Working of Regression Analysis

1

Prepared by Mutee-Ur-Rahman

Introduction: Econometrics is the application of mathematics and especially of statistical methods to

economics. It is especially concerned both with observational studies and with systems of equations. Econometrics has become strongly identified with Regression analysis. Regression analysis is used to check the impact of independent variable on the dependent variable. It tells the percentage change in independent variable due to change in independent variable. we take the data of 19 companies which are as follows:

Suraj Cotton Mills Ltd

Survays Textile Mills Ltd

Sapphire Fibres Ltd

Sana Industries Ltd

Salfi Textile Mills Ltd

Nishat (Chunian Ltd)

Nagina Cotton Mills Ltd

N.P Spinning Mills Ltd

Masood Textile Mills Ltd

Shakarganj Mills Ltd

Shahtaj sugar Mills Ltd

Shahmurad sugar Mills Ltd

Noon sugar Mills Ltd

Mirpurkhas sugar Mills Ltd

JDW sugar Mills Ltd

Husain sugar Mills Ltd

Habib sugar Mills Ltd

Adam sugar Mills Ltd First of all, we take two variables, profitability (independent variable) and dividend

payout (dependent variable) and check the impact of profitability on dividend payout. This can be stated in linear form as:

Y= b0 + b1X1 Where b0 = intercept b1= slope of regression line After including the error term in our regression model, the equation can be stated as

Y= + + u Secondly, we take firm size as independent variable and dividend payout as dependent variable and test the impact of firm size on dividend payout. This can be stated in linear form as:


2


Y= + + u Thirdly the impact of sales growth independent variable on dividend payout dependent variable is tested. This can be stated in linear form as:


Y= + + u Then we use the multiple regression analysis to test the impact of all three independent variables (profitability, firm size, sales growth) on the dependent variable dividend payout. This can be stated in multiple regression equation as:

= + + +

After including the error term in our multiple regression models, the equation can be stated as:

= + + + + u

The Methodology of Econometrics

Econometric research, in general, involves the following three stages:

1. Specification of the model or maintained hypothesis in explicit stochastic equation form,

together with the prior theoretical expectations about the sign and size of the parameters

of the function.

2. Collection of data on the variables of the model and estimation of the coefficients of the

function with appropriate econometrics techniques.

3. Evaluation of the estimated coefficients of the function on the basis of economic,

statistical and econometric criteria.

Simple Regression Analysis

The Two-variable linear model

The Two- variable linear model or simple linear regression models are used to check the impact

of an independent variable X on the dependent variable Y. It tells us the percentage change in

dependent variable due to the change in independent variable. Simple linear regression analysis usually

3


begins by plotting the set of XY values on a Scatter diagram and determining if there exists an

approximate linear relationship.

= +

The difference between estimated data points on the regression line and actual points is known as

disturbance or error term. So we include error term in our linear regression model.

= + +

It is the assumption of regression model that

1. The error term has Zero expected value or mean

2. Error term is normally distributed

3. Error term has constant variance

4. Error terms are uncorrelated

5. The explanatory variable has fixed values in repeated sampling

The Ordinary Least- Squares Method

It is a method in which we fir the best straight line into the sample of XY observations. It

minimizes the sum of squares deviations of point from the regression line

Min∑( )

Where

Actual observations

= corresponding fitted values

= = the residual

Table 1: The values of Y represent the percentage distribution of Dividend payout of 19 companies and

the values of X1 represent the percentage distribution of profitability of these companies from the

period of 2004-2009.

Y X1

0.12 1.960095

0.131455 1.308333

0 1.987874

0 3.414443

0 2.397895

0.032384 3.437208

0.232425 2.151762

0.41169 2.272126

0.11215 2.186051

0.275862 1.987874

0.212465 2.4681

4


0.350146 2.302585

0.23 1.667707

0.234242 2.370244

0 0.530628

0 1.856298

0.362676 1.547563

0 2.066863

0.157372 2.541602

0.146314 2.74084

0.301777 2.292535

0.3244 0.641854

0.208177 2.821379

0.253798 2.76001

0.2145 1.252763

0.146067 2.014903

0.439628 1.223775

0 2.557227

0 2.251292

0.234 -0.91629

0.134 1.667707

0 2.721295

0.070658 3.462606

0.32 2.186051

0.106136 2.24071

0.078701 3.459466

0 3.005683

0.226399 2.312535

0 1.902108

0 1.589235

0 1.163151

0.265487 1.335001

0.32 -1.60944

0 0.993252

0 -1.60944

0 -0.10536

0.148487 1.902108

0.315737 1.568616

0.525657 1.902108

0.447813 2.821379

0.615472 1.987874

0.563542 2.079442

0.50803 2.292535

0.398871 3.222868

0.224165 1.609438

5


0 1.360977

0 1.704748

0.066805 2.70805

0.234 2.66026

0 3.086487

0 1.029619

0.211665 1.252763

1.109285 1.410987

0.851852 1.458615

0.37309 2.406945

0.081803 2.906901

0.590018 2.028148

2.08326 2.04122

0 1.335001

0.544944 1.808289

1.047009 1.163151

0 -0.69315

1.577371 1.163151

0.303123 1.94591

0.698432 1.481605

0 0.916291

0 -1.60944

0.254 -1.20397

0.090111 2.631889

0.07937 2.322388

0.449761 1.193922

1.107843 -2.30259

0.786096 -1.60944

0 0.182322

0 1.280934

0.909091 0

1.178571 -0.22314

1.1 2.292535

0.23 1.481605

0.12 4.264087

0.588235 1.458615

1.020408 0.693147

0.65445 1.335001

0 2.332144

1.1 -0.10536

0.536111 1.871802

0.129812 2.451005

0.107085 2.639057

0.1638 2.727853

0.11 3.299534

6


Regression Statistics

Multiple R 0.240273

R Square 0.057731

Adjusted R Square 0.049318

Standard Error 0.353596

Observations 114

After doing the Data analysis in Excel, these results are obtained.

Coefficients

-3

-2

-1

0

1

2

3

4

5

1 7

13

19

25

31

37

43

49

55

61

67

73

79

85

91

97

10

3

10

9

Dividend Puyout

Profitability

0.067011 4.245634

0.1 3.049273

0.150588 1.960095

0.089147 2.415914

0.25 1.098612

0 0.832909

0 0.405465

0.118151 2.140066

0.277992 1.974081

0.223187 2.197225

0.254237 2.066863

0 2.985682

0.199557 2.014903

0.437601 1.223775

7


Intercept 0.412764

b1 -0.07165

Here

bo = 0.412764

b1 = - 0.07165

We put these values of b0 and b1 in linear regression equation and estimated regression line is

= 0.412764 – 0.07165X1

At different values of independent variable X1 , the following result is obtained:

Observation Predicted Y Residuals Standard Residuals

1 0.272322 -0.15232 -0.4327

2 0.319021 -0.18757 -0.53282

3 0.270332 -0.27033 -0.76793

4 0.168118 -0.16812 -0.47757

5 0.240954 -0.24095 -0.68447

6 0.166487 -0.1341 -0.38094

7 0.258589 -0.02616 -0.07433

8 0.249965 0.161725 0.459408

9 0.256133 -0.14398 -0.40901

10 0.270332 0.00553 0.015709

11 0.235924 -0.02346 -0.06664

12 0.247783 0.102363 0.290781

13 0.293272 -0.06327 -0.17974

14 0.242935 -0.00869 -0.02469

15 0.374744 -0.37474 -1.06453

16 0.27976 -0.27976 -0.79471

17 0.301881 0.060796 0.172701

18 0.264672 -0.26467 -0.75185

19 0.230657 -0.07329 -0.20818

20 0.216382 -0.07007 -0.19904

21 0.248503 0.053274 0.151336

22 0.366775 -0.04237 -0.12037

23 0.210611 -0.00243 -0.00691

24 0.215008 0.038789 0.110188

25 0.323003 -0.1085 -0.30822

26 0.268395 -0.12233 -0.34749

27 0.32508 0.114548 0.325396

28 0.229538 -0.22954 -0.65204

29 0.251458 -0.25146 -0.71431

30 0.478416 -0.24442 -0.69431

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31 0.293272 -0.15927 -0.45244

32 0.217782 -0.21778 -0.61865

33 0.164667 -0.09401 -0.26705

34 0.256133 0.063867 0.181427

35 0.252216 -0.14608 -0.41497

36 0.164892 -0.08619 -0.24484

37 0.197406 -0.19741 -0.56077

38 0.24707 -0.02067 -0.05872

39 0.276477 -0.27648 -0.78538

40 0.298895 -0.29889 -0.84906

41 0.329424 -0.32942 -0.93579

42 0.317111 -0.05162 -0.14665

43 0.528081 -0.20808 -0.59109

44 0.341597 -0.3416 -0.97037

45 0.528081 -0.52808 -1.50011

46 0.420313 -0.42031 -1.19398

47 0.276477 -0.12799 -0.36358

48 0.300372 0.015365 0.043647

49 0.276477 0.24918 0.70784

50 0.210611 0.237202 0.673815

51 0.270332 0.34514 0.980433

52 0.263771 0.29977 0.851552

53 0.248503 0.259527 0.737235

54 0.181844 0.217026 0.616503

55 0.297447 -0.07328 -0.20817

56 0.315249 -0.31525 -0.89552

57 0.290618 -0.29062 -0.82555

58 0.218731 -0.15193 -0.43157

59 0.222155 0.011845 0.033647

60 0.191616 -0.19162 -0.54432

61 0.338991 -0.33899 -0.96297

62 0.323003 -0.11134 -0.31628

63 0.311666 0.797619 2.265781

64 0.308254 0.543598 1.544189

65 0.240305 0.132784 0.377197

66 0.204483 -0.12268 -0.3485

67 0.267446 0.322571 0.916323

68 0.26651 1.81675 5.160807

69 0.317111 -0.31711 -0.90081

70 0.283199 0.261744 0.743532

71 0.329424 0.717585 2.03843

72 0.462428 -0.46243 -1.31361

73 0.329424 1.247947 3.545021

74 0.273339 0.029784 0.084606

75 0.306606 0.391825 1.113051

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76 0.347111 -0.34711 -0.98603

77 0.528081 -0.52808 -1.50011

78 0.499029 -0.24503 -0.69605

79 0.224188 -0.13408 -0.38087

80 0.246364 -0.16699 -0.47438

81 0.327219 0.122542 0.348102

82 0.577745 0.530098 1.50584

83 0.528081 0.258016 0.73294

84 0.399701 -0.3997 -1.13542

85 0.320985 -0.32098 -0.91181

86 0.412764 0.496327 1.409907

87 0.428752 0.749819 2.129997

88 0.248503 0.851497 2.418832

89 0.306606 -0.07661 -0.21761

90 0.107241 0.012759 0.036245

91 0.308254 0.279982 0.795339

92 0.3631 0.657308 1.867204

93 0.317111 0.33734 0.958274

94 0.245665 -0.24567 -0.69786

95 0.420313 0.679687 1.930774

96 0.278649 0.257462 0.731369

97 0.237149 -0.10734 -0.30491

98 0.223675 -0.11659 -0.33119

99 0.217312 -0.05351 -0.15201

100 0.176351 -0.06635 -0.18848

101 0.108563 -0.04155 -0.11804

102 0.194282 -0.09428 -0.26783

103 0.272322 -0.12173 -0.34581

104 0.239663 -0.15052 -0.42757

105 0.334048 -0.08405 -0.23875

106 0.353086 -0.35309 -1.003

107 0.383712 -0.38371 -1.09

108 0.259427 -0.14128 -0.40132

109 0.27132 0.006672 0.018953

110 0.255332 -0.03215 -0.09131

111 0.264672 -0.01044 -0.02964

112 0.198839 -0.19884 -0.56484

113 0.268395 -0.06884 -0.19555

114 0.32508 0.112521 0.319637

Tests of Significance of Parameter Estimates

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We have to test whether the parameter estimates of the regression are statistically significant

or not. So we calculate the variance of b0 and b1.

ANOVA

df SS MS F Significance

F

Regression 1 0.857965 0.857965 6.862052 0.010025

Residual 112 14.0034 0.12503 Total 113 14.86136

Coefficients Standard

Error t Stat P-value

Intercept 0.412764 0.057397 7.191367 7.69E-11

Profitability -0.07165 0.027352 -2.61955 0.010025

t Stat P-value

7.191367 7.69E-11

-2.61955 0.010025

Test of Goodness of Fit and Correlation

If the observations fall closer to the regression line, the residual will be smaller and variations in Y

"explained" by the estimated regression equation will be greater. Total variation in Y is comprised of two

types of variations.

1. Explained variation

2. Residual or unexplained variation

Total variation in Y is the sum of these two variations.

∑( ) = ∑( ) + ∑( )

Total variation Explained variation Residual variation

In Y in Y in Y

TSS RSS ESS

Dividing both sides by TSS gives

1 =

+

11


Where ranges in values from 0 (when the estimated regression equation explain none of the

variation in Y) to 1 (when all points lie on the regression line)

In our case the value of R2 = 0.057731

So explained variation in Y = 5.7731% of the total variation and 94.2269% variation is unexplained

variation and it is attributed to factors include in error terms.

The correlation coefficient is given by

r= √ = ( )

Correlation tells us the direction and strength of relation between two variables.

In our case the value of r = 0.240273 or 24.0273%, and is positive it means that profitability and dividend

payout are positively correlated.

Table 2: The values of Y represent the percentage distribution of Dividend payout of 19 companies and

the values of X2 represent the percentage distribution of firm size of these companies from the period

of 2004-2009.

Y X2

0.12 1.860655

0.131455 1.864227

0 1.953989

0 1.95146

0 1.97187

0.032384 1.966969

0.232425 2.015018

0.41169 2.02953

0.11215 2.05174

0.275862 2.077685

0.212465 2.094087

0.350146 2.109542

0.23 1.82292

0.234242 1.866844

0 1.902876

0 1.88877

0.362676 1.854008

0 1.87207

12


0.157372 2.057309

0.146314 2.103931

0.301777 2.146206

0.3244 2.184685

0.208177 2.196051

0.253798 2.224686

0.2145 1.917796

0.146067 1.920389

0.439628 2.006376

0 2.018638

0 2.037263

0.234 2.027607

0.134 1.931115

0 1.923709

0.070658 1.94211

0.32 1.933266

0.106136 1.964706

0.078701 1.97132

0 1.909498

0.226399 1.971881

0 2.019881

0 2.038558

0 2.02387

0.265487 1.998368

0.32 2.042492

0 2.06628

0 2.068337

0 2.074307

0.148487 2.089616

0.315737 2.082732

0.525657 1.859645

0.447813 1.89819

0.615472 1.892738

0.563542 1.894873

0.50803 1.921958

0.398871 1.962788

13


0.224165 2.166794

0 2.221738

0 2.237401

0.066805 2.22973

0.234 2.243455

0 2.230756

0 2.125235

0.211665 2.160534

1.109285 2.172764

0.851852 2.185963

0.37309 2.199011

0.081803 2.216132

0.590018 1.904076

2.08326 1.881771

0 1.866988

0.544944 1.834327

1.047009 1.867443

0 1.862328

1.577371 1.95012

0.303123 2.05734

0.698432 2.06488

0 1.959924

0 1.957278

0.254 1.968639

0.090111 2.125837

0.07937 2.199188

0.449761 2.21434

1.107843 2.217633

0.786096 2.229027

0 2.240069

0 1.922476

0.909091 1.980372

1.178571 2.044184

1.1 2.031344

0.23 2.09155

0.12 2.085545

14


0.588235 1.730411

1.020408 1.744023

0.65445 1.761213

0 1.770359

1.1 1.772405

0.536111 1.780045

0.129812 2.125486

0.107085 2.157698

0.1638 2.195926

0.11 2.230613

0.067011 2.235783

0.1 2.222126

0.150588 1.911512

0.089147 1.972544

0.25 1.977089

0 1.995032

0 2.034472

0.118151 2.014988

0.277992 2.058149

0.223187 2.063973

0.254237 2.097137

0 2.102073

0.199557 2.10637

0.437601 2.093944


Multiple R 0.173703

R Square 0.030173



Observations 114

After doing the Data analysis in Excel, these results are obtained.

Coefficients

Intercept 1.25743

15


x2 -0.47833

1.25743

-0.47833

We put these values of b0 and b2 in linear regression equation and estimated regression line is at

different values of independent variable X2, the following result is obtained:

=



1 0.367426 -0.24743 -0.6928

2 0.365718 -0.23426 -0.65594

3 0.322782 -0.32278 -0.9038

4 0.323992 -0.32399 -0.90719

5 0.314229 -0.31423 -0.87985

6 0.316573 -0.28419 -0.79574

7 0.29359 -0.06117 -0.17127

8 0.286649 0.125041 0.350119

9 0.276025 -0.16388 -0.45886

10 0.263615 0.012247 0.034293

11 0.255769 -0.0433 -0.12125

12 0.248377 0.101769 0.284957

13 0.385476 -0.15548 -0.43534

14 0.364466 -0.13022 -0.36463

15 0.347231 -0.34723 -0.97226

16 0.353978 -0.35398 -0.99115

17 0.370606 -0.00793 -0.0222

18 0.361966 -0.36197 -1.01352

19 0.273361 -0.11599 -0.32477

20 0.25106 -0.10475 -0.29329

21 0.230839 0.070938 0.198629

22 0.212434 0.111966 0.313509

23 0.206997 0.00118 0.003305

24 0.1933 0.060498 0.169395

25 0.340094 -0.12559 -0.35167

26 0.338854 -0.19279 -0.53981

27 0.297724 0.141905 0.397338

28 0.291859 -0.29186 -0.81721

29 0.28295 -0.28295 -0.79227

30 0.287568 -0.05357 -0.14999

31 0.333723 -0.19972 -0.55923

16


32 0.337266 -0.33727 -0.94435

33 0.328464 -0.25781 -0.72187

34 0.332695 -0.01269 -0.03555

35 0.317656 -0.21152 -0.59226

36 0.314492 -0.23579 -0.66022

37 0.344064 -0.34406 -0.96339

38 0.314224 -0.08783 -0.24591

39 0.291264 -0.29126 -0.81555

40 0.28233 -0.28233 -0.79053

41 0.289356 -0.28936 -0.81021

42 0.301554 -0.03607 -0.10099

43 0.280448 0.039552 0.110746

44 0.26907 -0.26907 -0.7534

45 0.268086 -0.26809 -0.75065

46 0.26523 -0.26523 -0.74265

47 0.257908 -0.10942 -0.30638

48 0.261201 0.054536 0.152703

49 0.367909 0.157748 0.441699

50 0.349472 0.098341 0.275357

51 0.35208 0.263392 0.737506

52 0.351059 0.212483 0.594958

53 0.338104 0.169927 0.4758

54 0.318573 0.080298 0.224836

55 0.220991 0.003173 0.008885

56 0.19471 -0.19471 -0.54519

57 0.187218 -0.18722 -0.52422

58 0.190888 -0.12408 -0.34743

59 0.184323 0.049677 0.139098

60 0.190397 -0.1904 -0.53312

61 0.24087 -0.24087 -0.67444

62 0.223986 -0.01232 -0.0345

63 0.218136 0.891149 2.495244

64 0.211823 0.640029 1.792101

65 0.205581 0.167508 0.469028

66 0.197392 -0.11559 -0.32365

67 0.346657 0.243361 0.681418

68 0.357326 1.725934 4.832667

69 0.364397 -0.3644 -1.02032

70 0.38002 0.164924 0.461792

71 0.364179 0.682829 1.911942

72 0.366626 -0.36663 -1.02656

73 0.324633 1.252738 3.507705

74 0.273347 0.029776 0.083374

75 0.26974 0.428692 1.20035

76 0.319943 -0.31994 -0.89585

17


77 0.321209 -0.32121 -0.89939

78 0.315775 -0.06177 -0.17297

79 0.240582 -0.15047 -0.42133

80 0.205496 -0.12613 -0.35316

81 0.198249 0.251512 0.704241

82 0.196674 0.91117 2.551302

83 0.191224 0.594873 1.665661

84 0.185942 -0.18594 -0.52064

85 0.337856 -0.33786 -0.94601

86 0.310162 0.598929 1.677019

87 0.279639 0.898932 2.517037

88 0.285781 0.814219 2.279838

89 0.256983 -0.02698 -0.07555

90 0.259855 -0.13985 -0.3916

91 0.429726 0.158509 0.443831

92 0.423215 0.597194 1.67216

93 0.414992 0.239458 0.670489

94 0.410617 -0.41062 -1.14974

95 0.409639 0.690361 1.933033

96 0.405985 0.130127 0.364358

97 0.24075 -0.11094 -0.31063

98 0.225342 -0.11826 -0.33113

99 0.207057 -0.04326 -0.12112

100 0.190465 -0.08047 -0.2253

101 0.187992 -0.12098 -0.33875

102 0.194525 -0.09452 -0.26467

103 0.3431 -0.19251 -0.53904

104 0.313907 -0.22476 -0.62933

105 0.311733 -0.06173 -0.17285

106 0.30315 -0.30315 -0.84883

107 0.284285 -0.28428 -0.79601

108 0.293604 -0.17545 -0.49128

109 0.27296 0.005033 0.014092

110 0.270173 -0.04699 -0.13156

111 0.254311 -7.3E-05 -0.00021

112 0.251949 -0.25195 -0.70547

113 0.249894 -0.05034 -0.14095

114 0.255838 0.181764 0.508944

18


Tests of Significance of Parameter Estimates:



ANOVA


F

Regression 1 0.448407 0.448407 3.484472 0.064562

Residual 112 14.41295 0.128687 Total 113 14.86136



Intercept 1.25743 0.519373 2.421055 0.017082

Firm Size -0.47833 0.256246 -1.86667 0.064562

Test of Goodness of Fit and Correlation:







∑( ) = ∑( ) + ∑( )


In Y in Y in Y

TSS RSS ESS


1 =

+



In our case the value of R2 = 0.030173 or 3.0173%

19


So explained variation in Y equal to 3.0173% of the total variation and 96.9827% variation is unexplained

variation and it is attributed to factors included in error terms.


r= √ = ( )


In our case the value of r = 0.173703 or 17.3703%, and is positive it means that profitability and firm size

are positively correlated.

Table 3:

The values of Y represent the percentage distribution of Dividend payout of 19 companies and the

values of X3 represent the percentage distribution of profitability of these companies from the period of

2004-2009

Y X3

0.12 3.109061

0.131455 1.252763

0 3.841601

0 3.493473

0 4.282206

0.032384 2.397895

0.232425 4.858261

0.41169 3.427515

0.11215 3.427515

0.275862 3.380995

0.212465 4.063885

0.350146 0.741937

0.23 4.317488

0.234242 1.974081

0 2.557227

0 3.499533

0.362676 3.165475

0 3.11795

0.157372 1.916923

0.146314 4.012773

20


0.301777 4.636669

0.3244 1.667707

0.208177 3.735286

0.253798 2.572612

0.2145 1.458615

0.146067 1.029619

0.439628 1.871802

0 3.328627

0 3.817712

0.234 3.280911

0.134 2.312535

0 4.028917

0.070658 4.366913

0.32 2.928524

0.106136 2.549445

0.078701 4.089332

0 2.451005

0.226399 0.09531

0 2.850707

0 3.08191

0 4.172848

0.265487 2.694627

0.32 2.833213

0 3.173878

0 4.329417

0 2.895912

0.148487 3.817712

0.315737 2.867899

0.525657 2.687847

0.447813 2.230014

0.615472 2.564949

0.563542 3.484312

0.50803 3.010621

0.398871 3.735286

0.224165 4.174387

0 3.723281

21


0 1.916923

0.066805 2.80336

0.234 3.246491

0 3.634951

0 2.509599

0.211665 1.667707

1.109285 3.994524

0.851852 2.923162

0.37309 3.490429

0.081803 3.65842

0.590018 3.511545

2.08326 3.600048

0 3.214868

0.544944 2.091864

1.047009 2.890372

0 0.262364

1.577371 0.262364

0.303123 4.11578

0.698432 3.261935

0 3.811097

0 2.844909

0.254 2.906901

0.090111 4.023564

0.07937 2.879198

0.449761 2.95491

1.107843 2.827314

0.786096 2.960105

0 2.197225

0 3.526361

0.909091 3.269569

1.178571 -0.91629

1.1 4.628887

0.23 2.797281

0.12 0.741937

0.588235 2.74084

1.020408 3.453157

22


0.65445 3.7612

0 3.020425

1.1 -0.22314

0.536111 3.440418

0.129812 3.157

0.107085 3.113515

0.1638 3.756538

0.11 2.533697

0.067011 3.005683

0.1 2.667228

0.150588 3.877432

0.089147 2.912351

0.25 4.136765

0 1.568616

0 2.451005

0.118151 2.74084

0.277992 4.062166

0.223187 3.139833

0.254237 3.754199

0 2.388763

0.199557 -2.30259

0.437601 2.772589


Multiple R 0.129653

R Square 0.01681



Observations 114

After doing the Data analysis in Excel, these results are obtained

Coefficients

Intercept 0.408098

X3 -0.04059

23


0.408098

-0.04059

We put these values of b0 and b3 in linear regression equation and estimated regression line is at

different values of independent variable X3, the following result is obtained:

=



1 0.281913 -0.16191 -0.45027

2 0.357253 -0.2258 -0.62793

3 0.252182 -0.25218 -0.7013

4 0.266311 -0.26631 -0.74059

5 0.234299 -0.2343 -0.65157

6 0.310777 -0.27839 -0.77419

7 0.210919 0.021506 0.059806

8 0.268988 0.142702 0.396845

9 0.268988 -0.15684 -0.43616

10 0.270876 0.004986 0.013866

11 0.24316 -0.0307 -0.08536

12 0.377986 -0.02784 -0.07742

13 0.232867 -0.00287 -0.00797

14 0.327978 -0.09374 -0.26067

15 0.30431 -0.30431 -0.84627

16 0.266065 -0.26606 -0.73991

17 0.279623 0.083053 0.230965

18 0.281552 -0.28155 -0.78298

19 0.330297 -0.17293 -0.48089

20 0.245234 -0.09892 -0.27509

21 0.219913 0.081865 0.227661

22 0.340412 -0.01601 -0.04453

23 0.256497 -0.04832 -0.13437

24 0.303685 -0.04989 -0.13873

25 0.348899 -0.1344 -0.37375

26 0.36631 -0.22024 -0.61248

27 0.332129 0.1075 0.29895

28 0.273001 -0.273 -0.7592

29 0.253151 -0.25315 -0.704

30 0.274938 -0.04094 -0.11385

31 0.314241 -0.18024 -0.50124

32 0.244579 -0.24458 -0.68016

24


33 0.230861 -0.1602 -0.44552

34 0.28924 0.03076 0.085541

35 0.304626 -0.19849 -0.55199

36 0.242127 -0.16343 -0.45448

37 0.308621 -0.30862 -0.85826

38 0.40423 -0.17783 -0.49454

39 0.292399 -0.2924 -0.81314

40 0.283015 -0.28301 -0.78705

41 0.238738 -0.23874 -0.66391

42 0.298733 -0.03325 -0.09246

43 0.293109 0.026891 0.074784

44 0.279282 -0.27928 -0.77667

45 0.232383 -0.23238 -0.64624

46 0.290564 -0.29056 -0.80804

47 0.253151 -0.10466 -0.29106

48 0.291701 0.024036 0.066843

49 0.299008 0.226649 0.630296

50 0.31759 0.130223 0.362142

51 0.303996 0.311476 0.866194

52 0.266683 0.296859 0.825546

53 0.285908 0.222122 0.617708

54 0.256497 0.142374 0.395934

55 0.238675 -0.01451 -0.04035

56 0.256984 -0.25698 -0.71466

57 0.330297 -0.3303 -0.91854

58 0.29432 -0.22752 -0.6327

59 0.276335 -0.04234 -0.11773

60 0.260569 -0.26057 -0.72463

61 0.306243 -0.30624 -0.85164

62 0.340412 -0.12875 -0.35804

63 0.245975 0.86331 2.400811

64 0.289458 0.562394 1.563982

65 0.266434 0.106655 0.296601

66 0.259616 -0.17781 -0.49449

67 0.265577 0.32444 0.902248

68 0.261985 1.821274 5.064849

69 0.277619 -0.27762 -0.77204

70 0.323197 0.221747 0.616663

71 0.290789 0.75622 2.103

72 0.39745 -0.39745 -1.10528

73 0.39745 1.179921 3.281286

74 0.241054 0.062069 0.17261

75 0.275708 0.422724 1.175568

76 0.25342 -0.25342 -0.70474

77 0.292634 -0.29263 -0.8138

25


78 0.290118 -0.03612 -0.10044

79 0.244796 -0.15469 -0.43017

80 0.291242 -0.21187 -0.5892

81 0.288169 0.161592 0.449376

82 0.293348 0.814495 2.26506

83 0.287958 0.498138 1.38529

84 0.318921 -0.31892 -0.8869

85 0.264976 -0.26498 -0.73688

86 0.275398 0.633693 1.762259

87 0.445287 0.733284 2.039217

88 0.220229 0.879771 2.446589

89 0.294567 -0.06457 -0.17956

90 0.377986 -0.25799 -0.71744

91 0.296858 0.291378 0.810303

92 0.267947 0.752461 2.092547

93 0.255445 0.399005 1.109609

94 0.28551 -0.28551 -0.79399

95 0.417155 0.682845 1.898949

96 0.268464 0.267647 0.744309

97 0.279967 -0.15015 -0.41757

98 0.281732 -0.17465 -0.48568

99 0.255634 -0.09183 -0.25539

100 0.305265 -0.19526 -0.54302

101 0.286109 -0.2191 -0.6093

102 0.299845 -0.19985 -0.55576

103 0.250727 -0.10014 -0.27848

104 0.289897 -0.20075 -0.55827

105 0.240202 0.009798 0.027248

106 0.344434 -0.34443 -0.95785

107 0.308621 -0.30862 -0.85826

108 0.296858 -0.17871 -0.49697

109 0.24323 0.034763 0.096672

110 0.280664 -0.05748 -0.15984

111 0.255729 -0.00149 -0.00415

112 0.311147 -0.31115 -0.86528

113 0.501552 -0.302 -0.83983

114 0.295569 0.142032 0.394983

Tests of Significance of Parameter Estimates:



26


ANOVA


F

Regression 1 0.249817 0.249817 1.914889 0.169173

Residual 112 14.61154 0.13046 Total 113 14.86136



Intercept 0.408098 0.09183 4.444056 2.09E-05

Sales Growth -0.04059 0.02933 -1.38379 0.169173

t Stat P-value

4.444056 2.09E-05

-1.38379 0.169173

Test of Goodness of Fit and Correlation:







∑( ) = ∑( ) + ∑( )


In Y in Y in Y

TSS RSS ESS


1 =

+



27


In our case the value of R2 = 0.01681 or 1.681%

So explained variation in Y = 1.681% of the total variation and 98.319% variation is unexplained variation

and it is attributed to factors include in error terms.


r= √ = ( )


In our case the value of r = 0.129653or 12.9563%, and is positive it means that profitability and sales

growth are positively correlated.

Classical Normal Linear Regression Model (CNLRM)

Using the method of OLS we were able to estimate the parameters b1, b2and . Under the

assumption of classical linear regression model, we were able to show that the estimators of

these parameters satisfy several desirable statistical properties. , such that unbiasness, minimum

variance, etc. Estimation is half the battle. Hypothesis testing is other half. In regression analysis

our objective is not only to estimate the sample regression function (SRF), but also to use it to

draw inferences about the population regression function.

The Probability Distribution of Disturbance

To find out the probability distribution of the OLS estimators, we proceed as follows.

Specifically, consider

= ∑

The Normality assumption for

The classical normal linear regression model assume that is distributed normally with

Mean: E ( ) = 0

Variance: E [ - E( ) ^2 = E( ) =

Cov ( , ): E[ - E( ) - E( ) ] = E( , ) = 0 i ≠j

Properties of OLS Estimators under the Normal Assumption

With the assumption that follows the normal distribution, the OLS estimators have the

following properties;

1. They are unbiased.

28


2. They have minimum variance. Combined with 1, this means that they are minimum-

variance unbiased or efficient estimators.

3. They have consistency; that is as sample size increases indefinitely, the estimators

converge to their true population values.

4. is normally distributed with Mean : E( ) and var( ):

or more

completely ≈ N( ,

), then by the property of normal distribution the variable Z,

which is defined as Z=

5. is normally distributed withMean : E( ) and var( ):

or more completely

≈ N( ,

), then by the property of normal distribution the variable Z, which is

defined as Z=

6. ( )( ) is distributed as ( chi square) distribution with degree of freedom (n-

2). This knowledge will help us to draw inferences about the true .

7. ( ) are distributed independently of .

8. Have minimum variance in the entire class of unbiased estimators, whether

linear or not.

Multiple Regression Analysis

Multiple regression analysis is used to test the impact of two or more independent

variables on the dependent. The four-variable linear regression model can be written as:

= + +

The additional assumption (to those of the simple regression model) is that there is no exact

linear relationship between the X values.

Ordinary least-squares (OLS) parameters estimates for the equation above can be obtained by

minimizing the sum of the squared residual:

∑ = ∑( )

We take all three independent variables (X1 = profitability, X2 = firm size, X3 = sales growth) and

dependent variable (Y = dividend payout) to perform multiple regression analysis and test the impact of

all three independent variables on dependent variable.

Y X1 X2 X3

Predicted

Y Residuals

0.12 1.960095 1.860655 3.109061 0.335925 -0.21593

0.131455 1.308333 1.864227 1.252763 0.425981 -0.29453

0 1.987874 1.953989 3.841601 0.275821 -0.27582

29


0 3.414443 1.95146 3.493473 0.195257 -0.19526

0 2.397895 1.97187 4.282206 0.230414 -0.23041

0.032384 3.437208 1.966969 2.397895 0.21688 -0.1845

0.232425 2.151762 2.015018 4.858261 0.212754 0.01967

0.41169 2.272126 2.02953 3.427515 0.237586 0.174104

0.11215 2.186051 2.05174 3.427515 0.233887 -0.12174

0.275862 1.987874 2.077685 3.380995 0.237044 0.038818

0.212465 2.4681 2.094087 4.063885 0.18125 0.031215

0.350146 2.302585 2.109542 0.741937 0.27483 0.075316

0.23 1.667707 1.82292 4.317488 0.337656 -0.10766

0.234242 2.370244 1.866844 1.974081 0.337758 -0.10352

0 0.530628 1.902876 2.557227 0.424477 -0.42448

0 1.856298 1.88877 3.499533 0.320403 -0.3204

0.362676 1.547563 1.854008 3.165475 0.363466 -0.00079

0 2.066863 1.87207 3.11795 0.324154 -0.32415

0.157372 2.541602 2.057309 1.916923 0.249569 -0.0922

0.146314 2.74084 2.103931 4.012773 0.161159 -0.01484

0.301777 2.292535 2.146206 4.636669 0.155466 0.146311

0.3244 0.641854 2.184685 1.667707 0.324738 -0.00034

0.208177 2.821379 2.196051 3.735286 0.12538 0.082798

0.253798 2.76001 2.224686 2.572612 0.148743 0.105055

0.2145 1.252763 1.917796 1.458615 0.401821 -0.18732

0.146067 2.014903 1.920389 1.029619 0.36369 -0.21762

0.439628 1.223775 2.006376 1.871802 0.355901 0.083728

0 2.557227 2.018638 3.328627 0.226572 -0.22657

0 2.251292 2.037263 3.817712 0.225213 -0.22521

0.234 -0.91629 2.027607 3.280911 0.445672 -0.21167

0.134 1.667707 1.931115 2.312535 0.346862 -0.21286

0 2.721295 1.923709 4.028917 0.236532 -0.23653

0.070658 3.462606 1.94211 4.366913 0.172543 -0.10189

0.32 2.186051 1.933266 2.928524 0.296334 0.023666

0.106136 2.24071 1.964706 2.549445 0.290044 -0.18391

0.078701 3.459466 1.97132 4.089332 0.168131 -0.08943

0 3.005683 1.909498 2.451005 0.266748 -0.26675

0.226399 2.312535 1.971881 0.09531 0.348555 -0.12216

0 1.902108 2.019881 2.850707 0.280703 -0.2807

30


0 1.589235 2.038558 3.08191 0.286706 -0.28671

0 1.163151 2.02387 4.172848 0.290592 -0.29059

0.265487 1.335001 1.998368 2.694627 0.329972 -0.06449

0.32 -1.60944 2.042492 2.833213 0.495771 -0.17577

0 0.993252 2.06628 3.173878 0.310771 -0.31077

0 -1.60944 2.068337 4.329417 0.444805 -0.4448

0 -0.10536 2.074307 2.895912 0.384997 -0.385

0.148487 1.902108 2.089616 3.817712 0.225822 -0.07734

0.315737 1.568616 2.082732 2.867899 0.275506 0.040231

0.525657 1.902108 1.859645 2.687847 0.351379 0.174278

0.447813 2.821379 1.89819 2.230014 0.289129 0.158684

0.615472 1.987874 1.892738 2.564949 0.335526 0.279946

0.563542 2.079442 1.894873 3.484312 0.304056 0.259485

0.50803 2.292535 1.921958 3.010621 0.292011 0.216019

0.398871 3.222868 1.962788 3.735286 0.19628 0.202591

0.224165 1.609438 2.166794 4.174387 0.202957 0.021208

0 1.360977 2.221738 3.723281 0.208214 -0.20821

0 1.704748 2.237401 1.916923 0.228433 -0.22843

0.066805 2.70805 2.22973 2.80336 0.143759 -0.07695

0.234 2.66026 2.243455 3.246491 0.129201 0.104799

0 3.086487 2.230756 3.634951 0.096815 -0.09682

0 1.029619 2.125235 2.509599 0.301941 -0.30194

0.211665 1.252763 2.160534 1.667707 0.295768 -0.0841

1.109285 1.410987 2.172764 3.994524 0.217985 0.8913

0.851852 1.458615 2.185963 2.923162 0.238325 0.613526

0.37309 2.406945 2.199011 3.490429 0.157177 0.215912

0.081803 2.906901 2.216132 3.65842 0.113687 -0.03188

0.590018 2.028148 1.904076 3.511545 0.302787 0.287231

2.08326 2.04122 1.881771 3.600048 0.308799 1.774461

0 1.335001 1.866988 3.214868 0.370323 -0.37032

0.544944 1.808289 1.834327 2.091864 0.38388 0.161064

1.047009 1.163151 1.867443 2.890372 0.389829 0.657179

0 -0.69315 1.862328 0.262364 0.581072 -0.58107

1.577371 1.163151 1.95012 0.262364 0.426364 1.151007

0.303123 1.94591 2.05734 4.11578 0.228359 0.074763

0.698432 1.481605 2.06488 3.261935 0.277835 0.420597

31


0 0.916291 1.959924 3.811097 0.342528 -0.34253

0 -1.60944 1.957278 2.844909 0.530711 -0.53071

0.254 -1.20397 1.968639 2.906901 0.498483 -0.24448

0.090111 2.631889 2.125837 4.023564 0.158754 -0.06864

0.07937 2.322388 2.199188 2.879198 0.178949 -0.09958

0.449761 1.193922 2.21434 2.95491 0.242612 0.207149

1.107843 -2.30259 2.217633 2.827314 0.467677 0.640166

0.786096 -1.60944 2.229027 2.960105 0.415182 0.370914

0 0.182322 2.240069 2.197225 0.316878 -0.31688

0 1.280934 1.922476 3.526361 0.34243 -0.34243

0.909091 0 1.980372 3.269569 0.407082 0.502009

1.178571 -0.22314 2.044184 -0.91629 0.507586 0.670985

1.1 2.292535 2.031344 4.628887 0.203196 0.896804

0.23 1.481605 2.09155 2.797281 0.279308 -0.04931

0.12 4.264087 2.085545 0.741937 0.159662 -0.03966

0.588235 1.458615 1.730411 2.74084 0.431704 0.156531

1.020408 0.693147 1.744023 3.453157 0.455717 0.564692

0.65445 1.335001 1.761213 3.7612 0.399378 0.255072

0 2.332144 1.770359 3.020425 0.35194 -0.35194

1.1 -0.10536 1.772405 -0.22314 0.593857 0.506143

0.536111 1.871802 1.780045 3.440418 0.365987 0.170124

0.129812 2.451005 2.125486 3.157 0.193761 -0.06395

0.107085 2.639057 2.157698 3.113515 0.169611 -0.06253

0.1638 2.727853 2.195926 3.756538 0.130824 0.032975

0.11 3.299534 2.230613 2.533697 0.11293 -0.00293

0.067011 4.245634 2.235783 3.005683 0.037748 0.029263

0.1 3.049273 2.222126 2.667228 0.128808 -0.02881

0.150588 1.960095 1.911512 3.877432 0.294202 -0.14361

0.089147 2.415914 1.972544 2.912351 0.265859 -0.17671

0.25 1.098612 1.977089 4.136765 0.315033 -0.06503

0 0.832909 1.995032 1.568616 0.393683 -0.39368

0 0.405465 2.034472 2.451005 0.380875 -0.38087

0.118151 2.140066 2.014988 2.74084 0.270508 -0.15236

0.277992 1.974081 2.058149 4.062166 0.227671 0.050321

0.223187 2.197225 2.063973 3.139833 0.235857 -0.01267

0.254237 2.066863 2.097137 3.754199 0.213914 0.040324

32


0 2.985682 2.102073 2.388763 0.190027 -0.19003

0.199557 2.014903 2.10637 -2.30259 0.376441 -0.17688

0.437601 1.223775 2.093944 2.772589 0.295426 0.142176

Resulting from the various use of profitability X1 and firm size X2 and sales growth X3 in

percentage from the year 2004-2009.Using the excel calculation we found the estimators as

1.314409

= -0.06378

-0.413172

-0.02692

The regression equation

= 1.314409 - 0.06378

Test of Significance of Parameter Estimates

In order to test for the statistical significance of the parameter estimates of the multiple

regressions, the variance of the estimate is required.



Intercept 1.314409 0.510604 2.574223 0.011376

Profitability -0.06378 0.027575 -2.31282 0.022593

Firm Size -0.41372 0.251868 -1.6426 0.103321

Sales Growth -0.02692 0.028902 -0.93135 0.353711

ANOVA


F

Regression 3 1.317196 0.439065 3.565903 0.01652

Residual 110 13.54417 0.123129 Total 113 14.86136

Interpretation of Results

Group Statistics:

33


Here the groups are identified, their sample sizes (N), their means, their standard deviations,

and their S.E.M.s is given. We should check that the right variables have been selected and the right

number of cases has been analyzed.

The Coefficient of Multiple Determinations ( )

The coefficient of multiple determinations is defined as the proportion of the total variation in

Y “explained” by the multiple regression of Y on X1, X2 and X3 it can be calculated as

∑

∑

= 1- ∑

∑


Multiple R 0.297712

R Square 0.088632



Observations 114

The value = 0.088632 or 8.8632% it means that the dependent variable is 8.8632% explained

by the independent variable and 91.1368% by other factors.

The Test of Overall Significance of the Regression

The overall significance of the regression can be tested with the ratio of the explained to

the unexplained variance. This follows an F distribution with k – 1 and n-k degrees of freedom,

where n is number of observations and k is number of parameters estimated.

=

⁄

⁄

34


If the calculated F ratio exceeds the tabular value of F at the specified level of significance and

degrees of freedom, the hypothesis is accepted that the regression parameters are not all equal to

zero and that is significantly different from zero.

In addition, the F ratio can be used to test any linear restriction of regression parameters by using

the form

=

∑ ∑

∑

Where p is the number of restriction being tested, ∑ indicates the sum of squared

residuals for the restricted regression where the restriction are assumed to be true, and

indicates the sum of squared residuals for the unrestricted regression ( i.e. the usual residuals).

The null hypothesis is that the p restriction are true, in which case the residuals from the

restricted and unrestricted models should be identical, and F would take the value of zero. If the

restrictions are not true, the unrestricted model will have lower errors, increasing the value of F.

If F exceeds the tabular value, the null hypothesis is rejected.

So by using the excel the calculated value of = 3.565903 exceed the table value of F 3.97

at 5% level of significance , the hypothesis is accepted that b1 and b2 are not both zero and that is

significantly different from zero.

Conclusion:

When we check the impact of profitability on dividend payout through linear regression

analysis, the value of = 0.057731. It means that 5.7731% change in dividend payout is due to

firm profitability.

When we check the impact of firm size on dividend payout through linear regression analysis,

the value of = 0.030173. It means that 3.0173% change in dividend payout is due to firm

size.

When we check the impact of sales growth on dividend payout through linear regression

analysis, the value of =0.01681. It means that 1.681% change in dividend payout is due to

sales growth.

When we check the impact of all three independent variables (profitability, firm size, sales

growth) on dependent variable (dividend payout) through multiple regression analysis, the

value of =0.088632. It means that these three independent variables bring 8.8632% change

in dividend payout and 91.1368% change is due to other factors not included in our study.

Documents

Working of Regression Analysis