“Measuring investors’ emotions using econometric models of trading volume of stock exchange indexes”

Traditional finance explains all human activity on the ground of rationality and suggests all decisions are rational because all current information is reflected in the prices of goods. Unfortunately, the development of information technology and a growth of demand for new, attractive possibilities of investment caused the process of searching new, unique signals supporting investment decisions. Such a situation is similar to risk-taking, so it must elicit the emotional reactions of individual traders. The paper aims to verify the question that the market risk may be the determinant of traders’ emotions, and if volatility is a useful tool during the investment process as the measure of traders’ optimism, similarly to Majewski’s work (2019). Likewise, various econometric types of models of estimation of the risk parameter were used in the research: classical linear using OLS, general linear using FGLS, and GARCH(p, q) models using maximum likelihood method. Hypotheses were verified using the data collected from the most popular world stock exchanges: New York, Frankfurt, Tokyo, and London. Data concerned stock exchange indexes such as SP500, DAX, Nikkei, and UK100.


INTRODUCTION 1
The project is financed within the framework of the program of the Minister of Science and Higher Education under the name "Regional Excellence Initiative" in the years 2019-2022, project number 001/RID/2018/19, the amount of financing PLN 10,684,000.00.
Stock exchanges are the most significant institutions forming virtual safety platforms of financial assets trading.They guarantee the safety of transactions and anonymity for investors, on the one hand, and transparency, on the other hand.It is one of the reasons why investors use the stock exchange market as a possibility of earning money.During years, investors have created some practices, which became quasi-official rules.Therefore, every ordinary investor knows, for example, the law that the stock exchange trading volume mirrors market trends and gives the confirmation signals obtained from technical analysis.So, every emotional decision of the stock exchange investor is immediately reflected in a trading volume.The article proposes calculating logarithmic changes in trading volume to observe not a state but the dynamics of investors' behavior.Just now, it is possible to see the excessive movement of trading volume changes supposing they describe the emotional

LITERATURE REVIEW
Investing in stock exchanges for the first look does not involve specialized knowledge from individual investors.Usually, they think that the most important is to have an intuition.The experimental work of Czekaj, Markiewicz, Kubinska, and Czupryna (2016) showed that new investors believe that technical analysis is not such an effective supporting investment decision regarding the market experience and the intuition.On the other hand, professional traders of futures market still support their investment decisions using technical analysis.In the paper, technical analysis is treated as proof for existing emotional reactions, belief in a self-fulfilling prophecy, and other market irrational phenomena.Czekaj, Kubinska, Markiewicz, and Czupryna (2016) conclude, based on their experiments, that the technical analysis is the most popular method supporting investment decisions between futures traders.
The technical analysis's major application is the anticipating of trends in the stock prices using historical data.According to Majewski (2019), "It relies on that markets discount everything except information generated by market action".It means that the anticipation process needs only data generated by the market (Sewell, 2007).
The problem of separating optimistic and pessimistic signals appears here.During the noticeable trends, one can assume that the beliefs of traders in their continuing are exactly described by trading volume.Therefore, this indicator allows uncovering such emotions as optimism and pessimism (Majewski, 2019).However, emotions accompany investors' every decision, not only during clear trends.Therefore, the authors consider additional various-time-length volatility representing the standard deviation (30-, 90-, and 180-days) to eliminate even little emotional reactions.
Many authors suggest that technical analysis should be improved by using automated algorithms and that the traditional patterns such as head-and-shoulders and rectangles, although sometimes efficient, are not optimal, like in the research of Lo, Mamaysky, and Wang (2002).Some of them claim that technical analysis today is neither more important nor less than before and could be called the "workhorse" of currency exchange, as in Gehrig and Menkhoff (2006).
The authors surveyed approximately 200 FX in Austria and Germany in 1992 and 2002.The analysis allowed them to create general five conclusions supporting using technical analysis as a decision-making method.The most important are as follows: • technical analysis has a complementary use; • technical analysis is an instrument for shortterm forecasting; • fund managers use the three types of information distinguished in a similar pattern as FX dealers but with longer horizons; • chartists still believe in the importance of market psychology.
However, there is also a big group of investors indicating a weakness in technical analysis.Very often, they list such kinds of problems as follows: • even the best method of technical analysis cannot reliably predict the future, and signals flowing from some indicators are strongly delayed (e.g., from moving averages); • tools are available to everyone so that nobody could gain an advantage over the other investors; Investment Management and Financial Innovations, Volume 17, Issue 3, 2020 http://dx.doi.org/10.21511/imfi.17(3).2020.21 • interpretation is more art than science (often it is based on the analyst's image than the transparent normative).
The answers to why so significant number of investors believe in technical analysis are usually hidden in the psychology of human activity.Sewell (2007) presents the major of them: 1. Communal reinforcement is the social construction consisting of forming common beliefs based on repeated opinions by stock exchange investors in isolation from empirical evidence, according to Carroll (2003).
2. Selective thinking is the process of evidence selection consisting of the preference of proofs confirming the thesis adopted earlier while ignoring this, which are inconsistent with it.
3. One of the heuristics, which is closely tied with the selective thinking process, is confirmation bias.It is a tendency seeking of information confirming existing beliefs and ignoring everything contrary to them.A literature review distinguishes two kinds of meanings for this term.The first meaning concerns confirmation acquired information, but the second is the verification method basing only on confirmatory cases ignoring these unfavorable (confirmatory strategy of testing hypothesis) like in the research of Majewski (2019).
4. Self-deception is considered a widespread phenomenon according to the work of Bortolotti and Mameli (2012).The term focuses beliefs acquired and held despite strong evidence, and which are motivated by desires and emotions, according to DeWeese-Boyd (2007).
It could be concluded that many investors' reasons for using technical analysis despite facts proving its low efficiency have their roots in human behavior.On the one hand, it is possible to explain the causes of the irrational behavior of investors.On the other hand, it is also a reason for other biases observed in behavioral economics.First of all, using technical analysis by investors drives to the illusion of control, known as a self-fulfilling prophecy like Murphy's work (1999).Generally, the term "illusion of control" is defined as overestimation of ex-pected the event likelihood relative to its objective probability, according to Langer (1975).Sometimes, investors find out that they are not efficient despite their believing in the rightness of their decisions.Then the illusion of control is replaced by faith in "the wheel of fortune" and "regression to the average".However, these beliefs are not supported by any reasonable shreds of evidence.At this point, it should be emphasized that both the causes and the effects of using technical analysis by investors belong to the same group of factors

METHODS AND HYPOTHESES
There are many different methods focused on sentiment measuring.Some of them were described in the work of Bandopadhyaya and Jones (2006).
Their article presents the essential works on the sentiment measuring divided into the parts according to research.They could be listed as follows: 1. Optimism/pessimism about the economy: where PO -put outstanding, CO -call outstanding; • Trin.statistic: • where VDI -volume of decline issues, NDI - number of decline issues, VAI -volume of de- cline issues, NAI -number of decline issues.
The statistic has not any academic references.

The riskiness of the stock market:
Insurance ratio by Baker and Wurgler (2006): where Ins -insurance [%], GAE -gross annual equities issued, GADE -gross annual debt and equities issued; • RIPO -average annual first-day returns on IPOs by Baker and Wurgler (2006); • Turnover by Baker and Wurgler (2006): where T -turnover, RSV -reported shares volume, ASL -average shares listed on NYSE (logged and detrended); where ANS -average number of shares; • NYSE seat prices -Trading volume or quoted bid-ask spread by Keim and Madhavan (2000).
4. The riskiness of the individual stock: • Beta ( )
There are also some other works taking into account optimism, but their point of view is not so general as in the case of measures standing above.There could be such measures as, for example: • analyst's optimism for earnings forecasts, according to Nianhang, Xuanyu, Kam, and Zhihong (2013): ,, , , , where FB -forecast bias, AF -analysts' forecast, EPS -actual earnings-per-share ratio, P -price; • managerial optimism, according to Yuehhsiang, Shing-yang, and Ming-shen (2005): , where I -investment, C -cash flow from the operation, /, Q MV BV = MV -market value, BV -book value, O -dummy variable (repre- senting optimism measure); • Testing optimism by measuring the relations between market ratios or parameters like stock dividends, splits, trading volume, and volatility, according to Crawford, Franz, and Lobo (2005).
It is worth noting that all of these studies are based on three groups of methods: • survey analysis; • descriptive statistics -statistical analysis of indicators; • correlation or regression analysis.
In the presented approach, there is a proposition to use econometric models to identify a correlation between some significant factors influencing individual and institutional investors' emotions.The research's core point is focused on the econometric analysis using classical econometric models and dynamic autoregressive models.The idea is based on the assumption that the volatilities of rates of return (risk parameters) and trading volume (logarithms of changes) are significantly correlated, according to Majewski (2019).In the paper was assumed that: • changes in trading volume illustrate investors' emotions; Investment Management and Financial Innovations, Volume 17, Issue 3, 2020 http://dx.doi.org/10.21511/imfi.17(3).2020.21 • volatility reflects the trading activity of stock exchange investors; • short-time (30 trading days) volatility has better cognitive features for econometric modeling than longer one; • every abnormal (emotional) activity of traders (expressed by volatility) has a significant influence on changes in trading volume; • one-day-in-a-week (calendar effects) are also taken into account in econometric models' specifications.
The study presents two research hypotheses: H1: 30-days trading volatility has the most significant influence on traders' emotions (measured by trading volatility).
H2: The best approximation describing changes in trading volume is obtained using (G) ARCH models regarding calendar effects.
The research was conducted in the following steps: • data collection; • model estimation; • selection of the best model.
The logarithmic changes of trading volume of stocks exchange indexes are described by: t R -stock exchange indexes' rates of return, - ti R - rates of return of the stock exchange index with i-days time-lag, Mo -Monday (dummy vari- able), Tu -Tuesday (dummy variable), We - Wednesday (dummy variable), Th -Thursday (dummy variable), Fr -Friday (dummy vari- able), 30 t V -30-days-volatility without time-lag, The best fits are usually obtained with (G)ARCH(p, q) models.Significant results are obtained for the first two types: ARCH(q) and GARCH(p, q).The basic ARCH(q) model is expressed as in the work of Engle (1982): where t h -conditional variance, ( ) The ARCH process is the special case of a more general model called GARCH.GARCH stands for Generalized ARCH, and it adds lags in the variance to equation (2).The GARCH(p, q) is expressed as in the work of Bollerslev (1986): The following parameters should comply with the conditions to guarantee the variance will be non-negative: The maximum likelihood methods of parameters estimation procedure is given by: ( ) ( ) Another kind of models, which could be used for identification of the relationship between analyzed variables, are vector autoregression models (VAR) described by Simms (1980): The estimation was carried out using the GRETL program.

RESULTS
During estimation, different types of econometric models were used.Based on it, it been established that the best results for every stock exchange index were obtained for autoregressive conditional heteroscedasticity models (GARCH).Tables 1-4 present the results of the best models' estimations.
The first model -for DAX index -indicates the negative relationship between the dependent variable (logarithm of trading volume) and dummy variable representing Mondays (calendar effect), the current return rate, and 1-day-lagged 30-days volatility.The volatility has the most substantial impact on the volume among these variables.Positive relationships were observed for the dummy variable representing Tuesdays, current, and 2-days-lagged 30-days volatility.It can be concluded that in this case, taking into account both the positive and negative impact on the dependent variable, volatility had the most significant impact on changes in the volume of trading.The best approximation was obtained for the ARCH(1) model.
The second model presented in the research is a model concerning relationships for the S&P500 index.The best approximation was obtained for the GARCH(1,1) model.Similarly, the DAX obtained negative rations between the dependent variable and Mondays and 1-day-lagged 30-days volatility, but the most substantial impact was observed for the current 30-days volatility.The other negative relation was observed for 2-days-lagged 30-days volatility.The following variables positively impact the changes in trading volume: dummy variables representing Tuesdays, Wednesdays, and Thursdays and current and 5-days-lagged 30-days volatility.Similarly, to DAX analysis, the most significant impact on trading volume was observed for the volatility.

DISCUSSION
The results of the estimation of econometric models for the DAX trading volumes were presented in Table 1.The best approximation was obtained for the ARCH(1) model (the highest value of maximum likelihood function with statistically significant parameters).The first fact is the occurrence of the Monday effect (already know in the literature).Every Monday, the logarithm of trading volume significantly decries and, every Tuesday, increases.However, the most significant changes in the index trading volume's logarithms are caused by changes in 1-day-lagged 30-days volatility.Much smaller reactions of trading volume were obtained for 30-days volatility without lags and 2-dayslagged 30-days volatility.There is also a significant negative relationship between the index's rates of return and trading volume.
The model describing relations for the S&P500 index was presented in Table 2.The best approximant for the econometric function was obtained for the GARCH(1,1) model in this case.The negative impact of the Monday effect was also observed for Standard and Poor's 500, but there were also relationships between the trading volume and Tuesdays, Wednesdays, and Thursdays (positive in all these cases).Similarly to the previous table, the highest impacts are caused by 30-days-volatility.The highest absolute value of parameter was obtained for volatility without any lags, and it was positive.The negative values of econometric models' parameters were obtained for 1-and 2-days-lagged volatility.The parameter for 5-days-lagged volatility was also positive.
The estimation of econometric models for the NIKKEI 225 index allows for the determination of some regularities (presented in Table 3).First, the best approximation was obtained for the ARCH (1) model.Second, the Monday effect exists on the Tokyo Stock Exchange.Third, the highest absolute value of parameter was obtained for 1-day-lagged 30-days volatility, and it was negative.The third negative parameter was obtained for the dummy variable representing Thursdays.Positive significant parameters were obtained for the 30-days volatility without lags and 2-and 4-days-lagged volatility.
Table 4 presents the results for the FTSE 100 index.The best results were obtained for the GARCH (1,1) model, and the following variables: Mondays (dummy variable), 30-days volatility (current, 1-and 3-days-lagged).The value of the parameter for Mondays was negative, as in other cases.The highest absolute value of the parameter was obtained for 1-day-lagged 30-days volatility, and it was negative.The parameters of the other two variables were positive.

CONCLUSION
As it has been assumed, changes in trading volume illustrate investors' emotions, and the volatility reflects the trading activity of stock exchange investors.Based on this research, it is possible to conclude that the volatility is the significant regressor of econometric models describing trading volume for analyzed stock exchanges having a substantial impact on this dependent variable.This hypothesis has been verified positively.So, it can be said that volatility could measure the emotions of stock exchange investors.
The analysis included three kinds of time-windows for the volatility of 30, 90, and 180 days and the best results have been obtained for the shortest period.An exciting result concerns time-lagged volatilityin every analyzed case, it indicated a negative relation for t-1 lag and positive relation for current volatility, but it was not stable for the other lags.
Additionally, it turned out that the one-day-in-week effect biased every analyzed stock exchange index, and every Monday, trading volume negatively deviates from the trend line.
days-volatility with i-days timelag, 90 t V -90-days-volatility without time-lag, days-volatility with i-days time-lag.The base equation for the estimation of logarithmic changes in trading volume is as follows: changes of trading volume of stock exchange indexes in t-time, kt X -value of regressor k in period t, t ε -random component normally distributed N(0,1).
(Thaler, 2005) can be classified as cognitive biases based on greed and fear.Many phenomena and heuristics influence investors, confirming their belief in technical analysis and hiding behind the concept of cognitive biases.Behavioral finance literature distinguishes, among the others, the following types of phenomena in this area(Thaler, 2005): and from 4 th March 2003 for NIKKEI (4,057 observations).Various types of econometric models were used, but the best approximations were obtained for dynamic econometric models of the type (G)ARCH.It is widely known in econometrics that different length of periods for standard deviation calculating influence on the results.
(5,ewski (2019)to verify the scientific problem that the market risk may be the determinant of traders' emotions, so it could be used during the investment process to indicate investors' emotions according to technical analysis (TA).Rules of TA dictate the confirmation of the truthfulness of signals deriving from historical prices by changes in trading volume.The convergence of changes' directions confirms continuity of trend and the divergencethe change of market trends.The major hypothesis is that stock exchange trading volume reflects investors' emotions supporting financial decisions by its econometric models.The assumption follows the work ofMajewski (2019), that "trading volume could be modeled by market risk" represented by "standard deviation of rates of return".The biggest and most popular stock exchanges were chosen to verify a raised hypothesis.Quotations of popular stock exchange indexes were used in the research.They are the following indexes: Standard and Poor's 500 (S&P500), Nikkei heikin kabuki (NIKKEI 225), Deutscher Aktienindex (DAX), Financial Times Stock Exchange (FTSE 100).Data for analysis were collected in the following periods: from 26 th May 1998 for S&P(5,411observations), 6 th June 2002 for DAX (4,392 observations), and FTSE (4,370 ob-Investment Management and Financial Innovations, Volume 17, Issue 3, 2020 http://dx.doi.org/10.21511/imfi.17(3).2020.21servations),