Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
An Empirical Examination of Heterogeneity and Switching in Foreign Exchange Markets* David Goldbaum† Remco C.J. Zwinkels‡ March 2010 Abstract Using a unique dataset of survey expectations, this paper examines the extent to which the classical fundamentalist – chartist dichotomy is valid for the foreign exchange market. By applying a recursive selection algorithm 1) respondents are classified into the two groups, and 2) the forecasting models are endogenously determined within the groups. We find that the largest part of the variation in expectations can be explained by the fundamentalist/chartist distinction. Switching, however, does not occur between fundamentalism and chartism but between a naïve and a sophisticated model that consists of both fundamentalist and chartist elements. The sophisticated model is increasingly used as the forecast horizon increases. Keywords: Heterogeneity, Discrete Choice, Foreign Exchange JEL Codes: F31, C35 * Paper prepared for Investing Strategies and Financial Market Inefficiency Paul Woolley Centre for Capital Market Dysfunctionality University of Technology, Sydney. Financial support from the Paul Woolley Centre is gratefully acknowledged. The paper was written in part while the second author was staying at the University of Technology Sydney, whose hospitality he gratefully acknowledges. The authors wish to thank Adrian Pagan, Paul de Grauwe, as well as participants of seminars at the University of Amsterdam, University of Technology Sydney, the second conference on heterogeneous agents in financial markets in Rotterdam, the 2009 SNDE meeting in Atlanta, …... † School of Finance and Economics, University of Technology Sydney; PO Box 123 Broadway; NSW 2007 Australia; email: [email protected]. ‡ Erasmus School of Economics, Erasmus University Rotterdam; PO Box 1738, 3000DR, Rotterdam, The Netherlands; email: [email protected]. 1 1. Introduction A substantial body of literature in economics and finance models investors as heterogeneous and adaptive. The heterogeneity allows for interactions between traders behaving differently to impact the market. The heterogeneity can exist in a market at equilibrium or may keep the market out of equilibrium. The adaptation allows traders to select behavior appropriate for the perceived, possibly changing, market setting. The sensitivity of the market to the behavior of the traders can produce market destabilizing feedback loops. Models based on adaptive heterogeneous agents have offered insight explaining a variety of market phenomena that are difficult to capture with representative agent models. In financial markets, these include fat tails in returns, volatility clustering without auto-correlation in returns, bubbles, excess volatility, and slow mean reversion; see e.g. Lux (1998), De Grauwe and Grimaldi (2006). Trader heterogeneity manifests in a variety of characteristics. A number of papers emerged to explore models of dynamic heterogeneity. Some models consider different levels of trader sophistication; for example Brock and Hommes (1998). A sophisticated trader might employ rational expectations when forecasting the behavior of market prices while other traders employ a more naïve strategy. Alternatively, a model might explore the heterogeneity in information. A fundamental approach might engage in research in order to gain a private signal about future value while the market-based approach attempts to extract information from the price, as in Goldbaum (2003) and De Grauwe and Grimaldi (2005, 2006). A theoretical foundation for sustainable market-based trading strategies (technical trading, charting) is rooted in Grossman and Stiglitz’s seminal “On the Impossibility of Informationally Efficient Markets” (1980). Their paper established an equilibrium market condition in which market-based (uninformed) traders coexist with and depend on fundamentalist (informed) traders. In Grossman and Stiglitz, the uninformed traders are fully rational, and their presence in the market is based on their ability to extract information from the price. Dynamics arise as traders switch between available information strategies or levels of sophistication. The agents of the Brock and Hommes (1997) and Brock and Hommes (1998) models consider the relative performance of different forecasting strategies. The models employ the random element in the discrete choice model of -2- Manski and McFadden (1981) to create heterogeneity in the individual level choice among the available options. The environment highlights the inherent instability of markets. The strategy that is in the minority performs better, but the superior performance attracts members of the population. Goldbaum (2005) introduces evolution in the strategies that are available to traders. The evolution reflects the effort by traders to improve the performance of inherently imperfect trading tools. A market populated by learning and adaptive traders has the potential of transitioning the market from one of price stability noisily reflecting the efficient market price, to a market in which the price is unstable and divergent from the fundamental value. While heterogeneous adaptive agents models provide intuitively appealing explanations for market phenomenon, do these explanations stand up empirically? If heterogeneity exists, is it dynamic and can the evolution be captured by a model of behavior? What dynamic model is most consistent with behavior? At present, there are two major classes of models, offering different behavior in the population. The parameters of these models, in particular the intensity of choice parameter, determine the existence, uniqueness, and stability of the market equilibrium. Estimating the parameters of these dynamic models can offer considerable insight into market behavior. Finally, are the strategies being employed by traders static, even as the proportion of the population employing them change, or are the strategies themselves also evolving? This paper contributes to the still emerging literature that empirically examines markets based on heterogeneous adaptive agent models. Only a handful of papers have sought to estimate these models. Included among these is Boswijk, Hommes, and Manzan (2007) that finds evidence of switching by traders between a trend following and mean reverting rule in the S&P500. Goldbaum and Mizrach (2008) model the distribution of new funds between active and passively managed mutual funds to estimate the intensity of choice model. The success of the model in capturing the shift towards passively managed funds is evidence in favor of adaptive heterogeneity. De Jong et al. (2009) present empirical evidence of behavioral heterogeneity in equity prices using a multi-market setting. Frijns et al. (2010) illustrate the importance of allowing a multitude of strategies in pricing options. -3- Evidence in favor of switching has also been found in experimental settings. Experiments involving market entry decisions often find a wide range of strategies have been employed by the participants that still combined to bring the market to the equilibrium number of entrants. Hommes et al (2007) have their participants forecast an endogenously determined price that is influenced by their own forecast and the forecast of the other participants. The participants are rewarded for accuracy the accuracy of their forecasts. The authors identify four rule of thumb strategies employed by participants. Hommes and Anufriev (2007) extend the analysis by modeling the switching between strategies. Branch (2004) empirically tests an adaptive heterogeneous agent model based on survey respondents’ reported inflation forecasts. Branch models the population as switching between three different models differentiated by there implicit level of sophistication. Again, evidence is found in support of a switching model where households respond to adopt the strategy that has performed well in the recent past. MacDonald and Marsh (1996) document, also on the basis of survey data, that market participants hold different beliefs on future price movements, and use different types of models to form expectations. A number of issues remain unresolved or are in need of empirical support. The current project also seeks to examine markets for evidence of adaptive heterogeneity and also to determine whether there is evidence in favor of the fundamentalist – chartist dichotomy in foreign exchange markets. De Grauwe and Grimaldi (2006) and De Grauwe and Markiewicz (2008) study heterogeneous agents and adaptation in foreign exchange markets and show that they are well capable of explaining the stylized facts. Similar to Branch, the current project seeks to model the reported forecast of survey participants. In this case, the data being employed is the exchange rate forecasts collected from participating international banks. Each period includes forecasts over a number of horizons for a number in individual institutions. Using the same data, Jongen et al. (2008) show that expectations are dispersed, and that panelists base expectations on fundamentalist/chartist types of considerations. The evaluation offers the possibility of explaining market deviations as they result from disequilibrium produced by competitive profit seeking in an environment of imperfect information and adaptation. The remainder of the paper is organized as follows. Section 2 presents the underlying model. Section 3 introduces the survey data used in the empirical section -4- and Section 4 explains the empirical methodology applied. In Section 5 we present the results, and Section 6 concludes. 2. Model Each of N traders in a market maximize an expected negative exponential utility function in next period’s wealth, Wt 1 , based on their information set, I ti . Formally, max E (U (Wi ,t 1 ) | I ti ) at ,bt subject to Wi ,t ai.t st bi.t Wi ,t 1 (1 rt* ) st 1 ai.t (1 rt )bi.t where U (Wt ) exp(Wt ) , st is the spot exchange rate, rt is the domestic interest rate, and rt* is the foreign exchange rate. Solving produces the optimal demand for the foreign currency, (1 rt* ) E ( st 1 | I ti ) (1 rt ) st ai.t i2,t (1) where i2,t (1 rt* ) 2 var( st 1 | I ti ) . A market clearing Walrasian equilibrium requires supply equals demand, N a i 1 Let xt X t / N and at 1 N i ,t Xt . (2) N a i 1 i ,t be the per capita supply and demand respectively for the foreign currency so that (2) can be expressed as at xt . Consider a market in which the population of traders is informed by two models of exchange rate determination. The fundamental approach presumes that the market is driven by fundamentals. This may include notions of purchasing power parity (PPP) or interest rate parity (IPP), among other fundamental determinants. A trader relying on fundamentals trades in the currency market seeking to take advantage of exchange rate deviations from the fundamentals. A chartist approach employs past exchange rate innovations as a predictor for future innovations. The chartist trades according to the predictions of the chartist approach. -5- 2.1 Fundamental model There is a fundamental exchange rate, st* . The realized market spot rate, st , can deviate from the fundamental. The market has a tendency to revert to the fundamental rate so that future innovations in the market spot rate are affected by the current deviation. The fundamental traders form expectations about future innovations accordingly, Et f (st 1 ) ( st st* ) . (3) Here, captures the rate at which the market reverts towards fundamentals. In a similar environment, DeGrauwe and Grimaldi (2006) model the fundamental rate as an exogenous process following a random walk. Their traders know the st 1 and st*1 as the most recently observed values of the spot and fundamental rates Capturing the forecast reports employed in the empirical section requires modeling the k period ahead forecast of exchange rates. Let Z t f represent the vector of time t fundamental information. Further, let t st k represents spot market innovation st k st . For integer k 1 , the fundamental forecast is captured by the following process: Et f ( t st k ) ' Z t f 1f Et 1 ( t 1 st k 1 ) 2f Et 1 ( t 1 st ) . (4) the second term on the right hand side is present to take advantage of the overlap in the prediction period from the forecast made in period t 1 and the current period t forecast. The third term controls for information in Et 1 ( t 1 st k 1 ) that is not useful in forecasting Et f ( t st k ) since st has already materialized such that potentially useful information in it is incorporated into Z t f ; this reduces noise and increases the usefulness in employing Et 1 ( t 1 st k 1 ) as a control variable that is reported in the survey of predictions. The first term is left to explain only the innovation in the forecast from the previous period. It is thus capturing the new component of the -6- forecast period, Et ( t k 1 st k ) as well as any change in the forecast of t st k 1 that flows from the new time t information. Individual trader fundamental forecasts are captured by the following: Ei ,ft ( t st k ) ' Z t f 1f Ei ,t 1 ( t 1 st k 1 ) 2f Ei ,t 1 ( t 1 st ) i ,t (5) There are thus two sources for heterogeneity among the fundamental traders. The idiosyncratic term, i ,t , captures trader specific differences between the forecasts of individual traders. These can be seen as the result of private information not available to the modeler, deviation in the objective function from the presumed utility function, or simply the result of randomness in the traders forecasting method. The presence of i ,t plus the fact that individuals can have different choice patterns cause the different traders to have individual forecasts histories that appear in the second and third term on the RHS of (5). 2.2 Chartist information Chartist information is composed on past market information, namely previous innovations in the exchange rate. Let Z tc represent the vector of time t chartist information. The chartist forecast is captured by the following process: Etc ( t st k ) ' Z tc 1c Et 1 ( t 1 st k 1 ) c2 Et 1 ( t 1 st ) (6) Individual forecasts are captured by Eic,t ( t st k ) ' Z tc 1c Ei ,t 1 ( t 1 st k 1 ) c2 Ei ,t 1 ( t 1 st ) i ,t , (7) thereby capturing the same sources of heterogeneity that exists among the fundamentalists. 2.3 Discrete choice -7- Equations (5) and (7) capturing the forecasts of individuals will be examined in a variety of settings. Included is an environment that allows each individual trader to choose which strategy to employ for each given period. In the empirical examination, a forecast is labeled as either a fundamentally derived forecast or a chartist forecast based on its relative proximity to systematic component of (5) or (7). Let i ,t 1 if the forecast by individual i is deemed to originate from the fundamental strategy, with i ,t 0 otherwise. Further, let Eˆ i ,ft ( t st k ) ' Z t f 1f Ei ,t 1 ( t 1 st k 1 ) 2f Ei ,t 1 ( t 1 st ) (8) Eˆ ic,t ( t st k ) ' Z tc 1c Ei ,t 1 ( t 1 st k 1 ) c2 Ei ,t 1 ( t 1 st ) (9) and represent the systematic components of each model. Thus, Ei ,t ( t st k ) i ,t ( ' Z t f 1f Ei ,t 1 ( t 1 st k 1 ) 2f Ei ,t 1 ( t 1 st )) (1 i ,t )( ' Z tc 1c Ei ,t 1 ( t 1 st k 1 ) c2 Ei ,t 1 ( t 1 st )) i ,t , (10) where 1 if E ( s ) Eˆ f ( s ) 2 E ( s ) Eˆ c ( s ) 2 i ,t t t k i ,t t t k i ,t t t k i ,t t t k . i ,t 2 2 0 if Ei ,t ( t st k ) Eˆ ic,t ( t st k ) Ei ,t ( t st k ) Eˆ i ,ft ( t st k ) (11) 3. Data 3.1 Survey Expectations To investigate the behavioral aspects of the forecasts of market participants, we use a unique database of survey-based exchange rate forecasts. The individual forecasts are obtained from a survey conducted by Consensus Economics of London on a monthly basis among leading market participants in the foreign exchange market, investment banks, and professional forecasting agencies. Examples of panelist companies are Morgan Stanley, Oxford Economic Forecasting, Deutsche Bank Research, and BNP -8- Paribas. The panelists companies are located worldwide, although they are all from developed economies. The forecasts are point forecasts for a large set of currencies against the U.S. dollar and are available for horizons of 1, 3 and 12 months ahead. The names of the panelist companies are revealed. Although survey participants have a few days time to return their forecasts, we learned that the vast majority send their responses by e-mail on the Friday before the publication day, which is typically the second Monday of the month. We consider this Friday to be the day on which the forecasts are formed and assume that the beliefs are translated one-to-one in a point forecast. To verify that the information sets of market participants are not too diverse, all of the analyses throughout this study were reestimated using spot data from various days surrounding this Friday, yet the overall results remain virtually unchanged. There may be reasons for panelists not to reveal their true beliefs, though. One motive may be that agents do not want to expose their (private) information to other market participants. This effect may be mitigated by the reputation effect that this survey can have. When the names of the forecasters are given in the survey publication (as is the case with our data), agents have an incentive to perform well in order to attract transactions. In this study we use the forecasts for the Japanese Yen and the Euro4 against the U.S. dollar (i.e., in foreign currency per US Dollar) from 31 respondents for the period of November 1995 through December 2007, which are 146 monthly observations.5 This period is of particular interest since it contains several financial crises, the introduction of a single monetary currency unit, and several large changes in the level of the exchange rates. The panel is unbalanced since the response rate of the individual market participants is less than 100 percent and since market participants left the panel and were replaced by others. Analyses are done on the 1, 3, and 12 months forecasting horizon in order to distinguish between the short-, middle-, and long-run. The 1 month forecasts are also used as a control variable for the models of the 3 and 12 months horizon (see Section 2). 4 Our database also contains U.K. Pound expectations. However, because due to to unknown reasons the U.K. Pound expectations are only reported every other month, we do not use this currency because of the limited number of observations. 5 Prior to January 1999 we use forecasts on the Deutschemark versus the U.S. Dollar. We transform these forecasts into Euro / U.S. dollar forecasts using the official conversion rate. -9- < Insert Table 1 Here > Table 1 presents the descriptive statistics of the survey data. Respondents are consistent across currencies and forecast horizons in answering the survey. The median response rate per period is 70%, which results in an average total number of observations per currency/horizon pair of approximately 2900. The descriptive statistics of the expected exchange rate returns in panel b) indicate there is a wide variety in answers, ranging from -30 to +45%. The kurtosis indicates that, as is the case for market returns, the distribution of expectations is heavy tailed. The expected returns are heavy auto correlated, which is inconsistent with actual FOREX returns. Partly, the autocorrelation in expectations is due to overlapping observations, i.e., the frequency of the data is higher than the forecast horizon. The autocorrelations for the one-month forecast horizon, however, show that this is not the complete story. 3.2 Forecasting Rules All exogenous data is retrieved from Datastream. The chartist information Z tc includes a constant and z1,c t , which is the most recent innovation in the spot rate, t 1 st st st 1 . The fundamentalist information Z t f includes a constant and a term representing the partial mean reversion towards the fundamental value, ( st st* ) . Important subsequent question is the choice of fundamental exchange rate st* . The issue here is not necessarily to select a model that actually performs well in forecasting the exchange rate, but that is perceived by the panellist to have a certain added value in forecasting the exchange rate. In addition, the model should be such that the panellists could have reasonably been able implement it practically. As such, we use a version of the monetary model introduced by Mark (1995) given by s t* (mt mt* ) ( y t y t* ) , - 10 - (12) in which mt is the home money supply, mt* the foreign money supply, yt the home income, and y t* the foreign income. The choice for this model is based on two arguments. First, the study by Mark (1995) is one of the most well known and persuasive studies illustrating the forecasting power of a fundamental model. Second, this fundamental value is relatively simple to implement; that is, it does not require any further estimation. This makes sure that we do not have to make any additional choices or assumptions regarding the estimation process that could take us (further) away from the fundamental value actually applied by panellists. Data wise, we use M2 for money supply m and industrial production for income y. Figure 1 displays the (log) exchange rates and fundamental exchange rates for both the Yen and the Euro. < Insert Figure 1 Here > 4. Methodology Unique to the current examination (to our knowledge) is the fact that different models under consideration are endogenous to the traders employing them. Branch (2004) for example, considers three exogenous models of inflation. His naïve expectation model has te1 t . The two more sophisticated models are a model of adaptive expectations and a VAR. In both cases, the parameters of the model are chosen to fit the data, so that the model is optimized to minimize the error of the forecast of inflation, rather than to capture the model employed by the forecaster. Our objective is to have those traders employing the model indicate the parameters of the model. This is accomplished by choosing the parameters to minimize the mean squared error of the forecast by those traders who employ the forecast. This involves some degree of simultaneity as the estimation of the model depends on how the individuals are sorted and the sorting depends on the model. Our solution is to estimate, sort and then re-estimate over a number of iterations until the sorting and the model parameters settle. Formally, the estimation procedure for the discrete choice model is as follows: The model is estimated in a single equation using simple OLS. The initial distribution of agents over groups (or initial determination of weights) is done by estimating the two expectation formation models individually per respondent. Based on best fit, each respondent is subsequently classified as either fundamentalist or chartist. There exists - 11 - a certain path dependency conditional on the initial distribution of agents. We feel, however, that this procedure yields the best results in that the final fit is maximized and the initial distribution is credible as it is based on individual estimates. Throughout, we perform a grid search to find the global optimum; we report the solution with the best fit. Next, the two rules are estimated in a single equation in a pooled setup, using the initial distribution of respondents. The distribution of respondents across groups is subsequently updated based on the new estimation results, and the system is again estimated. This procedure is repeated until convergence, i.e. until respondents do not change groups anymore and coefficient estimates of the rules are constant. Generally, this occurs within ten iterations, conditional on the complexity of the model. As such, the classification of agents and the actual expectation formation rules are being learned endogenously in the iteration process. The autocorrelation in the residuals due to the overlapping data issue is accounted for by calculating Newey-West standard errors, as in MacDonald (2000). 5. Results A benchmark version of the model is estimated without weights or division of respondents. Both rules are estimated for the full sample of respondents and time. The results are presented in Table 2. < Insert Table 2 Here > The estimation results in Table 2 generally indicate that both fundamentalist and chartist information sources are being used significantly in forming survey expectations. The model is able to capture a considerable amount of variation in the expectations. The fit increases notably on the longer forecast horizons. The fundamentalists act stabilizing vis-à-vis the fundamental exchange rate in all but one cases; α<0. In three cases, this effect is significant. For the Euro, the mean reversion clearly increases with the forecast horizon. For the Yen this is the case for the 1 and 3 months horizon, but not for the 12 months horizon. The chartist coefficient β is negative and highly significant for all currency-horizon combinations; this implies that panellists expect a strong reversion of previous returns and therefore act like contrarians. The reversion increases on the longer horizons. This is consistent - 12 - with the literature on survey expectations; see MacDonald (2000) for an overview6. The lagged and the 1-month expectations, γ1 and γ2, are both highly significant and carry the expected signs. This implies that panellists are depending heavily on last period’s expectation due to the fact that the sampling frequency is higher than the forecasting horizon. Also, expectations are being updated consistently with regards to the materialized previous 1-month expectation. 5.1 Discrete choice estimation Table 3 presents the results of the model with static discrete weights. That is, panellists are classified as either fundamentalist or chartist for the entire period covered by the survey. The modified model being estimated is Ei ,t ( t st k ) i ( ' Z t f 1f Ei ,t 1 ( t 1 st k 1 ) 2f Ei ,t 1 ( t 1 st )) (10’) (1 i )( ' Z tc 1c Ei ,t 1 ( t 1 st k 1 ) 2c Ei ,t 1 ( t 1 s t )) i ,t where T T 2 2 f ˆ ˆ c ( s ) 1 if ( ) ( ) ( ) E s E s E s E i t t t k i t t t k i t t t k i t t t k , , , , t 1 t 1 i (11’) T T 2 2 c f 0 if ˆ ˆ Ei ,t ( t st k ) Ei ,t ( t st k ) Ei ,t ( t st k ) Ei ,t ( t st k ) t 1 t 1 < Insert Table 3 Here > Comparing the results from Table 3 with those of the benchmark model in Table 2, there are a number of differences. First, with regards to the fundamentalists’ mean reversion, we observe that the effect size becomes notably stronger. The significance level, though, does not gain much. As for the chartist coefficient β, the effect size is also stronger compared to the benchmark model. The significance levels are equally high. The estimates for the lagged expectations, finally, remain comparable; 1f is typically somewhat smaller than 1c , while 2f is typically somewhat larger than 2c . Strikingly, the fit of the model has hardly improved after 6 On shorter horizons, expectations are found to be stabilizing or exhibit bandwagon effects. Apparently, the 1-month horizon is perceived to be the long run. - 13 - introducing static weights. The increase in R2 does not weigh up to the increased number of estimated coefficients. The fit of the model for the 3 months Euro forecast even decreases substantially. A final interesting observation is that the percentage of panellists using the fundamentalist rule actually decreases as the forecast horizon increases; from 35 to 13% for the Euro and from 21 to 5% for the Yen. < Insert Table 4 Here > Table 4 reports the results of estimating the original model captured by (10) and (11) in which panellists are allowed to update their forecasting strategy each period. Hence, instead of considering the average distance between the rule and the expectation over the full sample period, as in (11’) and Table 3, the selection procedure is applied per period. The increased flexibility changes the estimation results substantially. The fundamentalist mean reversion looses its significance for the Euro. For the Yen, on the other hand, we observe an increase in significance and effect size for both the 3 and the 12 months horizon. The chartist contrarian behaviour becomes stronger for both currencies and all horizons. The effect sizes of the auto-regressive terms for fundamentalists, 1f and 2f , decrease substantially compared to the static setup in Table 3. The autoregressive term for the 1 month Yen forecast also loses its significance. For chartists, the opposite occurs and the effect sizes of the autoregressive terms increase substantially. The fit of the model is substantially higher in the dynamic case compared to the static case. Whilst the model fit did not change when moving from the benchmark model to the model with static classification, the added flexibility of having dynamic classification does matter notably. The percentage of panellist using the fundamental rule is generally larger than in the static case, but still smaller than fifty percent. Fundamentalism is still less common for the long than the short horizon. The autocorrelation in the chosen rule is low. This means that strategies are chosen each period, independent of the choice in the previous period. - 14 - One can draw a number of conclusions from the estimation results in Tables 2 through 4. First of all, both the fundamentalist and the chartist forecasting rule are being used by the respondents in the survey. The fundamentalist-chartist dichotomy often put forward in the literature therefore appears a relevant classification, consistent with the findings of, among others, Allen and Taylor (1990, 1992), and Jongen et al. (2008). Another interesting finding is that panellists lean heavily on their previous period’s expectation, 1 0 . This makes sense as the period over which the expectation is formed coincides for two and eleven periods with previous period’s expectation for the 3 and 12 months forecast horizons. The same pattern, however, is found for the 1-month forecast horizon where the overlapping data issue is not present. Also, panellists’ expectations are consistently updated relative to previous period’s expectation by subtracting the 1-month expectation of period t-1; 2 0 . The flexibility of agents to change strategy is of great importance. There is a substantial improvement in the fit after introducing switching. This is direct evidence in favour of the heterogeneous agents models with switching, as introduced in Brock and Hommes (1997, 1998). A final important finding here is that chartism is dominant. More than half of the panellists are classified as being chartist. Again, this is consistent with Allen and Taylor (1992), who state that 90% of market participants use some sort of technical analysis. 5.2 Robustness Two results regarding the estimated behaviour of fundamentalists from the three different models are puzzling. First, the estimated α is typically negative, but the size and significance changes strikingly. This is especially the case for the Euro, where we find no significant α. Second remarkable observation is the fact that the percentage of fundamentalists decreases as the forecast horizon increases. This is inconsistent with the existing literature. In order to look deeper into the behaviour of the panellists, we proceed along two lines. First, to ascertain ourselves of the validity of the estimation procedure we simulate data and apply the estimation procedure on the generated numbers. Second, to check the functional forms of the different groups, we estimate the different models on the different groups separately. - 15 - The simulation model is calibrated to mirror the benchmark model of the 1month Yen forecasts. That is, we use the estimate coefficients from Table 2 and the mean and variance characteristics of the Japanese fundamental exchange rate, the market exchange rate, and the residuals from the benchmark model to simulate the s ) given by exchange rate expectations E ( ~ t t t 1 Et ( t ~ st 1 ) c f ( st s t* ) s t Et 1 ( t 1 ~ st ) t . (13) The model is simulated for 1,000 periods and 15 individuals. Subsequently, we estimate the model given by (10) and (11) using the data generated by (13). Because the data generating process is static, i.e., has no switching between rules, the econometric model is misspecified. Comparing the estimation results of the simulated data with those of the actual data gives a clue about the true data generating process underlying the survey expectations. Table 5 presents the estimation results of the models applied to simulated data. We do not only show the benchmark and switching models, but also the fundamental, chartist, and benchmark models estimated for the separate groups of fundamentalists and chartists. < Insert Table 5 Here > The first column representing the benchmark model is highly comparable to the estimation results of the 1 month Yen forecast in Table 2. This indicates that the simulations are correct. The second column subsequently presents the estimation results of the full switching model. As was the case for the actual 1-month Yen forecasts, the fit of the model increases substantially. Clearly different compared to the results for the actual data, is that α gains substantially in terms of significance compared to the benchmark model. Also, the γ2’s of the two groups are equal and not significantly different from the estimated value for the benchmark model. The latter two results are clear indications that the actual data is generated by a non-static data generating process. Estimating the fundamental, chartist, and benchmark models for the separate groups of fundamentalists and chartists, displayed in columns 3 to 8, yields very - 16 - similar results. That is, if we estimate the fundamental model for either the group of fundamentalists or the group of chartists (columns 3 and 4), gives the same parameter estimates. This is exactly as it should be, reflecting the simulated model. The same conclusion holds for the chartist model and the benchmark model. < Insert Table 6 Here > Performing the same analysis on the actual 1-month Yen forecasts gives a completely different picture. The first two columns of Table 6 give the estimation results of the fundamental model applied to the group of fundamentalists and chartists. First of all, the coefficients are not equal across groups, indicating that the selection procedure does not result in spurious results. Interestingly, though, it is not the fundamentalists for who we find a significant α, but the chartists. Also striking is the large difference between the groups in the estimated γ2 and the model fit. The same conclusions can be drawn after estimating the chartist model for the groups of fundamentalists and chartists separately, as presented in columns 3 and 4. The estimated β is significant for the chartists, and insignificant for fundamentalists. Again, γ2 is much smaller for fundamentalists. The final two columns, representing the benchmark model estimated for the two groups, basically combine the aforementioned results: significance and a high fit for chartists, not for fundamentalists. The final conclusion to be drawn from these analyses is twofold. First of all, we have ascertained ourselves of the fact that the selection and classification algorithm we apply does not yield spurious results. Second, the panellists in the survey do apply two different models, but they are not a fundamentalist and a chartist model. Instead, they are a naïve or random walk model and a sophisticated model, consisting of both fundamentalist and chartist characteristics. This directly explains the two puzzling results from Table 4: i) The size and significance of α varies greatly because it is estimated for the wrong group of panellists, and ii) the percentage of panellists using the fundamentalist model decreases as the forecast horizon increases because the naïve model is used predominantly for the shorter forecast horizons, and the sophisticated model is used increasingly as the forecast horizon increases. - 17 - 7. Conclusion A model has been developed to examine the behaviour of banks when forming forecasts of future exchange rate innovations over a variety of time horizons. The model allows for market participants to switch between different strategies for forming expectations. Two model based on two strategies is developed. The two strategies examined are a fundamental strategy by which predictions concerning future exchange rates are based on exchange rate fundamentals, and a chartist strategy by which market based information serves as a predictor of future exchange rates. The empirical analysis suggests that the switching model is useful for explaining the heterogeneity in the forecasts of the different banking institutions that took part in the survey. Allowing the banks to switch strategies during the sample period improved the fit of the model. It also provides an attractive narrative of market behaviour that is consistent with stylized facts. The switching, however, does not occur between the fundamental model and the chartist model, but between a naïve model and a sophisticated model consisting of both fundamental and chartist elements. Allen and Taylor (1990) document the use of chartist techniques among foreign exchange traders. Individual traders explain that it is not necessarily that they believe that charting captures fundamentals, but that the market can be driven by chartists since they are so plentiful in the foreign exchange markets. For this reason, it is important to include chartist tools when considering trades. Presumably, the same is true when forming predictions. The fact that bank forecasts appear to be driven, at times, by a chartist models may be a reflection of the fact that bank believe that the market based information is informative about market innovations away from fundamentals. The results could also be considered supportive of the notion that market based information is useful for predicting fundamental innovations supported by private information not available to the modeller. The latter interpretation is consistent with Grossman and Stiglitz (1980) and other papers that argue in favour of the use of chartists techniques to extract information from the market. - 18 - Bibliography Allen, H. and M.P. Taylor (1990). Charts, Noise and Fundamentals in the London Foreign Exchange Market, Economic Journal 100(400): 49-59. Boswijk, H.P., C.H. Hommes and S. Manzan (2007). Behavioral Heterogeneity in Stock Prices, Journal of Economic Dynamics and control 31: 1938-1970. Branch, W.A (2004). The Theory of Rational Heterogeneous Expectations: Evidence from Survey Data on Inflation and Expectations. The Economic Journal 114: 592–621 Brock, W. and C.H. Hommes (1997). A Rational Route to Randomness, Econometrica 69: 1059-1095. Brock, W. and C.H. Hommes (1998). Heterogeneous Beliefs and Routes to Chaos in a Simple Asset Pricing Model, Journal of Economic Dynamics and Control 22: 1231274. Taylor, M.P. and H. Allen (1992). The Use of Technical Analysis in the Foreign Exchange Market, Journal of International Money and Finance 11(3): 304-314. Hommes, C.H., Sonnemans, J., Tuinstra, J. and Velden, H. van de, (2007), Learning in cobweb experiments, Macroeconomic Dynamics 11 (Supplement 1), 8-33. MacDonald, R. and I.W. Marsh (1996). Currency Forecasters are Heterogeneous: Confirmation and Consequences, Journal of International Money and Finance 15(5): 665-685. Frijns, B., T. Lehnert, and R.C.J. Zwinkels (2010). Behavioral Heterogeneity in the Option Market, Journal of Economic Dynamics and Control, forthcoming. Goldbaum, D. (2005). Market Efficiency and Learning in an Endogenously Unstable Environment, Journal of Economic Dynamics and Control 29: 953-978. - 19 - De Grauwe, P. and M. Grimaldi (2005). Heterogeneity of Agents, Transaction Costs and the Exchange Rate, Journal of Economic Dynamics and Control 29: 691-719. De Grauwe, P. and M. Grimaldi (2006). Exchange Rate Puzzles: A Tale of Switching Attractors, European Economic Review 50(1): 1-33. De Grauwe, P. and A. Markiewicz (2008). Learning to Forecast the Exchange Rate: Two Competing Approaches, CESifo Working Paper 1747. Grossman, S.J., Stiglitz, J.E., 1980. On the impossibility of informationally efficient markets. The American Economic Review 70(3), 393-408. De Jong, E., W.F.C. Verschoor, and R.C.J. Zwinkels (2009). Behavioral Heterogeneity and Shift-Contagion: Evidence from the Asia Crisis, Journal of Economic Dynamics and Control 33(11): 1929 – 1943. Lux, T. (1998). The Socio-Economic Dynamics of Speculative Markets: Interacting Agents, Chaos and Fat Tails of Return Distributions, Journal of Economic Behavior and Organization 33: 143-165. Mark, N.C. (1995). Exchange Rates and Fundamentals: Evidence on Long-Horizon Predictability, American Economic Review 85(1): 201-218. Manski, C.F., McFadden, D., 1981. Structural Analysis of Discrete Data with Econometric Applications (MIT Press, Cambridge, MA). Jongen, R., C.C.P. Wolff, W.F.C. Verschoor, and R.C.J. Zwinkels (2008). Dispersion of Beliefs in Foreign Exchange, - 20 - CEPR Discussion Paper 6738. Tables and figures Table 1: Data 1 Month Euro 3 Months 12 Months 1 Month Yen 3 Months 12 Months a) # Observations Min. # panelists / period Max. # panelists / period Median # panelists / period 12 24 19 14 24 20 14 24 20 13 24 19 15 24 20 15 24 20 Min. # periods / panelist Max. # periods / panelist Median # periods / panelist 3 143 101 12 144 102 12 144 102 3 143 102 12 144 103 12 144 103 2825 2932 2930 2835 2941 2940 Total # observations b) Descriptive statistics Median -0.0036 -0.0118 -0.0397 -0.0030 -0.0060 -0.0343 Maximum 0.1136 0.1366 0.2110 0.1985 0.3331 0.4512 Minimum -0.1700 -0.2036 -0.2883 -0.1667 -0.1667 -0.3050 Standard deviation 0.0246 0.0370 0.0686 0.0301 0.0473 0.0857 Skewness -0.3430 -0.2080 0.0565 0.3216 0.3332 0.5532 Kurtosis 5.7290 3.9952 2.8560 6.1241 4.7625 4.3544 Autocorrelation (1st lag) 0.3900 0.5090 0.7550 0.4170 0.5710 0.7410 Notes: Table presents the number of observations per period and per respondent (Panel a) as well as the descriptive statistics of the expected log-changes in the exchange rate, i.e. ln( E i ,t s t 1 ) ln(s t ) over all panellists and periods (Panel b). - 21 - Table 2: Benchmark Model c α β 1 Month -0.0033*** (0.0004) Euro 3 Months -0.0052*** (0.0006) 12 Months -0.0058*** (0.0007) 1 Month -0.0020*** (0.0005) Yen 3 Months -0.0028*** (0.0006) 12 Months -0.0042*** (0.0008) 0.0012 (0.0029) -0.2468*** (0.0151) -0.0028 (0.0038) -0.3245*** (0.0195) -0.0147*** (0.0041) -0.5807*** (0.0204) -0.0079* (0.0042) -0.3560*** (0.0141) -0.0099* (0.0057) -0.4720*** (0.0189) -0.0061 (0.0069) -0.6862*** (0.0223) 0.7063*** (0.0249) -0.1982*** (0.0372) 0.8835*** (0.0098) -0.2055*** (0.0254) 0.7383*** (0.0244) -0.1556*** (0.0378) 0.8785*** (0.0106) -0.113*** (0.0283) 1 NA 2 0.4280*** (0.0175) NA 0.4728*** (0.0158) 0.2747 0.4509 0.8198 0.3938 0.5459 0.8026 adj. R2 # Obs. 2494 2515 2512 2510 2528 2527 Notes: Table presents estimation results for the benchmark model. Newey-West standard errors are given in parentheses. *, **, and *** represents significance at the 10, 5, and 1% level, respectively. - 22 - Table 3: Static Weights 1 Month Fundamentalists -0.0009 cf (0.0007) -0.0039 α (0.0048) 1f f 2 NA 0.4036*** (0.0310) Euro 3 Months 12 Months 1 Month -0.0013 (0.0009) -0.0100* (0.0063) 0.6263*** (0.0413) -0.2215*** (0.0639) 0.0009 (0.0016) -0.0205* (0.0113) 0.8700*** (0.0270) -0.3543*** (0.0676) -0.0012 (0.0010) -0.0030 (0.0096) -0.0076*** (0.0008) -0.4523*** (0.0236) 0.7250*** (0.0300) -0.2019*** (0.0443) -0.0063*** (0.0007) -0.6515*** (0.0215) 0.8931*** (0.0099) -0.1875 (0.0270) NA 0.4197*** (0.0457) Yen 3 Months 12 Months -0.0002 (0.0017) -0.0033 (0.0161) 0.7092*** (0.0666) -0.2237* (0.1171) -0.0030 (0.0038) -0.0875** (0.0437) 0.8242*** (0.0476) -0.4112*** (0.1126) -0.0032*** (0.0007) -0.5342*** (0.0201) 0.7373*** (0.0258) -0.1420*** (0.0397) -0.0041*** (0.0008) -0.7030*** (0.0224) 0.8824*** (0.0105) -0.0948*** (0.0292) Chartists cc β 1c c 2 -0.0046*** (0.0005) -0.3419*** (0.0183) NA 0.4273*** (0.0208) -0.0022*** (0.0005) -0.4286*** (0.0155) NA 0.4981*** (0.0163) adj.R2 0.2991 0.3614 0.8241 0.4136 0.5553 0.8039 % fun 34.70 15.71 12.94 20.96 13.88 4.59 Notes: Table presents estimation results for the model with static discrete weights. R2 is adjusted R2; standard errors in parenthesis; % fun is the percentage of panellists using the fundamentalist rule. 23 Table 4: Dynamic Weights 1 Month Fundamentalists -0.0010** cf (0.0004) 0.0004 α (0.0028) 1f f 2 NA 0.1135*** (0.0167) Chartists cc -0.0045*** (0.0005) β -0.7692*** (0.0170) 1c c 2 NA 0.8329*** (0.0182) Euro 3 Months 12 Months 1 Month -0.0075*** (0.0006) -0.0021 (0.0040) 0.3566*** (0.0252) -0.2457*** (0.0378) -0.0063*** (0.0008) -0.0029 (0.0056) 0.6956*** (0.0130) -0.5556*** (0.0323) 0.0012*** (0.0005) 0.0019 (0.0043) -0.0028*** (0.0005) -0.8411*** (0.0196) 1.0693*** (0.0241) -0.2124*** (0.0363) -0.0023*** (0.0006) -0.9683*** (0.0183) 0.9701*** (0.0080) -0.0093 (0.0225) -0.0050*** (0.0005) -0.7900*** (0.0147) NA 0.1924 (0.0152) NA 0.8390*** (0.0163) Yen 3 Months 12 Months 0.0002 (0.0007) -0.0237*** (0.0061) 0.3802*** (0.0257) -0.2041*** (0.0411) -0.0076*** (0.0012) 0.0404*** (0.0107) 0.3045*** (0.0171) -0.0169 (0.0414) -0.0032*** (0.0006) -0.9193*** (0.0171) 1.0520*** (0.0218) -0.1302*** (0.0329) -0.0014** (0.0006) -0.9124*** (0.0180) 1.0008*** (0.0081) -0.1155*** (0.0231) R2 0.6395 0.7281 0.9047 0.6961 0.7905 0.8999 % fun 52.25 43.42 29.66 51.91 41.18 22.52 AC -0.0300 -0.0320 0.0000 0.0010 0.0070 0.0450 Notes: Table presents estimation results for the model with dynamic weights. R2 is adjusted R2; standard errors in parenthesis; % fun is the percentage of fundamentalist panellists; AC is the autocorrelation in the fundamentalist/chartist classification. 24 Table 5: Estimation Results Simulated Data Model: Group: Benchmark All Switching All cf -0.0005 (0.0008) -0.0063 (0.0065) 0.4560*** (0.0249) -0.0233*** (0.0007) -0.0259*** (0.0059) 0.4076*** (0.0235) α f 2 cc β 2c R2 % fun AC NA -0.3756*** (0.0220) NA 0.3833 NA NA Fundamental model Fundamentalists Chartists -0.0233*** (0.0007) -0.0259*** (0.0061) 0.4076*** (0.0243) 0.0191*** (0.0007) -0.0226*** (0.0065) 0.3901*** (0.0246) 0.0178*** (0.0007) -0.2331*** (0.0207) 0.4022*** (0.0229) NA NA NA NA NA NA 0.7338 48.25 0.0590 0.3739 NA NA 0.3371 NA NA Chartist Model Fundamentalists Chartists NA NA NA NA NA NA -0.0203 (0.0007) -0.2214*** (0.0199) 0.4112*** (0.0220) 0.0178*** (0.0007) -0.2330*** (0.0200) 0.4022*** (0.0222) 0.4853 NA NA 0.4629 NA NA 25 Benchmark model Fundamentalists Chartists -0.0208*** (0.0007) -0.0177*** (0.0055) 0.4171*** (0.0219) 0.0176*** (0.0007) -0.0107* (0.0059) 0.4017*** (0.0221) NA NA -0.2125*** (0.0199) -0.2264 (0.0203) NA NA 0.4962 NA NA 0.4653 NA NA Table 6: Estimation Results Comparison Model: Fundamental model Group: Fundamentalists Chartists 0.0012*** (0.0005) 0.0019 (0.0042) 0.1924*** (0.0150) -0.0050*** (0.0009) -0.0376*** (0.0076) 0.7906*** (0.0301) cc NA NA β NA NA 2c NA NA R2 0.1120 0.3888 cf α 2f Chartist Model Fundamentalists Chartists NA NA NA NA NA NA 0.0012*** (0.0005) -0.0185 (0.0133) 0.1909*** (0.0148) -0.0050*** (0.0005) -0.7900*** (0.0149) 0.8390*** (0.0165) 0.1132 0.8132 26 Benchmark model Fundamentalists Chartists 0.0013*** (0.0005) 0.0027 (0.0043) 0.1924*** (0.0150) -0.0051*** (0.0005) -0.0107*** (0.0042) 0.8325*** (0.0166) NA NA -0.0196 (0.0134) -0.7854*** (0.0150) NA NA 0.1128 0.8140 Figure 1: Fundamental Exchange Rates Euro Japanese Yen 5.0 .2 .1 4.9 .0 4.8 -.1 4.7 -.2 4.6 -.3 4.5 -.4 96 97 98 99 00 01 s 02 03 04 05 06 96 07 97 98 99 00 01 s s* 27 02 03 s* 04 05 06 07