The population studied in this paper consisted of the stocks traded on the B3 between January 02, 2018, and June 28, 2019. The year 2018 was used to evaluate the set of parameters that produce the best performance of BVC, and 2019 was used to validate these parameters and compare the performances between BVC and TR. Only the 181 stocks traded every day over the period studied were included in the sample. We adopted the criterion of selecting only those assets that were traded on all days so that the construction of the BVC time or volume intervals would not be affected by external factors relating to long periods between transactions.

The next segmentation refers to the volume of shares traded for each asset. As this is the input used by the algorithms, each asset was allocated to a class referring to its average volume traded in 2018. Unlike Panayides et al. (2019), who segmented the assets into three classes with a similar amount in each, in the present paper, we chose to use the Fisher-Jenks algorithm to separate the assets into three volume classes: small, medium, and large. This algorithm was chosen because it allows for defining the threshold points and isolating the assets within their respective classes. This, in turn, decreases the variance between assets of the same class and increases the variance in relation to the assets of the other classes. The number of assets and the average volume traded in 2018 for each class of assets are displayed in Table 1.

Table 1

Class	Average volume	Number of assets
Small	287,688	99
Medium	1,287,740	39
Large	7,109,882	43

Source: Research data.

The average volume traded for small stocks is close to that reported by Panayides et al. (2019) in the European market; however, medium and large stocks were 44% and 30% lower than what was reported by these authors, which, in turn, points to the first difference between the Brazilian stock market activity and that in more developed countries.

One of the primary limitations of the application of PIN and VPIN is the possibility of misclassifying buy and sell orders. To contribute to the analysis of the classification algorithms’ performance, this study relied on actual data traded on the Brazilian market as a basis for comparison with the results generated by the TR and BVC algorithms. The data was collected from B3’s market data directory, which contains information about the issued orders of all stocks traded in B3 in the last two years, as well as the time, price, amount, and side that initiated the transaction.

The volume of data used in this paper totaled about 150 million rows, where each row represents a buy or sell order executed in the referenced period, averaging 2.6 million shares traded per day. In comparison, the sample used by Panayides et al. (2019) contained an average of 4 million shares traded daily.

Finally, several aggregations were performed for the application of BVC, reducing the volume by about 88% when using a 5-minute interval, which, in turn, shows the advantage of using aggregated data.

It is important to calibrate the BVC parameters to subsequently compare the performance of the TR and BVC algorithms. Following Panayides et al. (2019), and considering Easley et al. (2012b) ponderation that BVC performs differently for securities with different transaction volumes, several parameters were tested in 2018 to define the best set for each asset class. In addition, the parameters were tested with 2019 data to check whether the previous performance held and thus attest to the possibility of applying BVC to future data.

To select the best set of parameters for each asset, the BVC accuracy was calculated using equation (4).

AR=1-VB-V^Bmax⁡VB,V^B+VS-V^Smax⁡VS,V^S2 (4)

Wherein: V _B and V _S are the actual buying and selling volumes; and V^B and V^S  are the buying and selling volumes estimated by BVC, respectively. For each asset, the highest accuracy was selected. Table 2 shows the representativity of each parameter within the three analyzed classes, in percent.

Table 2

		Asset class
Grouping	Parameter	Small	Medium	Large
Time	1 minute	0.00%	0.00%	0.00%
Time	2 minutes	0.00%	0.00%	0.00%
Time	3 minutes	1.01%	0.00%	11.63%
Time	5 minutes	20.20%	66.67%	60.47%
Volume	1,000	0.00%	0.00%	0.00%
Volume	5,000	0.00%	0.00%	0.00%
Volume	10,000	6.06%	0.00%	0.00%
Volume	25,000	3.03%	0.00%	0.00%
Volume	50,000	9.09%	2.56%	0.00%
Volume	75,000	13.13%	2.56%	0.00%
Volume	100,000	12.12%	0.00%	0.00%
Volume	200,000	17.17%	12.82%	4.65%
Volume	500,000	18.18%	15.38%	23.26%

Source: Research data.

For all three classes, BVC showed higher accuracy when using a 5-minute interval. Interestingly, in line with previous studies (Easley et al., 2012b), assets with lower traded volume showed less consistency in terms of overall parameters, as assets were almost evenly split between the 5-minute time intervals and the volume intervals of 75, 100, 200 and 500 thousand shares.

This first evidence creates uncertainty regarding the applicability of BVC as a forecasting algorithm, considering that, among the assets with lower traded volume, the parameters do not present consistency. Indeed, other forms of clustering of assets were tested, and, among the 80 smallest assets, the parameters did not stabilize. This suggests that this phenomenon persists even when the traded volume is used to separate the stocks (a customary practice in the literature).

Another important feature in forecasting algorithms is the applicability of the parameters in various periods. The percentages of stocks for which the most accurate parameter was maintained in the years 2018 and 2019 differed between the groups analyzed, as follows: (i) 78% among high-volume (large) stocks; (ii) 74% among intermediate volume (medium) stocks; (iii) and 35% among low volume (small) stocks.

As before, inconsistency was more significant in lower-volume assets, which advises caution when using BVC for this asset class. Having established the best set of parameters for BVC (i.e., the 5-minute interval), subsequent results will use these values for estimating the buying and selling volumes.

Table 3 presents the results of the TR and BVC accuracy rates. It can be seen, for both methods, an improvement in performance as the assets grow in traded volume. The overall performance of TR was 80.82%, a higher value than those found by Omrane and Welch (2016) and Chakrabarty et al. (2015).

Table 3

	TR			BVC
Class	Minimum	Mean	Maximum	Minimum	Mean	Maximum
Small	62.99%	77.71%	92.63%	33.08%	51.60%	67.28%
Medium	74.95%	82.95%	91.48%	37.19%	62.18%	68.69%
Large	75.10%	86.05%	95.60%	40.89%	64.10%	70.90%
Total	62.99%	80.82%	95.60%	33.08%	56.85%	70.90%

Source: Research data.

The average performance of BVC was 56.85%, lower than the percentage found by Easley et al. (2012b), who analyzed the three most active futures contracts in the US market; and Omrane and Welch (2016), who analyzed foreign exchange contracts.

Overall, we can see that TR outperformed BVC. While the lowest accuracy rate of TR was 62.99%, BVC achieved a minimum rate of 33.08%. In addition, for all asset classes, TR performed above 90%, whereas BVC reached a maximum of 70.90% for the assets with the highest trading volume.

Figure 1 shows that the performance rate of TR is concentrated in the 80% range and the accuracies among the assets present symmetric behavior in relation to the median. BVC values are around 63%, the first quartile is around 45% and the third quartile is around 66%, slightly above the median (64%). As before, this asymmetry stems from the assets with lower trading volume, which typically deliver inferior performance. This result corroborates the evidence found by Easley et al. (2012b), that BVC performs better for stocks with higher trading volumes.

Figure 1 Source: Research data.

The preliminary results indicate that the TR algorithm outperforms the BVC algorithm. The next section shows the result of the practical application of the two methods based on a model that requires information on buying and selling volumes as its primary input.

To analyze the problems regarding the classification of trades when applying a method that requires the number of buy and sell trades, we proceeded to calculate VPIN using the actual data of transactions performed between January 02 and June 28, 2019, in addition to the volumes determined by TR and BVC. Grammig and Theissen (2002) and Hwang et al. (2013) draw attention to the problems concerning the misclassification of orders when estimating informational risk proxies.

Figure 2 shows the average VPIN of each set. At first, the disparity of VPINs between the largest and smallest stocks can be identified. This result has been reported by several authors (Easley et al., 1996; Mohanram & Rajgopal, 2009; Abad & Yagüe, 2012; Wei et al., 2013) and points to the existence of a negative correlation between VPIN and the company’s market value.

Figure 2 Source: Research data.

From these results, we can see that VPIN_ACTUAL and VPIN_TR do not differ significantly, particularly for medium and large stocks. This evidence is reinforced by the results shown in Table 4, in which the difference between TR and the actual data varied around 2% or 3%.

The VPIN_BVC calculated for small stocks must be highlighted. Although the lower accuracy of BVC was evidenced for this asset class, its VPIN was the closest to the actual one among the three classes. This result is explained by the particularity of the VPIN methodology, in which the imbalance of orders is taken into consideration. If the buy and sell orders are incorrectly estimated but their imbalance is close to the actual estimate, the VPIN will be close to the one calculated with the actual data. This may advise additional caution when using BVC, as it may indicate promising results from incorrect data; this, in turn, may render it inapplicable to other methods that require the buying and selling data as an input.

Table 4

Class	VPINACTUAL	VPINBVC	VPINTR
Small	58.79%	59.36%	55.82%
Medium	37.32%	46.76%	35.98%
Large	34.17%	45.11%	32.74%

Source: Research data.

To analyze the characteristics of the estimated VPINs, we calculated, for each asset class, the correlation between VPIN_ACTUAL and VPIN_TR, and between VPIN_ACTUAL and VPIN_BVC. The weakest correlation, the average, and the strongest correlation between each asset class, as well as the mean, were also calculated (Table 5).

Table 5

	TR			BVC
Class	Minimum	Mean	Maximum	Minimum	Mean	Maximum
Small	0.3551	0.7817	0.9732	-0.1769	0.2872	0.7630
Medium	0.5733	0.8636	0.9855	0.1018	0.5018	0.8563
Large	0.6889	0.8544	0.9713	0.2231	0.4508	0.8632

All correlations showed p-values equal to zero.

For all asset classes, VPIN_TR presented a strong correlation with VPIN_ACTUAL, averaging roughly 80%. These numbers reinforce the accuracy verified for TR. In contrast, analysis of the correlation of VPIN_BVC with VPIN_ACTUAL points to a lower average, of 50% for medium-class assets at most.

The maximum correlations achieved by BVC are close to the average correlation of TR. For small stocks, there were even cases of negative correlation, which indicates that the imbalance reported by BVC showed an opposite sign to the actual data. This means that, while VPIN_ACTUAL indicates moments of increased informational risk (alerting to the orders’ imbalance), VPIN_BVC may indicate the opposite. Therefore, this contradicts the very purpose of VPIN, which, according to Easley et al. (2012a), is to alert investors of moments of volume imbalance and thus avoid illiquidity events that result in stock market crashes, such as the Flash Crash.

As a way of showing the consequence of using BVC to classify trade orders, Figures 3 and 4 show the behavior of VPINs for the stocks that presented, respectively, stronger and weaker correlation with the actual data for each class.

Figure 3 Source: Research data.

Figure 4 Source: Research data.

The figures show that VPIN_BVC presents more extreme values than VPIN_ACTUAL even for the stocks with a stronger correlation. As for the stocks with weaker correlation, VPIN_BVC reached values close to 90% of VPIN_ACTUAL on occasion. If used as an indicator of liquidity problems, this method would present multiple false positives when compared to the actual value, which could lead to problems in its practical use. The unbalanced behavior of BVC will be analyzed in the section dedicated to the analysis of the properties of this method.

The results presented in this section indicate that BVC is not an effective trade classification algorithm compared to actual data. This evidence is corroborated by the application of VPIN, which shows that the values estimated by BVC differ substantially from those obtained from actual data. On the other hand, VPIN_TR showed similar values to VPIN_ACTUAL for all stocks analyzed in this study. In the next sections, we will identify and analyze the points when TR and BVC misclassify orders, which would explain the differences in VPIN estimates.

The TR algorithm is based on the economic principle that a buy (sell) order increases (decreases) the demand for a given stock, which leads to an increase (decrease) in its price. To verify in which situations this economic principle holds, we analyzed the frequency of the order signs given the price changes of the trades. That is, we checked, for each value of ΔP _t , the number of buy (for ΔP _t >0) and sell orders (for ΔP _t <0) in relation to the total. Finally, we calculated the number of times the order was repeated for moments when ΔP _t =0. Thus, equations (5), (6), and (7) represent the calculations performed.

PBt∆Pt=P+) (5)

PSt∆Pt=P-) (6)

PXt=Xt-1∆Pt=0) (7)

Where: B _t and S _t are, respectively, a buy and sell order at the moment t; and P⁺ and P_ are the positive and negative values for the price variations between transactions, respectively. Lastly, X_t represents the sign of the order posted at the moment t, and it can be either a buy (B) or sell order (S). Equation (7) represents the case where the price change equals zero and we intend to check how often the side that initiated the order in t is equal to the side that initiated the previous order.

The results of equations (5) and (6) are shown in Table 6. There is consistency in the frequency of the transaction side, both for positive and negative price changes. Even in more pronounced price changes, above 0.20 currency units, the percentage of buy or sell orders remained at the same level (around 88%).

The results in Table 6 show why TR performs well for classifying trades. In general, the basis of this algorithm holds for the sample analyzed, that is, positive price changes point to buy orders, while negative changes indicate sell orders.

For the result of equation (7) related to price changes equal to zero, it was found that PXt=Xt-1∆Pt=0)=0,9531 . That is, for the sample analyzed, in 95.31% of the cases, when there was no price change, the transaction at the moment t was the same as at the moment t-1, as recommended by TR.

Table 6

ΔP t	P(B t | ΔP t = P + )	ΔP t	P(S t | ΔP t = P - )
0.01	88.43%	-0.01	88.74%
0.02	89.15%	-0.02	89.77%
0.03	88.35%	-0.03	89.22%
0.04	87.83%	-0.04	88.85%
0.05	86.94%	-0.05	88.17%
0.06	87.27%	-0.06	88.48%
0.07	87.39%	-0.07	88.60%
0.08	87.40%	-0.08	88.31%
0.09	87.04%	-0.09	87.77%
0.10	85.21%	-0.10	86.79%
0.11	86.31%	-0.11	88.09%
0.12	87.10%	-0.12	87.75%
0.13	87.93%	-0.13	88.76%
0.14	87.96%	-0.14	88.52%
0.15	87.49%	-0.15	87.61%
0.16	87.27%	-0.16	87.86%
0.17	89.05%	-0.17	88.45%
0.18	87.84%	-0.18	89.02%
0.19	87.76%	-0.19	88.61%
0.20	86.23%	-0.20	87.14%
> 0.20	88.11%	< -0.20	89.27%

Source: Research data.

To further analyze the situations in which TR misclassifies trades, we verified that five variables influence the performance of this method, namely the price change (ΔP _t ); the sign of the order being sorted; the sign of the previous order; the difference in time between the two trades; and whether the buying and selling brokers are the same as in the previous transaction.

Table 7 presents the situations and frequencies in which TR initiates a sequence of misclassified transactions. Most of the TR errors derive from situations where the price change is positive, but the order is classified as a sale preceded by another sale. In this case, the brokerage firm involved in the sales at t and t-1 is the same, whereas the buyer is different. Therefore, the situation described is one in which: (i) a given brokerage firm places a sell order at t-1; when this order is executed, (ii) another sell order is placed by the same brokerage firm with a difference of zero (0) seconds, executed by a different buyer than the one who sent the order. In this case, the second order (at t) has a higher price than the trade at t-1, which is probably due to the fast execution of the sell order. This suggests that there is liquidity for the stock at that moment and that its demand is high, which explains the increase in the sale price.

This is also true for the situation where the current and previous orders are both buy orders, but the price varied negatively between trades (row 2 of Table 7). In this case, the broker sending the buy orders is the same for both transactions but is not the seller. The second buy order is executed faster than the first (zero-second difference), indicating that many traders are interested in selling the stock (i.e., the supply is high). This, in turn, leads to a reduction in the price of the transaction, which is executed even at a lower price than the previous transaction.

The scenarios described above were those that presented the highest frequencies of trades misclassified by TR. When one of these situations occurs, a sequence of misclassifications can follow if there are no further price changes. This is because, in this case, TR continues to misclassify the transaction from the sign of the previous order that had been misclassified before.

The next two situations involving TR errors are those in which the orders at t are buys (sells), and in which the orders at t-1 are sells (buys), but with an equal price change and time between transactions equal to zero, and the same selling (buying) broker. As in this case the TR repeats the classification, a sequence of errors is initiated. Indeed, the time between transactions and the brokers involved in them play a significant role in defining which side initiated the buy or sell order. Since trades are practically instantaneous, two phenomena can influence the TR classification. The first has to do with order splitting, that is, the same order is divided into several smaller orders so that the market does not notice that a trader is moving a high volume of shares. This type of strategy can be detected by analyzing the time between transactions, the broker executing the order, and the volume traded, since many times the order is split into multiple orders of equal volume. The second phenomenon happens because since the time between trades is practically zero, the market does not adjust in time for the price variation to reflect the supply and demand of the stock in question.

In general, the results in Table 7 show that the scenarios present similar frequency for the orders’ signals, therefore pointing to symmetry, besides revealing that the market behaves equivalently regardless of whether the trade in question is initiated by a buyer or a seller. Finally, this analysis points to the opportunity to build a more complex model that can capture the relationship between the variables, thus reducing the initiation of the error sequence.

Table 7

Time difference	ΔP t	Current order	Previous order	Buying Broker	Selling Broker	Frequency
0	+	S	S	≠	=	10.97%
0	-	B	B	=	≠	10.70%
0	0	B	S	≠	=	7.82%
0	0	S	B	=	≠	7.80%
+	+	S	S	≠	≠	4.67%
+	-	B	B	≠	≠	4.61%
0	0	B	S	≠	≠	4.32%
0	0	S	B	≠	≠	4.20%
+	0	B	S	≠	≠	4.16%
+	0	S	B	≠	≠	4.08%
0	+	S	S	≠	≠	3.15%
0	0	S	B	=	=	2.94%
0	-	B	B	≠	≠	2.87%
0	0	B	S	=	=	2.75%
+	0	B	S	≠	=	2.40%
+	0	S	B	=	≠	2.25%
0	-	B	B	=	=	2.07%
0	+	S	S	=	=	2.06%
+	+	S	S	≠	=	1.84%
+	-	B	B	=	≠	1.67%
0	0	S	B	≠	=	1.45%
0	0	B	S	=	≠	1.37%

Note: ‘+,’ ‘-’ and ‘0’ respectively mean positive and negative price changes (of any magnitude) and no changes. The values in the ‘Buying Broker’ and ‘Selling Broker’ columns, ‘=’ and ‘≠,’ represent whether the broker is the same or different from the previous transaction.

Having identified the scenarios in which TR misclassifies trade signs, we proceed to analyze the properties of BVC in the next section.

By using the normal distribution to compute the buy and sell percentage for each time interval, it is possible to compare them with the actual buy and sell percentages within the same interval. Figure 5 shows how the percentage of buying evolves with price variation compared to the percentage signaled by BVC. There were no major disparities between asset classes, so the values reported in Figure 5 represent the complete (total) analyzed sample.

Evidence shows that one of the primary characteristics of BVC is corroborated; that is, when price does not vary, the percentage of buying and selling within the same interval is roughly 50% (51.88% of buying volume in the sample analyzed). In practice, this makes BVC perform satisfactorily for intervals that do not present price variation (about 22% of the intervals).

However, as the price variation moves away from zero, the percentage signaled by BVC increases more rapidly than is found in practice. This characteristic derives from the distribution defined in the model design. Through the actual data, we can see that, on average, the percentage of buying stabilizes near an absolute price variation of about 0.05 currency units. Given the distribution chosen in the application of BVC, this stabilization does not occur within the first 0.1 currency unit of absolute change.

This behavior explains why VPIN_BVC has more frequent peaks than VPIN_ACTUAL or VPIN_TR. Since BVC assigns a higher percentage of buying or selling even for low price variations, it is natural that the volume imbalance presented by this model will be higher as well, leading to undetected peaks when relying on actual data to calculate VPIN.

Moreover, as observed in the analysis of the TR properties, when the price varies, in about 88% of the cases the trade occurs in the direction of the variation; that is, the price increase indicates a buy, and the price decrease indicates a sell. This holds when the analysis is performed transaction by transaction, which led TR to achieve about 80% accuracy in the sample analyzed. In contrast, BVC groups transactions into intervals and uses the last price as a supply or demand indicator. This implies that all the information content present within the interval (captured by TR) is discarded when BVC is used. This also explains why intervals calculated with longer periods or a larger number of aggregate transactions have inferior performance since the last price contains scarce information about the variations occurring within the time interval.

Finally, the performance of BVC in classifying buy and sell orders for assets traded on B3 may have been significantly inferior to that found in studies with data collected from more developed markets, due to the higher volatility of the Brazilian market. Sharp price variations are not properly captured by this method, which suggests the need to modify its calculation basis in addition to the calibration of its parameters.

Figure 5 Source: Research data.