# Tests Of Weak Form Efficiency Accounting Essay

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Efficient Market Hypothesis (EMH) is a very controversial and highly disputed theory nowadays. EMH was formulated by Eugene Fama in 1970. It is believed that it is impossible to find undervalued stocks or predict any trend in the stock market. There are three identified classifications of theÂ EMH (Strong efficiency, Semi-strong efficiency, Weak efficiency) which have as an aim to reflect the degree to which it can be applied to markets.

The principal objective of this individual report is to perform some tests for the weak form efficiency for three selected stocks (MARSHALL & ILSLEY CORP, ABERCROMBIE & FITCH CO, ANNALY CAPITAL MANAGEMENT INC) and two deciles indices (NYSE/AMEX/NASDAQ index capitalisation-based Deciles 1 and 10).

The structure of this report is organised as follows:

Section I gives a brief description about the companies.

Section II describes the data and the methodology used for the tests of the weak form efficiency

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Section III discusses the descriptive statistics of the data we used

Section IV presents the results

## SECTION I: THE DESCRIPTION OF THE COMPANIES

## MARSHALL & ILSLEY CORP (MI)

Marshall & Ilsley Corporation (M&I) is a bank holding company. M&I provides its subsidiaries with financial and managerial assistance, in budgeting, tax planning, auditing, compliance, asset and liability management, investment administration and portfolio planning, business development, advertising and human resources management. M&I also provides diversified financial services to a variety of corporate, institutional, government and individual customers. M&I operates in four business segments: Commercial Banking, Community Banking, Wealth Management and Treasury. M&I provides banking services, which include lending to and accepting deposits from commercial and community banking customers Source:Â Worldscope

## ABERCROMBIE & FITCH CO (ANF)

Abercrombie & Fitch Co. (A&F) through its subsidiaries, is a specialty retailer that operates stores and direct-to-consumer operations selling casual sportswear apparel, including knit and woven shirts, graphic t-shirts, fleece, jeans and woven pants, shirts, sweaters, outerwear, personal care products, and accessories for men, women and kids under the Abercrombie & Fitch, abercrombie kids, and Hollister brands. In addition, the Company operates stores and direct-to-consumer operations offering bras, underwear, personal care products, sleepwear and at-home products for women under the Gilly Hicks brand. Source:Â Worldscope

## ANNALY CAPITAL MANAGEMENT INC (NLY)

Annaly Capital Management, Inc. owns, manages and finances a portfolio of real estate related investments, including mortgage pass-through certificates, collateralized mortgage obligations (CMOs), Agency callable debentures, and other securities representing interests in the obligations backed by pools of mortgage loans. The Company also invests in Federal Home Loan Bank (FHLB), Freddie Mac and Fannie Mae debentures. The Company's wholly owned subsidiaries offer diversified real estate, asset management and other financial services. It is self-advised and self-managed.

Source:Â Worldscope

## SECTION II: DATA AND METHODOLOGY

## DATA

We use both daily and monthly returns for the three stocks and two deciles indices over the period January 2007 to December 2010. All the daily data have 1007 observations and the monthly data have 47 observations. We have also used logarithmic (log) returns and arithmetic returns for the descriptive statistics of each stock

## METHODOLOGY

## B.1.AUTO CORRELATION ANALYSIS

Correlation is a statistical relationship between 2 or more random variables or observed data values. The correlation of a series with its own lagged values is called autocorrelation. The most known measure of dependence is the correlation coefficient and is defined as:

Î¡k is the autocorrelation at lag k

If the autocorrelation is positive or +1 then we have positive linear relationship (correlation). In case it is -1 then we have negative linear relationship (anticorrelation). If the value is between âˆ’1 and 1 indicates the degree of linear dependence between the variables. If it is close to zero then we say that the data are uncorrelated. If the coefficient is closer to either âˆ’1 or 1, then the correlation is stronger between the variables. We have used the autocorrelation coefficients up to 10 lags and they are computed for each stock and decile index.

## B.2. DAY OF THE WEEK EFFECT

Seasonal effects in financial markets have been perceived and they are often called as "calendar anomalies". Some examples are the day-of-the-week effect, open- or close-of-market, January or bank holiday and many others. We may find that the stock returns are statistically significant different during some calendar periods compared to others. If stock returns follow a random walk then there mustn't be any difference between returns in different calendar periods. If we have decided to detect seasonality in returns, the most common way is to use a "Dummy Variable" regression. The coefficients are interpreted as the average return on each day of the week

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Always on Time

Marked to Standard

Where is the return at time t

1 if it is a Monday return

D1t =

0 otherwise

## B.3. VARIANCE RATIO TESTS

The variance ratio Test was used by Lo and MacKinlay (1988) to check if the data we used, follow random walk. Price movements usually follow random walk. The test statistic derived by Lo and MacKinlay to test if a series of price movements follows a random walk is robust to many forms of heteroscedasticity and nonnormality. The variance test is defined as :

If the VR (q) is equal to 1 then we can say that we have the random walk null hypothesis. When VR (q) is bigger than 1 and the null hypothesis is rejected then the return is positively correlated. If the VR (q) is smaller than 1 and the null hypothesis is again rejected then we can determine that the return is negatively correlated.

## B.4 VOLATILITY MODELLING

Generally, when we have to test for heteroscedasticity in econometric models, the best test is the White test. However, when we have to deal with time series data, we have to use the GARCH model. GARCH (generalized autoregressive conditional heteroscedasticity) model was introduced by Bollerslev in 1986.

In that case, the GARCH (p, q) model (where p is the order of the GARCH terms ~\sigma^2 and q is the order of the ARCH terms ~\epsilon^2 ) and it is given by:

\sigma_t^2=\alpha_0 + \alpha_1 \epsilon_{t-1}^2 + \cdots + \alpha_q \epsilon_{t-q}^2 + \beta_1 \sigma_{t-1}^2 + \cdots + \beta_p\sigma_{t-p}^2 = \alpha_0 + \sum_{i=1}^q \alpha_i \epsilon_{t-i}^2 + \sum_{i=1}^p \beta_i \sigma_{t-i}^2

The EGARCH (exponential general autoregressive conditional heteroskedastic) model was introduced by Nelson in 1991 and it is another form of the GARCH model.

An EGARCH(p, q) is defined as:

\log\sigma_{t}^2=\omega+\sum_{k=1}^{p}\beta_{k}g(Z_{t-k})+\sum_{k=1}^{q}\alpha_{k}\log\sigma_{t-k}^{2}

\sigma_{t}^{2} Is the conditional variance

If the \log\sigma_{t}^{2} is negative, then there are no restrictions on the parameters.

## SECTION III: DISCUSSION OF THE DESCRIPTIVE STATISTICS

We have examined the descriptive statistics from the Table I and Table II so we can identify if the distribution of the returns are normal. If they are normal, then they must have followed the random walk model. At Table I and Table II are reported the daily and monthly logarithmic returns (r) and the daily and monthly arithmetic returns (R).

For the daily frequencies, all the mean and median for the returns are positive except from the MI (-0.00194,-0.00121) and the ANF (-0.00021). The lowest minimum returns are DC1 (-0.0561) and DC10 (-0.095184) which both give the lowest maximum returns 0.062526 and 0.111735 respectively. In contrast with them, MI and NLY show the highest maximum returns. As a measure of dispersion we take the standard deviation which reveals that DC1 and DC10 are the least volatile from all the data. MI is the most volatile. NLY is the only one which is positive skewed, which means that the return distribution has a high possibility of earning positive returns. In contrast MI, ANF, DC1 and DC10 are negatively skewed which means that they have a high possibility of earning negative returns. All three stocks and the two decile indices have positive kurtosis which means that they have leptokurtic return distributions

Histogram of the return distribution of MI

Time Series of the return Distribution of D1

For the monthly frequencies, the stock and the decile indices have in general the same values as in daily frequencies. On the measures of central tendency, measures of dispersion, and the measures of skewness and kurtosis, they have the same lowest or highest values. We can see on Table II that there is difference if we take the logarithmic returns or the arithmetic returns. The values are totally different.

Taking into account the skewness, the three stocks and the two decile indices are negatively skewed. On the other hand, kurtosis has large values. Beginning with NLY (17.38134) and MI (11.05521) and they are followed by DC10 (9.767838), ANF (8.844107) and DC1 (8.116648). Except from MI and NLY, all the other assets are approximately normal. We configure this by the Jarque - Bera test from which we can say that they reject to fail the null hypothesis of normality.

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Examples of our workTable I

Descriptive Statistics of Daily Returns

This table shows the descriptive statistics of the daily returns for 5 stocks and 2 deciles indices over the period January 2007 to December 2010. The lower case r denotes a logarithmic (log) return and the upper-case later R denotes an arithmetic (simple) return.

MI-MARSHALL & ILSLEY CORP. ANF- ABERCROMBIE & FITCH CO. NLY- ANNALY CAPITAL MANAGEMENT INC. DC1- NYSE/AMEX/NASDAQ Decile1. DC10- NYSE/AMEX/NASDAQ Decile10. Max stands for Maximum. Min. stands for Minimum. Std. Dev. Stands for Standard Deviation. JB STANDS FOR Jarque-Bera. Obs stands for Observations.

MI

ANF

NLY

DC1

DC10

## Â

r

R

r

R

r

R

r

R

r

Mean

-0.00194

-0.00042

-0.00021

0.000324

0.000244

0.000708

0.000409

0.000467

0.00000032

Median

-0.00121

-0.00121

0.000369

0.000369

0.001116

0.001117

0.001086

0.001087

0.000899

Max.

0.329341

0.390052

0.175582

0.19194

0.274437

0.315789

0.062526

0.064523

0.111735

Min.

-0.30146

-0.26026

-0.23221

-0.20722

-0.19842

-0.17998

-0.0561

-0.05455

-0.095184

Std. Dev.

0.055064

0.055315

0.032708

0.032558

0.030397

0.030676

0.010712

0.010711

0.017264

Skewness

-0.10445

0.72514

-0.42307

-0.05985

0.278683

1.064435

-0.14067

-0.0256

-0.242333

Kurtosis

11.05521

11.79593

8.844107

8.148605

17.38134

21.0768

8.116648

8.267992

9.767838

JB

2724.358

3334.5

1463.068

1112.838

8690.982

13900.92

1101.794

1164.527

1931.7

JB p-value

0

0

0

0

0

0

0

0

0

Obs

1007

1007

1007

1007

1007

1007

1007

1007

1007

Table II

Descriptive Statistics of Monthly Returns

This table shows the descriptive statistics of the monthly returns for 3 stocks and 2 deciles indices over the period January 2007 to December 2010. The lower case r denotes a logarithmic (log) return and the upper-case later R denotes an arithmetic (simple) return.

MI-MARSHALL & ILSLEY CORP. ANF- ABERCROMBIE & FITCH CO. NLY- ANNALY CAPITAL MANAGEMENT INC. DC1- NYSE/AMEX/NASDAQ Decile1. DC10- NYSE/AMEX/NASDAQ Decile10. Max stands for Maximum. Min. stands for Minimum. Std. Dev. Stands for Standard Deviation. JB STANDS FOR Jarque-Bera. Obs stands for Observations.

## MI

## ANF

## NLY

## DC1

## DC10

## Â

r

R

r

R

r

R

r

R

r

Mean

-0.04079

-0.02098

-0.00686

0.003159

0.005589

0.008179

0.010004

0.013051

-0.000445

Median

-0.00753

-0.0075

-0.00476

-0.00475

0.011141

0.011203

0.009105

0.009147

0.01316

Max.

0.367885

0.444676

0.225666

0.253158

0.122739

0.130589

0.175197

0.191481

0.09284

Min.

-0.87079

-0.58138

-0.40426

-0.33253

-0.30049

-0.25955

-0.16079

-0.14853

-0.19825

Std. Dev.

0.208479

0.184455

0.145081

0.139956

0.07361

0.07047

0.077933

0.078693

0.060218

Skewness

-1.3345

-0.24794

-0.58973

-0.25698

-1.56381

-1.12723

-0.12042

0.103217

-0.844232

Kurtosis

6.893326

4.041347

2.989927

2.502646

7.723667

5.906776

2.94561

2.95443

3.872671

JB

43.63478

2.605157

2.724526

1.001727

62.8528

26.50002

0.11938

0.08752

7.074418

JB p-value

0

0.27183

0.256081

0.606007

0

0.000002

0.942057

0.957184

0.029094

Obs

47

47

47

47

47

47

47

47

47

Table III

Autocorrelation and Partial Autocorrelation Tests Log Returns

This table show the autocorrelation and the partial autocorrelation coefficients up to 10 lags for the log-returns of each stock and each decile index. AC stands for autocorrelation. PAC stands for partial autocorrelation. MI-MARSHALL & ILSLEY CORP. ANF- ABERCROMBIE & FITCH CO. NLY- ANNALY CAPITAL MANAGEMENT INC DC1 NYSE / AMEX / NASDAQ Decile1. DC10 - NYSE / AMEX / NASDAQ Decile10

PANEL A: Results for Daily Log -Returns

## ANF

## MI

## NLY

## D1

## D10

Lag.

ACF

PAC

Q Stat.

ACF

PACF

Q Stat.

ACF

PACF

Q Stat.

ACF

PACF

Q Stat.

ACF

PACF

1

0.023

0.023

0.5239

-0.029

-0.029

0.8335

-0.251

-0.251

63.726

0.271

0.271

74.388

-0.111

-0.111

2

-0.032

-0.033

1.56

-0.026

-0.027

1.5077

-0.037

-0.107

65.111

0.217

0.154

121.86

-0.093

-0.107

3

0.064*

0.066

5.7251

-0.047

-0.049

3.7553

-0.036

-0.079

66.423

0.214

0.135

168.01

0.086

0.064

4

-0.027

-0.032

6.4883

0.03

0.026

4.6646

-0.047

-0.089

68.621

0.115

0.008

181.4

-0.033

-0.026

5

-0.017

-0.011

6.7799

-0.018

-0.019

4.9941

0.053*

0.009

71.479

0.071*

-0.011

186.56

-0.032

-0.025

6

-0.037

-0.043

8.1715

-0.094

-0.097

14.05

-0.03

-0.028

72.395

0.041

-0.018

188.27

0.022

0.004

7

-0.021

-0.017

8.639

-0.033

-0.037

15.137

-0.004

-0.023

72.412

0.047*

0.02

190.56

-0.021

-0.02

8

0.023

0.022

9.1587

0.041

0.032

16.861

0.024

0.015

73.019

-0.015

-0.044

190.78

0.018

0.02

9

0.072

0.074

14.384

-0.001

-0.009

16.862

-0.026

-0.018

73.715

0.03

0.035

191.7

0.006

0.003

10

-0.006

-0.008

14.415

0.097*

0.101

26.409

-0.011

-0.027

73.836

0.043

0.035

193.61

0.035

0.044

PANEL B: Results for Monthly Log -Returns

## ANF

## MI

## NLY

## D1

## D10

Lag.

AC

PAC

Q Stat.

ACF

PACF

Q Stat.

ACF

PACF

Q Stat.

ACF

PACF

Q Stat.

ACF

PACF

1

0.137

0.137

0.9345

0.026

0.026

0.0339

-0.102

-0.102

0.5188

0.372

0.372

6.9116

0.294

0.294

2

0.143

0.127

1.9814

-0.11

-0.111

0.6522

-0.191

-0.203

2.3788

0.182

0.051

8.6016

-0.037

-0.136

3

0.037

0.003

2.0546

-0.172

-0.168

2.205

0.131

0.091

3.2747

0.274

0.222

12.518

0.166

0.244

4

0.154

0.136

3.3306

-0.03

-0.037

2.2543

-0.052

-0.07

3.4185

0.216

0.056

15.021

0.281

0.166

5

-0.021

-0.064

3.3552

0.036

0

2.3246

-0.147

-0.124

4.6057

-0.052

-0.213

15.17

-0.061

-0.201

6

-0.154

-0.19

4.6798

-0.046

-0.086

2.443

-0.061

-0.133

4.8149

-0.241

-0.289

18.432

-0.323

-0.275

7

0.033

0.088

4.7431

0.017

0.01

2.459

0.142

0.091

5.9807

-0.079

0.04

18.793

-0.012

0.102

8

0.042

0.057

4.8475

0.137

0.134

3.5745

0.004

0.017

5.9819

0.105

0.282

19.447

0.072

-0.015

9

-0.097

-0.123

5.4216

-0.054

-0.079

3.751

-0.179

-0.149

7.9186

-0.201

-0.179

21.898

-0.187

-0.088

10

-0.184

-0.134

7.5348

-0.099

-0.074

4.3557

0.045

-0.045

8.0455

-0.138

0.007

23.076

-0.135

0.106

Table IV

Autocorrelation and Partial Autocorrelation Tests for Squared Log Returns

This table show the autocorrelation and the partial autocorrelation coefficients up to 10 lags for the log-returns of each stock and each decile index. AC stands for autocorrelation. PAC stands for partial autocorrelation. MI-MARSHALL & ILSLEY CORP. ANF- ABERCROMBIE & FITCH CO. NLY- ANNALY CAPITAL MANAGEMENT INC DC1 NYSE / AMEX / NASDAQ Decile1. DC10 - NYSE / AMEX / NASDAQ Decile10

PANEL A: Results for Daily Squared Log -Returns

## ANF

## MI

## NLY

## D1

## D10

Lag.

AC

PAC

Q Stat.

ACF

PACF

Q Stat.

ACF

PACF

Q Stat.

ACF

PACF

Q Stat.

ACF

PACF

1

0.153

0.153

23.599

0.188

0.188

35.555

0.364

0.364

134.1

0.247

0.247

61.744

0.186

0.186

2

0.144

0.124

44.707

0.312

0.287

133.74

0.293

0.184

220.72

0.281

0.235

141.84

0.393

0.371

3

0.158

0.125

69.991

0.148

0.06

156.04

0.304

0.179

314.25

0.17

0.066

171.03

0.16

0.054

4

0.137

0.088

88.886

0.126

0.012

172.1

0.264

0.099

384.75

0.097

-0.013

180.5

0.31

0.17

5

0.127

0.071

105.24

0.156

0.091

196.77

0.267

0.106

456.91

0.147

0.086

202.29

0.353

0.287

6

0.202

0.147

146.7

0.201

0.144

237.65

0.221

0.036

506.48

0.195

0.144

240.85

0.325

0.16

7

0.097

0.016

156.26

0.176

0.076

269.09

0.276

0.126

583.83

0.109

-0.003

252.83

0.333

0.136

8

0.125

0.053

172.19

0.123

-0.012

284.5

0.349

0.187

707.56

0.155

0.05

277.17

0.198

-0.011

9

0.131

0.054

189.71

0.163

0.066

311.69

0.307

0.094

803.71

0.123

0.044

292.58

0.308

0.089

10

0.2

0.131

230.34

0.111

0.031

324.13

0.251

0.015

867.78

0.149

0.073

315.29

0.278

0.101

PANEL B: Results for Monthly Squared Log- Returns

## ANF

## MI

## NLY

## D1

## D10

Lag.

AC

PAC

Q Stat.

ACF

PACF

Q Stat.

ACF

PACF

Q Stat.

ACF

PACF

Q Stat.

ACF

PACF

1

0.49

0.49

12.027

-0.017

-0.017

0.0142

0.008

0.008

0.0034

0.613

0.613

18.789

0.243

0.243

2

0.442

0.266

22.045

-0.026

-0.026

0.0482

0

0

0.0034

0.321

-0.087

24.066

-0.056

-0.122

3

0.183

-0.15

23.807

-0.051

-0.052

0.183

0.165

0.165

1.4352

0.307

0.236

28.992

0.061

0.114

4

0.129

-0.018

24.695

0.03

0.027

0.2306

-0.089

-0.094

1.8608

0.351

0.127

35.588

0.274

0.24

5

-0.023

-0.09

24.725

0.014

0.013

0.2418

-0.037

-0.035

1.9348

0.292

0.002

40.255

0.072

-0.055

6

-0.12

-0.136

25.528

-0.045

-0.046

0.3552

0.192

0.172

3.9983

0.343

0.242

46.87

0.084

0.141

7

-0.07

0.106

25.812

0.167

0.17

1.9597

-0.09

-0.073

4.4654

0.232

-0.21

49.963

-0.055

-0.158

8

-0.137

-0.067

26.92

-0.081

-0.081

2.3451

-0.079

-0.08

4.8331

0.01

-0.199

49.968

-0.123

-0.143

9

-0.148

-0.11

28.245

0.11

0.117

3.0744

0.068

0.016

5.1106

-0.164

-0.25

51.596

0.019

0.083

10

-0.194

-0.061

30.586

-0.105

-0.096

3.7649

-0.079

-0.026

5.4953

-0.057

0.097

51.801

-0.101

-0.255

In the case when autocorrelation of squared daily logarithmic returns and absolute value of daily logarithmic return is calculated the autocorrelation results and partial autocorrelation results in all cases are much stronger and statistically important in 5% level for the first 3 lags. For the 4th and 5th lags there is a mixed image between the assets as in ASX and RHT autocorrelation and partial autocorrelation effects are beginning to faint and they presenting statistically equal to zero while in indices deciles only partial autocorrelation is statistically zero in the last two lags. This evidence is being enhanced by the Lhung-Box test results show evidence of serial correlation in all series of our sample. Thus the Weak form efficiency market hypothesis is being rejected in this sample.

Table IV

Autocorrelation and Partial Autocorrelation Tests for the Absolute Value of the Log Returns

This table show the autocorrelation and the partial autocorrelation coefficients up to 10 lags for the log-returns of each stock and each decile index. AC stands for autocorrelation. PAC stands for partial autocorrelation. MI-MARSHALL & ILSLEY CORP. ANF- ABERCROMBIE & FITCH CO. NLY- ANNALY CAPITAL MANAGEMENT INC DC1 NYSE / AMEX / NASDAQ Decile1. DC10 - NYSE / AMEX / NASDAQ Decile10

PANEL A: Results for Daily Absolute Value of Log -Returns

## ANF

## MI

## NLY

## D1

## D10

Lag.

AC

PAC

Q Stat.

ACF

PACF

Q Stat.

ACF

PACF

Q Stat.

ACF

PACF

Q Stat.

ACF

PACF

1

0.162

0.162

26.512

0.315

0.315

100.31

0.115

0.115

0.6602

0.304

0.304

93.342

0.236

0.236

2

0.19

0.168

62.829

0.339

0.266

216.29

0.033

0.02

0.716

0.31

0.239

190.29

0.36

0.322

3

0.2

0.156

103.5

0.267

0.126

288.54

0.316

0.314

5.9393

0.28

0.159

269.76

0.286

0.18

4

0.184

0.118

137.76

0.277

0.127

366.5

-0.075

-0.162

6.2404

0.226

0.073

321.48

0.329

0.185

5

0.191

0.112

174.76

0.272

0.113

441.36

0.022

0.054

6.2658

0.267

0.128

393.63

0.404

0.267

6

0.185

0.094

209.4

0.329

0.171

551.16

0.271

0.185

10.401

0.274

0.124

469.71

0.354

0.178

7

0.16

0.059

235.48

0.304

0.113

645.13

-0.113

-0.126

11.133

0.237

0.06

526.85

0.387

0.181

8

0.194

0.093

273.9

0.261

0.037

714.55

-0.084

-0.096

11.553

0.237

0.057

584.2

0.29

0.051

9

0.178

0.069

306.03

0.292

0.092

801.48

0.181

0.105

13.547

0.217

0.038

632.03

0.317

0.043

10

0.184

0.072

340.71

0.237

0.018

858.91

-0.109

-0.042

14.284

0.249

0.085

695.11

0.306

0.029

PANEL B: Results for Monthly Absolute Value of Log -Returns

## ANF

## MI

## NLY

## D1

## D10

Lag.

AC

PAC

Q Stat.

ACF

PACF

Q Stat.

ACF

PACF

Q Stat.

ACF

PACF

Q Stat.

ACF

PACF

1

0.489

0.489

11.966

-0.017

-0.017

0.0142

0.115

0.115

0.6602

0.552

0.552

15.262

0.194

0.194

2

0.447

0.274

22.212

-0.026

-0.026

0.0482

0.033

0.02

0.716

0.379

0.106

22.604

-0.009

-0.049

3

0.316

0.033

27.442

-0.051

-0.052

0.183

0.316

0.314

5.9393

0.375

0.188

29.959

0.09

0.106

4

0.284

0.059

31.773

0.03

0.027

0.2306

-0.075

-0.162

6.2404

0.446

0.241

40.624

0.326

0.302

5

0.059

-0.217

31.961

0.014

0.013

0.2418

0.022

0.054

6.2658

0.287

-0.11

45.142

0.123

0.013

6

0.011

-0.083

31.968

-0.045

-0.046

0.3552

0.271

0.185

10.401

0.313

0.158

50.633

0.1

0.106

7

0.043

0.119

32.074

0.167

0.17

1.9597

-0.113

-0.126

11.133

0.263

-0.05

54.617

-0.057

-0.147

8

-0.125

-0.187

33.003

-0.081

-0.081

2.3451

-0.084

-0.096

11.553

0.11

-0.214

55.33

-0.15

-0.253

9

-0.166

-0.08

34.672

0.11

0.117

3.0744

0.181

0.105

13.547

-0.094

-0.263

55.86

-0.003

-0.002

10

-0.193

-0.041

36.994

-0.105

-0.096

3.7649

-0.109

-0.042

14.284

0.038

0.105

55.951

-0.041

-0.144

TABLE V

Correlation Matrix Table

The table shows the correlation between assets for a four year period (1st January 2007 to 31st December 2010). MI-MARSHALL & ILSLEY CORP. ANF- ABERCROMBIE & FITCH CO. NLY- ANNALY CAPITAL MANAGEMENT INC DC1 NYSE / AMEX / NASDAQ Decile1. DC10 - NYSE / AMEX / NASDAQ Decile10

PANEL A:Results for Daily Log Returns

ANF

MI

NLY

DC10

ANF

1

MI

0.394939

1

NLY

0.400766

0.478346

1

DC10

0.631216

0.621608

0.584086

1

DC1

0.382347

0.399492

0.218922

0.585987

PANEL B:Results for Monthly Log Returns

ANF

NLY

MI

DC1

ANF

1

NLY

0.187049

1

MI

0.330498

0.12031

1

DC1

0.457612

0.203086

0.243027

1

DC10

0.700746

0.2563

0.465772

0.741112

The correlation between the assets indicates a positive, semi-strong correlation among them. The correlation coefficients are varying between 0.35 and 0.67 with N/A/M 1 and N/A/M 1 having the strongest dependence in their returns and ASX and RHT the weakest. In general assets returns turn to move towards the same direction as in the case of stocks they all belong in the same sector and are being affected from the movements of the market which are being captured by the indices deciles.

TABLE VI

Day- of- the week Regression

The table shows the day of the week regression results of the daily logarithmic returns for a four Year period (1st January 2007 to 31st December 2010). MI-MARSHALL & ILSLEY CORP. ANF- ABERCROMBIE & FITCH CO. NLY- ANNALY CAPITAL MANAGEMENT INC DC1 NYSE / AMEX / NASDAQ Decile1. DC10 - NYSE / AMEX / NASDAQ Decile10

## ANF

## MI

## NLY

## D1

## D10

## Â

Coefficient

Prob.

Coefficient

Prob.

Coefficient

Prob.

Coefficient

Prob.

Coefficient

Monday

-0.002027

0.3673

-0.007186

0.0885

-0.001404

0.4865

-0.000137

0.873

-0.000495

0.002248

0.004214

0.002017

0.000858

0.001431

Tuesday

0.000323

0.8751

-0.001972

0.6272

0.003621

0.1294

0.00003780

0.9601

0.00059

0.002052

0.004058

0.002386

0.000755

0.001255

Wednesday

0.001039

0.5956

-0.000902

0.7852

-0.002854

0.1235

0.000537

0.4499

0.000269

0.001958

0.003309

0.001851

0.000711

0.001204

Thursday

0.001309

0.6418

-0.003096

0.4341

-0.001788

0.4921

0.000146

0.8594

-0.000272

0.002814

0.003957

0.002602

0.000825

0.001184

Friday

-0.001848

0.4339

0.003215

0.3895

0.003612

0.0379

0.001448

0.0392

-0.00014

0.002361

0.003734

0.001738

0.000701

0.000943

R Squared

0.001875

0.003628

0.008565

0.002761

0.000509

Adjusted R Squared

-0.00211

-0.00035

0.004608

-0.00122

-0.003481

TABLE VI

The Results of the Variance Ratio Test (excluding outliers)

The table shows the Variance ratio test results of the daily logarithmic returns for a four Year period (1st January 2007 to 31st December 2010). MI-MARSHALL & ILSLEY CORP. ANF- ABERCROMBIE & FITCH CO. NLY- ANNALY CAPITAL MANAGEMENT INC DC1 NYSE / AMEX / NASDAQ Decile1. DC10 - NYSE / AMEX / NASDAQ Decile10

PANEL A: Results for daily Log - Returns

Number of Observations

Lag Orders

## Â

2

4

8

16

ANF

1006

0.5291

0.2644

0.1267

0.0649

-14.937

0

-12.4718

0

-9.3643

0

-6.7379

-8.4812

0

-7.5364

0

-6.0992

0

-4.5901

MI

1006

0.4996

0.2371

0.1181

0.0627

-15.8724

0

-12.9337

0

-9.4562

0

-6.7536

-7.0907

0

-6.1388

0

-4.9198

0

-3.8135

NLY

1006

0.415

0.2103

0.0987

0.0506

-18.5536

0

-13.3889

0

-9.6638

0

-6.8408

-6.2083

0

-5.1192

0

-4.2436

0.00002

-3.2924

D1

1006

0.5386

0.3053

0.1763

0.0849

-14.6347

0

-11.7781

0

-8.8322

0

-6.5936

-9.0642

0

-7.6095

0

-5.9124

0

-4.5532

D10

1006

0.493

0.2338

0.112

0.0526

-16.0795

0

-12.9908

0

-9.5215

0

-6.8268

## Â

-7.6834

0

-6.5415

0

-5.0433

0

-3.6543

TABLE VII

The Results of the Variance Ratio Test

The table shows the Variance ratio test results of the daily logarithmic returns for a four Year period (1st January 2007 to 31st December 2010). MI-MARSHALL & ILSLEY CORP. ANF- ABERCROMBIE & FITCH CO. NLY- ANNALY CAPITAL MANAGEMENT INC DC1 NYSE / AMEX / NASDAQ Decile1. DC10 - NYSE / AMEX / NASDAQ Decile10

Number of Observations

Lag Orders

2

4

8

16

ANF

956

1.3652

1.9168

3.1072

5.0306

11.2906

0

15.1519

0

22.026

0

28.3123

4.591

0

6.6971

0

10.0172

0

12.4287

MI

956

1.302

1.8725

2.7829

4.2356

9.3372

0

14.4203

0

18.6358

0

22.7283

3.713

0.0002

5.2844

0

7.1512

0

9.3274

NLY

956

1.3616

1.9955

2.9451

4.8223

11.1797

0

16.4532

0

20.3315

0

26.8491

4.8072

0

7.2772

0

9.3297

0

12.2183

D1

956

1.3579

1.9185

2.6447

4.3758

11.0663

0

15.1796

0

17.1915

0

23.7132

3.1444

0.00166

4.3912

0.00001

5.4345

0

8.7244

D10

956

1.4544

2.1468

3.1658

5.1018

14.0512

0

18.9527

0

22.6387

0

28.8128

5.1171

0

6.8725

0

8.886

0

12.1863

TABLE VIII

The Table of the Garch model

## Â

## MI

## ANF

## NLY

## D1

## D10

Î´1

-0.001592

-0.000616

0.00424

0.000118

0.000599

0.002498

0.5239

0.001482

0.6775

0.001242

0.0006

0.000563

0.8337

0.000808

Î´2

0.002670

-0.000085

-0.00481

-0.00053

-0.00066

0.002662

0.3158

0.002339

0.971

0.001664

0.0038

0.000807

0.5087

0.001228

Î´3

-0.001734

0.002536

-0.00255

0.000914

0.001127

0.003114

0.5777

0.00199

0.2024

0.001491

0.0876

0.000609

0.1338

0.001184

Î´4

0.000127

0.005807

-0.0058

-0.00019

-0.00073

0.003317

0.9695

0.002759

0.0353

0.001988

0.0035

0.000749

0.7999

0.001114

Î´5

0.002220

-0.001027

-0.00335

0.001329

-0.00037

0.003169

0.4836

0.002365

0.6643

0.001498

0.0251

0.000739

0.072

0.0011

Î³1

0.000880

-0.000098

0.000029

0.000039

0.000066

0.000012

0

0.000051

0.0558

0.000024

0.2184

0.000002

0

0.000003

Î³2

-0.001098

0.000140

0.000134

-0.000043

-0.000068

0.000083

0

0.000068

0.0398

0.000070

0.0541

0.000008

0

0.000032

Î³3

-0.000872

0.000034

-0.000121

-0.000049

-0.000096

0.000018

0

0.000124

0.783

0.000056

0.0312

0.000007

0

0.000020

Î³4

-0.000850

0.000417

0.000047

-0.000031

-0.000056

0.000067

0

0.000131

0.0015

0.000046

0.3098

0.000005

0

0.000009

Î³5

-0.000803

-0.000085

-0.000104

-0.000038

-0.000063

0.000102

0

0.000118

0.4704

0.000038

0.0067

0.000005

0

0.000011

Î±1

0.294763

0.049335

0.241057

0.247822

0.118943

0.091634

0.0013

0.012352

0.0001

0.086798

0.0055

0.051893

0

0.025517

Î²1

0.718643

0.946772

0.760615

0.698386

0.844913

0.064228

0

0.012854

0

0.062008

0

0.047951

0

0.030823

TABLE IX

Table of the Egarch Model

## Â

## MI

## ANF

## NLY

## D1

## D10

Î´1

-0.00061

-0.00106

0.002278

0.000184

0.000713

0.001618

0.7054

0.001487

0.4779

0.001125

0.0428

0.000566

0.7449

0.000797

Î´2

0.000829

0.000481

-0.00296

-0.00075

-0.00175

0.003325

0.803

0.002222

0.8287

0.002021

0.1428

0.000717

0.2988

0.001178

Î´3

-0.00087

0.00233

-0.00236

0.000629

0.000843

0.002202

0.6939

0.001992

0.2422

0.00155

0.1282

0.000715

0.3787

0.001039

Î´4

-0.00179

0.005382

-0.00521

-0.00049

-0.00074

0.002463

0.4685

0.002437

0.0272

0.001864

0.0052

0.000665

0.4635

0.001082

Î´5

-0.00104

0.000449

-0.00121

0.001284

-0.00074

0.002311

0.6543

0.002299

0.8451

0.00146

0.4067

0.000748

0.0861

0.001057

w1

-0.45491

-0.10447

-0.64022

-0.42291

-0.30839

0.255134

0.0746

0.099794

0.2952

0.297218

0.0312

0.121026

0.0005

0.094188

w2

0.316505

0.002048

0.414284

0.368262

0.099563

0.102343

0.002

0.013394

0.8785

0.079915

0

0.038549

0

0.024333

w3

-0.06217

-0.07096

-0.08823

-0.01211

-0.15092

0.049522

0.2093

0.010691

0

0.065372

0.1771

0.01609

0.4517

0.01976

w4

0.979579

0.999218

0.959608

0.964866

0.980731

0.010713

0

0.001409

0

0.018725

0

0.008323

0

0.00376

w5

0.38086

-0.00614

0.673158

-0.19826

0.346997

0.224362

0.0896

0.187484

0.9739

0.354439

0.0575

0.155508

0.2023

0.162194

Î±

-0.17668

0.049627

-0.63799

-0.36042

-0.28028

0.252313

0.4838

0.165931

0.7649

0.21631

0.0032

0.134325

0.0073

0.166646

Î²

0.098961

0.46383

0.515914

-0.14214

0.251806

0.22034

0.6533

0.12404

0.0002

0.315153

0.1016

0.1345

0.2906

0.137409

Î³

0.1827

-0.03576

-0.44443

-0.22915

-0.00772

0.402244

0.6497

0.173104

0.8364

0.289316

0.1245

0.146435

0.1176

0.141933

## SECTION IV RESULTS

## AUTOCORRELATION TESTS

## A.1 TESTS FOR LOG-RETURNS

Table III presents the autocorrelation analysis results for logarithmic returns for 3 stocks and 2 decile indices. We have selected 10 lags for the AC and the PAC. From the results we can notice that autocorrelation is in the first lag for NLY, DC1 and DC10, in the third lag of ANF and in the sixth and tenth lags for MI. We can observe similar results for partial autocorrelation test. Overall we can conclude that Weak form efficiency is being rejected for all the data we have.

## A.2 TESTS FOR SQUARED LOG-RETURNS

Table VI presents the results of the autocorrelation analysis of squared log returns.