Asymmetry in the EMS: New evidence based on non-linear forecasts by Oscar Bajo-Rubio Simón Sosvilla-Rivero Fernando Fernández-Rodríguez DOCUMENTO DE TRABAJO 97-24

November, 1997

The authors wish to thank Gabriel Quirós for providing us with the data set, as well as Julián Andrada-Félix for very able computing assistance. A first version of this paper was presented at the V Jornadas de Economía Internacional (Pamplona, June 1997). Financial support from the Spanish Ministry of Education, through DGICYT Project PB94-0425, is also gratefully acknowledged.

Universidad Pública de Navarra.

*** FEDEA and Universidad Complutense de Madrid.

**** Universidad de Las Palmas de Gran Canaria.

ABSTRACT

In this paper we provide new evidence on the hypothesis of German leadership and asymmetric performance in the EMS, using daily data. Given the evidence about non-linearity in financial series, we apply non-linear forecasting methods based on the literature on complex dynamic systems. Our analysis covers nine EMS countries, and includes the more recent events in the EMS history, such as the German reunification, the monetary turmoil at the end of 1992, and the broadening of fluctuation bands in 1993.

JEL codes: C53, E43, F33

Key words: Interest rate, European Monetary System, Non-linear forecasting.

1. Introduction

As it is well known, fixed exchange rate systems face the so-called n-1 problem: there are n countries pegging their exchange rates but only n-1 exchange rates among them, which gives the system one degree of freedom when setting money supply and the interest rate. This degree of freedom can be used either in an asymmetric (i. e., hegemonic) way, by enabling one country to become the leader and settle monetary policy independently, with the other countries following its stance; or, alternatively, in a symmetric (i. e., co-operative) way, so that all countries are allowed to decide jointly over the implementation of monetary policy [see De Grauwe (1994)].

The European Monetary System (EMS) means no exception to this problem. However, and despite the initial objectives of the founders of the EMS, a general consensus has emerged that the system works in an asymmetric way, with Germany assuming the leading role and the remaining countries passively adjusting to German monetary policy actions. In its turn, the follower countries may find beneficial to behave in such a way, since they can take advantage of the firmly established anti-inflation credibility of the German Bundesbank [see, e. g., Giavazzi and Pagano (1988) or Mélitz (1988)]. On the other hand, these countries would have retained some degree of monetary autonomy by resorting to capital controls, which would have allowed them to dissociate the evolution of domestic (i. e., onshore) interest rates from those prevailing in the Euromarket (i. e., offshore) [see Rogoff (1985) or Giavazzi and Giovannini (1989)].

Some evidence along these lines has been provided by Giavazzi and Giovannini (1987, 1989), who find a higher volatility for French and Italian offshore interest rates as compared to the German ones during the EMS period, reflecting the higher autonomy of German monetary policy. Similarly, Mastropasqua, Micossi and Rinaldi (1988) analyse foreign exchange intervention and sterilization for four EMS countries (Germany, Belgium, France and Italy) along the period 1979-1987, and obtain that interventions are fully sterilized only in Germany, which is again interpreted in terms on Germany's leadership in the system. Also, by means of Granger causality tests, Karfakis and Moschos (1990) conclude that interest rate changes for Belgium, France, Italy, and the Netherlands can be predicted using information on the past evolution of German interest rates, but not conversely.

These conclusions, however, have been challenged by other authors. So, for instance, Cohen and Wyplosz (1989), Fratianni and von Hagen (1990), von Hagen and Fratianni (1990), Cherubini, Ciampolini, Hamaui and Sironi (1992), and Kutan (1992), estimate equations for monetary aggregates or interest rates in several EMS countries, finding that monetary variables in every country depend on German variables, but also on those concerning the other countries. Comparable results have been obtained, using different methods, by De Grauwe (1989) and Koedijk and Kool (1992).

A common result to most of the above quoted studies is the finding that, both in terms of size and persistence, the effect is stronger from German variables to the other EMS countries' variables, rather than the other way round. In other words, whereas monetary policy in the other EMS countries would be affected not only by German actions but by the other EMS partners also, German monetary policy would operate rather independently. This would point to a special role of Germany within the EMS, even though the hypothesis of German leadership or dominance might appear too strong. In von Hagen and Fratianni's words: "(I)n the short run, the EMS is best portrayed as an interactive web of monetary policies, where Germany is an important player, but not the dominant one (...) (I)t is tempting to conclude that many observers have mistaken German dominance with the relative strength of Germany and the relative weakness of France in the EMS" [von Hagen and Fratianni (1990), p. 373].

Unlike the previous papers, which make use either of quarterly or monthly data, we can quote two more recent studies which use instead high frequency (i. e., daily) data on interest rates, and also address the issue of the influence of German reunification on the hypothesis of the asymmetric behaviour of the EMS. However, a common feature to both papers is that they only deal with the cases of France and Germany.

So, by computing the impulse effects of unit shocks to interest rates from VAR estimates, Gardner and Perraudin (1993) find the effect of French innovations on Germany to be significant, although smaller than the German effect on France. They also detect the presence of a structural break coinciding with German reunification, with Germany losing its leadership role after then but recovering it thereafter.

Finally, by using several concepts of causality, Henry and Weidmann (1995) confirm the hypothesis of German dominance vis-à-vis France in the short-run dynamics, unlike the long run where some evidence of interdependence is found for the period before German reunification. Again, according to their results, after that date German dominance would have become even stronger.

In this paper we will try to provide some additional evidence on the hypothesis of German leadership and asymmetric performance in the EMS, by using high frequency (i. e., daily) data. Unlike the abovementioned papers, we will make use of non-linear forecasting methods (based on the literature on complex dynamic systems), which can be justified given the evidence about non-linearity present in financial series [see, e. g., Mills (1996)]. At the same time, we extend the analysis to nine EMS countries, and include in our sample the more recent events in the EMS history, such as the German reunification, the monetary turmoil at the end of 1992, and the broadening of fluctuation bands in 1993.

The rest of the paper is organised as follows. Our testing strategy is discussed in section 2, whereas the data and empirical results are presented in section 3. The main conclusions are summarised in section 4.

2.- A procedure for testing causality using non-linear forecasting methods

The traditional method for testing causality in economic time series makes use of the well-known Granger definition of causality [see Granger (1969)]. Given two variables, x and y, x is said to Granger-cause y if the latter can be predicted better by past values of x and y, rather than by past values of y alone. From this definition, the test proceeds by using x and y as the independent variables, so that four results are possible: x Granger-causes y, y Granger-causes x, two-way Granger-causality, and no Granger-causality.

In practice, the criterion for assessing Granger-causality consists of comparing the prediction errors (PE) from both information sets. Formally, denoting by , the prediction of , if

\[\mathrm{PE} (y _ {i} ^ {f} | Y _ {i - 1} \cup X _ {i - 1}) < \mathrm{PE} (y _ {i} ^ {f} | Y _ {i - 1})\]

then x Granger-causes y, where and are, respectively, the set of all past information on variables x and y existing at time t.

Starting from this approach, we will form predictions for the variable y by means of the nearest neighbour forecasting technique proposed by Farmer and Sidorowich (1987), together with the bivariate case presented in Fernández-Rodríguez, Sosvilla-Rivero and Andrada-Félix (1997).

The basic idea behind these predictors, inspired in the literature on forecasting in non-linear dynamic systems, is that pieces of time series sometime in the past might have a resemblance to pieces in the future. In order to generate predictions, similar patterns of behaviour are located in terms of nearest neighbours. The time evolution of these nearest neighbours is exploited to yield the desired prediction. Therefore, the procedure only uses information local to the points to be predicted and makes no attempt to fit a function to the whole time series. The general ideas behind this procedure can be found in Bajo-Rubio, Fernández-Rodríguez and Sosvilla-Rivero (1992a,b) and Fernández-Rodríguez, Sosvilla-Rivero and Andrada-Félix (1997).

More specifically, the procedure runs as follows. Beginning with the univariate case, let ( ) be a finite time series. To detect behavioural patterns in this series, segments of equal length are considered as vectors of m observations sampled from the original time series:

\[Z _ {1} ^ {m} = \left(z _ {1}, z _ {t - 1}, \dots , z _ {t - (m - 1)}\right), t = m, m + I, \dots , n\tag{1}\]

with m referred to as the embedding dimension. These m-dimensional vectors are often called m-histories, while the m-dimensional space is referred to as the phase space of time series.

The sequence of m-histories makes up a m-dimensional object that can, for a big enough m, mimic the data generation process (Takens, 1981). The proximity of two m-histories in the phase space allows us to talk of “nearest neighbours” in the dynamic behaviour of two segments in the time series .

This approach does not require stationarity in the time series , the local predictions being generated by analysing the historical paths of the vectors around the last available vector

\[Z _ {n} ^ {m} = \left(z _ {n}, z _ {n - 1}, z _ {n - 2}, \dots , z _ {n - (m - l)}\right).\tag{2}\]

Segments with similar dynamic behaviour are detected and used to produce the forecast, which is computed as some average of actually observed terms next to the segments involved. Therefore, in order to construct a local predictor we have considered the k m-histories

\[Z _ {i _ {1}} ^ {m}, Z _ {i _ {2}} ^ {m}, Z _ {i _ {3}} ^ {m}, \dots , Z _ {i _ {k}} ^ {m},\]

most similar to . The future short-term evolution of the time series will then be obtained using the information contained in the nearest neighbours found in the past.

In order to establish nearest neighbours to , we look for the closest k vectors (3) in the phase space , which maximise the function:

\[\rho (Z _ {i} ^ {m}, Z _ {n} ^ {m})\]

so that the chosen -histories present the highest serial correlation with respect to the last one, .

Once the nearest neighbours to have been established, we consider predictors of the future evolution of . A predictor is simply a rule for obtaining an estimate of , i.e., a prediction of the next observation . This is made by using some extrapolation of the observations

\[z _ {i _ {1} \cdot 1}, z _ {i _ {2} \cdot 1} \dots , z _ {i _ {k} \cdot 1}\]

subsequent to the k nearest neighbours m-histories chosen, that is to say:

\[z _ {n + 1} ^ {f} = F (z _ {i _ {1} + 1}, z _ {i _ {2} + 1}, \dots , z _ {i _ {k} + 1})\]

When generating nearest neighbours predictions, locally adjusted linear autoregressive predictions are usually employed. The procedure involves the regression by ordinary least squares of the future evolution of the k nearest neighbours chosen (5) on their current values (3). Then, the fitted coefficients are used to generate predictions for any as follows:

\[z _ {n, 1} ^ {f} = \hat {a} _ {0} z _ {n} + \hat {a} _ {1} z _ {n - 1} + \dots + \hat {a} _ {m - 1} z _ {n - (m - 1)} + \hat {a} _ {m}\]

The above approach was extended to the bivariate case by Fernández-Rodríguez, Sosvilla-Rivero and Andrada-Félix (1997). Let us consider a set of two time series: and .

We are interested in making predictions for an observation of one of these series (e.g., ), by simultaneously considering nearest neighbours in both series. To that end, we embed each of these series in the vectorial space , paying attention to the following vector:

\[(Z _ {n} ^ {m}, W _ {n} ^ {m}) \in \mathbb {R} ^ {m} \times \mathbb {R} ^ {m}\]

which gives us the last available m-history for each time series.

To establish nearest neighbours to the last m-histories , we can look for the closest k points that maximise:

\[\rho (Z _ {i} ^ {m}, Z _ {n} ^ {m}) + \rho (W _ {i} ^ {m}, W _ {n} ^ {m}), \quad i \cdot m, m + 1, \dots , n.\]

In this way, we obtain a set of k simultaneous m-histories in both series:

\[Z _ {i _ {1}} ^ {m}, Z _ {i _ {2}} ^ {m}, Z _ {i _ {3}} ^ {m}, \dots , Z _ {i _ {k}} ^ {m},\]

\[W _ {i _ {1}} ^ {m}, W _ {i _ {2}} ^ {m}, W _ {i _ {3}} ^ {m}, \dots , W _ {i _ {k}} ^ {m},\]

The predictions for and can be obtained from a linear autoregressive predictor estimated by ordinary least squares as in (7):

\[Z _ {n + 1} ^ {f} = \hat {\boldsymbol {a}} _ {0} Z _ {n} + \hat {\boldsymbol {a}} _ {1} Z _ {n - 1} + \dots + \hat {\boldsymbol {a}} _ {m - 1} Z _ {n - (m - 1)} + \hat {\boldsymbol {a}} _ {m}\tag{10a}\]

\[w _ {n - 1} ^ {f} = \hat {\boldsymbol {b}} _ {0} w _ {n} + \hat {\boldsymbol {b}} _ {1} w _ {n - 1} + \dots + \hat {\boldsymbol {b}} _ {m - 1} w _ {n - (m - 1)} + \hat {\boldsymbol {b}} _ {m}\tag{10b}\]

The difference between this predictor and that presented in (7) is that now the nearest neighbours are established using information from both series.

3. Data and empirical results

The above local predictors have been applied to daily three-month interbank interest rates of the seven countries participating at the exchange rate mechanism (ERM) of the EMS from its start: Belgium, Denmark, France, Germany, Ireland, Italy, and the Netherlands. The sample period for this group of countries runs from 13 March 1979 (the date the ERM started to operate) to 17 February 1997 (i. e., more than 6000 observations). In addition, we have repeated the exercise for the three newcomers to the ERM of the EMS: Spain, the United Kingdom and Portugal, with the sample period running from the date of their accession: 19 June 1989 for Spain, 8 October 1990 for the United Kingdom, and 9 April 1992 for Portugal. Finally, we have computed our test both for the whole sample, and also before and after the German reunification.

It should be noticed that multi-country analysis of financial series requires a special treatment for high-frequency (daily) data. Despite a relatively large number of observations, holiday effects, which differ across countries, may distort the outcomes. To avoid such a possibility, the data have been purged of this holiday effect, matching day by day the interest rate series and eliminating observations when there is no trading in any of the countries under study. Therefore, the univariate prediction results for Germany change according to the sample size used for the reference country.

Since our predictors depend on the values of the embedding dimension m and the number of k closest points in the phase space , we chose them according to Casdagli's (1991) algorithm, obtaining in our case an embedding dimension m=6 and a number of nearest neighbourhood points equal to 2% of the sample. Notice that the results are robust to the choice of m and k, since other values for these parameters gave similar qualitative results.

In the case of the seven founding members, the forecasting period runs from the last realignment in the EMS before the monetary turmoil, 13 January 1987, to the end of the sample. Our local predictors are used to produce forecasts for every change in interest rates since 14 January 1987. Every time a forecast is produced after a change in interest rates, the observation for this date is added to the sample, the models are re-estimated, and new forecasts are recursively generated for the next change in all the series until the end of the sample. In the case of Spain, the United Kingdom and Portugal, we follow the same recursive process from their respective joining dates.

The forecasting performance is measured by the root mean square error (RMSE), which is defined as follows:

\[R M S E = \sqrt {\frac {\sum_ {i = 1} ^ {r} \left(z _ {i} ^ {f} - z _ {i}\right) ^ {2}}{T}}\tag{11}\]

where z is the current value of the variable to be predicted (in our case the interest rate), is the predicted value, and T is the number of forecasts in the prediction period.

Table 1 shows the forecasting performance, measured by the RMSE, of our predictors in both versions (univariate and bivariate), for the whole period. In the bivariate case, the interest rate of Germany is used for establishing occurring analogues for each of the remaining countries, and vice versa. Then, by comparing the RMSEs, the last column reports the result of the causality test, so that if the RMSE in the bivariate case is lower (higher) than the RMSE in the univariate case, there is (there is not) causality from the first country to the second.

[Table 1 here]

As can be seen in that table, the interest rates in all the countries considered can be predicted better by adding German interest rates to the past values of the interest rates in every country, rather than by past values of national interest rates alone. On the other hand, causality is also found running from interest rates in Belgium, Denmark and the Netherlands to those in Germany.

Note also that, when two-way causality is found, the reduction in RMSEs is greater for forecasts of national interest rates using information about German interest rates than in the cases of German interest rates using information about other national rates. This could be taken as a first indication that, in these cases, the German influence on the other country is stronger than the other way round.

In order to further evaluate forecasting accuracy, in the sense of testing whether the differences between RMSEs obtained in Table I are statistically significant or not, we have used a test recently proposed by Diebold and Mariano (1995). Let and denote forecasts of the variable based on the univariate and bivariate local predictors, respectively, and let denote the loss differential between the two forecasting errors. In this way, Diebold and Mariano suggest a test of the null hypothesis that the mean loss differential is zero with an appropriate correction for serial correlation in the series:

\[S = \frac {\tilde {d}}{\sqrt {\frac {2 \pi \hat {f} _ {d} (0)}{T}}},\tag{12}\]

where is a consistent estimate of the spectral density of the loss differential at frequency 0, T is the number of forecasts and S is asymptotically distributed as a N(0,1). Therefore, a significant and positive (negative) value for S would indicate a significant difference between the two forecasting errors, which would mean a better accuracy of the bivariate (univariate) predictor.

The results are shown in Table 2. As can be seen, we reject the hypothesis of equal expected squared error (i. e., bivariate local predictors would be statistically significantly better predictors than univariate local predictors) when predicting interest rates in all the countries considered by adding the information content of the German interest rates. On the other hand, except for Belgium, Denmark and the Netherlands, we do not reject the hypothesis of equal expected squared error when predicting German interest rates based on national interest rates (i. e., bivariate local predictors would not be statistically significantly better predictors than univariate local predictors), which would suggest that forecasting accuracy for German interest rates cannot be gained by considering also the information content of national interest rates. Therefore, these results reinforce our earlier conclusion from Table 1.

[Table 2 here]

As in Gardner and Perraudin (1993) and Henry and Weidmann (1995), we have checked the possible effects on the previous results following German reunification. To this end, we have divided the sample in two parts, the breaking point being 29 November 1990 [as in Henry and Weidmann (1995)]. The reason for taking this particular point is that, although the German reunification process actually started in the end of 1989, it was only in late November 1990 when the German interest rate series registered a jump.

Tables 3 and 4 offer the results for the RMSEs and the Diebold-Mariano test, respectively. Notice that Portugal is not included, since she entered into the ERM of the EMS after the breaking point. As can be seen in Table 3, for the first subperiod we obtain similar results than those for the whole period. A different picture emerges, however, after the German reunification. Now, causality is only found running from Germany to all other countries, except for the Netherlands, where two-way causality is still detected. Again, these results are supported when performing the Diebold-Mariano test, as shown in Table 4.

[Tables 3 and 4 here]

According to these results, it would seem that German leadership in the EMS would have increased after reunification. In this way, we confirm, using different methods, earlier findings by Gardner and Perraudin (1993) and Henry and Weidmann (1995) for France, as well as extending them to the rest of the countries in our sample, except for the Netherlands.

4. Conclusions

In this paper we have provided some new evidence on the hypothesis of German leadership and asymmetric performance in the EMS. The main contribution of this paper has been the use of non-linear forecasting methods when performing causality tests, given the non-linearities detected in most financial series. On the other hand, we have used daily data on the interest rates of nine ERM members, extending the number of countries considered as compared with previously available studies. Moreover, we analyse a longer period, covering from the start of the EMS for its founding members or the date of entry for the newcomers, until 17 February 1997.

Our results suggest that, for the whole period of analysis, there is two-way causality between, on the one hand, interest rates in Belgium, Denmark and the Netherlands, and, on the other hand, those in Germany. However, for the cases of France, Ireland, Italy, Portugal, Spain and the United Kingdom, causality is only found running from Germany to those countries. It is also interesting to note that, when two-way causality is found, the forecasting improvement is greater when national interest rates are predicted using information on German interest rates, than when German interest rates are predicted using information on national rates. Overall, these results could be taken as a first indication of the special role played by Germany within the EMS, even though we cannot talk of “dominance” in a strict sense.

Next, and following previous studies on the subject, we have analysed the robustness of our results by dividing the sample in two parts, before and after German reunification. Although our conclusions are not modified for the first half of the sample, they are modified after German reunification, since two-way causality is only found in the case of the Netherlands. Therefore, this would indicate a reinforcement of German leadership in the working of the EMS following this major asymmetric shock. What this means for the future of the EMS and for the creation of the European Monetary Union remains to be seen.

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FEDEA - D.T. 97-24 by O. Bajo-Rubio, S. Sosvilla-Rivero and F. Fernández-Rodríguez

FEDEA - D.T. 97-24 by O. Bajo-Rubio, S. Sosvilla-Rivero and F. Fernández-Rodríguez

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