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Question:
Describe the Collecting, manipulating and preparing data for statistical inference

Introduction:

The report discusses about the volatility and risk of stock market, return and market return. Usually, it is observed that stock market is volatile and unsteady (Berenson et al. 2012). Therefore, it is hard for evaluating the market performance. The report focuses to indicate the method of evaluating the company’s Price indexes through its market price movements. The price indexes for Boeing and International Business Machines (IBM) has been chosen for demonstrating (Freed, Bergquist and Jones 2014). However, the historical price movements of the individual price indexes from 1st December 2010 and 31st May 2016 cannot depict the appropriate outputs. Therefore, the historical index prices of S&P500 index and the 10 years’ US Treasury Bill are involved in the evaluation method for computing the effective outcomes. S&P500 index represents the summarisation of the market and the 10 years’ US Treasury Bill refers the risk-free return of the market.

The method of price index evaluation is divided in some parts. They are the compared and their market returns are calculated in the first part. In the next part, hypotheses are tested and CAPM is calculated using linear regression model. The whole evaluation is incorporated based on Capital Asset Pricing Model. 

 
Discussion and Data Analysis:
Line Charts of Close prices:

The movement of the price indexes for a defined time-period depicts trends of price indexes. The first line chart involves all the three types of trend lines of price indexes. The second, third and fourth line charts involve the line charts individually. The price indexes of IBM and BA in the line charts are shown below: 

It could be inferred from the above line charts that the price indexes of both S&P500 and BA have increased from 01/12/2010 to 31/05/2016. The IBM price index has increased and then decreased within this period. It has better stationary trend in case of IBM price index than BA price index.

Calculation with return prices:
Calculation of returns:

 

Price Indexes

 

 

 

Price Returns

 

Date

S& P 500 price

Boeing (BA) price

IBM price

T-Bill price

S&P 500 price return

Boeing (BA) price return

IBM Price return

12/1/2010

1257.64

65.26

146.76

3.305

S&P 500 price return

Boeing (BA) price return

IBM Price return

1/1/2011

1286.12

69.48

162

3.378

2.239296944

6.265966274

9.879776981

2/1/2011

1327.22

72.01

161.88

3.414

3.145657708

3.576604148

-0.074098434

3/1/2011

1325.83

73.93

163.07

3.454

-0.104786202

2.631367353

0.73242485

4/1/2011

1363.61

79.78

170.58

3.296

2.809693855

7.615413437

4.502480722

5/1/2011

1345.2

78.03

168.93

3.05

-1.359291933

-2.217947863

-0.972002012

6/1/2011

1320.64

73.93

171.55

3.158

-1.842618552

-5.397465574

1.539040029

7/1/2011

1292.28

70.47

181.85

2.805

-2.170835635

-4.793159804

5.830741764

8/1/2011

1218.89

66.86

171.91

2.218

-5.846750175

-5.258620558

-5.621109711

9/1/2011

1131.42

60.51

174.87

1.924

-7.446710096

-9.979228837

1.707170481

10/1/2011

1253.3

65.79

184.63

2.175

10.23065919

8.365925872

5.431103736

11/1/2011

1246.96

68.69

188

2.068

-0.507155381

4.313579104

1.808811334

12/1/2011

1257.6

73.35

183.88

1.871

0.849656569

6.563881998

-2.21585647

1/1/2012

1312.41

74.18

192.6

1.799

4.2660044

1.125209474

4.633213239

2/1/2012

1365.68

74.95

196.73

1.977

3.978734502

1.032661246

2.121667944

3/1/2012

1408.47

74.37

208.65

2.216

3.085147563

-0.776850947

5.882596833

4/1/2012

1397.91

76.8

207.08

1.915

-0.752569988

3.215200416

-0.755297659

5/1/2012

1310.33

69.61

192.9

1.581

-6.469930807

-9.829642967

-7.093331288

6/1/2012

1362.16

74.3

195.58

1.659

3.879271978

6.520274255

1.37976243

7/1/2012

1379.32

73.91

195.98

1.492

1.251888486

-0.526280118

0.204307969

8/1/2012

1406.58

71.4

194.85

1.562

1.957060987

-3.455029356

-0.578253022

9/1/2012

1440.67

69.6

207.45

1.637

2.394711887

-2.553335875

6.266026974

10/1/2012

1412.16

70.44

194.53

1.686

-1.998784252

1.199677342

-6.430394465

11/1/2012

1416.18

74.28

190.07

1.606

0.284267279

5.3080411

-2.319392548

12/1/2012

1426.19

75.36

191.55

1.756

0.704336761

1.443492052

0.775642462

1/1/2013

1498.11

73.87

203.07

1.985

4.919779205

-1.996981139

5.840189485

2/1/2013

1514.68

76.9

200.83

1.888

1.099992755

4.019907037

-1.109199265

3/1/2013

1569.19

85.85

213.3

1.852

3.535529388

11.00956615

6.024085053

4/1/2013

1597.57

91.41

202.54

1.675

1.792416591

6.275336138

-5.17622762

5/1/2013

1630.74

99.02

208.02

2.164

2.055020242

7.996689421

2.669688481

6/1/2013

1606.28

102.44

191.11

2.478

-1.511292881

3.395546122

-8.478506442

7/1/2013

1685.73

105.1

195.04

2.593

4.827772796

2.5634978

2.035544611

8/1/2013

1632.97

103.92

182.27

2.749

-3.179826758

-1.129090574

-6.771550366

9/1/2013

1681.55

117.5

185.18

2.615

2.931559129

12.28169805

1.583916104

10/1/2013

1756.54

130.5

179.21

2.542

4.362997098

10.49348932

-3.27699456

11/1/2013

1805.81

134.25

179.68

2.741

2.766329003

2.833050663

0.261911042

12/1/2013

1848.36

136.49

187.57

3.026

2.328947407

1.654765511

4.297469436

1/1/2014

1782.59

125.26

176.68

2.668

-3.623140749

-8.585979544

-5.981199928

2/1/2014

1859.45

128.92

185.17

2.658

4.221337488

2.880044837

4.693416414

3/1/2014

1872.34

125.49

192.49

2.723

0.690824862

-2.696598325

3.876991905

4/1/2014

1883.95

129.02

196.47

2.648

0.618164318

2.774140392

2.046552269

5/1/2014

1923.57

135.25

184.36

2.457

2.081219595

4.715745562

-6.361937971

6/1/2014

1960.23

127.23

181.27

2.516

1.887899662

-6.112842199

-1.690271874

7/1/2014

1930.67

120.48

191.67

2.556

-1.519468737

-5.451270678

5.578747831

8/1/2014

2003.37

126.8

192.3

2.343

3.696364432

5.112727671

0.328153535

9/1/2014

1972.29

127.38

189.83

2.508

-1.563543608

0.456365573

-1.292772286

10/1/2014

2018.05

124.91

164.4

2.335

2.293639895

-1.958121139

-14.38264992

11/1/2014

2067.56

134.36

162.17

2.194

2.423747367

7.292926219

-1.365729073

12/1/2014

2058.9

129.98

160.44

2.17

-0.419738458

-3.314221049

-1.072510203

1/1/2015

1994.99

145.37

153.31

1.675

-3.153277922

11.19016204

-4.545805109

2/1/2015

2104.5

150.85

161.94

2.002

5.343887644

3.700382334

5.476390633

3/1/2015

2067.89

150.08

160.5

1.934

-1.754919723

-0.51175068

-0.893196604

4/1/2015

2085.51

143.34

171.29

2.046

0.848472245

-4.594909592

6.506404307

5/1/2015

2107.39

140.52

169.65

2.095

1.043672976

-1.98695468

-0.962052978

6/1/2015

2063.11

138.72

162.66

2.335

-2.123555928

-1.289233562

-4.207529853

7/1/2015

2103.84

144.17

161.99

2.205

1.954968325

3.853562471

-0.412752153

8/1/2015

1972.18

130.68

147.89

2.2

-6.46247309

-9.824161178

-9.106588838

9/1/2015

1920.03

130.95

144.97

2.06

-2.679873126

0.206401488

-1.994191606

10/1/2015

2079.36

148.07

140.08

2.151

7.97193795

12.28696342

-3.431312911

11/1/2015

2080.41

145.45

139.42

2.218

0.050474186

-1.785281788

-0.472275654

12/1/2015

2043.94

144.59

137.62

2.269

-1.768565854

-0.593024094

-1.299471827

1/1/2016

1940.24

120.13

124.79

1.931

-5.206761723

-18.53276586

-9.786389491

2/1/2016

1932.23

118.18

131.03

1.74

-0.413690564

-1.636557884

4.879396445

3/1/2016

2059.74

126.94

151.45

1.786

6.390499042

7.150566372

14.48292207

4/1/2016

2065.3

134.8

145.94

1.819

0.269576165

6.00776711

-3.705992562

5/1/2016

2096.95

126.15

153.74

1.834

1.520836665

-6.632052979

5.20673044

Summary Statistics:

Boeing (BA) Price return

IBM Price return

 

 

 

 

 

Mean

1.0139883

Mean

0.071483586

Standard Error

0.7426766

Standard Error

0.626657713

Median

1.1996773

Median

-0.074098434

Standard Deviation

5.9876504

Standard Deviation

5.052276003

Sample Variance

35.851957

Sample Variance

25.52549281

Kurtosis

0.6987056

Kurtosis

0.655346526

Skewness

-0.4699036

Skewness

-0.136800306

Range

30.819729

Range

28.86557199

Minimum

-18.532766

Minimum

-14.38264992

Maximum

12.286963

Maximum

14.48292207

Sum

65.909237

Sum

4.646433103

Count

65

Count

65

Confidence Level (95.0%)

1.4836671

Confidence Level (95.0%)

1.251892683

The average return of Boeing (BA) is greater than average returns of IBM (1.0139883>0.071483586). The risk is determined by standard deviation of returns of close rates of price index. The risk in terms of standard deviation shows that Boeing return is more volatile than IBM return (5.9876504>5.052276003).

The risk is relatively greater for Boeing price return for its greater variability in terms of standard deviation.    

Jarque - Bera test of normality:

Jerque-Bera test is carried out for testing the normality of price indexes that are Boeing and IBM.

The Jerque-Bera test statistic (JB) is given as-

JB = n *

Jarque-Bera test

 

 

 

 

 

 

 

 

Skewness

Kurtosis

n

JB

α

χ2 (0.05,2)

Decision

Boeing (BA)

-0.469903619

0.698706

65

16.7353157

0.05

5.991464547

Normality is Rejected

IBM

-0.136800306

0.655347

65

15.0915299

0.05

5.991464547

Normality is Rejected

Firstly, the JB test statistics of both the price indexes are calculated. For BA price return and IBM price return, they are 16.7353157 and 15.0915299. Then applying significant test statistic, we have tested Chi-square tests at 5% level of significance (χ2 (0.05, 2) = 5.99). For both one and two-tail Chi-square tests, Boeing and IBM price returns failed to attain normality. Hence, none of the price returns is normally distributed at 95% confidence limit.

Testing of average return price of Boeing (BA):

One sample t-test

Boeing Close return (BA)

 

 

Average (X-bar) =

1.01398826

hypothetical mean (μ) =

3%

(X-bar - μ) =

0.98398826

Standard deviation =

5.987650369

sample size (n) =

65

degrees of freedom=

64

Standard error =

0.742676624

t-statistic =

1.324921544

T(critical) =

1.997729633

Decision making =

Null hypothesis rejected

A one-sample t-test determines whether the average price return of Boeing Close return (BA) is at least 3%.   The t-statistic is - . The t-statistic is 1.324921544. At 5% level of significance, we reject the null hypothesis of average price return greater than or equal to 0.03 as T0.05 < Tcric.

Therefore, the average price return of Boeing is not at least 3%.

Comparison of risk associated to each of the BA and IBM price returns:

 

Boeing (BA) return

IBM return

Variance

35.85195694

25.52549281

Degrees of freedom

64

64

F-statistic

1.404554937

 

p-value of F-statistic

0.088449703

 

level of significance

0.05

 

decision making

Null hypothesis accepted

 

The riskiness of returns of two price returns could be more effectively compared by F-test of two samples variances. The F-test for comparing the riskiness of the price returns of IBM and GE are conducted here.

Hypotheses:

Null hypothesis (H0): σ12 = σ22

Alternative hypothesis (HA): σ12 ≠ σ22

The F value for two-tail test is computed as F = F1-α/2, N1-1, N2-1

Here, α=0.05, N1-1=64 and N2-1=64.

The risk associated with each of the two price returns is compared with the help of F-statistic. The calculated F-statistics (F = is 1.404554937.

For Boeing and IBM price returns, p-value of the F-statistic is 0.088449703. It is greater than 0.05. The null hypothesis is accepted at 5% level of significance.

Hence, it could be depicted that level of volatility of the two price returns for the given period are almost equal to each other (Groebner et al. 2008).

Comparison of average returns of each of the two investing price returns:

The average return is indicated by the mean of returns of the price returns. Hence, for comparing the average return of Boeing (BA) and IBM price returns, two sample z-test (for unequal samples) and two sample t-test (for equal samples) can be conducted on the calculated returns of the two price returns.

Hypotheses:

Null hypothesis (H0): μBA = μIBM

      Alternative hypothesis (HA): μBA ≠ μIBM

The z-statistic is given as z and t-statistic is given as .

Z-test of equality of means of two samples:

z-Test: Two Sample for Means

 

 

 

Boeing (BA) returns

IBM returns

Mean

1.01398826

0.071483586

Known Variance

35.8519

25.5254

Observations

65

65

Hypothesized Mean Difference

0

 

z

0.969920863

 

P(Z<=z) one-tail

0.16604297

 

z Critical one-tail

1.644853627

 

P(Z<=z) two-tail

0.33208594

 

z Critical two-tail

1.959963985

 

decision making

Null hypothesis accepted

 

For comparing the average returns of each of the two investing price returns, a z-test is applied. The variances are known for each of the price returns. The calculated z-statistic is 0.9699. The p-value for two-tail z-statistic is 0.332 (>0.05). Therefore, we can reject the null hypothesis of equality of averages of returns of two price returns at 5% level of significance.

Two-sample t-test of equality of means for unequal variances:

t-Test: Two-Sample Assuming Unequal Variances

 

 

 

Boeing (BA) return

IBM return

Mean

1.01398826

0.07148359

Variance

35.85195694

25.5254928

Observations

65

65

Hypothesized Mean Difference

0

 

df

124

 

t Stat

0.96991968

 

P(T<=t) one-tail

0.166987311

 

t Critical one-tail

1.657234971

 

P(T<=t) two-tail

0.333974621

 

t Critical two-tail

1.979280091

 

decision making

Null hypothesis accepted

 

The t-test assuming equal variances of BA and IBM price returns gives the t-statistic 0.96991968. The p-value of the two-tail t-test is found to be 0.333974621. The level of significance is 5%, which is lesser than calculated p-value. Therefore, we cannot reject the null hypothesis of equality of averages of both the price returns.

Inference:

According to the price return averages and price return standard deviations (risk), an equality is established. Hence, we cannot draw firm decision to choose any one price returns between BA and IBM. Hence, we further proceed with both of them. Next, we are willing to excess price return, excess market return and CAPM of both the price returns. With the help of these, we can find the volatility of both the price returns. The preferable price return would be distinguished after that.

Calculation of Excess Return and Excess Return:

Excess Return

Excess Return

Excess Market Return

Boeing Excess return (BA)

IBM Excess return

 

BA

IBM

 

ytBA

ytIBM

xt

2.887966274

6.501776981

-1.138703056

0.162604148

-3.488098434

-0.268342292

-0.822632647

-2.72157515

-3.558786202

4.319413437

1.206480722

-0.486306145

-5.267947863

-4.022002012

-4.409291933

-8.555465574

-1.618959971

-5.000618552

-7.598159804

3.025741764

-4.975835635

-7.476620558

-7.839109711

-8.064750175

-11.90322884

-0.216829519

-9.370710096

6.190925872

3.256103736

8.055659186

2.245579104

-0.259188666

-2.575155381

4.692881998

-4.08685647

-1.021343431

-0.673790526

2.834213239

2.4670044

-0.944338754

0.144667944

2.001734502

-2.992850947

3.666596833

0.869147563

1.300200416

-2.670297659

-2.667569988

-11.41064297

-8.674331288

-8.050930807

4.861274255

-0.27923757

2.220271978

-2.018280118

-1.287692031

-0.240111514

-5.017029356

-2.140253022

0.395060987

-4.190335875

4.629026974

0.757711887

-0.486322658

-8.116394465

-3.684784252

3.7020411

-3.925392548

-1.321732721

-0.312507948

-0.980357538

-1.051663239

-3.981981139

3.855189485

2.934779205

2.131907037

-2.997199265

-0.788007245

9.157566153

4.172085053

1.683529388

4.600336138

-6.85122762

0.117416591

5.832689421

0.505688481

-0.108979758

0.917546122

-10.95650644

-3.989292881

-0.0295022

-0.557455389

2.234772796

-3.878090574

-9.520550366

-5.928826758

9.666698047

-1.031083896

0.316559129

7.951489318

-5.81899456

1.820997098

0.092050663

-2.479088958

0.025329003

-1.371234489

1.271469436

-0.697052593

-11.25397954

-8.649199928

-6.291140749

0.222044837

2.035416414

1.563337488

-5.419598325

1.153991905

-2.032175138

0.126140392

-0.601447731

-2.029835682

2.258745562

-8.818937971

-0.375780405

-8.628842199

-4.206271874

-0.628100338

-8.007270678

3.022747831

-4.075468737

2.769727671

-2.014846465

1.353364432

-2.051634427

-3.800772286

-4.071543608

-4.293121139

-16.71764992

-0.041360105

5.098926219

-3.559729073

0.229747367

-5.484221049

-3.242510203

-2.589738458

9.515162035

-6.220805109

-4.828277922

1.698382334

3.474390633

3.341887644

-2.44575068

-2.827196604

-3.688919723

-6.640909592

4.460404307

-1.197527755

-4.08195468

-3.057052978

-1.051327024

-3.624233562

-6.542529853

-4.458555928

1.648562471

-2.617752153

-0.250031675

-12.02416118

-11.30658884

-8.66247309

-1.853598512

-4.054191606

-4.739873126

10.13596342

-5.582312911

5.82093795

-4.003281788

-2.690275654

-2.167525814

-2.862024094

-3.568471827

-4.037565854

-20.46376586

-11.71738949

-7.137761723

-3.376557884

3.139396445

-2.153690564

5.364566372

12.69692207

4.604499042

4.18876711

-5.524992562

-1.549423835

-8.466052979

3.37273044

-0.313163335

CAPM calculation by linear regression method:

The Capital Asset Pricing Model (CAPM) is known as CAPM, which is one of the fundamental models in the financial field. The CAPM elaborates variability in the rate of return (rt) as a function of the rate of return on a market portfolio (rM,t) consisting all publicly traded price returns. Usually, the rate of return of any price return can be measured using opportunity cost that is the return on a risk free asset (rf,t). The difference between the return and risk free rate is known as “risk premium” as it is the reward or punishment for performing a risky investment (Peirson et al. 2014). In accordance to CAPM, the risk premium on a security (rt –rf,t) is proportional to the risk premium on the market portfolio (rM,t – rf,t). According to CAPM,

(rt –rf,t) = βM*(rM,t – rf,t)   ……………….(1)

Equation (1) is called economic model as it describes association between excess price returns and excess market return.

The CAPM beta is crucial from the viewpoints of investors as it discloses the volatility of market price returns. Particularly, the bête (slope) measures the sensitivity of variation of given return of security in the whole price market. Value of beta defines whether the price return is a defensive, a neutral price index or an aggressive price index. Including an intercept (β0) and an error term (ut) in the model, we have a simple linear regression model –

(rt - rf,t) = β+ βM (rM,t - rf,t) +ut    ………………..(2)

 
Estimation of CAPM using linear regression:

Boeing (BA) Excess return:

SUMMARY OUTPUT

 

 

 

 

 

 

 

 

 

 

 

 

Regression Statistics

 

 

 

 

 

Multiple R

0.63626059

 

 

 

 

 

R Square

0.40482754

 

 

 

 

 

Adjusted R Square

0.39538036

 

 

 

 

 

Standard Error

4.65767923

 

 

 

 

 

Observations

65

 

 

 

 

 

 

 

 

 

 

 

 

ANOVA

 

 

 

 

 

 

 

df

SS

MS

F

Significance F

 

Regression

1

929.6231383

929.6231

42.85167

1.22694E-08

 

Residual

63

1366.720475

21.69398

 

 

 

Total

64

2296.343613

 

 

 

 

 

 

 

 

 

 

 

 

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

0.4017602

0.629404328

0.638318

0.52558

-0.856003975

1.659524372

xt

1.11931665

0.170989356

6.546119

1.23E-08

0.777621689

1.461011607

IBM Excess return: 

SUMMARY OUTPUT

 

 

 

 

 

 

 

 

 

 

 

 

Regression Statistics

 

 

 

 

 

Multiple R

0.487837632

 

 

 

 

 

R Square

0.237985555

 

 

 

 

 

Adjusted R Square

0.225890087

 

 

 

 

 

Standard Error

4.424020742

 

 

 

 

 

Observations

65

 

 

 

 

 

 

 

 

 

 

 

 

ANOVA

 

 

 

 

 

 

 

df

SS

MS

F

Significance F

 

Regression

1

385.0900093

385.09

19.6756

3.757E-05

 

Residual

63

1233.03345

19.57196

 

 

 

Total

64

1618.123459

 

 

 

 

 

 

 

 

 

 

 

 

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

-1.12349158

0.597829448

-1.87928

0.064833

-2.3181584

0.07117523

xt

0.720411483

0.162411454

4.435718

3.76E-05

0.3958581

1.04496487

Interpretation of Coefficients:

The calculated β-value for Boeing excess return and Excess market return is 1.11931665. The calculated β-value for IBM excess return and Excess market return is 0.720411483. The calculated β-values define that BA price indexes are 111.93% less volatile than the market, whereas the volatility level of IBM compared to the market is 72.04%. Therefore, it can be stated that Boeing (BA) is highly volatile than IBM. Therefore, Boeing (BA) is considered to be more profitable than IBM price returns.

Interpretation of R2:

The linear regression tables describe that the values of R2 of BA and IBM are 0.40482754 and 0.237985555. The R2 indicates the relationship of the dependent variable with the independent variable. Hence, from the values of multiple R2 of the two price returns it could be stated that Boeing (BA) excess return (40.48%) is more associated than the association of IBM (23.80%).

Construction of 95% confidence interval for Slope Efficient:

Confidence Interval of IBM Price Return:

  1. For Boeing (BA) price return, slope (β1) = 1.11931665, Standard Error = 170989356, d.f. = 64, t-value = 6.546119. Hence, the 95% confidence interval for the slope coefficient would be (0.777621689, 1.461011607).
  2. For IBM price return, slope (β1) = 0.720411483, Standard Error = 162411454, d.f. = 64, t-value = 4.435718. Hence, the 95% confidence interval for the slope coefficient would be (0.3958581, 1.04496487). 
 
Preferable neutral Price Return: 

The testing of aggressiveness of the excess price returns needs the following hypothesis:

Null hypothesis (H0): β1 = 1

Alternative hypothesis (H1): β1 < 1

For BA price returns, β1 is 1.11931665 along with the standard error (SE) 0.170989356. The “residual degrees of freedom” is 63 and calculated p-value is 0.0. Hence, t = β1/ SE = 6.546119.

For IBM price indexes, β1 is 0.720411483 along with the standard error (SE) 0.162411454. The “residual degrees of freedom” is 63 and calculated p-value is 0.0. Hence, t = β1/ SE = 4.435718.

For both the excess price returns, the p-values are positive t-value and equal degrees of freedom 64. The 95% confidence intervals for beta values of both BA and IBM price returns are (0.777621689, 1.461011607) and (0.3958581, 1.04496487). The confidence intervals near to 0 refers more neutral nature for price excess return. The confidence intervals of t-statistics indicate that IBM price return is more neutral (Moffett, Stonehill and Eiteman 2014).

Normal Probability Plot in OLS:

IBM Ecess Price return residual plot: 

The method of ordinary least squares (OLS) helps to establish the normality with diagram. The error terms in the model are graphically shown in normal probability plot. It shows that the error terms are not following normal distributions for IBM price indexes. The distributions of residual values are not symmetric for both the market return values.  

Jarque-Bera test

 

 

 

 

 

 

 

 

Skewness

Kurtosis

n

JB

α

χ2 (0.05,2)

Decision

IBM

-0.039955804

0.40782718

65

18.21556

0.05

5.99146455

Normality is Rejected

Besides, we perform a Jarque-Bera test for examining the normality of the residual values. The JB statistic of IBM (18.21) refers that normality of residual values of the regression is rejected at 5% level of significance. 

 
Annotated Bibliography:

Berenson, M., Levine, D., Szabat, K. A., & Krehbiel, T. C. (2012). Basic business statistics: Concepts and applications. Pearson Higher Education AU.

Freed, N., Bergquist, T., & Jones, S. (2014). Understanding business statistics. John Wiley & Sons.

Groebner, D.F., Shannon, P.W., Fry, P.C. and Smith, K.D., 2008. Business statistics. Pearson Education.

Moffett, M. H., Stonehill, A. I., & Eiteman, D. K. (2014). Fundamentals of multinational finance. Pearson.

Peirson, G., Brown, R., Easton, S., & Howard, P. (2014). Business finance. McGraw-Hill Education Australia.

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