Abstract
The impact of financial transaction taxes are used to assess how well a model of behavioural heterogeneity developed by Hommes and in’t Veld (2015) reflects the expected behaviour of market participants. The model is characterised by two types of behavioural heterogeneous and boundedly rational agents. We assume that all agents calculate a fundamental value for a risky asset according to a dynamic Gordon model, but that agents differ in their beliefs regarding how this may deviate in price from its fundamental benchmark value. We characterise agent beliefs into two regimes with agents switching between the two based upon an evolutionary selection rule determined by the relative historic performance of the beliefs. The model is applied to empirical market data for the S&P500, CAC40 and DAX30 indices and parameters for the beliefs and the degree of behavioural heterogeneity are estimated for each market. We assume that financial transaction taxes will reduce the levels of speculative in behaviour in markets that are affected. Levels of speculative behaviour for each market are calculated from the model and then used to assess if the model reflects the introduction of a financial transaction tax. We find that while there is suggestive evidence the model of Hommes and in’t Veld may manage to identify the introduction of a financial transaction tax on agent behaviour this is far from conclusive. We additionally find that the impact of financial transaction taxes on investor behaviour may not be fully understood and could lead to an unintended increase market instability by increasing levels of behavioural heterogeneity in investment strategies.
Masters Thesis
MSc Economics – Behavioural Economics and Game Theory track
Prepared by: Nicholas Rees
Student number: 11088052
Thesis Supervisor: Prof. J.H. Sonnemans
Submitted: 15
thAugust 2016
2 Statement of Originality
This document is written by Student Nicholas Rees who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in
creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
3 1. Introduction and Background
Since the financial crash of 2008 economists have come under renewed criticism for their failure to identify risk factors and accurately predict financial crashes. Whilst there were some notable exceptions, this criticism is certainly justified when levelled against many main stream economists. The failure of economists en-masse to see what with hindsight was obvious has led to one of the most significant global economic catastrophes in the last century, plunging the world’s major economies into a recession, the impact of which we are still experiencing today.
In the aftermath of the crash many economists rushed to address the failures of their models, simply applying sticking plasters to their theories. Others have attempted to divert attention by blaming the behaviour of bankers and politicians (among others) for the failure of their models (Lucas, 2009). While such criticisms are undoubtedly popular, both within the media and with the public, they are not always entirely fair. Models that fail to reflect the behaviour of an economic or financial system, regardless of how badly bankers behaved, have undeniably failed. There is however another group of economists who, instead of attempting to rehabilitate existing models, have gone in search of new ones. The range of new models being developed is diverse but, as noted by Buttonwood for The Economist (2012), of particular interest is the growing literature on the impact of behaviour in markets.
Behavioural models have been proposed to account for some of the inconsistencies between financial markets and classical economic theory. One of the first contributions, and perhaps the most well-known, is the observation of Shiller among others: that stock markets exhibit excess volatility (1981; 2000). Simply put, stock price fluctuations are observed to be significantly greater in magnitude than classical economic and financial analysis would suggest as being expected from movements in underlying asset fundamentals such as changes in earnings expectations or dividend pay-outs for example. Perhaps unsurprisingly, this effect is particularly pronounced during stock market bubbles and the subsequent inevitable crashes. This excess volatility is suggested by Shiller as being the result of behavioural amplification (1981), whereby the reaction of market participants to movements in fundamental values amplifies the resulting change in stock prices. An increasingly popular way of generating this kind of behavioural amplification in models is through the use of models with behaviourally heterogeneous agents.
These models run counter to classical economics thought where market participants are all presumed to act as homo economics: a single perfect agent who can (among other things) form entirely perfect rational expectations over future events. In a heterogeneous agents model however this is not the case. Agents can form beliefs over future events in more than just one way. Learning behaviour and agents shifting strategies between beliefs then produce positive expectations feedback leading to large price swings. Clearly not every market participant acts in the same way in real markets so this is an obvious improvement in modelling realism over a model where all agents possess rational expectations, but these models can also nicely reflect the effect of behavioural amplification described by Shiller. Much work has already been done on the subject of heterogeneous agent’s models (HAMs) and an overview is provided by Chen et al. (2012).
This area of research is particularly exciting as it begins to unify the ideas presented by authorities such as Soros (1987) of reflexivity in financial markets, and Minsky (1975, 1992) on financial instability,
4 with mainstream economic literature. Whilst interest in the work of such figures has undergone a resurgence post 2008, behaviourally heterogeneous models are perhaps some of the most promising avenues for investigating these theories in practice. As identified by Hommes (2013), some HAMs exhibit reflexive properties consistent with the ideas of Soros and attempts at modelling Minsky’s financial instability hypothesis have been made using a HAM framework by Williams (2012) and Chiarella and Guilmi (2011) among others.
However, very little work has in fact been done to verify if these new models are useful in modelling empirical data. In particular there remains the question of whether the parameters of these model actually represent what their developers claim? The empirical work carried out to date such as that found in Chiarella and Guilmi (2011) has largely focused on comparisons of simulations with empirical market data. While these comparisons invariably do a good job of reflecting historic behaviour this is perhaps unsurprising given that the authors are at liberty to pick parameters that will allow their models to do just that. As a consequence, these models are of little use for the most important task of anticipating future market behaviour. A better solution is to estimate parameters in the model being used from historic market data; and then to assess the significance and robustness of such values over time. Some good examples of estimating heterogeneous agents models in this way is the work of Hommes and in’t Veld (2015) for the S&P 500 and Frijns et. al. (2010) using data for DAX30 options. But this method too is by no means perfect. Inherent in the estimation of non-linear models is the fact that multiple sets of values can be chosen to minimise a regression performed on the model. Many authors may suffer from a confirmation bias, assuming simply that the values closest to their expectations are true, but with no way of guaranteeing if they are actually correct.
An alternative method for investigating the usefulness of behaviourally heterogeneous models in representing real world effects is to test if these models and the parameters contained in them behave in the expected manner when estimated from empirical data. This is subtly, but crucially, different to simply estimating a model and noting that major market events such as bubbles and crashes are observable, although this should also still be the case. For these models to represent an improvement over those that they are supposed to supplant, they should not only be able to identify key market events but also other behavioural changes that are less apparent from raw market data.
Therefore in order to properly test this family of models it is necessary to look for specific instances in financial markets where a behavioural changes may have occurred and to see if these models and the parameters in change as expected manner when estimated from empirical data. One such event that would be expected to impact a market is the introduction of a financial transaction tax. Financial transaction taxes are often introduced with the intention of curbing speculative behaviour in markets and there is indeed some suggestion (Summers L. & Summers V., 1984; McCulloch, 2010) that they produce changes in investor and so aggregate market behaviour.
5 Accordingly, for the remainder of this paper we will focus on a specific model of behavioural heterogeneity developed by Hommes and in’t Veld (2015) and investigate if we can use it to identify the expected behavioural effects when estimated from real market data. This particular model utilizes boundedly rational traders within a heterogeneous agents pricing model framework developed by Brock and Hommes (1997, 1998). The model of Hommes and in’t Veld has previously been applied to data for the S&P 500 with some success and applying the model to index level data instead of individual company information largely avoids the problem of company and sector level idiosyncrasies that could otherwise skew results. Additionally, the model nicely encapsulates some stylised facts to characterise core market behaviour without being overly complex and therefore more difficult to estimate. Additionally, the model is defined in terms of deviations from a fundamental value rather than absolute prices meaning it is suitable for performing comparisons across markets. Importantly for the purposes of this paper the model has two type behavioural heterogeneity using time vary proportions of speculators and fundamentalists to account for market instability and boom bust economic cycles. As speculators in this model exhibit the type of market behaviour that financial transaction taxes are designed to curb we will be able to see directly if such an impact is visible on the levels of speculators in our model over time.
We will focus on three separate markets for our analysis: the S&P500 from 1900 to present, the DAX30 and the CAC40, both from 1996 to present. This is done so that results obtained for each market can be contrasted and comparisons between them can be made. Both the S&P500 and the CAC40 have been subject to a financial transaction tax for some of the period over which we are studying them, whilst the DAX30 has not.
The remainder of the paper is organised in the following way in section 2 we introduce the theory, a description of the model of Hommes and in’t Veld that we use and our hypotheses. Section 3 briefly describes the data and gives some information as to how the model is estimated in practice. In section 4 we cover the main methodology used to estimate the model and the results that we have obtained. Section 5 contains a discussion of the results that we obtain in section 4. We present our conclusions in section 6 and references are included in section 7.
2. Theory and Model Description 2.1 Model Overview
The heterogeneous agents model that we use for our analysis, developed by Hommes and in’t Veld (2015), provides a mechanism for reflexive behaviour through the use of behavioural amplification and outperforms a comparable rational agent model according to Hommes and in’t Veld. For the model to work and retain a sense of realism, it requires that all agents are boundedly rational in their anticipation of future prices. To do this we will assume that all agents can calculate and agree upon a fundamental value for a risky asset that they expect prices to return to in the long run, in this case a stock market index, and the current price deviation of the market from that fundamental. To calculate the fundamental value we use a dynamic Gordon model developed from the static Gordon model (1962) by Boswijk et al. (2007) using the approach of Poterba and Summers (1988)to form a benchmark value.
6 Where agents in the model are permitted to differ is in their short run expectations. In the model agent behaviours are split into two regimes, speculators and fundamentalists, an approach that’s is widely used in related literature (Chen et al. 2012). For the sake of clarity in this paper we define speculators as those agents that believe prices will continue to increase in deviation from fundamental values, whilst fundamentalists believe that in the near future there will be a slow reversion towards fundamentals. Agents can gradually switch between the two different investment behaviours based upon an evolutionary learning process that compares the historic performance for the respective strategies. This switching leads to agents that underperform when compared to their peers gradually switching strategy to the belief type that has generated better returns in the recent past.
For the switching behaviour to work in a realistic manner the concept of market memory was introduced by Hommes and in’t Veld (2015) into the model. The use of limited memory is consistent with the idea of boundedly rational agents by allowing them to forget market historic performance over time, encouraging gradual herding towards the belief types that have been successful most recently. Combined with switching behaviour, this allows markets to become dominated by one type of investor for short periods and therefore leading to self-reinforcing regimes of mean reversion or speculation. This self-reinforcing positive expectations feedback behaviour that is characterised here provides a mechanism for persistent deviations of the market from the fundamental benchmark value. Importantly these deviations can occur endogenously within our model providing an explanation for market bubble formation and collapse.
Estimating the heterogeneous agents model developed by Hommes and in’t Veld requires a two-step procedure. In the first stage the raw data must be used to construct growth and returns figures for each quarter. From these figures an AR(1) rules for updating future growth and returns expectations for a dynamic Gordon fundamental value can then be estimated. These values are then used to construct a time series for a dynamic Gordon fundamental for the market in question. In the second stage we calculate deviations of the price dividend ratio from the Gordon fundamental value. Finally, a non-linear least squares regression is then used to fit the behavioural model to the calculated deviations and estimate the model parameters that we wish to study.
2. 2 Model Assumptions
To construct a tractable model a series of simplifying assumptions are made in line with the literature. While these assumptions will be explained in brief here, more information can be found in the work of Chen et al. (2012) and Hommes and in’t Veld (2015) who originally developed this model. Firstly we assume that all agents can calculate and agree upon a fundamental value for a particular market and that they can additionally calculate the deviation of the current position of the market from that fundamental value. Agents then form expectations of future price deviations from fundamental values: , that are linear in the previous observation as shown below in equation 1. This forecasting rule allows for a simple form of behavioural heterogeneity that is sufficient to capture the desired behavioural effect whilst remaining possible to estimate from empirical data.
7 Work by Aoki (2002) suggests using a theoretical model that behaviour of many different types of market participants can often be clustered into as few as two separate groups. In this model we label the different belief types with the index h and the time period with index t. Therefore, by utilising h = {1,2} belief types, we characterise the two different investing strategies: and that should be sufficient to accurately capture on an aggregate level the behaviour of a majority of the market participants. In this instance we expect values for each of these groups such that 0 < <1 and 1 < represent fundamentalist and speculative behaviour respectively. The interpretation of these parameters is that investors with belief type we expect deviations from fundamental values to slowly decrease towards zero over time, whereas for belief type deviations are expected to increase. This may seem counterintuitive, however we do not require agents in this model to be rational. Instead we impose the weaker assumption that agents must only be boundedly rational. In this instance it is possible for agents searching for short-run profits to believe that prices may continue to deviate from fundamental valuations in the near term even if they expect them to return to fundamental levels over a longer time period. Agents that follow this strategy are the key difference between this model and a more typical rational expectation models where all agents behave in a similar manner and will prove to be important in driving boom-bust cycles.
The second assumption is that given agents can anticipate future prices based upon their particular strategies forecasting rule, when these predictions deviate too far from the realised values agents may choose to switch strategies. In the model agents are permitted to gradually switch strategies, reflecting a process of evolutionary learning. Following Hommes and in’t Veld (2015) we use the multinomial logit model of Brock and Hommes (1997) to characterise this strategy switching behaviour. As with the first assumption, choosing h = {1,2} types of belief, we can describe the fraction of agents with beliefs 1 and 2 in any given period as , and , respectively with equation 2 shown below. ,= , ∑ , (2)
By using this form it allows for switching and in some cases herding behaviour, meaning that agents can at certain points in time converge on just one belief type as an optimal investment strategy. Hommes and in’t Veld (2015) found that the intensity of choice variable β for the multinomial logit switching cannot be precisely determined in this model due to the relatively small number of distinct time periods that will be used. After some testing they do however suggest that any value greater than one will be sufficient to allow the model function as it is supposed to, and that changes in the value of β will in fact have little impact of the final results of the model. For this reason we will not attempt to estimate values of β but will be setting the value of β equal to 1 for each of the models estimated in this paper in order to be consistent across all the models.
Third we will assume profit expectations for the agents are proportional to the product of the agents excess demand as shown in the first set of parentheses, and the realised excess returns shown in the second set of parentheses.
, = (, − ∗
8 Here the parameter , represents the realised profits at time t+1 for belief type h and ∗ is the expected effective discount rate for advancing one time period. This means that provided agents can correctly anticipate the direction in which the market moves they will return a positive profit.
Finally, a parameter ω is introduced to account for the limited memory of the agents in accordance with Hommes and in’t Veld (2015). This memory parameter determines the balance of weighting between the most recent observation and all observations previous to that. Parameterising the model in this way allows agents to account for realised profits for more than one period, and the mathematical form is presented below in equation 4.
,= (1 − "),+ " , (4)
Allowing for market memory is important for evolutionary learning in the model because it allows agents to ‘remember’ and therefore learn from previous experience. However the form used in this instance provides for agents that are boundedly rational and does not allow for perfect memory. This is done by giving decreasing weight to performance in previous time periods with less weight being given to periods further back in time. Using market memory in this way has the effect of representing fading memory with some agents ‘forgetting’ some of the historic profit performance each time period. Ultimately this means that agents can repeat historic mistakes such as engaging in speculative behaviour which is not possible when using perfect memory.
The recursive nature of these equations and the non-linearity of the full model means that fitting for small values of t is imprecise, however the fit will improve as the value of t increases.
2.3 The Full Model
Whilst there are a range of methods for estimating the fundamental value of a company or index, one of the most commonly used is the standard Gordon model, sometimes referred to as the dividend discount model. Under discrete time the solution for the value of a company in terms of its PD ratio for the Gordon model is given as:
$∗= %
& % (5)
Where the static fundamental price-dividend ratio $∗ is a function of the dividend growth rate g, and the risk free rate r. As this method is commonly used in finance to assess the value of companies or indices, the Gordon method will be used to calculate a measure for a commonly agreed upon fundamental value in our model. However, clearly this value for the Gordon fundamental value should not remain constant and is likely to change over time. Therefore in this paper we will use a variation of the standard Gordon model known as the dynamic Gordon model. In this version of the model, agents estimate future values for g and r based upon the historic data for these variables available to them. In this model agents use an AR(1) rule to update their beliefs for future values of r and g as shown in equations 6 and 7:
'()* = ( + +)((
− () (6)
',)* = , + -)(,
9 Following Hommes and in’t Veld (2015) we use the approach of Poterba and Summers (1988) for calculating the time varying fundamental value of the dynamic Gordon model $∗ to a first order Taylor approximation as shown in the formula below:
$∗= %& %+(& %)/& .(%)0.(%) ((− () +(& %)/& 1(%)01(&) (,− ,) (8)
Where r is the static risk free rate, g the static dividend growth rate, ( the dynamic risk free rate and , the dynamic dividend growth rate calculated each quarter. The parameters ρ and τ are the AR(1)
coefficients used for updating agent expectations of ( and ,.
By taking this approach we can model deviations of the price-dividend ratio around a changing fundamental value. These deviations are calculated according to the formula below:
= $− $∗ (9)
Where $ is the price-dividend (PD) ratio 2 3
4 .
As above, setting h = {1,2} in equations (1) to (4) captures the two-type behaviour required to model speculative and fundamentalist beliefs. Combining equations (1) to (9) allows us to then construct the full behavioural model of Hommes and in’t Veld described by the following three equations.
=5∗/, + , 0 + 6 (10)
,= (1 + exp :( − ) ∑ <)=>")(1 − ") ; )/ )− ∗ )0 ) (11)
,= 1 − , (12)
The model is presented in this mathematical form for clarity. However, for the purposes of further analysis a re-arranged but mathematically equivalent form is used for ease of use. Note that in equation (11) there is a term − . This is the level of behavioural heterogeneity in the model and acts as a destabilising factor, with larger values of heterogeneity describing more unstable markets.
2. 4 Hypotheses
The core of the research contained in this paper will be in estimating the heterogeneous agent’s model of Hommes and in’t Veld (2015) to see if we can identify changes in agent behaviour. As set out in section 1, the implementation of a financial transaction tax in a country is designed to reduce speculative investing behaviour in markets. Furthermore there is some evidence to suggest that they do indeed change investing behaviour. Accordingly we attempt to see if it is possible to identify the effect of introducing a financial tax in the results of models for a number of different national stock market indices. We will do this in two parts.
The first part will be a comparison of the DAX30 and the CAC40. The Cac40 is considered the premier stock index in France, comprised of the top 40 companies by market cap, and a bell weather for the French economy. On August 1st 2012 the French government introduced a financial transaction tax on
10 euros. If the general premise of our argument is correct then we expect speculative behaviour to be reduced relative to what it would have been if no financial transaction tax had been introduced. However, given that we only have 16 periods after the introduction of the tax, this is unlikely to be sufficient to identify any secular trend from modelling the CAC40 alone. For this reason we will model the DAX30 alongside the CAC40. Like the CAC40, the DAX30 is also a premier national stock market index, in this case for Germany and consists of the top 30 German companies by market cap. Both indices are compiled using cap weighting, with both economies being similar and using the same currency we assume that the results for the two indices should be broadly comparable. The one big difference over this period is that unlike France, Germany did not introduce a financial transaction tax. Therefore we can use the results for the model of the DAX30 to gain an insight into how the CAC40 may have behaved if not financial transaction tax had been introduced, and so draw conclusions as to whether the model can indeed identify a behavioural change as a result of introducing a financial transaction tax.
Therefore we form our first hypothesis: that if the introduction of financial transaction taxes decrease speculative behaviour as expected and the behaviour of the DAX30 is a good proxy for the behaviour of the CAC40 then for the period after the introduction of as financial transaction tax we expect to see a decrease in speculative behaviour in the CAC40 relative to the DAX30 as estimated by our model. The second approach to be employed is analysis of long-run stock market data for S&P 500 taken from the online data set provided by Robert Shiller U.S. Stock Markets 1871-Present and CAPE Ratio. A portion of the data set, from 1950 to 2015 was previously used by Hommes and in’t Veld (2015) to estimate the model in their original paper. For the purpose of this research however, we will use a longer time period. This is due to the fact that a financial transaction tax was in place in the US between 1914 and 1966. By comparing the estimated model behaviour between those dates with model behaviour before and afterwards, it is expected to see a difference in the levels of speculative behaviour.
This leads to our second hypothesis: that for the period 1914 to 1966 we expect the model to identify a lower level of speculative agent behaviour than before and after that period.
Both of these approaches individually have limitations that may mean that they could struggle to identify a change in behaviour resulting from the use of a financial transaction tax. With the first approach, there is only a limited length of time for which there is data available, from 1996 onwards, meaning that it may be difficult to accurately estimate the models. In the case of the second approach, while there is lots of data, it may be difficult to attribute or even identify any change in behaviour between 1914 and 1966 solely due to the introduction of an FTT as there are many other forces that may influence market behaviour, such as two world wars and the great depression. By using the results of both approaches however, we attempt to mitigate the impact of the key weaknesses of each individual approach.
11 3. Data and Statistical Analysis
As was mentioned above we use data for three separate markets in order to estimate levels of speculative behaviour for each market. The first two markets that we study are the CAC40 and the DAX30. For these markets we will use quarterly data from Q1 1996 to Q2 2016. Stock price and dividend yield data are obtained from DataStream and bond yield and CPI data are obtained from the datasets published by the OECD. Additionally we will also study the S&P 500 using quarterly data for the period Q1 1900 to Q2 2016. All the raw data required for this model is acquired from the online dataset produced by Robert Shiller U.S. Stock Markets 1871-Present and CAPE Ratio.
A brief qualitative examination of the price-dividend (PD) ratios obtained from the raw data for the CAC40 and DAX30 is shown below in figure 1. As the PD ratio is often used as an indicator for the valuation of a market this an insight as to whether a market is over or under prices at various points in time. From this plot we
can see that for the majority of the period 1998 Q1 to 2016 Q2 the CAC40 and DAX30 have very similar PD ratios, although these begin to diverge around 2012. This supports our initial assumption that the DAX30 can provide a good approximation for the behaviour of the CAC40 after the introduction of a financial transaction tax. A similar examination of the PD ratio shown in figure 2 for the S&P500
also yields some
interesting observations. The tech bubble can clearly be seen around the year 2000 as can the stock market crash of 2008. These events are both far more pronounced for the S&P500 than they appear for the CAC40 and DAX30.
0 20 40 60 80 100 120 Q1 1988 Q1 1991 Q1 1994 Q1 1997 Q1 2000 Q1 2003 Q1 2006 Q1 2009 Q1 2012 Q1 2015 P ri ce -d iv id e n d r a ti o Date Cac40 Dax30
Figure 1: A comparison of the price-dividend ratios for the CAC40 and DAX30 indices for the period 1988 Q1 to 2016 Q2. It should be noted that the ratios are broadly similar for both indices for the majority of this period 0 10 20 30 40 50 60 70 80 90 100 1 8 9 8 1 9 0 3 1 9 0 8 1 9 1 3 1 9 1 8 1 9 2 3 1 9 2 8 1 9 3 3 1 9 3 8 1 9 4 3 1 9 4 8 1 9 5 3 1 9 5 8 1 9 6 3 1 9 6 8 1 9 7 3 1 9 7 8 1 9 8 3 1 9 8 8 1 9 9 3 1 9 9 8 2 0 0 3 2 0 0 8 2 0 1 3 P ri ce -d iv id e n d r a ti o Date
Figure 2: A plot of the price-dividend ratio for the S&P500 for the period 1898 Q1 to 2016 Q2. The effect of the tech bubble around 2000 and the financial crisis of 2008 are clearly visible on the right hand side of the graph.
12 Additionally we will briefly examine a graph of the inflation rate for the United States for the period 1900 Q1 to 2016 Q2 shown below in figure 3. We see that the inflation rate is significantly more volatile up until the 1950’s. Because the rate of inflation plays an important role in determining the fundamental Gordon value for the market we would therefore also expect that fundamental value to be more volatile in this period.
For the purposes of model estimation, the majority of the model is constructed in excel however the statistical package that will be used to conduct the nonlinear least squares regression analysis is Stata MP 13. For the purposes of running the analysis we eliminate , from the regression equation so as to reduce the number of RHS variables, allowing us to maximise the overall number of memory terms used and improve the convergence behaviour for the regressions. However, the restriction on the number of variables Stata can take will still force us to truncate the number of memory terms used in the analysis of the S&P 500 data.
4. Estimation methodology and results 4.1 Outline of the Estimation Procedure
In line with theory as seen in section two, our model estimation procedure follows a two-step process. First we must estimate the fundamental value of the Gordon model with a constant risk premium. Secondly, we will estimate the heterogeneous agents model outlined in equations (10), (11) and (12) using a non-linear least squares method. This will be applied to deviations of the index price-dividend ratio from the fundamental values calculated in the first step and will give us estimates of the parameters , , and ω. The difference between and is the level of behavioural heterogeneity of the market and as mentioned in the theory, this is a destabilising factor in the model so we additionally identify its magnitude. Because each model is different the methodology in constructing each model also differs to some degree. For this reason we present the model building process for each index separately along with some of the intermediary results for each index. Once we have estimated values for , , and ω we can then use these to construct a time series of estimates for the fractions of agents that follow fundamentalist and speculative strategies , and
-15.00 -10.00 -5.00 0.00 5.00 10.00 15.00 20.00 25.00 A n n u a li se d r a te o f in fl a ti o n Date
13 , respectively as described by equations (11) and (12). Doing so we can then see how agents change
their strategies over time, switching from fundamentalist to speculative strategies and back again. This allows us to identify periods in the market where one particular strategy dominates over another. 4.2 Estimating the Gordon fundamental value for the DAX30 and CAC40 indices
We start with estimating the fundamental index values according to the Gordon model for the DAX30 and CAC40 indices. Using daily raw data for the relevant indices we first construct quarterly average closing prices and dividends for each of the indices. Doing this helps to prevent introducing excess volatility into the model that would occur if raw quarterly closing values were used. Taking this data we then construct the static Gordon model for the indices as shown in equation (8). The static quarterly values and the yearly equivalents of the relevant parameters are shown below in Table 1. Table 1: Estimation of the static fundamental value for the CAC40 and Dax30 indices
Frequency
K
D/P
g
r
i
RP
R*
δ*
Cac40
Yearly 1.4766 2.3212 6.5818 9.0376 2.1474 6.8902 1.0230 43.3999 Quarterly 0.3671 0.5803 1.6063 2.1866 0.5326 1.6540 1.0057 175.0927Dax30
Yearly 1.3321 2.5641 9.5617 12.3335 1.9794 10.3540 1.0253 39.5279 Quarterly 0.3314 0.6410 2.3092 2.9502 0.4912 2.4590 1.0063 159.6038 Values calculated using CAC40 and DAX30 data between Q1 1996 and Q2 2016 used in estimating both static Gordon models. All values are given as percentages, with the exception of ∗ which is given as a decimal and δ* which is a ratio. K is the average inflation rate over the period calculated from the quarterly CPI index for both countries and used to deflate the other nominal variables. D/P is the average price-dividend yield: 3⁄2 . The average growth rate of dividends over theperiod is given by g and the risk free rate, given by r and is equal to D/P +g. The average real return on long term debt is given by i (calculated from OECD data for long run debt yields). RP is the investment risk premium and is equal to r-i. The expected effective discount rate R* is calculated as (1+r)/(1+g). The static value of the Gordon fundamental is shown as δ* and is given by the formula (1+g)/(r-g).
Although we will use quarterly values for all the model calculations for accuracy, we also show here the yearly equivalents calculated using geometric progressions of the related quarterly values for the readers ease of comparison. From Table 1 we see that the values for both indices are broadly similar. In the case of the static fundamental price-dividend ratio δ* we see that the CAC40 has a slightly higher value than the DAX30 suggesting that returns from the CAC40 are slightly less generous than those of the DAX30.
We then estimate the AR(1) rules used to update agents expectations for future values of dividend yield growth rate and the risk free rate as shown in equations (6) and (7). For the CAC40 we estimate a persistence of the risk free rate ρ = 0.10 and a persistence of the dividend growth rate τ = - 0.11. For the DAX30 we find the persistence of the risk free rate to be ρ = - 0.69 and persistence of the dividend growth rate τ = 0.09. Clearly these values are very different from each other and in both cases different from the values that would normally be expected. This is likely due to the short time period of only 80 quarters available to estimate the parameters, however in the case of both of these models the effect variations in these parameters was checked and does not have a significant impact on the final model so the disparities between the estimated values and expectations is not of concern.
14 Using the persistence
parameters and the static values for the risk free rate and dividend growth rate derived above we calculate fundamental values for the time varying price dividend ratio $∗ according to equation (8). Plots of the price-dividend ratio PD and the dynamic Gordon fundamental benchmark $∗ for each index are shown Figures 4 and 5.
We observe in both figure 4 and 5 that the calculated fundamental values are around 40 and vary significantly less than the actual price dividend ratio for each market and in both cases has a very slight negative trend. We also observe that while some movements in the price-dividend ratio are reflective of movements in the fundamental value they tend to be smaller in magnitude, and there are others that appear to be relatively unrelated to movements in the fundamental at all.
In the case of the CAC40 we find the price-dividend ratio has a range [17.85,103.07] while the fundamental value has a range [18.31,47.01] For the DAX30 the price-dividend ratio has a range [20.57,77.59] and the fundamental value has a range [37.20,46.76]. In both cases the DAX30 has a smaller range but they are similar enough that the two markets are still largely comparable. 0 20 40 60 80 100 Q1 1996 Q1 2000 Q1 2004 Q1 2008 Q1 2012 P ri ce -d iv id e n d r a ti o Date
Price-dividend ratio Gordon fundamental ratio Figure 4: A plot of the price-dividend ratio (blue) and the calculated Gordon benchmark (red) of the CAC40 for the period Q1 1996 to Q2 2016 0 20 40 60 80 100 Q1 1988 Q1 1992 Q1 1996 Q1 2000 Q1 2004 Q1 2008 Q1 2012 P ri ce -d iv id e n d r a ti o Date
Price-dividend ratio Gordon fundamental ratio Figure 5: A plot of the price-dividend ratio (blue) and the calculated Gordon benchmark (red) of the DAX30 for the period Q1 1988 to Q2 2016
Note that in while both figure 4 and 5 display a similar trend s from 1996 onwards, the CAC40 rises above its calculated fundamental value from 2012 onwards whereas the DAX30 does not.
15 4.3 Estimation of the Heterogeneous agents models for the DAX30 and CAC40 indices
After building the Gordon fundamental benchmark we are able to construct deviations from that benchmark according to equation (9) for both indices. We then perform a Non-linear least squares estimation of the heterogeneous agents model described in equations (10) (11) and (12) to estimate the parameters , , and ω for both the CAC40 and DAX30. As explained in the theory, values for β are set to one for the purposes of the regression procedure. The results are given in Table 2.
Table 2: Estimation of the belief coefficients , and memory parameter ω of the heterogeneous agents
model for both the CAC40 and DAX30 Indices for the period Q1 1997 to Q2 2016. Parameters are estimated using NLLS regression with the exception of the intensity of choice parameter β which is fixed at β = 1. Significance at the 10%, 5% and 1% levels is indicated by one two and three asterisks respectively.
CAC40
DAX30
0.7853*** 0.8842** (0.0680) (0.0576)
0.9868 0.9793 (0.0830) (0.0525)
β
1 1 - -ω
0.8622** 0.8587 (0.0799) (0.1861)Δ
0.2015** 0.0951 (0.1073) (0.0779)
0.7796 0.8919
Clearly while estimates of for both markets are between 0 and 1 as expected, it is surprising to find that in both cases is not statistically significantly different from 1. This is more consistent with agents forming beliefs with naïve expectations, where they believe that prices in the short term will maintain their level of deviation from fundamentals rather than the magnitude of the deviation increasing or decreasing. While this belief type is perfectly possible within our model it is in contrast to the results of Hommes and in’t Veld who found evidence for speculative behaviour in the S&P500, albeit over a longer time period. Based upon the estimated parameters shown in table 4 the models seem to differ in their estimation of parameter but be very similar in their estimation of . We find similar values for the memory parameter ω, suggesting that agents in both markets treat historic market information in a similar way, although in the case of the DAX30 this falls just short of being significant at the 90% level. We also find that there is evidence for behavioural heterogeneity Δϕ in the CAC40 index but once again the parameter estimated for the DAX30 this falls slightly short of being significant at the 90% level. The lack of statistically significant results for the DAX is likely due to the difficulty in fitting a model over a relatively short data range with only a small degree of behavioural heterogeneity. We also find that the R value that indicates the predictive power of the model for the DAX30 is higher than that for the CAC40 which is to be expected with a lower level of estimated behavioural heterogeneity.
16 While it would have been equally possible to estimate the model for which is greater than and β =-1, when conducting the estimation procedure we found that convergence was superior with coefficient less than and β =1. When checked we found that almost identical values can be estimated using β =-1 to those found in the table above with and switched, as would be expected. The small discrepancies in the estimations and better convergence for β =1 is simply due to the behaviour of the non-linear least squares estimation algorithm used by Stata.
Using the values of , , and ω previously estimated for both the CAC40 ad DAX 30 we can then calculate the time series of the fraction of agents that invest with a naïve belief type , each quarter using equations (11) and (12). As mentioned in the theory from section 2, because parts of the model are recursive in nature their convergence properties are low for small values of t, leading to imprecise estimated market fractions during those periods. We found that it took around 10 time periods for the model to stabilise so for that reason we have omitted the first 10 time periods for our analysis of market fractions. A comparison of the fractions of investors with naïve belief types , for both markets is shown below in Figure 7. As with the plot of the price-dividend ratios shown in section 2 we see that the plots for both markets are very similar up until 2012.
From our initial hypothesis and assumptions we would have expected that the number of investors with naïve beliefs for the CAC40 would have fallen below that of the DAX30 around 2012 and subsequently followed a similar trend. From Figure 7 we can see that this clearly is not the case. While the number of investors with naïve beliefs does indeed appear to fall in 2012 for the CAC40, the fraction of agents pursuing this strategy subsequently becomes significantly more volatile than for the DAX30. Because of this level of volatility we cannot apply a statistical test to assess in this instance if the fraction of investors with naïve beliefs is significantly different after the introduction of a financial transaction tax for the CAC40 with any confidence. Qualitatively however the change in behaviour of investors in the CAC40 when compared to the DAX30 is interesting, and is suggestive that the model may indeed be accurately identifying a change in behaviour in line with our hypothesis, albeit not in the way that was expected, however cannot be conclusively be attributed to the effect of introducing a financial transaction tax.
Figure 7: A plot of the fraction of investors following a naïve investment strategy in the CAC40 (blue) and the DAX30 (red) for the period Q3 1999 to Q2 2016. Note the deviation beginning around 2011-2012.
0.00 0.10 0.20 0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 Q3 1999 Q3 2003 Q3 2007 Q3 2011 Q3 2015 F ra ct io n o f n a iv e i n ve st o rs i n t h e m a rk e t Date Cac40 Dax30
17 4.4 Estimating the Gordon fundamental value for the S&P500
To estimate the Gordon fundamental benchmark for the S&P500 we use the raw data provided online by Robert Shiller U.S. Stock Markets 1871-Present and CAPE Ratio for the period Q1 1900 to Q2 2016. As this is in the correct form for the model we are able to construct the static Gordon model for the indices as shown in equation (8) without reformatting the raw data. The static quarterly values and the yearly equivalents of the relevant parameters are shown below in Table 3.
Table 3: Estimation of the fundamental value for the S&P 500 Index
Frequency
K
D/P
g
r
i
RP
R*
δ*
Yearly
3.0665 4.1861 1.6319 5.9359 1.6934 4.2425 1.0423 23.6136Quarterly
0.7580 1.0465 0.4055 1.4520 0.4207 1.0314 1.0104 95.9415 Values calculated using S&P 500 data between Q1 1900 and Q2 2016 for estimating a static Gordon models. All values are given as percentages, with the exception of ∗ which is given as a decimal and δ* which is a ratio. As before K is the average inflation rate over the period calculated from the quarterly CPI index for both countries and used to deflate the other nominal variables. D/P is the average price-dividend yield: 3⁄2 . The average growth rate of dividends over the period is given byg and the risk free rate, given by r and is equal to D/P +g. The average real return on long term debt is given by i (calculated
from 10 year T bills). RP is the investment risk premium and is equal to r-i. The expected effective discount rate R* is calculated as (1+r)/(1+g). The static value of the Gordon fundamental is shown as δ* and is given by the formula (1+g)/(r-g). As previously we use quarterly values for all the model calculations, but we again show here the yearly equivalents calculated using geometric progressions of the related quarterly values for ease of comparison. Despite using a longer time period than was originally used by Hommes and in’t Veld (2015) we notice that the values we obtain here are broadly similar to those obtained in their paper. We also find a value for the static price-dividend ratio δ* of 23.6 which is within the range that would typically expected for a long run estimation, although given the significantly different time scales this is not directly comparable to the values previously obtained for the CAC40 and the DAX30.
We again estimate the AR(1) rules used to update agents expectations for future values of dividend yield growth rate and the risk free rate as before. For the S&P500 we find a persistence of the risk free rate ρ = 0.42 and a persistence of the dividend growth rate τ = 0.71. While these values are somewhat different to the values found by Hommes and in’t Veld, they are within the kind of range that would be expected. The difference between these results and those of Hommes and in’t Veld is likely due to the use of a different time period used for the estimation.
Using the estimated persistence parameters and the static values for the risk free rate and dividend growth rate derived above we then calculate fundamental values for the time varying price dividend ratio $∗ in the same way we did for the CAC40 and DAX30. A plot of the price-dividend ratio PD and the dynamic Gordon fundamental benchmark $∗ for the S&P500 is shown in Figure 8.
18 As expected from the qualitative analysis of the interest rate data performed in section 2 we see that the fundamental value is significantly more volatile in early time periods. The effect this will have on the model can be seen more clearly in the plot of the quarterly price-dividend deviations from the dynamic Gordon fundamental shown below in Figure 9. Here we observe that the earlier periods exhibit significantly more volatility than later ones, making estimating the full model in these periods more difficult. 0 10 20 30 40 50 60 70 80 90 100 P ri ce -d iv id e n d r a ti o Date
Price-dividend ratio Dynamic Gordon fundamental value
Figure 8: A plot of the price-dividend ratio (blue) and the calculated Gordon benchmark (red) of the S&P500 for the period Q1 1900 to Q2 2016
Figure 9: A plot of the quarterly price-dividend deviations from the dynamic Gordon fundamental benchmark for the period 1900 Q1 to 2016 Q2
-40.00 -20.00 0.00 20.00 40.00 60.00 80.00 D e v ia ti o n o f th e p ri ce -d iv id e n d r a ti o f ro m th e G o rd o n f u n d a m e n ta l Date
19 4.5 Estimation of the heterogeneous agents model for the S&P 500
Following our hypothesis we expect identify with our model the effect of a financial transaction tax that was in place on US securities for the period 1914-1966. Unlike the previous analysis of the CAC40 and DAX30 however we do not have another market to compare the S&P500 against. For this reason we will instead compare the behaviour of agents investing in the S&P500 across different time periods. Following the same method as before, we are able to construct deviations from that benchmark according to equation (9) and a Non-linear least squares estimation of the heterogeneous agents model described in equations (10) (11) and (12) can then be performed. In this case however we estimate models for a number of different periods. The first interval runs from 1914 Q1 to 1966 Q1 the period for which the US had a financial transaction tax in place. The second model is for the period 1966 Q1 to 1996 Q1 so that we can make a comparison with the first interval without being concerned with the effects of the dotcom bubble which as noted in section 2 significantly distorts the S&P500 from 1996 onwards. A third model is estimated for the period 1996 Q1 to 2016 Q2 to allow is to make comparisons with the results of the CAC40 and DAX30. Finally there is a model for 1900 Q1 to 2016 Q2 that we use to construct a plot of the fractions of investors with naïve beliefs over the entire period. As before, values for β are again set to 1 and estimates for the parameters , , and ω are shown below in table 4.
Table 4: Estimation of the belief coefficients , and memory parameter ω of the heterogeneous agents
model for the S&P 500 index for the periods Q1 1914 to Q1 1966, Q1 1966 to Q1 1996, Q1 1996 to Q2 2016 and Q1 1900 to Q2 2016 . Parameters are estimated using NLLS regression with the exception of the intensity of choice parameter β which is fixed at β = 1. Significance at the 10%, 5% and 1% levels is indicated by one two and three asterisks respectively.
1914-1966
1966-1996
1996-2016
1900-2016
0.7515*** 0.9522 0.9793 0.8945*** (0.0473) (0.0565) (0.0216) (0.0214)
1.0159 1.0458 1.0266* 1.0202 (0.1345) (0.0933) (0.0202) (0.0219)
β
1 1 1 1 - - - -ω
0.7811*** 0.9634 0.8300* 0.8074*** (0.0722) (0.0671) (0.1093) (0.0416)Δ
0.2644** 0.0936 0.0478* 0.1257*** (0.1426) (0.1091) (0.0296) (0.0306)
0.6182 0.9222 0.9840 0.8915
Similar to our findings for the CAC40 and DAX30 we find that estimates for in each period are very similar and in all but one case are not statistically significantly different from 1 and therefore agents with this belief type are most appropriately described as having naïve expectations. We also find that for the periods that wholly or partially contain the use of a financial transaction tax, 1914 to 1966 and 1900 to 2016, produce significantly lower estimates for than those that do not encompass a time span where a financial transaction tax is in place. Interestingly this is again similar to the result found
20 for the CAC40 and DAX30, where the CAC40 is estimated over a time period in which there is a financial transaction tax in place for a portion, also has a lower estimate for than the DAX30.
We find that market memory ω is broadly similar for all periods with the exception of 1966-1996 and that these values are similar to those obtained for the CAC40 and DAX30. We also find that for every instance other than 1966-1996 there is statistically significant behavioural heterogeneity present in the market. For the case of 1966-1996 we do not find any statistically significant results which is likely due to the difficulty of fitting the model. As with the CAC40 and DAX30 the predictive power of the model given by R is lower for those periods where behavioural heterogeneity is greater.
Ideally we would have also included an estimate for the model on the data for the period 1900 to 1914 however the short data range and the high volatility in market deviations from the fundamental resulting from large fluctuations in the interest rate for this period meant that it was not possible to fit the model accurately.
We then use the estimated parameters , and ω obtained for the period 1900 to 2016 to calculate the fraction of agents that invest with a naïve belief type , each quarter using equations (11) and (12). Unlike the method used previously we have sufficient data prior to 1900 to stabilise the estimation of the market fraction , and because of that we do not have to omit the first 10 terms from the time period. The resulting plot is shown below in Figure 10.
With persistence parameters and the static values for the risk free rate and dividend growth rate, we now calculate fundamental values for the price dividend ratio $∗ according to equation (8). A plot of the price dividend ratio PD and $∗ is shown below in Graph 3.
We see from graph 3. That the fundamental value does not fluctuate significantly remaining within Whilst we can identify qualitatively that the behaviour shown by the plot in Figure 10 reflects what we would expect for some key events such as the dotcom bubble in 2000 we cannot immediately see any difference between periods where a financial transaction tax is in place and where it is not. To do so we must use additional statistical tools to analyse the data. Because we do not know the distribution of errors for our market fractions we require a non-parametric test to compare the difference in behaviour between the periods 1914-1966 and 1966-1996 for the data used in Figure 10. For both periods we again use the estimated parameters calculated for the period 1900-2016 so that we can compare the two periods directly.
In order for us to test for differences between these time periods we have to make several strong assumptions. First we assume that as both of these periods are relatively long the effects of market cycles will be minimal over these intervals, and second that investment behaviour should not change significantly over this time except in reaction to a financial transaction tax. With those assumptions we can use a Kolmogorov-Smirnov test to compare the time series of the fractions of investors
Figure 10: A plot fraction of investors with naïve a belief type in the S&P500 for the period 1900 Q1 to 2016 Q2 0.00 0.20 0.40 0.60 0.80 1.00 F ra ct io n o f n a iv e i n ve st o rs in t h e m a rk e t Date
21 following a naïve investment strategy (,) in both periods. The values for , are calculated using the same parameters and as estimated over the period 1900-2016 so that we can compare solely the difference in the fraction agents with each belief type between the two time periods. The results of the test are shown in Table 5.
Smaller Group
Test statistic
P-value
1914 − 1966
0.0502 0.5961966
−1996
-0.2557*** 0.000From table 5 we see the test indicates that investors with naïve beliefs make up a larger proportion of the market for the period 1914-1966 than the period 1966-1996 with a p value of 0.000. While this does seem to indicate there is a change in behaviour between the two periods, this result is the opposite of what we would have expected from our second initial hypothesis.
We also perform a Wilcoxon rank-sum test to verify the results found with the Kolmogorov-Smirnov test. Again we use values for , that are calculated using the same parameters and as estimated over the period 1900-2016. The test comparing the periods 1914-1966 and 1966-1996 returns a statistic of 4.481 with a p value of 0.000 allowing us to reject the null hypothesis that the two periods have medians are not statistically different at any level. This reinforces the result found with the Kolmogorov Smirnov test that the periods 1966-1996 has lower levels of speculative behaviour than the period 1914-1966.
5. Discussion
The methodology we use involves first estimating parameters for , and ω; then taking these values and using them to calculate the market fractions , and , representing the proportion of fundamentalists and speculators in the market respectively according to equations (11) and (12). In the case of both the DAX30 and the S&P500 between 1966 and 1996 we were not able to estimate statistically significant parameters for , and ω .This is a problem inherent in the type of non-linear estimation required for this type of model and one of the biggest weaknesses of these types of models. This can is normally be addressed by using longer time periods or simply improving the data resolution. This was not possible with the data available to us for this paper as the regression intervals are constrained in time and we could not find all the raw data necessary to estimate the model with
Table 5: A Kolmogorov-Smirnov test comparing the distribution of agents following an investment strategy with naïve expectations.
The Kolmogorov-Smirnov test is a one sided test used to identify is the values drawn from one sample are smaller than the values drawn from another. The first row in the table above shows that
the fraction of investors with a naïve belief type in the period 1914-1966 is not statistically
significantly smaller than those in the period 1966-1996. However in the second row we see that for
the period 1966-1996 the fraction of investors with naïve belief type is statistically significantly
smaller than for the period 1914-1996. This means that for the period 1966-1996 there is a smaller proportion of agents in our model showing naïve investment behaviour than for the period 1914-1996.
22 lower than quarterly resolution. However, if such data can be found it would offer an improvement in the estimation of the models and likely produce significant effects for both the DAX30 and the S&P500 between 1966 and 1996.
In the cases of all the models presented in this paper we found multiple sets of values that they could converge on although the majority can be discarded as they produce estimates that are not economically rational (it would not for example make sense to have a value for the memory parameter ω greater than one). The values presented here are the most likely candidates for the correct regression equilibria based upon their economic rationality, statistical significance and the total sum of squares for each regression. Despite this it is still possible that some of the sets of values may be erroneous. This is perhaps the largest and unfortunately an unavoidable problem with non-linear estimations. However, studying a series of models and ensuring that their values are relatively consistent as we have done in this paper is an improvement over previous methodology as it reduces the risk of identifying an incorrect set of parameters that in isolation may otherwise look reasonable. In all of our models we found estimates for that indicated agents with this belief type behaved with naïve expectations regarding future prices or in the case of the S&P500 between 1996 and 2016 had only slight speculative tendencies. This is most consistent with the explanation that instead of engaging in speculative behaviour as was found by Hommes and in’t Veld (2015), agents with this belief type tend to naively forecast that prices will not change in the short term. This behaviour would be more consistent with a form of behavioural anchoring where agents are using the current price as a reference point for anticipating short term market performance. Interestingly we did not observe any large differences in the estimated behaviour type between those models that contained periods with a financial transaction tax, and those that did not. This is contrary to what we expected from our initial hypotheses.
However for estimates of we did observe that for models wholly or partially containing periods with a financial transaction tax lower values for were identified. This therefore leads to higher estimated values for behavioural heterogeneity for these periods. This may suggest that instead of affecting the behaviour of those agents with naïve expectations, introducing a financial transaction tax instead influences those agents who believe that short term future prices will return to fundamental values, to now believe that prices will return to fundamentals more quickly. This would support our hypotheses to the extent that the model can identify changes in market behaviour but that financial transaction taxes may actually affect behaviour in way that we (or policy makers) did not anticipate.
In our comparison of the CAC40 and DAX30 we found both the price-dividend ratios and the market fractions are broadly similar up until 2012. We also found that, despite the estimated parameters for the DAX30 falling just short of significance at the 90% level the parameters we estimated were fairly close, with the sole exception of the parameter for fundamentalist beliefs. This supports our initial assumption that the DAX30 provides a good sense of how the CAC40 may have behaved if a financial transaction tax had not been introduced. From 2012 onwards the data clearly shows that behaviour in the two markets differs significantly, with a dramatic increase in the volatility for the fraction of naïve investors. This is not what we expected from our initial hypothesis and as such we could not make a direct statistical comparison.
23 However these results may be consistent with other work on the impact of financial transaction taxes that suggests financial transaction taxes may in fact increase market volatility and negatively impact on price discovery (Habermeier & Kirilenko, 2001). In this context we would expect an increase in the volatility of market fractions as agents struggle to adapt their strategies. Under such an assumption it may be likely that the behavioural results we observe are actually consistent with the introduction of a financial transaction tax suggesting that the model may well be accurately capturing the market behaviour. It should be noted that we cannot definitively attribute this change in behaviour found by the model to the introduction of a financial transaction tax. It is entirely possible that some or all of the behavioural change is driven by an entirely unrelated factor, and more work would need to be done to conclusively disprove this.
The analysis of the S&P500 found that qualitatively there is no obvious difference shown in the model between periods during which a financial transaction tax is implemented and those where it is not. We performed a Kolmogorov-Smirnov test to try and identify if there was a significant difference in behaviour between the period 1914-1966 when there was a financial transaction tax and the period 1966-1996. This was further reinforced by similar the results of a Wilcoxon rank-sum test. In order to do so we had to make several very strong assumptions regarding the data that may mean such tests are not in fact appropriate. If there are other tests available that do not require such assumptions it would be a significant improvement to the methodology.
In performing the Kolmogorov-Smirnov and Wilcoxon rank-sum tests we found that levels of naïve investing behaviour are higher during the period when there was a financial transaction tax instituted when compared to when one was not in place. This was somewhat surprising and is the opposite to the prediction of our second hypothesis. However, this may again be better explained by the idea that financial transaction taxes may increase market volatility and negatively impact on price discovery (Habermeier & Kirilenko, 2001). As mentioned in the theory our agents all agree that price dividend ratios will eventually return to fundamental values, but differ in their short term expectations as to when this might happen. Therefore, due to an increase in uncertainty for being able to anticipate when short when values will return towards fundamentals resulting from a financial transaction tax, agents may well increasingly choose to use the current value an anchor point for anticipating short term performance.
Of significant concern to policy makers we find that where financial transaction taxes are introduced, our models show they serve to increase levels of heterogeneity in the market. In the context of this type of model, increased behavioural heterogeneity serves as a destabilising factor that leads to the formation of asset bubbles. This is again consistent with the view that financial transaction taxes in fact have an unintended destabilising impact on markets
6. Conclusions
Given the results that we have found it is not possible for us to definitively conclude that the model is able to detect changes in investor behaviour as the result of a financial transaction tax. While we do find some evidence that is suggestive that this is the case, the results do not confirm our initial hypotheses and instead indicate that financial transaction taxes may impact market behaviour in a way that both ourselves, others and policy makers did not expect them to. The difficulties experienced
24 in estimating our model show that there are significant limitations in the usefulness of this class of models, especially where only short periods of relevant data is available.
In order to determine if models of behavioural heterogeneity are indeed useful and can correctly identify behavioural changes in markets more work should be done in a similar way to what we have begun in this paper. It is possible that a comparison of markets, similar to our examination of the CAC40 and DAX3, but using policy other than a financial transaction tax to change market behaviour would be more successful at confirming or repudiating the effectiveness of the model. In the absence of such a test repeating this methodology in a decade when more data is available for the CAC40 and DAX3 may yield more fruitful results.
What this research does however indicate is that models such as the one used here could offer new insights into the behavioural implications of market policies such as financial transaction taxes that would not otherwise be observed. The model shows that the use of financial transaction taxes can lead to an increase in behavioural heterogeneity. This would support the view that financial transaction taxes can negatively affect market stability. If the model does indeed accurately reflect the behaviour of market participants then the results found here add to the literature by providing a more fundamental understanding of how the behavioural effect on investors of a financial transaction taxes can destabilise markets. This has obvious and significant implications for policy makers and for this reason alone it would be of great benefit to continue developing, testing and verifying the usefulness of these models with empirical data.
