The mathematics behind blockchain In a previous Math Investor blog, we described the emerging world of blockchain, emphasizing how it might influence the financial services and investment world. Already numerous firms, including several startup organizations, are pursuing blockchain to facilitate and streamline many types of financial transactions.

# Social signals and algorithmic trading of Bitcoin

The availability of data on digital traces is growing to unprecedented sizes, but inferring actionable skill from large-scale data is far from being trivial. This is especially significant for computational finance, where digital traces of human behaviour suggest a fine potential to drive trading strategies. We contribute to this by providing a consistent treatment that integrates various datasources in the design of algorithmic traders. This permits us to derive insights into the principles behind the profitability of our trading strategies. We illustrate our treatment through the analysis of Bitcoin, a cryptocurrency known for its large price fluctuations. In our analysis, we include economic signals of volume and price of exchange for USD, adoption of the Bitcoin technology and transaction volume of Bitcoin. We add social signals related to information search, word of mouth volume, emotional valence and opinion polarization as voiced in tweets related to Bitcoin for more than three years. Our analysis exposes that increases in opinion polarization and exchange volume precede rising Bitcoin prices, and that emotional valence precedes opinion polarization and rising exchange volumes. We apply these insights to design algorithmic trading strategies for Bitcoin, reaching very high profits in less than a year. We verify this high profitability with sturdy statistical methods that take into account risk and trading costs, confirming the long-standing hypothesis that trading-based social media sentiment has the potential to yield positive comebacks on investment.

Our online society generates data on the digital traces of human behaviour at unprecedented scales and resolutions. This produces a *data deluge*, in which researchers are confronted with a vast amount of observational data that is not the product of cautiously designed experiments [1]. One of the main challenges of the scientific community is to develop methods to extract meaningful skill from that data beyond mere descriptive analyses [Two]. This is particularly significant in financial trading: data can be available to all financial agents, but it is the analysis and its applications which makes a difference. Within computational finance, the field of algorithmic trading [Three] deals with the implementation and evaluation of automatic trading strategies, which are often kept in private companies and away from publicly accessible research. The most common kind of algorithmic trading is based on the principles of *technical analysis* [Four], using the time series of prices to formulate predictions about comes back. Technical analysis is often insufficient to derive satisfactory comebacks [Five], motivating the inclusion of large-scale social signals and the evaluation through data-driven simulations on historical data, called *backtesting* [6,7]. In this article, we present a set of methods to derive stylized facts from the analysis of multidimensional economic and social signals, and to apply that skill in the design and evaluation of algorithmic trading strategies. We illustrate an application of our treatment to algorithmic trading of the Bitcoin cryptocurrency, using a broad multiplicity of digital traces about economic and social aspects of the Bitcoin ecosystem.

Bitcoin (BTC) is a digital currency designed to operate in a distributed system without any central authority, based on a cryptographic protocol that does not require a trusted third party [8]. Introduced in a two thousand eight paper written under the pseudonym of Satoshi Nakamoto [9], Bitcoin serves as a technology to transfer money quickly for negligible fees [Ten]. One of the very first markets to adopt Bitcoin was the *Silk Road*, a website where illegal commerce became possible thanks to the relative anonymity of Bitcoin [11], in line with the evidence in search trends that relates Bitcoin usage to computer expertise and illegal activities [12]. Since then, the use of Bitcoin has widely expanded beyond criminal activities: at the time of writing, Bitcoin is accepted by many legal merchants and charities [13], including large businesses like Dell [14]. Bitcoin-accepting businesses, exchange markets and wallet services compose the *Bitcoin ecosystem* [8], where different kinds of agents interact, trade and communicate through digital channels. The enhancing adoption of Bitcoin and its online nature permit us to at the same time monitor its social and economic aspects. Every purchase of goods or services in Bitcoin leaves a trace in a public ledger called the *Block Chain*, creating a publicly accessible economic network [15]. Bitcoin’s delocalized technology aligns with the online interaction of its users through social networks and forums, motivating its adoption by fresh users through word-of-mouth [16]. Previous research has shown how search trends and Wikipedia views are related to price switches [17] and to the speculative and monetary aspects of Bitcoin [Legitimate], leading to dynamics that combine search interest, user adoption, word-of-mouth and prices [16].

### 1.1 Contributions of this article

Based on established principles of time-series analysis and financial trading, we present a framework to derive general skill from multidimensional data on social and economic aspects of a market. We apply a general statistical model to detect temporal patterns in the co-movement of price and other signals. Those patterns are tested through a method sturdy to the empirical properties of the analysed data, formulating concise principles on which signals precede market movements. We combine those principles to produce tractable trading strategies, which we evaluate over a leave-out sample of the data, quantifying their profitability. Our treatment, rather than focusing on improving a particular method, takes a multidisciplinary stance in which we combine principles from social psychology and economics with methods from information retrieval, time-series analysis and computational finance.

We apply our framework to the Bitcoin ecosystem, monitoring the digital traces of Bitcoin users with daily resolution. We combine *economic signals* related to market growth, trading volume, and use of Bitcoin as means of exchange, with *social signals* including search volumes, word-of-mouth levels, emotional valence and opinion polarization about Bitcoin. Our results expose which signals precede switches of Bitcoin prices, a skill that we use to design algorithmic trading strategies. We evaluate the power of our strategies through backtesting data-driven simulations, comparing comebacks with technical analysis strategies. As a consequence, we test the hypothesis that social media sentiment predicts financial comes back in the Bitcoin ecosystem.

### 1.Two Social signals in finance

Understanding the role of social signals in finance not only has the potential to generate significant profits, but also has scientific relevance as a research question [Nineteen]. Two different research approaches give insights to this question: one is the *statistical analysis* of social and financial signals in order to test the existence of temporal correlations that lead financial markets. The 2nd one applies these signals in *prediction scripts*, measuring their accuracy as a validation of the underlying behaviour of the system, but not necessarily of their profitability. The statistical analysis of search engine data exposes that search trends can predict trading volumes of individual stocks [20]. In addition, stock prices in S&P five hundred are correlated with tweet volumes [21], but the applicability of these patterns into trading strategies is yet to be evaluated.

Sentiment in social media is closely related to socio-economic phenomena, including public opinion [22]. This motivates the application of sentiment indicators in the statistical analysis of financial data. Early works on the sentiment in specialized forums gave negative results about their influence on comes back [23]. Further research displayed that emotions in private instant messaging inbetween workers of a trading company precede stages of market volatility [24]. The expression of anxiety in publicly accessible data from general blogs precedes trading peaks and price drops in the S&P five hundred [25], and sentiment in Twitter can be used to predict movements in large-scale stock indices [26]. It is significant to note that, to date, there is no evidence that such sentiment-based predictions produce significant comes back on investment [Nineteen].

### 1.Trio Online polarization

While most previous works on sentiment in financial markets concentrate on dimensions of valence or mood, the collective phenomenon of polarization of opinions is often overlooked. The emergence of polarization in a society gives early warnings on political and economic phenomena: polarization in social networks of Swiss politicians precedes controversial elections [27], and polarization patterns in the Eurovision Song Contest show up before states of distrust in the European economy [28]. With respect to financial markets, speculation theories point to the role of diverse beliefs in financial transactions [29], leading to the hypothesis that polarization and disagreement influence trading volumes and prices [30]. In this line, the empirical analysis of polarization in stock message boards shows that states of disagreement lead to enhanced volatility [31].

### 1.Four The missing link

To date, there is a significant skill gap inbetween the analysis and application of social signals to trading scripts. Findings from statistical analyses alone are not assured to lead to profitable strategies at all [25]. For example, movements of the Dow Jones Industrial Average (DJIA) can be predicted with mass media sentiment [32] and Twitter mood [26], but to date no research has shown that such prediction methods can be profitable in trading scripts. Similarly, the analysis of discussion patterns in specialized blogs predict comebacks of some technology companies [33], but it is still open to evaluate the potential comes back of such a predictor. The application of methods that process arbitrarily large datasets lead to results difficult to apply, for example the predicting power of search volumes of the query *‘moon patrol’* [34] in backtesting over the DJIA [6]. Furthermore, analyses of Twitter discussions about companies can be applied in a portfolio strategy, yet its evaluation through backtesting leads to very moderate comes back and their statistical significance is not assessed [35]. In addition, no previous research has proposed a prediction technology that derives significant comes back on investment from online sentiment data [Nineteen]. Our research aims at closing the gap inbetween these lines of research. To do so, we unify the statistical analysis and its application to design and evaluate trading strategies, based on tractable principles with potential influence in the finance community.

## Two. Trading strategy framework

To design and evaluate trading strategies, we present a framework that uses a set of economic and social signals related to the agents of the market under scrutiny. Among those signals, the only required one is an economic signal of prices of an asset, namely a stock, currency, or tradable index. To understand profitability, we convert the price time-series *P*(*t*) into a comeback time series: Ret ( t ) = P ( t ) − P ( t − one ) P ( t − one ) , Two.1 which quantifies proportional switches in the price at every time step. The data on these signals are divided in an analysis period and a leave-out period, as depicted in figure 1. The division in these periods needs to allocate enough data in the leave-out sample to provide the testing power to assess the statistical significance of strategy profits. For daily trading, a leave-out period of about one year is usually sufficient, but this ultimately depends on the expected profitability and variance of the trading strategies.

Framework for analysis of social and economic signals and trading strategy design and evaluation.

### Two.1 Multidimensional analysis

The very first step in our framework concentrates on the analysis period, applying a multidimensional model of vector auto-regression (VAR) [36], which is commonly used in the analysis of multidimensional time series in finance [16,23,37]. A VAR models multidimensional linear relations with given lags, which in our analysis we set to one day. Thus, given the vector of signals *V* (*t*), we fit the equation V ( t ) = ϕ V ( t − one ) + r × t + c + ϵ , Two.Two where *ϕ* is a matrix of weights of the linear relations inbetween variables, *r* is a deterministic trend vector, *c* is the vector of constant intercepts and *ϵ* is a vector of uncorrelated errors. While more advanced models can be considered, including longer lags and nonlinear terms, we choose the VAR model of lag one for its general character and its proved power to expose patterns in finance [16,23]. More sophisticated models might have higher power to expose nuance patterns, but at the expense of a loss of generality owing to the concentrate on particular systems.

We include all the time series in a single model to avoid the false positives associated with pairwise Granger tests. To ensure the correct application of the VAR model, we need to verify that our analysis is consistent with its fundamental assumptions: (i) that the elements of *V* (*t*) do not have a unit root, and (ii) that the error term *ϵ* has no temporal nor structural correlations. We verify the very first set of assumptions on the properties of *V* (*t*) by applying a set of tests and transformations prior to the application of the VAR model. We ensure that our conclusions are sturdy to the 2nd set of assumptions by correcting for correlations in the noise term, as explained in the Material and methods section.

### Two.Two Impulse analysis

The VAR weights *ϕ* are only informative when there are no correlations in the error term *ϵ* of equation (Two.Two), which is usually not the case in practice. To extract stylized facts that can be used in the design of trading strategies, we perform an impulse analysis by measuring impulse response functions (IRF) [38] while correcting for correlations in the empirical error. This method simulates the system dynamics when it receives a shock in one of the variables, applying the VAR dynamics of equation (Two.Two) to reproduce the switches in the rest of the variables through time. By recording the switches in each variable, we can estimate the total size and the timespan of the perturbation produced by the shock. In essence, the IRF method creates a computational equivalent of the system under scrutiny, to test its reaction to exogenous impulses in each of its elements.

### Two.Trio Trading strategy design and evaluation

The output of the impulse analysis step, shown in figure 1, is a set of patterns of Granger-type ‘causation’, i.e. it tests the null hypothesis of the absence of temporal correlations among the variables. We use these patterns as stylized facts that indicate which variables precede switches in price comebacks. For example, if variable *Y* (*t*) has a significant influence on Ret(*t*) in the impulse analysis, we will include *Y* (*t*) in our trading strategy design with sign *s*_{Y}, which takes the value one if the response of Ret(*t*) to *Y* (*t*) was positive, and −1 otherwise. Thus, a predictor based on *Y* (*t*) would be sign ( Ret ( t + one ) ) = sign ( s Y × ( Y ( t ) − Y ( t − one ) ) ) . Two.Trio This way, we predict increases (decreases) in price inbetween time *t* and *t*+1 if signals with positive responses increase (decrease) inbetween time *t*−1 and *t*, and vice versa for signals with negative responses. Since our multidimensional analysis is sturdy to confounds inbetween numerous time series, the findings of impulse analysis can be integrated in a *Combined* strategy based on a voting mechanism. The *Combined* strategy applies the other predictors and formulates a prediction corresponding to the sign of the sum of their outputs, i.e. the majority vote.

We evaluate the profitability of the designed strategies in comparison to the benchmark of standard strategies, based on the backtesting over the leave-out sample as indicated in figure 1. For each strategy, we make a data-driven simulation of a trader following that strategy, and we record the profits of that trader on a daily basis. Details on the computational simulation of financial traders can be found in the Material and methods section.

Social signals in Bitcoin trading, Open Science

# Social signals and algorithmic trading of Bitcoin

The availability of data on digital traces is growing to unprecedented sizes, but inferring actionable skill from large-scale data is far from being trivial. This is especially significant for computational finance, where digital traces of human behaviour suggest a excellent potential to drive trading strategies. We contribute to this by providing a consistent treatment that integrates various datasources in the design of algorithmic traders. This permits us to derive insights into the principles behind the profitability of our trading strategies. We illustrate our treatment through the analysis of Bitcoin, a cryptocurrency known for its large price fluctuations. In our analysis, we include economic signals of volume and price of exchange for USD, adoption of the Bitcoin technology and transaction volume of Bitcoin. We add social signals related to information search, word of mouth volume, emotional valence and opinion polarization as voiced in tweets related to Bitcoin for more than three years. Our analysis exposes that increases in opinion polarization and exchange volume precede rising Bitcoin prices, and that emotional valence precedes opinion polarization and rising exchange volumes. We apply these insights to design algorithmic trading strategies for Bitcoin, reaching very high profits in less than a year. We verify this high profitability with sturdy statistical methods that take into account risk and trading costs, confirming the long-standing hypothesis that trading-based social media sentiment has the potential to yield positive comes back on investment.

Our online society generates data on the digital traces of human behaviour at unprecedented scales and resolutions. This produces a *data deluge*, in which researchers are confronted with a vast amount of observational data that is not the product of cautiously designed experiments [1]. One of the main challenges of the scientific community is to develop methods to extract meaningful skill from that data beyond mere descriptive analyses [Two]. This is particularly significant in financial trading: data can be available to all financial agents, but it is the analysis and its applications which makes a difference. Within computational finance, the field of algorithmic trading [Three] deals with the implementation and evaluation of automatic trading strategies, which are often kept in private companies and away from publicly accessible research. The most common kind of algorithmic trading is based on the principles of *technical analysis* [Four], using the time series of prices to formulate predictions about comebacks. Technical analysis is often insufficient to derive satisfactory comes back [Five], motivating the inclusion of large-scale social signals and the evaluation through data-driven simulations on historical data, called *backtesting* [6,7]. In this article, we present a set of methods to derive stylized facts from the analysis of multidimensional economic and social signals, and to apply that skill in the design and evaluation of algorithmic trading strategies. We illustrate an application of our treatment to algorithmic trading of the Bitcoin cryptocurrency, using a broad multitude of digital traces about economic and social aspects of the Bitcoin ecosystem.

Bitcoin (BTC) is a digital currency designed to operate in a distributed system without any central authority, based on a cryptographic protocol that does not require a trusted third party [8]. Introduced in a two thousand eight paper written under the pseudonym of Satoshi Nakamoto [9], Bitcoin serves as a technology to transfer money quickly for negligible fees [Ten]. One of the very first markets to adopt Bitcoin was the *Silk Road*, a website where illegal commerce became possible thanks to the relative anonymity of Bitcoin [11], in line with the evidence in search trends that relates Bitcoin usage to computer expertise and illegal activities [12]. Since then, the use of Bitcoin has widely expanded beyond criminal activities: at the time of writing, Bitcoin is accepted by many legal merchants and charities [13], including large businesses like Dell [14]. Bitcoin-accepting businesses, exchange markets and wallet services compose the *Bitcoin ecosystem* [8], where different kinds of agents interact, trade and communicate through digital channels. The enlargening adoption of Bitcoin and its online nature permit us to at the same time monitor its social and economic aspects. Every purchase of goods or services in Bitcoin leaves a trace in a public ledger called the *Block Chain*, creating a publicly accessible economic network [15]. Bitcoin’s delocalized technology aligns with the online interaction of its users through social networks and forums, motivating its adoption by fresh users through word-of-mouth [16]. Previous research has shown how search trends and Wikipedia views are related to price switches [17] and to the speculative and monetary aspects of Bitcoin [Eighteen], leading to dynamics that combine search interest, user adoption, word-of-mouth and prices [16].

### 1.1 Contributions of this article

Based on established principles of time-series analysis and financial trading, we present a framework to derive general skill from multidimensional data on social and economic aspects of a market. We apply a general statistical model to detect temporal patterns in the co-movement of price and other signals. Those patterns are tested through a method sturdy to the empirical properties of the analysed data, formulating concise principles on which signals precede market movements. We combine those principles to produce tractable trading strategies, which we evaluate over a leave-out sample of the data, quantifying their profitability. Our treatment, rather than focusing on improving a particular method, takes a multidisciplinary stance in which we combine principles from social psychology and economics with methods from information retrieval, time-series analysis and computational finance.

We apply our framework to the Bitcoin ecosystem, monitoring the digital traces of Bitcoin users with daily resolution. We combine *economic signals* related to market growth, trading volume, and use of Bitcoin as means of exchange, with *social signals* including search volumes, word-of-mouth levels, emotional valence and opinion polarization about Bitcoin. Our results expose which signals precede switches of Bitcoin prices, a skill that we use to design algorithmic trading strategies. We evaluate the power of our strategies through backtesting data-driven simulations, comparing comes back with technical analysis strategies. As a consequence, we test the hypothesis that social media sentiment predicts financial comebacks in the Bitcoin ecosystem.

### 1.Two Social signals in finance

Understanding the role of social signals in finance not only has the potential to generate significant profits, but also has scientific relevance as a research question [Nineteen]. Two different research approaches give insights to this question: one is the *statistical analysis* of social and financial signals in order to test the existence of temporal correlations that lead financial markets. The 2nd one applies these signals in *prediction scripts*, measuring their accuracy as a validation of the underlying behaviour of the system, but not necessarily of their profitability. The statistical analysis of search engine data exposes that search trends can predict trading volumes of individual stocks [20]. In addition, stock prices in S&P five hundred are correlated with tweet volumes [21], but the applicability of these patterns into trading strategies is yet to be evaluated.

Sentiment in social media is closely related to socio-economic phenomena, including public opinion [22]. This motivates the application of sentiment indicators in the statistical analysis of financial data. Early works on the sentiment in specialized forums gave negative results about their influence on comes back [23]. Further research demonstrated that emotions in private instant messaging inbetween workers of a trading company precede stages of market volatility [24]. The expression of anxiety in publicly accessible data from general blogs precedes trading peaks and price drops in the S&P five hundred [25], and sentiment in Twitter can be used to predict movements in large-scale stock indices [26]. It is significant to note that, to date, there is no evidence that such sentiment-based predictions produce significant comes back on investment [Nineteen].

### 1.Trio Online polarization

While most previous works on sentiment in financial markets concentrate on dimensions of valence or mood, the collective phenomenon of polarization of opinions is often overlooked. The emergence of polarization in a society gives early warnings on political and economic phenomena: polarization in social networks of Swiss politicians precedes controversial elections [27], and polarization patterns in the Eurovision Song Contest emerge before states of distrust in the European economy [28]. With respect to financial markets, speculation theories point to the role of diverse beliefs in financial transactions [29], leading to the hypothesis that polarization and disagreement influence trading volumes and prices [30]. In this line, the empirical analysis of polarization in stock message boards shows that states of disagreement lead to enhanced volatility [31].

### 1.Four The missing link

To date, there is a significant skill gap inbetween the analysis and application of social signals to trading scripts. Findings from statistical analyses alone are not assured to lead to profitable strategies at all [25]. For example, movements of the Dow Jones Industrial Average (DJIA) can be predicted with mass media sentiment [32] and Twitter mood [26], but to date no research has shown that such prediction methods can be profitable in trading screenplays. Similarly, the analysis of discussion patterns in specialized blogs predict comes back of some technology companies [33], but it is still open to evaluate the potential comebacks of such a predictor. The application of methods that process arbitrarily large datasets lead to results difficult to apply, for example the predicting power of search volumes of the query *‘moon patrol’* [34] in backtesting over the DJIA [6]. Furthermore, analyses of Twitter discussions about companies can be applied in a portfolio strategy, yet its evaluation through backtesting leads to very moderate comebacks and their statistical significance is not assessed [35]. In addition, no previous research has proposed a prediction technology that derives significant comes back on investment from online sentiment data [Nineteen]. Our research aims at closing the gap inbetween these lines of research. To do so, we unify the statistical analysis and its application to design and evaluate trading strategies, based on tractable principles with potential influence in the finance community.

## Two. Trading strategy framework

To design and evaluate trading strategies, we present a framework that uses a set of economic and social signals related to the agents of the market under scrutiny. Among those signals, the only required one is an economic signal of prices of an asset, namely a stock, currency, or tradable index. To understand profitability, we convert the price time-series *P*(*t*) into a comeback time series: Ret ( t ) = P ( t ) − P ( t − one ) P ( t − one ) , Two.1 which quantifies proportional switches in the price at every time step. The data on these signals are divided in an analysis period and a leave-out period, as depicted in figure 1. The division in these periods needs to allocate enough data in the leave-out sample to provide the testing power to assess the statistical significance of strategy profits. For daily trading, a leave-out period of about one year is usually sufficient, but this ultimately depends on the expected profitability and variance of the trading strategies.

Framework for analysis of social and economic signals and trading strategy design and evaluation.

### Two.1 Multidimensional analysis

The very first step in our framework concentrates on the analysis period, applying a multidimensional model of vector auto-regression (VAR) [36], which is commonly used in the analysis of multidimensional time series in finance [16,23,37]. A VAR models multidimensional linear relations with given lags, which in our analysis we set to one day. Thus, given the vector of signals *V* (*t*), we fit the equation V ( t ) = ϕ V ( t − one ) + r × t + c + ϵ , Two.Two where *ϕ* is a matrix of weights of the linear relations inbetween variables, *r* is a deterministic trend vector, *c* is the vector of constant intercepts and *ϵ* is a vector of uncorrelated errors. While more advanced models can be considered, including longer lags and nonlinear terms, we choose the VAR model of lag one for its general character and its proved power to expose patterns in finance [16,23]. More sophisticated models might have higher power to expose nuance patterns, but at the expense of a loss of generality owing to the concentrate on particular systems.

We include all the time series in a single model to avoid the false positives associated with pairwise Granger tests. To ensure the correct application of the VAR model, we need to verify that our analysis is consistent with its fundamental assumptions: (i) that the elements of *V* (*t*) do not have a unit root, and (ii) that the error term *ϵ* has no temporal nor structural correlations. We verify the very first set of assumptions on the properties of *V* (*t*) by applying a set of tests and transformations prior to the application of the VAR model. We ensure that our conclusions are sturdy to the 2nd set of assumptions by correcting for correlations in the noise term, as explained in the Material and methods section.

### Two.Two Impulse analysis

The VAR weights *ϕ* are only informative when there are no correlations in the error term *ϵ* of equation (Two.Two), which is usually not the case in practice. To extract stylized facts that can be used in the design of trading strategies, we perform an impulse analysis by measuring impulse response functions (IRF) [38] while correcting for correlations in the empirical error. This method simulates the system dynamics when it receives a shock in one of the variables, applying the VAR dynamics of equation (Two.Two) to reproduce the switches in the rest of the variables through time. By recording the switches in each variable, we can estimate the total size and the timespan of the perturbation produced by the shock. In essence, the IRF method creates a computational equivalent of the system under scrutiny, to test its reaction to exogenous impulses in each of its elements.

### Two.Trio Trading strategy design and evaluation

The output of the impulse analysis step, shown in figure 1, is a set of patterns of Granger-type ‘causation’, i.e. it tests the null hypothesis of the absence of temporal correlations among the variables. We use these patterns as stylized facts that indicate which variables precede switches in price comebacks. For example, if variable *Y* (*t*) has a significant influence on Ret(*t*) in the impulse analysis, we will include *Y* (*t*) in our trading strategy design with sign *s*_{Y}, which takes the value one if the response of Ret(*t*) to *Y* (*t*) was positive, and −1 otherwise. Thus, a predictor based on *Y* (*t*) would be sign ( Ret ( t + one ) ) = sign ( s Y × ( Y ( t ) − Y ( t − one ) ) ) . Two.Trio This way, we predict increases (decreases) in price inbetween time *t* and *t*+1 if signals with positive responses increase (decrease) inbetween time *t*−1 and *t*, and vice versa for signals with negative responses. Since our multidimensional analysis is sturdy to confounds inbetween numerous time series, the findings of impulse analysis can be integrated in a *Combined* strategy based on a voting mechanism. The *Combined* strategy applies the other predictors and formulates a prediction corresponding to the sign of the sum of their outputs, i.e. the majority vote.

We evaluate the profitability of the designed strategies in comparison to the benchmark of standard strategies, based on the backtesting over the leave-out sample as indicated in figure 1. For each strategy, we make a data-driven simulation of a trader following that strategy, and we record the profits of that trader on a daily basis. Details on the computational simulation of financial traders can be found in the Material and methods section.