Modelling markets as ecosystems with the help of maximum entropy

The French stock broker Jules Augustin Frédéric Regnault was the first who suggested a modern theory of stock price changes, based on statistical and probabilistic analyses, in his Calcul des Chances et Philosophie de la Bourse (1863). His hypotheses were later used by the French mathematician Louis Bachelier to found the science of investing or mathematical finance with his 1900 doctoral thesis, Théorie de la speculation (The Theory of Speculation). It was an attempt to create a formula to capture the movement of bonds in the Paris Bourse, or stock exchange. Bachelier developed the theory of what we nowadays call Brownian motion , named after the Scottish botanist Robert Brown.

In 1827, Brown was struck by the jittery motion of small particles of pollen floating on the surface of water he observed through a microscope. Since pollen was organic, Brown's first thought was that it might be exhibiting signs of life as it jumped around the surface. But when Brown later saw the pollen's behavior replicated by inorganic matter like coal dust, he ruled out the hypothesis that the effect was life-related. Brown couldn't provide an explanation of this frenetic dance.

In fact, it was Einstein in 1905 who developed the theory for the Brownian motion by modelling the motion of the pollen particles as being displaced by collisions with individual water molecules. The direction of the force of atomic bombardment is constantly changing, and at different times the particle is hit more on one side than another, leading to the seemingly random nature of the motion. Importantly, this explanation of Brownian motion served as convincing evidence that atoms and molecules exist. The many-body interactions that yield the Brownian motion cannot be solved by taking into account every involved molecule. Instead, Einstein adopted a statistical mechanics approach, in terms of a probabilistic description of the molecular effects on the grain of pollen. He arrived at a theory formally identical to the one devised five years earlier by Bachelier for the movement of bonds. So here we have another formal equivalence linking statistical mechanics with finance economics.

When treated as a discrete-space (integers) and discrete-time model, Brownian motion becomes the epitome of a stochastic or random process: the random walk . In other words, essentially, Bachelier's formula tells us that the price of a security rises or falls accordingly to the result of flipping a coin. This random walk is also called the drunkard's walk . The simplest example is one-dimensional; let us imagine a drunkard who exits a bar attempting to return home. He is so intoxicated that he only remembers his house and the bar are both on the same street, but he has no idea whether it is to the left or to the right of the bar. So he starts by arbitrarily choosing the right and giving a step toward this direction. But then he hesitates and before every next step he faces a binary decision: right or left? This concept was introduced into science by Karl Pearson in a letter to Nature in 1905: 'A man starts from a point O and walks l yards in a straight line; he then turns through any angle whatever and walks another l yards in a second straight line. He repeats this process n times.' Pearson then required the probability that after these n stretches the man is at a distance d from his starting point O. Interestingly, the solution to this problem was provided in the same volume of Nature by Lord Rayleigh (1905), who told him that he had solved this problem 25 years earlier when studying the superposition of sound waves of equal frequency and amplitude but with random phases. Pearson concluded that 'The lesson of Lord Rayleigh's solution is that in open country the most probable place to find a drunken man who is at all capable of keeping on his feet is somewhere near his starting point!'

However, Bachelier's theory remained unnoticed gathering dust on library bookshelves for 60 years until the American economist Paul Samuelson discovered it. Samuelson expanded and refined Bachelier's work. As Bachelier did, he also assumed that market prices are the best estimates of value and that price changes follow random patterns, so future news and stock prices are unpredictable. A problem with Bachelier's modelling was that security prices can take negative values, but this is nonsense from a financial point of view. This drawback was overcome by Samuelson (1965a, 1965b) who proposed the so-called Geometric Brownian motion, in which the natural logarithm of the price is assumed to walk a random walk, usually a random walk with drift. That is, the changes in the natural log from one period to the next are assumed to be independent and identically normally distributed.

Therefore, the so-called random walk hypothesis (RWH) (Cootner 1964, Fama 1965, Malkiel 1973) is a financial theory stating that stock market prices evolve according to a random walk (so price changes are random) and thus cannot be predicted.

Since prices move as a random walk, Bachelier set forth the revolutionary conclusion that there is no useful information contained in historical price movements of securities. In his own words:

'The mathematical expectation of a speculator is nil' (Bachelier 1900).

Between the time Bachelier proposed his theory and the 1960s when it was rediscovered, the Wall Street Crash of 1929 occurred and a severe worldwide economic depression, the so-called Great Depression, took place during the 1930s. These remarkable economic events stimulated some fruitful investigation. For example, James Case in his 2008 Competition: The Birth of a New Science tells the findings of the American economist and businessman Alfred Cowles III supporting the difficulties of predicting market movements:

'In 1928, he purchased subscriptions to twenty-four of the most widely circulated newsletters and monitored their performance for four eventful years−... To his surprise, Cowles discovered that exactly none of twenty-four publications had managed to foretell either the crash of 1929 or the steady market decline that followed'

This idea that financial market returns are difficult to predict is the basis of the modern efficient-market hypothesis (EMH) (Samuelson 1965a, 1965b, Fama 1970), which states that asset prices reflect all available information and as a consequence that it is impossible to 'beat the market' consistently since market prices should only react to new information. In other words, stocks always trade at their fair value on exchanges, making it impossible for investors to purchase undervalued stocks or sell stocks for inflated prices. Therefore, it should be impossible to outperform the overall market through expert stock selection or market timing. The RWH is consistent with the EMH.

EMH is somehow related with the rational choice theory we discussed in chapter 4, both assume that there is a well-defined model of economic behavior and that rational investors would all follow it. EMH added another step. In the strong version of the theory, financial markets, because they are populated by a multitude of rational and competitive players, would always set prices that reflected all available information in the most accurate possible way.

Proponents of EMH posit that investors benefit from investing in a low-cost, passive portfolio. In fact, data compiled by Morningstar Inc., in its August 2019 Active/Passive Barometer study, supports the EMH. This report measures the performance of U.S. active funds against passive peers in their respective Morningstar Categories. The Active/Passive Barometer spans more than 4000 unique funds that account for approximately $12.5 trillion in assets, or about 64% of the U.S. fund market. Morningstar compared active managers' returns in all categories against a composite made of related index funds and exchange-traded funds (ETFs). The study found that over a 10 year period beginning June 2009, only 23% of active managers were able to outperform their passive rivals (Morningstar 2019).

In addition, while a percentage of active managers do outperform passive funds at some point, the challenge for investors is being able to identify which ones will do so over the long term. Less than 25% of the top-performing active managers can consistently outperform their passive manager counterparts over time (Downey 2021).

Economist Burton Malkiel mocked the financial services industry. He famously quipped, 'A blindfolded monkey throwing darts at a newspaper's financial pages could select a portfolio that would do just as well as one carefully selected by the experts.' Actually, not a monkey but a hamster is currently proving such a claim. And it's not stocks: it's cryptocurrencies. This crypto-trading hamster that goes by the name of Mr Goxx is a social media sensation in Germany. Named after Mt Gox—a Tokyo-based cryptocurrency exchange company founded in 2010 that was responsible for more than 70% of bitcoin transactions at its peak until it was hacked and declared bankruptcy in 2014—the rodent is equipped with a trading office attached to his regular cage. Mr Goxx goes on the wheel where he selects which cryptocurrency he would like to trade by simply spinning it. The office has two tunnels: one for buying and one for selling, and every time he goes through one of them, the desired transaction is completed. The hamster livestreams his crypto tradings on video live streaming service twitch. As of 24 September 2021, the hamster's performance was up more than 16% ¹ since it began trading in June 2021 (Protos 2021), an impressive feat for anyone. Bitcoin and the S&P 500 have climbed about 14% and 5% respectively, over the same time period. In other words, the hamster is beating not only bitcoin, but the S&P 500, as well as many professional traders and funds since it started trading.

There have been several criticisms of EMH. Some of the most common are as follows.

First, real markets exhibit inefficiencies and there are some markets that are less efficient than others. An inefficient market is one in which an asset prices do not accurately reflect their true or fair value. This may occur for several reasons. Market inefficiencies may exist due to information asymmetries, lack of competition, tax distortions, lack of buyers and sellers (i.e., low liquidity), high transaction costs or delays, market psychology, and human emotion, among other reasons (Downey 2021). In particular, psychology and emotions are key drivers of market sentiment. That is, the overall consensus of investors about a stock or the stock market as a whole is not always based on fundamentals. Day traders and technical analysts also rely on market sentiment, as it influences the technical indicators they utilize to measure and profit from short-term price movements often caused by investor attitudes toward a security. In A Study of History, Arnold Toynbee (1946) warns to beware of what he calls the 'apathetic fallacy', whereas 'Ruskin warned his readers against the 'pathetic fallacy' of imaginatively endowing inanimate objects with life', when addressing a system composed of human beings we need to avoid the converse error of blindly applying a scientific method devised for the study of inanimate nature.

Second, Eugene Fama never imagined that his efficient market would be 100% efficient all the time. That would be impossible, as it takes time for stock prices to respond to new information. The EMH, therefore doesn't give a strict definition of how much time prices need to revert to fair value. Third, the EMH assumes a sort of homogeneity of the market participants; i.e., all investors perceive all available information in precisely the same manner. But this is hardly the case. For example, there are many different methods for analyzing and valuing stocks. If one investor focuses on value, and then looks for undervalued market opportunities, while another focuses on growth and evaluates the same stock on the basis of its growth potential, these two investors may arrive at a different assessment of the stock's fair market value. Therefore, since investors value stocks differently, it is impossible to determine what a stock should be worth under an efficient market. Fourth, under the EMH, no investor should ever be able to beat the market or the average annual returns that all investors and funds are able to achieve using their best efforts. This is why proponents of the EMH conclude investors may profit from investing in a low-cost, passive portfolio, i.e., purchasing a representative benchmark, such as the S&P 500 index, and hold it over a long time horizon. But there are many investors who have consistently beaten the market. Warren Buffett is one of those who's managed to outpace the averages year after year (Dhir 2021).

In addition, there is evidence against the RWH for the logarithm of wealth relatives of small-firms portfolios (Lo et al 2002).

Opponents of EMH believe that it is possible to beat the market and that stocks can deviate from their fair market values. In particular, behavioral finance challenges the notion that companies always trade at their fair value, pointing out that investors are not always rational and stocks do not always trade at their fair value during financial bubbles, crashes, and crises. Thus behavioral economists attempt to explain stock market anomalies through psychology-based theories. This is the case of the adaptive market hypothesis (AMH), an alternative economic theory that combines principles of the well-known and often controversial EMH with behavioral finance (Lo 2004).

AMH argues that people are motivated by their own self-interest, make mistakes, and tend to adapt and learn from them. In fact, Andrew Lo, the theory's founder, believes that people are mainly rational, but sometimes can overreact during periods of heightened market volatility (Liberto 2021). He postulates that investor behaviors—such as loss aversion, overconfidence, and overreaction—are consistent with evolutionary models of human behavior, which include actions such as competition, cooperation, adaptation, and natural selection (Lo 2004). In the next subsection we will briefly elaborate on this apparent dichotomy of competition/cooperation in markets.

Competition has been for a long time the keyword of financial markets. There are several kinds of competition. There is competition between investment funds for money, or between individual investors, and between firms to sell their products. Indeed, for several years strategic management researchers focused on competitive relationships between firms (Porter 1980).

However, businesses also cooperate with each other, in various ways and forms. Since the early 1980s, we have been able to observe a soaring interest in alliances, joint ventures, collusions, federations, clusters and so on (Czakon 2010), collectively called interorganizational relationships (for a review see: Oliver and Ebers 1998). Actually, cooperation and competition characterize the inter-firm relationships in strategic alliances (Clarke-Hill et al 2003). Therefore, the competitive paradigm (Rumelt 1991) as well as the cooperative paradigm (Dyer and Singh 1998) both offer incomplete pictures of a more complex tangle of relationships between firms. Instead, the neologism 'Coopetition' (Yami et al 2010), coined to describe cooperative competition, seems to better capture the reality of inter-firm relationships. Inter-firm dynamics can be classified as competitive, cooperative, mixed (competitive-cooperative), or neither/nor (Chen 2002). A competitive action may elicit a cooperative response (from players in either the same industry or a different industry from where the action is taken), and cooperation between two firms often provokes competitive actions. As we have seen in chapter 3, this mix of competition and cooperation is quite common in ecological communities thus suggesting that a more realistic picture of markets is provided by networks ecosystems (Astley and Fombrun 1983, Emary and Fort 2021, Oliver and Ebers 1998).

Interestingly, coopetition is a dynamic process; a firm's competition with another firm at one level, at a given time and within one organizational unit, usually may turn cooperation with this same firm in different circumstances and vice versa (Chen 2002).

Often coopetition takes place when companies that are in the same market work together in the exploration of knowledge and research of new products, at the same time that they compete for market share of their products and in the exploitation of the knowledge created. In this case, the interactions occur simultaneously and in different levels in the value chain. For instance, on 17 March 2020, Pfizer Inc. and BioNTech SE announced a collaboration to jointly develop a COVID-19 vaccine (Pfizer 2020). This is an example of coopetition agreement between the two companies that increased the manufacturing capacity to meet the global supply for the vaccine. In this way the companies were able to produce millions of vaccine doses by the end of 2020 and hundreds of millions of additional doses in 2021. later on, in June 2021, the companies announced another deal with the government to provide 500 million more doses of the vaccine to support some of the poorest countries. The agreement states that 60% of the doses are to be purchased during the first half of 2022 (Pfizer 2021). BioNTech contributed the vaccine candidates, while Pfizer contributed the clinical research and development as well as the manufacturing and distribution capabilities of the company.

At any rate, whatever the pattern of inter-firm relationships, at the end of the day money tends to flow from firms that underperform to those that outperform the market. This suggests that the concepts of population dynamics and organizational ecology we discussed in chapter 4 can be useful to describe market dynamics (Dosi and Nelson 1994, Moore 1993a, 1993b, Nelson and Winter 1982), which is our goal, as we explain in the next section. For example, Lakka et al (2013) analyze the evolutionary and competitive dynamics of the highly concentrated desktop/laptop operating systems market as if companies were species of an ecosystem and draws conclusions that may be valuable inputs for managerial decisions and strategic planning to the players of software markets.

To model the dynamics of the chosen set of firms, representative of the NYSE market, we use a finite difference version of the RDE equation (4.10), with time steps coinciding with days, given by:

$$\begin{eqnarray}&&{x}_{i}(t+1)-{x}_{i}(t)={x}_{i}(t)\left(\displaystyle \sum _{j=1}^{38}{P}_{ij}{x}_{j}-\phi ({\bf{x}};t)\right),\,i=\mathrm{1,2},\ldots ,38,\end{eqnarray} \tag{ 7.3 }$$

where P = [P_ij ] is a 38×38 payoff matrix, that is, P_ij is the payoff received by company i from its interaction with company j. So our first task is to estimate this matrix. To do this we will use once again the recipe developed in chapter 5, i.e., to infer through the MaxEnt principle a PME model. The interaction matrix ${\bf{I}}=[{I}_{ij}]$, given by equation (5.10), in turn yields a payoff matrix given by equation (4.23). Thus we finally arrive at the set of equations for the RDPME method:

$$\begin{eqnarray}&&{x}_{i}(t+1)-{x}_{i}(t)={r}_{i}{x}_{i}(t)\left(\displaystyle \sum _{j=1}^{38}{I}_{ij}{x}_{j}-\phi ({\bf{x}};t)/{r}_{i}\right),\,i=\mathrm{1,2},\ldots ,38,\end{eqnarray} \tag{ 7.4 }$$

From covariance matrices to interaction matrices

As is customary in time series forecasting analysis, we partition the data set of 38 time series of length T into an earlier training period T_tr and a later validation period T_v (such that T_tr + T_v = T) (Shmueli and Lichtendahl 2016). The training period is used to estimate the parameters of the model (in our case, the RDE), and then this model with these estimated parameters is used to generate forecasts to be compared with data corresponding to the validation period. In this way we assess the predictive performance of a model on new data. Varying the training period T_tr allows one to develop multiple model instances. Thus, the validation partition is used to assess the performance of each of these models for different time frames.

Specifically, we obtain the covariance matrix Σ = [Σ_ij ] whose entries are given by (see equation (5.5b)):

$$\begin{eqnarray}&&{{\rm{\Sigma }}}_{ij}=\displaystyle \sum _{t=1}^{{T}_{{\rm{tr}}}}{\mathit{\unicode[Book Antiqua]{x76}}}_{i}(t) {\mathit{\unicode[Book Antiqua]{x76}}}_{j}(t) /{T}_{{\rm{tr}}}-\displaystyle \sum _{t=1}^{{T}_{{\rm{tr}}}}{\mathit{\unicode[Book Antiqua]{x76}}}_{i}(t)\displaystyle \sum _{s=1}^{{T}_{{\rm{tr}}}}{\mathit{\unicode[Book Antiqua]{x76}}}_{j}(s)/{{T}_{{\rm{tr}}}}^{2},\end{eqnarray} \tag{ 7.5 }$$

and then, from equation (7.5), we first obtain matrix ${\boldsymbol{M}}={{\boldsymbol{\Sigma }}}^{-1}$ through equation (5.9b) and next matrix I through equation (5.10).

Some general properties of matrix I are:

• I.  
  It has positive as well as negative elements; this is in agreement with what we mentioned in section 7.5.1.
• II.  
  Furthermore, positive elements dominate over negative elements. Indeed, the same happens for the covariance matrix, which is a known result (Chan et al 1999, Engle 2009, Vasicek and McQuown 1972).
• III.  
  Despite most of its entries being in (−0.5,+0.5), there is a large dispersion, as shown in panel (b) of figure 7.1 which shows this matrix I for a training period T_tr = 1000.
• IV.  
  As can be seen from panel (a) of figure 7.1, all the large departures to these values fall in the upper triangle part; that is the interaction strength of smaller cap firms (high firm numbers) over larger firms (low firm numbers).

Property IV can be understood from the covariance matrix with firms ranked by their market value as follows:

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First, from equation (7.5), the larger entries of Σ occur at the upper left corner (panel (a) of figure 7.2). Second, since ${\boldsymbol{M}}=-{{\boldsymbol{\Sigma }}}^{-1}$, the larger entries of this MaxEnt matrix will concentrate at the bottom right corner (panel (b) of figure 7.2). This can be shown by using that ${{\boldsymbol{\Sigma }}}^{-1}$ = adj(Σ)/det(Σ) (equation (vi) of Box 5.1). Thus, for firms with larger market values adj(Σ) ≪ det(Σ); conversely, for firms with smaller market values adj(Σ) ≫ det(Σ). And this explains the places where the smaller (larger) values of M do occur.

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Second, to obtain the MaxEnt interaction matrix I the entries of each row i of M are divided by M_ii ; since M_ii increases with i this implies that the stronger interaction coefficients will appear in the upper right corner (panel (c) of figure 7.2).

We will use the RDPME equation (7.4) to predict the market shares for T_tr + 1 ⩽ t ⩽ T = T_tr + T_v with an interaction matrix I estimated for t = T_tr. It is worth remarking that this matrix depends of the training period. Figure 7.3 shows the matrices I obtained for different training periods; in the left column T_tr = 100, 300 and 500 days, while in the right column T_tr =1000, 1200 and 1400.

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By simple visual inspection it seems that I changes faster from lower values of T_tr. In the appendix at the end of this chapter we propose a metric to quantify the variation rate of an arbitrary matrix M, and that indicates that after T_tr = 600 the variation rate for matrix I stabilizes thus suggesting the entrance of the market into a kind of stationary state.

Growth rates

To compare predictions from the RDE against the observed x_i (t) for t > T_tr, we still have to estimate the growth rate r_i parameter. There are different alternative procedures to obtain this estimate from equation (7.4). One direct way is by taking:

$$\begin{eqnarray}&&{r}_{i}=\,\max \,\left(0,\displaystyle \frac{{x}_{i}({T}_{{\rm{tr}}})-{x}_{i}(1)}{\displaystyle \sum _{t=1}^{{T}_{{\rm{tr}}}-1}{x}_{i}(t)\left(\displaystyle \sum _{j=1}^{38}{P}_{i}{x}_{j}(t)j-\phi ({\bf{x}};t)\right)}\right)\end{eqnarray} \tag{ 7.6 }$$

The maximum in equation (7.6) is because the sign of x_i (t + 1) − x_i (t) in the replicator dynamics equation (7.3) must be the same as the one of $\left(\displaystyle \sum _{j=1}^{38}{P}_{ij}{x}_{j}-\phi ({\bf{x}};t)\right)$, i.e., it must be determined by whether the fitness of firm i is above or below the weighted average fitness, and thus r_i must be ⩾ 0 for all i. Therefore, for a given training period there are firms with growth rates r_i >0 and others with r_i = 0. Typically, half of the firms have r_i >0. The RDPME reduces for firms with r_i = 0 to a 'null' model which predicts that the share of company i remains constant, i.e., ${x}_{i}^{{\rm{pred}}}(t)$ = constant = x_i (T_tr) for all t>T_tr. An alternative estimate of r_i from equation (7.4) is by using least squares, however, this method yields larger errors.

The random walker as standard benchmark

According to the RWH, stock market prices evolve as a random walk. As we have seen, this hypothesis is consistent with the EMH, which implies that it is impossible to 'beat the market' consistently.

We have also noticed that both the RWH and the EMH have been challenged by economists and investors who claim that stock market prices may move in trends and that the market is predictable to some degree. Indeed, there have been several economic studies and tests supporting this view (Lo 1999, 2004, Fromlet 2001, Lo et al 2002).

In any case, the standard model for stock prices is the geometric Brownian motion (Ross 2014). Therefore, here we use the discrete-time version of the geometric Brownian motion, aka geometric random walk (grw), as a model to compare the RDPME. Denoting by w_i the natural logarithm of the market value of company number i, it is easy to show that if it evolves as a random walk with probability p_i and 'step' s_i , then its average at time step t will be given by

$$\begin{eqnarray}&&\bar{{w}_{i}^{\text{rw}}}(t)={w}_{i}^{} ({t}_{0})+(\text{2}p -i1){s}_{i}(t-{t}_{0}),\end{eqnarray} \tag{ 7.7 }$$

where w_i (t₀) is its initial value at time step t₀. Then, the training period is used to estimate parameters p_i and s_i of equation (7.7) as:

And the grw predictions of market values for t ⩽ T_v are thus obtained as:

$$\begin{eqnarray}&&\bar{{\nu }_{i}^{\text{rw}}}({T}_{\nu })={\nu }_{i}^{}({T}_{\mathrm{tr}})\exp [(2\displaystyle \sum _{t\text{'}=1}^{{T}_{\mathrm{tr}}-1}\frac{\text{sign}({w}_{i}(t\text{'}+1)-{w}_{i}(t\text{'}))}{{T}_{\mathrm{tr}}-1}-1)\frac{|({\nu }_{i}({T}_{\mathrm{tr}})-{\nu }_{i}(1))|}{{T}_{\mathrm{tr}}-1}(t-{T}_{\mathrm{tr}})].\end{eqnarray} \tag{ 7.9 }$$

Therefore, the corresponding shares are given by:

$$\begin{eqnarray}&&{x}_{i}^{\text{rw}}(t)=\frac{{\nu }_{i}^{}({T}_{\mathrm{tr}})\exp [(\text{2}\displaystyle \sum _{t\text{'}=1}^{{T}_{\mathrm{tr}}-1}\frac{\text{sign}({w}_{i}(t\text{'}+1)-{w}_{i}(t\text{'}))}{{T}_{\mathrm{tr}}-1}\text{\hspace{0.17em}}-1)\frac{|({\nu }_{i}({T}_{\mathrm{tr}})-{\nu }_{i}(1))|}{{T}_{\mathrm{tr}}-1}(t-\mathrm{tr})]}{\displaystyle \sum _{i=1}^{S}{\nu }_{i}^{}({T}_{\mathrm{tr}})\exp [\text{(2}\displaystyle \sum _{t\text{'}=1}^{{T}_{\mathrm{tr}}-1}\frac{\text{sign}({w}_{i}(t\text{'}+1)-{w}_{i}(t\text{'}))}{{T}_{\mathrm{tr}}-1}\text{\hspace{0.17em}}-1)\frac{|({\nu }_{i}({T}_{\mathrm{tr}})-{\nu }_{i}(1))|}{{T}_{\mathrm{tr}}-1}(t-{T}_{\mathrm{tr}})]}.\end{eqnarray} \tag{ 7.10 }$$

Quantifying the accuracy of predictions through the percentage errors of share trajectories

To compare accuracy for predicting future fractions x_i of the RDPME, equation (7.4), versus the one of the grw benchmark forecasting, equation (7.10), the procedure is as follows:

First, the mean absolute percentage error (MAPE) are computed over the validation period T_v, for each firm i with the two formulas. Denoting by x^pred predictions using either the RDPME or the grw, the MAPE for firm i is given by:

$$\begin{eqnarray}&&{{\rm{M}}{\rm{A}}{\rm{P}}{\rm{E}}}_{i}=100\displaystyle \frac{\displaystyle \sum _{t=1}^{{T}_{v}}\displaystyle \left|x(t{)}_{i}^{{\rm{p}}{\rm{r}}{\rm{e}}{\rm{d}}}-{x}_{i}\right|/{x}_{i}(t)}{{T}_{{\rm{v}}}},\,i=1,2,\ldots ,S.\end{eqnarray} \tag{ 7.11 }$$

This metric quantifies the error of the predicted 'trajectory' for the share x_i of firm i between days T_tr + 1 and T _{$\mathit{\unicode[Book Antiqua]{x76}}$} with respect to the empirical trajectory it has followed.

Second, to obtain a global assessment of the accuracy of the method, the average of MAPE_i over all firms is taken thus producing a global MAPE denoted as GMAPE:

$$\begin{eqnarray}&&{\rm{G}}{\rm{M}}{\rm{A}}{\rm{P}}{\rm{E}}=\displaystyle \sum _{i=1}^{S}\displaystyle \frac{{{\rm{M}}{\rm{A}}{\rm{P}}{\rm{E}}}_{i}}{S}=100\displaystyle \sum _{i=1}^{S}\displaystyle \frac{\displaystyle \frac{\displaystyle \sum _{t=1}^{{T}_{v}}\displaystyle \left|x(t{)}_{i}^{{\rm{p}}{\rm{r}}{\rm{e}}{\rm{d}}}-{x}_{i}\right|/{x}_{i}(t)}{{T}_{{\rm{v}}}}}{S}.\end{eqnarray} \tag{ 7.12 }$$

The smaller this global metric the more accurate is the method as a whole.

Let us first analyze the accuracy of RDPME short-term and long-term predictions for each company. This is done by considering a relatively short validation period of T _{$\mathit{\unicode[Book Antiqua]{x76}}$} = 50 days (and for 14 training periods, T_tr = 100, 200, ..., 1300, 1400 days) and a relatively long validation period of T _{$\mathit{\unicode[Book Antiqua]{x76}}$} = 500 days (and for five training periods, T_tr = 500, 600, 700, 800 and 900 days).

Results for both validation periods T _{$\mathit{\unicode[Book Antiqua]{x76}}$} = 50 days and T _{$\mathit{\unicode[Book Antiqua]{x76}}$} = 500 days (and for the corresponding different training periods T_tr) are depicted in figure 7.4. In this figure it is also indicated for which firms the estimated r_i >0, and thus the RDPME predicts a variable share, different from the empirical share at t = T_tr. In panels (a) and (c) if the cell at (T_tr,i) is black it means that for this company i and this training period T_tr equation (7.6) produces a positive growth rate parameter r_i . Conversely, if the cell is white it means that the obtained r_i by equation (7.6) is 0 and thus the RDPME predicts a market share of this firm ${x}_{i}^{{\rm{pred}}}(t)$ = constant = x_i (T_tr) for all t>T_tr. Indeed, roughly half of the cells in panels (a) and (c) are black, meaning that the growth rate parameter r_i produced by equation (7.6) is positive 50% of the times. For example, notice that there are two companies for which r_i >0 for all training periods: MSFT (company #2) and WMT (company #8). Additionally, there is one firm, MCK (company #34), such that r_i = 0 for all training periods. Colored cells in panels (b) and (d) denote the MAPE for cell (T_tr,i) given by equation (7.11). Blue (red) indicates small (large) MAPE. A pattern which can be observed by direct inspection is that larger MAPE occurs for smaller cap firms. This pattern is particularly evident for T _{$\mathit{\unicode[Book Antiqua]{x76}}$} = 500 days, where the orange-red cells occur mostly in the lower part of panel (d).

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Regarding forecasting trajectories of shares of companies x(t), figure 7.5 shows the empirical trajectories for the shares of 20 companies (in black) and those predicted by RDPME, equation (7.4) (red), and grw, equation (7.11) (blue), for T_tr = 200 and T_v = 50 days. The 20 companies are those such that equation (7.7) produces, for T_tr = 200 days, an intrinsic growth rate r_i >0 and thus equation (7.4) leads to a share x_i that varies with time. For the other 18 firms equation (7.7) produces an intrinsic growth rate r_i = 0 and thus the prediction of RDPME through equation (7.4) is a horizontal line x^pred(t) = x_i (200) for all t such that 200 <t⩽ T _{$\mathit{\unicode[Book Antiqua]{x76}}$} = 50 days (not shown in the figure). Notice that while empirical shares describe wiggled curves, those predicted by RDPME or grw do not. This difference is because the RD is a deterministic equation and the grw predictions are mean values of a stochastic process (which yields also deterministic results).

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As we can see from figure 7.5, for this particular partition of T_tr = 200 and T_tr = 50, the RDPME model outperforms the grw benchmark (i.e., it achieves a smaller MAPE) for 16 out of 20 = 80% of these companies, it underperforms the grw for 2 in 20 = 10% and both tie for 2 in 20 = 10%. When considering all the 38 companies, including those with r_i = 0, these percentages become: 63%, 26% and 11%.

Figure 7.6 is the same as figure 7.5 but for a much longer training period; T_tr = 700 days. In this case there are 16 such that equation (7.7) produces an intrinsic growth rate r_i >0 and thus equation (7.4) leads to a share x_i that varies with time.

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Now the RDPME model outperforms the grw benchmark for 10 out of 16 = 63% of the companies; these companies are MSFT, BRK.B, WMT, HD, UNH, CMSA, COST, IBM, LOW and PSX. When considering all the 38 companies, including those with r_i = 0, the percentage of wins becomes 53%. For the great majority of partitions of T_tr−T_tr, these percentages are qualitatively similar.

It is worth remarking that the MAPEs of both methods for this larger validation period are in general much larger (${\overline{{\rm{MAPE}}}}_{{\rm{RDPME}}}$ = 26.1%, ${\overline{{\rm{MAPE}}}}_{{\rm{grw}}}$ = 28.1%). This increase in the errors is an expected result, we have to take into account that the validation period in this latter case comprises more than two years of trade.

Table 7.2 shows the GMAPE, both for the RDPME and grw, for different partitions of T_tr and T_v (in days).

Remarks:

• i.  
  For the great majority of partitions the RDPME yields a smaller GMAPE than the grw.
• ii.  
  For each training period the RDPME achieves always the minimum GMAPE.
• iii.  
  Most of the minimum GMAPE correspond to T_tr = 200, thus suggesting a sort of trade-off implying that very large training periods do not provide larger accuracy.
• iv.  
  As expected, for a given training period, the GMAPE grows with the validation period (i.e., the larger the forecast horizon the less accurate the predictions).