The Impact of the Current European Short Selling Regulation on Market Efficiency Danielle Giusti Machado
Abstract
This paper gives a recent evaluation of the impact of short selling regulations on market efficiency. This analysis is performed assessing daily stock data from May 2012 to May 2013 corresponding to 6 months prior and 6 months after the official implementation of the short selling regulation in all EU Member States. The results of the panel regression that combines cross-sectional and time-series variants for the period in consideration, conclude the following effects: 1) The short selling regulation has a positive impact on bid-ask spreads of stock prices; 2) Small but positive impact on volatilities, and 3) Increased amount in volume traded for the period. Overall, the hypothetical damages of the short selling restrictions, forecasted by previous studies, seem to not outperform the positive externalities, like the increase in consumer confidence that took place along the time of the regulation.
Short selling plays an important role for investors in international stock markets (Fleischer, 2011). Being one of the most potentially lucrative and versatile investing techniques of nowadays, short selling is also one of the most
misunderstood (Taulli, 2004; Teague, 2013). The practice of short selling plays an essential role in increasing the liquidity of markets, allowing investors to allocate resources efficiently, and revealing bad management and weaknesses of companies (Kerr, 2012; Kolasinski, Reed, and Thornock, 2012). Nonetheless, its activity is often looked upon as ruthless and greedy endeavor for betting against companies or even whole markets. Short selling is also associated with “bear raids”: bear raids occur when stocks are sold short, after which negative rumors about the companies are spread in order to lower their stock prices, and subsequently the stocks are purchased back at a lower price (Selling stocks short: Ever controversial, 2009; Davies, 2010).
As part of a broad set of implemented rules and regulations in response to the financial crisis in American markets in 2008, the SEC issued a temporary directive restricting the practice of short selling. With claims that aggressive short selling was a key factor in the failure of financial giant Lehman Brothers, the SEC extended and expanded the rules to remove exceptions for certain asset classes and to cover all sectors (Marcy, 2008).
This paper is particularly relevant given the current Regulation (EU) No 236/2012 established by the European Parliament and of the Council that came into force in November 2012 (Overview of EU Short Selling Regulation, n.d). The three main categories of this regulation include 1) requirements about the reporting and disclosure of net short shares positions and sovereign debt 2) restrictions on
European sovereign credit default swap (Resource Guide: EU Short Selling Regulation Compliance, n.d.).
Previous research papers on the subject are by no means unanimously
positive about the effect that short selling regulation has on market efficiency. Miller (1977) argues that in the absence of short selling the demand for equities originates from an optimistic crowd. Consequently, there will exist some securities that have been bid up to excessive ranks, leading to overpricing and thus contradicting the hypothesis of efficient markets. In contrast, Diamond and Verrecchia (1987) claim that if traders form “rational expectations” about the future, the effect of short selling restrictions will lower the speed by which prices adjust but not systematically overvalue equity prices. Nonetheless, Diamond and Verrecchia (1987) predict that the full ban of short sales would lead to an increase in the bid-ask spread and volatilities of stocks, which in turn suggest market inefficiency.
In their study, Bris, Goetzmann, & Zhu (2003) found that the prohibition of naked short selling was associated with a decline in the liquidity of stocks. They make use of two measures of market efficiency described as the relative movement of individual stock returns with the market and cross-autocorrelation between market returns and individual stock returns. They conclude that in countries where short selling is allowed and observed, markets are significantly more efficient on both measures.
A more recent paper by Battalio, Mehran, & Schultz (2012) suggests that short selling constraints can cause more harm than good. They find that the bans imposed on short selling during 2008 did little to limit the decline in stock prices. Adversely, the bans generated negative side effects such as a decline in market liquidity and rise in trade costs.
Allen and Gale (1991) performed one of the few studies with positive findings on short selling restriction, where they suggest that the prohibition of short sales leads firms to innovate and to behave competitively. Their argument is that even on frictionless markets, where short selling is permitted and firms’ values are maximized, markets are still incomplete due to the costs associated with innovation and free riding problems. Hence, asymmetric information, short selling restrictions, and other frictions are found to be incentives to innovative steps that firms take.
Most studies available today base their research on data originated from the years between 2008 and 2009. During this period of turbulent global financial markets, several countries adopted some kind of restriction to short selling. This paper aims to give a more recent evaluation of the subject at hand by assessing data from 2012-2013 in order to investigate the impact of the current European short selling regulation on market efficiency. This objective is achieved with the use of a panel regression that combines cross-sectional and time-series variants for the period of May 2012- May 2013 (6 months before and 6 months after the official EU
implementation of short selling regulation in all Member States). The regression makes use of a dummy variable for the regulatory regime with fixed-effects for time and stocks, to assess the impact of the directive on individual stocks’ daily bid-ask spreads, intraday volatilities, and trading volumes. Together, these variables form a proxy for assessing market efficiency that is in line with prior research such as that performed by Bernal, Henrickx, & Szafar (2013), Beber and Pagano (2013), and Diamond and Verrecchia (1987).
The remainder of this paper is structured as follows: Section 1 presents the data and descriptive statistics. Section 2 poses the methodology. Section 3 discusses the results. Section 4 concludes.
1. The Setting 1.1 Data
The data collected originates from a selection of countries that fit the following criteria: 1) short selling is experienced; 2) daily stock information is available; 3) countries that officially implemented all scopes of the European short selling regulation on November 1st 2012, and 4) countries that at the moment the directive was imposed, had not yet adopted a flagging system distinguishing short sales transaction from regular ones. Among the 27 EU countries where the new regulation applies, 13 countries fit the proposed conditions. This set of countries includes: Austria, Belgium, Denmark, Finland, France, Germany, Ireland, Italy, the Netherlands, Portugal, Spain, Sweden, and the United Kingdom. For each of these countries daily stock data is gathered from the National DataStream Global Equity Index, which can be considered highly representative, as the index covers 97.68% of each market’s investable market capitalization (Global Equity Indices, n.d.).
The analysis of panel data with a cross-sectional character requires
heterogeneity in the scope of the regulation adoption. Therefore, this paper adopts the period of May 2012- May 2013, resulting in 255 days of observable daily data. During the first part of this period, thus before the official implementation date of the EU directive on short selling, some countries chose for an early adoption of the regulation and others chose to adopt it within a specific group of stocks or sector (see Table 1). This paper’s regression analysis manages to capture the impact of the heterogeneity in regulation, distinguishing all the individual stocks that were already under regulatory control on November 1st 2012.
Table 1: Short sales rules across EU Member States/ March 2012
Table 1 shows the scope of the adoption of the regulation, before the official date of implementation.
Country Scope of short selling regulation prior to the official EU directive Austria
Belgium Limited to 4 Belgian financial institutions Denmark
France All shares listed in French regulated markets or MTF’s1 Finland
Germany 10 German financial stocks
Ireland Italy
Netherlands All Dutch financial institutions
Portugal All shares listed in Portuguese regulated markets or MTF’s Spain All shares listed in regulated markets
Sweden
United Kingdom Financial sector companies as well as those companies conducting rights issues.
Note. Adapted from Draft technical standards on the Regulation (EU) 236/2012 of the European Parliament and of the Council on short selling and certain aspects of credit defaults swaps, 2012, European Securities and Markets Authority.
1.2 Descriptive statistics
This study focuses on the effects of the European short selling regulation on three different dimensions of market efficiency Firstly, the daily bid-ask spread on stock prices. Secondly, the intraday volatility that measures the stock price variation during the trading day. Finally, the volume, which is the total number of daily traded shares. For comparability these definitions follow methodologies used in previous research (Boehmer et al., 2011; Beber and Pagano, 2013; Bernal et al., 2013; FSA, 2009).
Tables 2, 3, and 4 show the mean and medians per country for all three variables. In line with Beber and Pagano (2013), the data is winsorized with the
1 MTF is a trading system that facilitates the exchange of financial instruments between multiple
elimination of the outliers corresponding to the top 1% of all variables2 and
exclusion of negative bid-ask spreads. Additionally, only stocks with a trading volume of at least 200 shares per day and with a positive volume of trade for at least 177 out of the 253 days are include. These last criteria are included to circumvent limitations to liquidity that is experienced on stocks with low traded volume, conforming to Bernal et al (2013).
Table 2: Descriptive statistics for variable: Spread
Table II shows the descriptive statistics of variable Spread for the period of May 1st 2012 to April 31st 2013 explained by country. Spread is the difference between ask and bid prices at the market close, divided by the quote-midpoint. The number of Stock Day Observations represents the original figure of observed data with containing only non-negative figures. The Final Stock Observations illustrate the observations that satisfy the condition: 1) Value contained in the 99Th percentile range.
Spread (%) (Outliers Included) Spread (%) (Outliers Excluded) Country Mean Median # Stock day Obs Mean Median Final # stock day Obs
All 0.47% 0.17% 237624 0.41% 0.17% 235240 Austria 0.57% 0.39% 6906 0.57% 0.39% 6896 Belgium 0.52% 0.30% 12182 0.52% 0.30% 12166 Denmark 0.38% 0.19% 8609 0.36% 0.19% 8587 Finland 0.52% 0.27% 13241 0.50% 0.27% 13200 France 0.25% 0.14% 29742 0.24% 0.14% 29680 Germany 0.38% 0.17% 27323 0.33% 0.16% 27123 Ireland 1.03% 0.45% 4048 0.72% 0.42% 3891 Italy 0.67% 0.19% 20913 0.51% 0.18% 20369 Netherlands 0.19% 0.10% 8134 0.19% 0.10% 8123 Portugal 0.49% 0.33% 3568 0.49% 0.33% 3567 Spain 2.43% 1.91% 10961 1.91% 1.76% 9815 Sweden 0.52% 0.29% 28996 0.50% 0.29% 28923 UK 0.18% 0.09% 63001 0.17% 0.09% 62900
2 The values equivalent to the top 1% are the ones exceeding 4,46% for Spread, 50,528,700 for Volume and, 8,39% for Volatility.
Table 3: Descriptive statistics for variable: Volume
Table 3 shows the descriptive statistics of variable Volume for the period of May 1st 2012 to April 31st 2013 explained by country. Volume is the number of shares traded daily expressed in thousands. The number of Stock Day Observations represents the original figure of observed data. The Final Stock Observations illustrate the observations that satisfy the conditions: 1) At least 177 positive trading days out of 253; 2) Trading volume of at least 200 shares per day; 3) Value contained in the 99Th percentile range.
Volume ('000) (Outliers) Volume ('000) (No Outliers) Country Mean Median # Stock day Obs Mean Median Final # stock day Obs
All 2906.5 329.2 237216.0 1698.2 320.2 234836.0 Austria 192.0 52.1 6914.0 191.8 52.0 6908.0 Belgium 200.1 31.8 12221.0 200.1 31.8 12221.0 Denmark 363.8 105.0 8604.0 363.8 105.0 8604.0 Finland 1192.5 142.8 13219.0 959.9 141.2 13182.0 France 1430.8 251.8 29829.0 1319.8 251.0 29788.0 Germany 959.3 183.1 26890.0 956.9 183.1 26889.0 Ireland 3700.9 186.5 4025.0 1568.6 179.8 3945.0 Italy 10782.2 588.8 20912.0 3385.5 491.0 19746.0 Netherlands 2276.9 578.8 8160.0 2147.5 575.5 8146.0 Portugal 10131.5 738.5 3570.0 3204.3 658.8 3399.0 Spain 4971.4 758.1 10958.0 3505.3 719.7 10736.0 Sweden 855.9 130.9 28925.0 855.9 130.9 28925.0 UK 3555.6 909.4 62989.0 2412.2 892.6 62347.0
Table 4: Descriptive statistics for variable: Volatility
Table 4 shows the descriptive statistics of variable Volatility for the period of May 1st 2012 to April 31st 2013 explained by country. Volatility is the difference between the highest and lowest daily prices, divided by the price at closing. The number of Stock Day Observations represents the original figure of observed data. The Final Stock Observations illustrate the observations that satisfy the condition: 1) Value contained in the 99Th percentile range.
Volatility (%) (Outliers) Volatility (%) (No Outliers)
Country Mean Median # Stock day Obs Mean Median Final # stock day Obs
All 2.48% 2.07% 237646 2.38% 2.06% 235263 Austria 2.46% 2.20% 6914 2.45% 2.20% 6893 Belgium 2.17% 1.81% 12221 2.11% 1.80% 12136 Denmark 2.07% 1.73% 8604 2.02% 1.72% 8555 Finland 2.46% 2.11% 13219 2.40% 2.10% 13113 France 2.35% 2.02% 29829 2.31% 2.01% 29657 Germany 2.30% 1.98% 27320 2.25% 1.97% 27170 Ireland 2.51% 2.10% 4025 2.42% 2.09% 3983 Italy 3.27% 2.75% 20912 3.05% 2.70% 20373 Netherlands 2.11% 1.78% 8160 2.07% 1.78% 8127 Portugal 2.86% 2.35% 3570 2.71% 2.33% 3507 Spain 3.40% 2.81% 10958 3.09% 2.74% 10623 Sweden 2.41% 2.00% 28925 2.29% 1.98% 28593 UK 2.36% 2.01% 62989 2.30% 2.00% 62533
2. Methodology
The prohibition of naked short sales and disclosure requirements are classified as restrictions and not as full bans. Indeed, these particular constraints leave the practice of short sales accessible, but impose extra costs to it, such as substantial costs for systems and controls, extra borrowing costs, and exposure of trading strategies (Bernal et al., 2013; FSA, 2009).
The impact of short selling regulation has different effects on markets according to the scope of the directive. Boehmer, Jones, and Zhang (2009) predict deterioration in liquidity under short selling constraints. Kolasinki, Reed, and Thornock (2012) find a similar effect on liquidity and a heavier burden on uninformed market participants under short sales bans, since sophisticated investors could still make use of options to create synthetic short positions. Nonetheless, Diamond and Verrecchia (1987) claim that due to the presence of informed and uninformed investors, the overall effect of short sales restrictions is ambiguous.
In order to test the theoretical predictions, this paper makes use of the following fixed effects3 panel regression with cross-countries and time-series dimensions:
𝑌!,! = 𝛽!+ 𝛽! 𝑅𝑒𝑔!,!+ 𝛽! 𝜒!,!+ 𝛼! + 𝛾!+ 𝜀!,!
Where 𝑖 represents the stock and 𝑡 the date. Y describes successively spread, volatility, and volume (measured in logarithms) of stock 𝑖 at day 𝑡. The value 𝛽!
3 The Hausman test was performed where the null hypothesis is that the preferred model uses random effects vs. the alternative hypothesis of fixed effects (results are reported in the Appendix).
denotes the intercept. 𝑅𝑒𝑔 defines the dummy variable that takes a value of 1 when the regulation is in place and 0 otherwise. The variable 𝜒 acts as lags4 and other control variables. Finally, 𝛼! and 𝛾!, characterize stock and time-specific effects.
3. Panel Regressions
This section aims to provide and discuss the outcomes of the panel regressions for each of the three dimensions of market efficiency, successively.
4 The inclusion of lagged variables as dependent variables is used to provide robust estimates of the effect of independent variables. The presence of additional lags increases the accuracy of the parameters (Wilkins, n.d.).
3.1. Bid-Ask Spreads
Table 5: Panel regressions – Bid-Ask Spreads
The dependent variable Spread is the difference between ask and bid prices at the market close, divided by the quote-midpoint. Regulation is a dummy variable that equals one if the regulation is in place and zero otherwise. Volatility is the difference between the highest and lowest daily prices, divided by the price at closing. Spread (-1) is the 1 day lagged spread. Spread (-2) is the 2 days lagged spread. Spread (-3) is the 3 days lagged spread. The estimates reported in parentheses are t-statistics. The three asterisks mark the coefficients that are significantly different from zero at the 1% level.
Dependent variable: Spread
Variables (1) (2) Intercept 0.0031*** 0.0025*** (188.76) (119.28) Regulation -0.0011*** -0.0009*** (-38.81) (-35.43) Spread (-1) 0.1693*** 0.1653*** (110.27) (107.86) Spread (-2) 0.1201*** 0.1185*** (79.27) (78.55) Spread (-3) 0.0508*** 0.0496*** (42.53) (41.72) Volatility 0.0224*** (41.51) Stock fixed effects
Day fixed effects
Yes Yes Yes Yes 𝑅! 0.5586 0.5568 Number of observations 227,384 227,227
The coefficients in column one indicate that the impact of the regulation is associated with a decrease of 0,11 percentage points on the bid-ask spread of stocks. In column two, where the regression is expanded to control for volatility, the
coefficient weakens indicating a decrease of 0,09 percentage points. The positive coefficient for volatility is consistent with the notion that increasing risk should induce larger bid-ask spreads, hence a less negative coefficient for the regulation dummy variable. These values are significantly high compared to the 0,41% average
spread of the sample. All coefficients are significantly different from zero at the 1% level.
In the studies performed by Beber and Pagano (2013), and Bernal et al. (2013), where prohibition of naked short selling and disclosure requirements are analyzed separately, their outcome is that these two restraints have opposite effects on spreads. They find that the prohibition of naked short selling increases the bid-ask spreads, while the disclosure requirements decrease them. Since the current European regulation engages both aspects simultaneously, the reduction on the bid-ask spreads suggests that the disclosure requirement dominates the facets of the regulation and possibly reduces adverse selection problems in the market due to short sellers’ asymmetric information. As these short sellers must now disclose their positions to authorities and other market participants, there is less hostile trading on negative information.
3.2. Volume
Table 6: Panel regressions – Volume
The dependent variable LogVolume is the natural logarithm of the amount of shares of stock i traded at day t. Regulation is a dummy variable that equals one if the regulation is in place and zero otherwise. Volatility is the difference between the highest and lowest daily prices, divided by the price at closing. LogVolume (-1) is the 1 day lagged LogVolume. LogVolume (-2) is the 2 days lagged LogVolume. LogVolume (-3) is the 3 days lagged LogVolume. LogVolume (-4) is the 4 days lagged LogVolume. The estimates reported in parentheses are t-statistics. The three asterisks mark the coefficients that are significantly different from zero at the 1% level.
Dependent variable: LogVolume
Variables (1) (2) Intercept 2.7989*** 2.6874*** (203.30) (207.65) Regulation 0.0859*** 0.1643*** (14.55) (32.15) LogVolume (-1) 0.2603*** 0.2603*** (141.05) (131.57) LogVolume (-2) 0.1022*** 0.0948*** (47.06) (46.48) LogVolume (-3) 0.0651*** 0.0631*** (30.19) (31.17) LogVolume (-4) 0.0318*** 0.0310*** Volatility (15.79) (16.43) 13.6801*** (172.76) Stock fixed effects
Day fixed effects
Yes Yes Yes Yes 𝑅! 0.9120 0.8951 Number of observations 222,949 222,949
The results from Table 6 show the impact of the regulation on volumes, which are measured by the logarithm of the daily amount of traded shares. In column one an increase of 8,59 percent change in volume is indicated, which is also statistically significant at the 1% level. Column two presents the coefficients of an expanded model that controls for volatility5 and predicts a rise of 16,43 percent
5 Equity prices and volatilities interact in a very complex matter with volume on high frequency data (Karpoff, 1987).
change in volume. Estimations in both columns are presented by highly explanatory models, with 𝑅! values of approximately 0.91 and 0.90 respectively.
These results seem counterintuitive, since the additional costs imposed by all aspects of the regulation are expected to diminish the number of traded shares. Nonetheless, external factors play an important role here. The easing fears of a Eurozone crisis and low interest rates led to growing investor optimism in the stock market during 2013 (Jackson, 2013). Therefore, one possible reason for this increase in volume is due to the upsurge in consumer confidence that generated a growth in demand for stocks in European markets.
3.3. Intraday Volatility
Table 7: Panel regressions – Intraday Volatilities
The dependent variable Volatility is the difference between the highest and lowest daily prices, divided by the price at closing. Regulation is a dummy variable that equals one if the regulation is in place and zero otherwise. Volatility (-1) is the 1 day lagged volatility. Volatility (-2) is the 2 days lagged volatility. Volatility (-3) is the 3 days lagged volatility. The estimates reported in parentheses are t-statistics. The three asterisks mark the coefficients that are significantly different from zero at the 1% level.
Dependent variable: Volatility
Variables (2) Intercept 0.0174*** (241.12) Regulation -0.0027*** (-34.74) Volatility (-1) 0.1723*** (104.98) Volatility (-2) 0.0760*** (46.23) Volatility (-3) 0.0632*** (39.54) Stock fixed effects
Day fixed effects
Yes Yes
𝑅! 0.2494
Number of
The results from Table 7 show an impact of decreasing volatilities on stock prices of 0.27 percentage points, which is statistically significant at the 1% level. In comparison to the mean volatility of 2.38%, the coefficient for the dummy variable is small, but economically significant. Nonetheless, the fall in volatility plays an important role in markets, especially for large investors, meaning that a trade will be made at a price very similar to the price at the time of execution and reducing uncertainty on the stock price level (FSA, 2009).
The results obtained in this paper contradict the ones from Bernal et al. (2013), who found a positive impact for both naked short sales constraints and disclosure requirements on volatilities. One possible explanation for this ambiguity may be due to a potential endogeneity6 and lack of explanatory variables, also
reflected in the low 𝑅! figures. The decline in volatility presented in this study, may
also reflect the overall improvement of macroeconomic fundamentals, besides a movement towards an economic stabilization for most European countries in consideration during the period of study. Additionally, in their study, Bernal et al. (2013) work with data that originates from 2008-2009, when financial markets were under heavy distress. Under distressed conditions, markets operate alongside great price instability, which in turns induces uncertainty on stock price levels and reflects on increasing volatility levels.
6 In the fixed effects model the variables not explicitly measured are controlled for, but their effects are not estimated (Williams, n.d.).
4. Conclusion
This paper analyzed the impact of the Short Sales Regulation (EU) No 236/2012 on European markets. A series of panel regressions on thirteen countries suggest that the directive has an overall positive influence on stocks’ bid-ask spreads, and a small but positive effect on intraday volatilities. Evidence suggests that traded volume increases, which seems contradictory under short selling
restrictions. Nonetheless, It is important to take note of independent factors that took place simultaneously to the adoption of the European short sales regulation. For instance, the improvement of all aspects of consumer confidence in European markets (Discretionary Spending Intentions Increasing around the World, 2013). Furthermore, the development towards a steadier economic state and low interest rates helped lead to a boost in demand for equities. These factors in turn, result in an increased amount of shares traded independently of the restrictions to short selling.
Consequently, in order to avoid false inferences of causality, this paper concludes that the current regulation has ambiguous effects on market efficiency. On the one hand, previous literature suggests some detriment to market efficiency in the existence of short selling restrictions. At times of market turbulence, the costs of short selling restrictions seem to outweigh the benefits, and possibly worsen market conditions. On the other hand, the presence of a positive economic environment seems to overcome the potential impairments of the short selling regulation.
The limitations of this study may serve as a recommendation for future research. In order to address the matter of the causal relationship of the regulation on market efficiency, future studies may choose to adopt instruments that control for causality. Since the regulation is imposed on a market-level, the instrument should
also make use of market-level variables. One possible candidate instrument may be the multifaceted financial stress index proposed by Balakrishnan et al. (2009)
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Appendix
Panel Analysis for Spread
𝑆𝑝𝑟𝑒𝑎𝑑!,! = 𝛽!+ 𝛽! 𝑅𝑒𝑔!,! + 𝛽!𝑆𝑝𝑟𝑒𝑎𝑑!,!!!+ 𝛽!𝑆𝑝𝑟𝑒𝑎𝑑!,!!!+ 𝛽! 𝑆𝑝𝑟𝑒𝑎𝑑!,!!! + 𝜀!,!
Table 8: Hausman test for fixed effects – Panel Analysis for Spread The Hausman tests for the distinction of random effects from fixed effects models. The null hypothesis is that the preferred model uses random effects vs. the alternative hypothesis of fixed effects.
(V_b-V_B is not positive definite) Prob>chi2 = 0.0000
= 21071.19
chi2(4) = (b-B)'[(V_b-V_B)^(-1)](b-B) Test: Ho: difference in coefficients not systematic
B = inconsistent under Ha, efficient under Ho; obtained from xtreg b = consistent under Ho and Ha; obtained from xtreg spread3 .1218163 .2158222 -.094006 .0003798 spread2 -.0515031 -.0006789 -.0508242 . spread1 .4716773 .5663571 -.0946798 .0003858 d -.0007788 -.0000764 -.0007025 .0000142 fixed random Difference S.E.
(b) (B) (b-B) sqrt(diag(V_b-V_B)) Coefficients
Panel Analysis for Volume
𝐿𝑜𝑔𝑉𝑜𝑙𝑢𝑚𝑒!,! = 𝛽!+ 𝛽! 𝑅𝑒𝑔!,!+ 𝛽!𝐿𝑜𝑔𝑉𝑜𝑙𝑢𝑚𝑒!,!!!+ 𝛽!𝐿𝑜𝑔𝑉𝑜𝑙𝑢𝑚𝑒!,!!! + 𝛽! 𝐿𝑜𝑔𝑉𝑜𝑙𝑢𝑚𝑒!,!!!+ 𝛽! 𝐿𝑜𝑔𝑉𝑜𝑙𝑢𝑚𝑒!,!!!+ 𝜀!,!
Table 9: Hausman test for fixed effects – Panel Analysis for Volume The Hausman tests for the distinction of random effects from fixed effects models. The null hypothesis is that the preferred model uses random effects vs. the alternative hypothesis of fixed effects.
(V_b-V_B is not positive definite) Prob>chi2 = 0.0000
= 42558.42
chi2(5) = (b-B)'[(V_b-V_B)^(-1)](b-B) Test: Ho: difference in coefficients not systematic
B = inconsistent under Ha, efficient under Ho; obtained from xtreg b = consistent under Ho and Ha; obtained from xtreg logvol4 .0310578 .1411944 -.1101366 . logvol3 .0645355 .1580204 -.0934848 . logvol2 .1017952 .2111244 -.1093292 . logvol1 .2976631 .4682676 -.1706045 .0001072 d .0438254 .0153218 .0285036 .0006976 fixed random Difference S.E.
(b) (B) (b-B) sqrt(diag(V_b-V_B)) Coefficients
Panel Analysis for Volatility
𝑉𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦!,! = 𝛽!+ 𝛽! 𝑅𝑒𝑔!,! + 𝛽!𝑉𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦!,!!!+ 𝛽!𝑉𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦!,!!! + 𝛽! 𝑉𝑜𝑙𝑎𝑡𝑖𝑙𝑖𝑡𝑦!,!!!+ 𝜀!,!
Table 10: Hausman test for fixed effects – Panel Analysis for Volatility The Hausman tests for the distinction of random effects from fixed effects models. The null hypothesis is that the preferred model uses random effects vs. the alternative hypothesis of fixed effects.
(V_b-V_B is not positive definite) Prob>chi2 = 0.0000
= 19864.02
chi2(4) = (b-B)'[(V_b-V_B)^(-1)](b-B) Test: Ho: difference in coefficients not systematic
B = inconsistent under Ha, efficient under Ho; obtained from xtreg b = consistent under Ho and Ha; obtained from xtreg volatility3 .0756456 .1551757 -.0795301 .0002718 volatility2 .0938341 .16388 -.0700458 . volatility1 .2308283 .3116173 -.080789 .0002891 d -.0019567 -.0007069 -.0012498 .0000285 fixed random Difference S.E.
(b) (B) (b-B) sqrt(diag(V_b-V_B)) Coefficients
