Can microstructure noise explain the MAX effect?

Accepted Manuscript

Can microstructure noise explain the MAX effect? Xindong Zhang , Lixu Xie , Yue Zhai , Dong Wang PII: DOI: Reference:

S1544-6123(17)30369-0 10.1016/j.frl.2018.01.006 FRL 850

To appear in:

Finance Research Letters

Received date: Revised date: Accepted date:

30 June 2017 13 December 2017 19 January 2018

Please cite this article as: Xindong Zhang , Lixu Xie , Yue Zhai , Dong Wang , Can microstructure noise explain the MAX effect?, Finance Research Letters (2018), doi: 10.1016/j.frl.2018.01.006

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Highlights  

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Provide a new perspective of market microstructure noise on the explanation of the MAX effect. The market microstructure noise, though not able to completely eliminate, can effectively moderate the MAX effect. The five-factor model is better than the Fama-French (1993)-Carhart (1997) four-factor model in explaining the MAX effect.

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Can microstructure noise explain the MAX effect? Xindong Zhanga, Lixu Xiea,b,*, Yue Zhaia, Dong Wangc a. School of Economics and Management, Shanxi University, Taiyuan, 030006,China b. School of Management Science and Engineering, Anhui University of Technology,

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Maanshan, 243032, China c.School of Mathematics and Physics, Anhui University of Technology, Maanshan, 243032, China

The first author : Xindong Zhang

Co-authors: Yue Zhai, Dong Wang

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Correspond author: Lixu Xie Email: [email protected]

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Abstract Bali et al. (2011) first find and investigate the MAX effect using raw returns

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(calculated by recorded closing prices), which include microstructure noise from bid-ask measurement errors. Motivated by this, we use noise-adjusted returns (which

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remove bid-ask errors) to examine the MAX effect and find microstructure noise is an

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important source of the effect. Average monthly return and five-factor alpha differences between the highest and lowest MAX stock portfolios aren’t significant in

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statistics. Most importantly, the negative five-factor alpha differences have no negative significance both in economics and statistics over equal-weighted portfolios.

Keywords: MAX; lottery; microstructure noise; bid-ask errors; five-factor Classification JEL Codes: G12, G14

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Can microstructure noise explain the MAX effect? Abstract Bali et al. (2011) first find and investigate the MAX effect using raw returns (calculated by recorded closing prices), which include microstructure noise from

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bid-ask measurement errors. Motivated by this, we use noise-adjusted returns (which remove bid-ask errors) to examine the MAX effect and find microstructure noise is an important source of the effect. Average monthly return and five-factor alpha

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differences between the highest and lowest MAX stock portfolios aren’t significant in statistics. Most importantly, the negative five-factor alpha differences have no negative significance both in economics and statistics over equal-weighted portfolios.

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Keywords: MAX; lottery; microstructure noise; bid-ask errors; five-factor

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Highlights: Provide a new perspective of market microstructure noise on the

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explanation of the MAX effect. The market microstructure noise, though not able to completely eliminate, can effectively moderate the MAX effect. The five-factor model

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is better than the Fama-French (1993)-Carhart (1997) four-factor model in explaining

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the MAX effect.

Classification codes: G120

1. Introduction According to the Efficient Market Hypothesis (EMH, Fama, 1970), the average excess return of every stock or portfolio during a period should be equal to zero. 3

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However, a number of recent empirical results show that there commonly exist price abnormal phenomena deviating from EMH, which are called market anomalies. Bali et al. (2011) propose and investigate a new prominent anomaly of the MAX effect that expected returns and the maximum daily return in the prior month (MAX) are

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significantly and negatively related in the U.S. stock market. Using raw returns for stocks, calculated by transaction prices, Bali et al. (2011) document average monthly return and Fama-French-Carhart four-factor alpha differences between the highest and

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lowest MAX stock portfolios are notably negative. After that, many relevant studies have been done in the aim of explaining the MAX effect from different angles. In this paper, we attempt to test the MAX effect from a new perspective of market

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microstructure noise. Our goal is to explore whether market microstructure noise

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caused by bid-ask measurement errors is attributable to the MAX effect. Bali et al. (2011) owe the MAX anomaly to the existence of those investors who

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are not well-diversified and have a strong tendency towards lottery-like stocks. It’s

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well-known that lottery-type stocks have high idiosyncratic volatility and skewness.

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However, the MAX effects aren’t notably attenuated through controlling for idiosyncratic volatility (Bali et al., 2011; Annaert et al.,2013; Zhong and Gray, 2016) and skewness (Bali et al., 2011; Annaert et al.,2013), respectively. From the aspect of investor behavior, Fong et al. (2014) document investor sentiment, especially the high sentiment, plays an important role on the MAX effect; Zhong and Gray (2016) demonstrate mispricing can partially explain the MAX effect. 4

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In fact, stock raw returns, used in most literatures, contain somewhat microstructural measurement bias (i.e., bid-ask effect), which is often ignored in empirical studies. Moreover, microstructure-induced noise would lead to inefficient financial markets or market anomalies (Black, 1986), such as upward biased closing

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prices (Blume and Stambaugh, 1983), short-term price reversals and spurious volatility in transaction returns (Kaul and Nimalendran, 1990). Dennis and Mayhew (2009) empirically explore how microstructure bias affects the tests of option pricing

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models. Rosenhaum (2011) proposes a new microstructure noise index from low to high frequencies to reproduce the facts observed on the data.

Motivated by the points above, in this paper, we use the noise-adjusted returns

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(which exclude the microstructure noise or bid-ask errors) to examine the MAX effect

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in the U.S. stock market. Besides the Fama-French-Carhart four-factor model, we set a precedent to apply the newest Fama-French (2015) five-factor model to study the

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potential impact of the microstructure noise on the MAX effect.

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The rest of the paper is organized as follows. Section 2 describes the source of

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our data and constructs the variables of noise-adjusted and risk-adjusted returns. Section 3 analyzes the empirical results. Section 4 is a robustness check, and we draw a simple conclusion in section 5.

2. Data and variables construction 2.1 Data 5

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The closing prices and quote prices of all common stocks are from Thomson Reuters DataStream in the New York Stock Exchange (NYSE), American Stock Exchange (AMEX) and NASDAQ. The sample period of closing prices covers from January 1993 to December 2014. The daily raw returns are computed by the closing

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price. When a firm goes bankrupt, it may compensate the stockholders after liquidation, and this compensation should be included in the returns of the stock. So besides stock splits, reverse splits, dividends and so on，we also consider the impact of

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delisting on the daily returns for individual stock. Due to the non-availability of the data of the quote prices before 1993, the data of quote prices is from January 1993 to December 2014. The noise-adjusted daily and monthly returns are calculated by the

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bid and ask prices. The average returns in each month are computed on daily returns.

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The data of the risk-free interest rate and six factors in asset pricing models1 comes from the personal website of Kenneth French2. The average monthly returns are

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filtered out if the numbers of trading days are below 15.

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2.2 Construction of noise-adjusted returns In recent literature, high-frequency data in studying the market microstructure

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noise are widely popular, and it does solve many issues. However, Kociński (2014) argues that the low-frequency data over a long period of time also play an important role in the research of microstructure noise due to human input errors，computer 1

The six factors include market factor, size factor, book-to-market factor, profitability factor, investment factor and momentum factor. 2 The personal website of Kenneth French is as follows: http://mba.tuck.dartmouth.edu /pages/faculty/ken.french /data_library.html 6

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system errors and material limitation for the high-frequency data. For instance, Blume ＆Stambaugh (1983) show that the full-year size effect is cut half based on upward bias-extracted returns, Roll (1984) demonstrates that price reversals and negative autocorrelation are caused by the bid-ask spread in stock returns, and Kaul &

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Nimalendran (1990) document that the bid-ask errors lead to apparent short-term price reversals and about half of daily return volatility. In this paper, following Blume and Stambaugh (1983) and Kaul and Nimalendran (1990), we use the midpoint of

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bid-ask prices and bid prices respectively in each day to calculate the noise-adjusted returns (thereafter, we call mid-bid-ask-adjusted returns and bid-to-bid-adjusted returns, respectively) to eliminate the microstructural bid-ask errors contained in the

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daily returns of individual stocks.

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2.3 Construction of risk-adjusted returns

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The risk-adjusted returns are obtained through the cross-sectional regression of the Fama-French (1993)-Carhart (1997) four-factor model and the Fama-French

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(2015) five-factor model. The four-factor model is expressed as follows: (1)

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Ri ,t  rf ,t  iFF 4  i MKTt  si SMBt  hi HMLt  iUMDt，

where subscript i denotes a stock and t denotes a day. R is a stock’s average return, rf denotes the risk-free interest rate,  FF 4 is the risk-adjusted return and we call it four-factor alpha. MKT , SMB , HML and UMD represent market factor, size factor, book-to-market ratio factor and momentum factor, respectively. 7

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The five-factor model is represented as follows:

Ri,t  rf,t  iFF 5  i MKTt  si SMBt  hi HMLt  i RMWt   iCMAt，

(2)

where  FF 5 denotes the risk-adjusted return and we call it five-factor alpha. RMW

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and CMA represent investment factor and profitability factor, respectively.

3. Empirical analysis

In this section we will conduct a deep and comprehensive portfolio-level analysis

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on the influence of microstructure noise to the MAX effect. We sort all the stocks into deciles by MAX. Decile 10 is the portfolio including the highest MAX stocks, and decile 1 is the portfolio with the lowest MAX stocks.

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3.1 Analysis on average returns in each month in the lowest and highest MAX deciles Table 1 Statistical results of average monthly returns in the lowest and highest MAX deciles

Raw

Mid-bid-ask

Bid-to-bid

1.06%

1.08%

0.49%

9.22%

Bid-to-bid

0.51%

0.56%

0.86%

0.37%

-2.54%

-2.31%

-2.17%

11.36%

13.18%

33.39%

31.05%

32.69%

1.51

8.63

11.3

5.98

7.28

5.59

140110

137350

137366

140181

137421

137416

median

0.82%

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1.21%

Skewness

Raw

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Number

Highest MAX Mid-bid-ask

mean

Std_dev

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Lowest MAX

Table 1 shows the statistical results of average returns in the subsequent month

calculated on the closing prices, mid-bid-ask prices and bid prices respectively for the lowest and highest MAX portfolios. From the distribution of returns, it’s easy to see the lowest MAX stock portfolios have higher future returns than the highest ones no matter whether microstructure noise is removed. Both the mean and median future 8

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returns for the lowest MAX portfolios are positive, but the median future returns for the highest MAX portfolios are negative and the means are positive, which indicates more than half of firms in the highest MAX portfolios own negative future returns. Not surprisingly, the magnitudes of noise-adjusted monthly return differences

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between the lowest and highest MAX portfolios are much smaller than that of the raw monthly return difference, with mid-bid-ask-adjusted return difference of 0.5%, bid-to-bid-adjusted return difference of 0.2% and raw return difference of 0.7%. It

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initially seems that there exists apparent MAX effect and microstructure noise may play some role in explaining the MAX effect.

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3.2 Portfolio-level analysis on average monthly returns and alphas 3.2.1 Reexamination of the MAX effect

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In this subsection, using raw returns, we reexaminate the MAX effect found in

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Bali et.al (2011) by extending the sample period to December 2014. Table 2 reports

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the average monthly returns and alphas for stocks in different MAX deciles over value-weighted and equal-weighted portfolios. We obtain consistent results with Bali

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et.al (2011).We can see from the table that the MAX effect significantly exists: average monthly return and risk-adjusted return differences in portfolios between decile 10 and decile 1 are significantly negative. Aside from raw returns in the equal-weighted portfolios, alphas and other average returns show apparent short-term reversals in the higher MAX stock portfolios. 9

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Table 2 Raw returns and test of the MAX effect VW Portfolios Average Decile

αFF4

EW Portfolios αFF5

Average

return

αFF4

αFF5

return

Average MAX

0.921

0.022

-0.048

1.082

0.182

0.033

1.213

2

0.976

0.043

-0.044

1.324

0.305

0.171

2.507

3

1.003

0.034

-0.063

1.438

0.387

0.241

3.359

4

1.048

0.071

0.064

1.418

0.357

0.227

4.188

5

1.032

0.086

0.052

1.364

0.300

0.200

5.094

6

1.067

0.049

0.071

1.335

0.261

0.201

6.161

7

0.893

-0.060

0.113

1.188

0.176

0.153

7.504

8

0.781

-0.231

-0.034

1.138

0.144

0.138

9.377

9

0.447

-0.494

-0.271

0.900

-0.113

-0.058

12.565

H-MAX

-0.033

-0.985

-0.631

0.329

-0.616

-0.466

25.585

H-L

-0.954

-1.007

-0.583

-0.753

-0.798

-0.498

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L-MAX

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Newey-West (-2.39) (-4.78) (-3.03) (-2.21) (-3.96) (-2.68) t-statistic Note: The returns and alphas are all in percent and the numerical values in parentheses are

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Newey-West adjusted t-statistic, the same below.

3.2.2 Microstructure noise and the MAX effect

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As demonstrated in the introduction, microstructure noise included in raw returns

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may cause somewhat measurement bias induced by the bid-ask spread. In this

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subsection we explore the relation between MAX and noise-adjusted returns, in which the bid-ask errors are extracted. A. Mid-bid-ask-adjusted returns and test of the MAX effect Table 3 reports the average monthly returns and alphas in different MAX deciles over mid-bid-ask-adjusted return stock portfolios. As shown in table 3, the absolute values of the Newey-West t-statistic of the average return and five-factor alpha 10

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Table 3 Mid-bid-ask-adjusted returns and test of the MAX effect VW portfolios Average Decile

αFF4

EW portfolios αFF5

return

Average

αFF4

αFF5

return

Average MAX

1.024

0.257

0.257

1.067

0.427

0.276

1.284

2

0.947

0.078

-0.058

1.293

0.467

0.315

2.402

3

1.057

0.176

0.028

1.295

0.413

0.280

3.191

4

0.926

0.040

-0.071

1.262

0.371

0.236

3.966

5

0.834

-0.120

-0.149

1.228

0.314

0.241

4.816

6

0.747

-0.266

-0.270

1.203

0.300

0.262

5.827

7

0.888

-0.089

0.116

1.163

0.270

0.286

7.102

8

0.707

-0.381

-0.105

1.145

0.268

0.372

8.874

9

0.550

-0.455

-0.097

1.059

0.271

0.401

11.849

H-MAX

0.482

-0.549

0.081

0.533

-0.190

0.193

24.880

H-L Newey-West t-statistic

-0.542

-0.806

-0.176

-0.534

-0.617

-0.083

(-0.73)

(-2.37)

(-0.54)

(-0.82)

(-1.90)

(-0.28)

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L-MAX

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are all smaller than 0.9, and the short-run reversals both in average returns and

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five-factor alphas for the highest MAX stock portfolios disappear. These are

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consistent with the conclusion in Kual and Nimalendran (1990) that the noise-adjusted returns, which exclude bid-ask errors and a lot of spurious volatility in

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raw returns, can attenuate apparent short-run price reversals.

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As shown in table 3, the average four-factor alpha differences between the highest and lowest MAX portfolios are still significant. However, the five-factor alpha differences have only economic significance. Especially, the five-factor alpha difference has no negative significance both in economics and statistics over the equal-weighted portfolios. It means that, using mid-bid-ask-adjusted returns, the 11

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MAX effect doesn’t survive any more based on the five-factor model for the equal-weight portfolios. Overall, the five-factor model is better than the four-factor model in explaining the MAX effect.

B. Bid-to-bid-adjusted returns and test of the MAX effect

VW Portfolios Average Decile

αFF4

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Table 4 Bid-to-bid-adjusted returns and test of the MAX effect EW Portfolios αFF5

return

Average return

αFF4

αFF5

Average MAX

0.976

0.210

0.208

1.104

0.439

0.275

1.260

2

0.921

0.062

-0.049

1.314

0.450

0.294

2.479

3

1.115

0.213

0.052

1.326

0.416

0.287

3.304

4

0.830

-0.070

-0.143

1.322

0.403

0.259

4.105

5

0.856

-0.121

-0.154

1.312

0.365

0.279

4.986

6

0.775

-0.291

-0.292

1.412

0.427

0.288

6.033

7

0.806

-0.201

0.076

1.207

0.275

0.288

7.361

8

0.739

-0.311

-0.026

1.234

0.329

0.420

9.207

9

0.512

-0.557

-0.188

1.165

0.347

0.473

12.337

H-MAX

0.513

-0.558

0.061

0.992

0.393

0.693

46.898

H-L Newey-West t-statistic

-0.463

-0.768

-0.1463

-0.113

-0.046

0.419

(-0.62)

(-2.19)

(-0.43)

(-0.14)

(-0.13)

1.16

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PT

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L-MAX

In order to confirm the previous empirical results, we employ the

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bid-to-bid-adjusted returns to test the MAX effect in this subsection. The results shown in table 4 are similar with the results in table 3. The short-term reversals of the average returns and five-factor alphas in the highest MAX deciles also don’t exist. The average monthly return and five-factor alpha differences between the highest and lowest MAX portfolios are not significant in statistics. Especially, the four-factor 12

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alpha difference doesn’t survive (0.05% with t-statistic of -0.13) and the five-factor alpha difference even appears positive (0.42 with t-statistic of 1.16) over the equal-weighted portfolios. It provides further evidence that market microstructure noise, though not able to completely eliminate, can effectively moderate the MAX

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effect. 4 A robustness check

In this section we examine the relationship between future returns and the mean

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of the five largest daily returns in the prior month (denoted as MAX(5)) employing the same method in section 3. The average returns and alphas in different deciles for raw and noise-adjusted return stock portfolios are reported in tables 5-7 in the

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attached list. The similar results are obtained as section 3. We find apparent MAX(5)

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effect (the significantly negative relation between average future monthly returns and MAX(5)) based on the raw returns. The MAX(5) effect are attenuated effectively

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based on the noise-adjusted returns.

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As shown in figure 1, the average monthly returns and MAX(5) exist nonlinear

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relationship. The spreads between adjusted-returns and raw returns are very small from decile 1 to 9 in both equal-weighted and value-weighted portfolios, but the spreads in decile 10 are very large, except for the equal-weighted and mid-bid-ask-adjusted return portfolio in panel B. It suggests that the MAX effect obviously exists chiefly due to higher deciles and microstructure noise attenuates the MAX effect mainly by narrowing the spread in decile 10. On the other hand, Kual and 13

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Nimalendran (1990) theoretically and empirically demonstrate that market microstructure noise would cause substantial spurious volatility in raw returns and is the main source of the price reversals. So it seems that market microstructure noise can moderate the MAX effect, partly as noise-adjusted returns can exclude a lot of

The similar results are found in the MAX effect.

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Panel B

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Panel A

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spurious volatility in raw returns and alleviate the effect of short-run price reversals.

Figure 1 The relation between average monthly returns and MAX(5)

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5. Conclusion and perspective for research In this paper we prove that the microstructure noise caused by measurement errors of the bid-ask effect is an important source of the MAX effect. We use the

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noise-adjusted returns to test it. We find average monthly return and five-factor alpha differences between the lowest and highest MAX stock portfolios are economically but not statistically significant. Most importantly, the negative five-factor alpha difference aren’t notable both in economics and statistics over the equal-weighted

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portfolios. The five-factor model is better than the four-factor model in explaining the MAX effect. The noise-adjusted returns could extract substantial spurious volatility in raw returns and attenuate the short-run price reversals effectively (Kaul and

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Nimalendran, 1990), which seems one of the reasons why market microstructure

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noise is an important source of the MAX effect.

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However, there still exist unsolved problems---the unexplained alpha differences in the four-factor model. So the natural extension of this paper could be considering

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what could explain the remaining alpha differences of the four-factor model over

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noise-adjusted portfolio returns. Moreover, it’s interesting and meaningful to consider the explanatory power of idiosyncratic volatility, skewness, size or other variables in the case of market microstructures.

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Funding This work is supported by the National Natural Science Foundation of China [grant

of Education [grant number 13YJA790154].

References

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number 71371113] and the Humanities and social science research project of Ministry

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Annaert, J., DeCeuster, M., Verstegen, K., 2013. Are extreme returns priced in the stock market? European evidence. Journal of Banking and Finance 37, 3401–3411.

Bali, T. G., Cakici, N., Whitelaw, R. F., 2011. Maxing out: stocks as lotteries and the cross-section

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of expected returns. Journal of Financial Economics 99, 427-446.

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Black, F., 1986. Noise. The Journal of Finance 41, 529-543. Blume, M. E., Stambaugh, R. F., 1983. Biases in computed returns an application to the size effect.

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Journal of Financial Economics 12, 387-404.

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Carhart, Mark M., 1997. On persistence in mutual fund performance. Journal of Finance 52,

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57-82.

Dennis, P., Mayhew, S., 2009. Microstructural biases in empirical tests of option pricing models. Review of Derivatives Research 12, 169-191.

Fama, E. F., 1970. Efficient Capital Markets: A review of theory and empirical work. The Journal of Finance 25, 383-417. Fama, E. F., French, K. R.,1993. Common risk factors in the returns on stocks and bonds. Journal 16

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of Financial Economics 33, 3-56. Fama, E. F., French, K. R., 2015. A five-factor asset pricing model. Journal of Financial Economics 116, 1-22. Fong, W., Toh, B., 2014. Investor sentiment and the MAX effect. Journal of Banking and Finance

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46, 190–201.

Kaul, G., Nimalendran, M., 1990. Price reversals: bid-ask errors or market overreaction? Journal of Financial Economics 28, 67-93.

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Kociński, M. A., 2014. On low-frequency estimation of bid-ask spread in the stock market. Quantitative Methods in Economics 65, 135-143.

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heteroskedasticity and autocorrelation consistent covariance matrix. Econometrica 55, 703-708.

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Roll, Richard, 1984, A simple implicit measure of the effective bid-ask spread in an efficient market. Journal of Finance 39, 1127-1140.

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Rosenhaum, M., 2011. A new microstructure noise index. Quantitative Fiance 11, 883-899.

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Zhong, A., Gray, P., 2016. The MAX effect: An exploration of risk and misprcing explanations.

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Journal of Banking & Finance 65, 76–90.

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Attached list Table5 Raw returns and test of the MAX(5) effect VW Portfolios Decile

Average return

EW Portfolios

αFF4

αFF5

Average return

αFF4

αFF5

Average MAX

1.040

0.113

0.052

1.073

0.202

0.045

0.730

2

1.034

0.107

-0.002

1.395

0.386

0.231

1.480

3

1.034

0.075

-0.015

1.464

0.419

0.265

1.998

4

1.025

0.102

0.014

1.464

0.415

0.284

2.477

5

1.021

0.018

-0.002

1.417

0.346

0.227

2.983

6

0.974

0.039

0.105

1.380

0.327

0.254

3.562

7

1.017

-0.051

0.160

1.270

0.246

0.220

4.266

8

0.594

-0.337

-0.131

1.098

0.077

0.088

5.215

9

0.360

-0.616

-0.272

0.779

-0.205

-0.130

6.745

H-MAX

-0.269

-1.210

-0.770

0.179

-0.831

-0.640

12.225

H-L Newey-West t-statistic

-1.309

-1.324

-0.822

-0.893

-1.033

-0.685

(-2.94)

(-5.29)

(-3.66)

(-2.60)

(-4.81)

(-3.49)

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L- MAX

Table 6 Mid-bid-ask-adjusted returns and test of the MAX(5) effect VW Portfolios

Average

αFF5

1.046

0.353

0.277

0.991

0.116

3

1.062

4

Average

Average

αFF4

αFF5

1.064

0.460

0.328

0.740

0.017

1.351

0.558

0.399

1.402

0.203

0.036

1.350

0.493

0.335

1.875

1.026

0.074

-0.064

1.310

0.420

0.283

2.313

5

0.794

-0.070

-0.230

1.306

0.424

0.288

2.773

6

0.918

-0.050

-0.032

1.234

0.323

0.257

3.301

7

0.688

-0.355

-0.145

1.191

0.307

0.327

3.947

8

0.782

-0.293

-0.038

1.066

0.175

0.287

4.802

9

0.478

-0.585

-0.070

1.049

0.214

0.355

6.135

H-MAX

0.189

-1.039

-0.404

0.328

-0.463

0.002

10.774

H-L

-0.858

-1.392

-0.680

-0.736

-0.923

-0.326

Newey-West

(-1.15)

(-3.66)

(-1.88)

(-1.08)

(-2.73)

(-1.08)

Decile L- MAX

return

AC

CE

2

PT

αFF4

EW Portfolios return

18

MAX

ACCEPTED MANUSCRIPT t-statistic

Table 7 Bid-to-bid-adjusted returns and test of the MAX(5) effect VW Portfolios Average Decile return

EW Portfolios

αFF4

αFF5

Average return

αFF4

αFF5

Average MAX

0.995

0.280

0.212

1.092

0.460

0.317

0.721

2

1.022

0.167

0.046

1.389

0.568

0.389

1.433

3

0.978

0.089

-0.047

1.371

0.469

0.306

1.934

4

1.013

0.085

-0.075

1.375

0.483

0.330

2.389

5

0.808

-0.098

-0.213

1.305

0.372

0.258

2.867

6

0.731

-0.237

-0.164

1.449

0.464

0.332

3.417

7

0.800

-0.336

-0.055

1.282

0.324

0.333

4.090

8

0.680

-0.358

-0.133

1.242

0.359

0.424

4.983

9

0.480

-0.643

-0.078

1.127

0.241

0.392

6.392

H-MAX

0.231

-0.975

-0.371

0.755

0.104

0.476

15.537

H-L Newey-West t-statistic

-0.764

-1.255

-0.583

-0.336

-0.356

0.159

(-0.99)

(-3.18)

(-1.55)

(-0.45)

(-0.95)

0.450

AC

CE

PT

ED

M

AN US

CR IP T

L- MAX

19