works with a filtered time series, when we deny this hypotheses this means that our system is nonlinear (because it is not independent, but we have suppressed any linear dependence). In our three time series we confirm the idea of nonlinearity (but not clearly in the case of the Premium Risk Residuals).

After showing the possible existence of nonlinear dynamics in our time series, we are going to compute de maximum Lyapunov exponent in order to know if there is chaotic dynamics. We work with two different algorithms: Wolf algorithm (a direct method of estimation) and Ellner algorithm (a Jacobian method of estimation).

Table 1.

Dominant Lyapunov Exponent. Worlf algorithm

Table 2.

Dominant Lyapunov Exponent. Ellner algorithm

Then, using Wolf method, the maximum Lyapunov Exponents are, in all the cases, positive and their values are similar if we compare similar products (german bund with Spanish bono and DAX index with IBEX index). But, using Ellner algorithm the maximum Lyapunov Exponents are, instead, negative. This confirms the idea of a nonlinar dynamical system with noise added.

1. Evidence shows that we cannot assure clearly the existence of chaos but there are evidences that confirm the presence of nonlinear relationships.

2. Existence of relatively short data sets can, in part, influence this results. This is a common problem in economic time series, which are usually short, aggregated and contaminated with noise. This question is really a handicap to apply the usual methods.

3. The existence of nonlinear relationships invalidate the use of linear models to analyse financial markets.

4. The behaviour of the correlation dimension and the estimation of Lyapunov exponents seem to confirm the presence of noise. In financial markets, systems creates its own noise called systematic noise: expectatives of investors. Besides, psychological elements are implicit in these markets.

5. We are starting so our intention is to describe and understand behaviours, not to propose a model. It is clear that nonlinear instruments have got economics to walk one step ahead. Traditional instruments do not seem to work well in economics: we can fit models, but we can not make good predictions. This change of paradigm is very important in economics because it opens new possibilities in the study of economic behaviour.