Dedication To my father and mother who always picked me up on time and encouraged me to go on every adventure especially this one To my brothers and sister

Pola

I

Acknowledgements First I would like to thank the Almighty God for helping me to accomplish my work, and I am grateful to my supervisor, Dr. Akhterkhan Sabir Hamad, whose expertise generous guidance and support made it possible for me to work on a topic.

I would like to thank the dean of the college of administration and economics Asst. Professor Dr. Kawa Mohammad Jamal Rashid and best thanks go to the head of Department of Statistics, Asst. Professor Dr. Mhammad Mahmod Faqe.

I extend my gratitude to the Asst. Professor Dr. Nawzad Mohammad, Professor Dr. Monem Aziz, Asst. Professor Dr. Shawnim Abdulkader, and Asst. Professor Dr. Mhammad Mahmod Faqe for the many courses that they have taught me during my years of study as an undergraduate and graduate student. My sincere thanks go to librarians in Administration and Economics College.

II

Abstract The purpose of this study is to represent one of the most important problems that affect the accuracy of standard error of the parameters estimates of the linear regression models. This problem named heteroscedasticity.

According to the ordinary least square, estimation of the parameter of the linear model does not possess the best linear unbiased estimator. To treat this problem, weighted least square and transformation methods were used, on the other hand wavelet shrinkage was proposed also to treat this problem.

In wavelet shrinkage, the sure thresholding method has been used to obtain the level of thresholding parameter, so as applying soft thresholding rules to treat the wavelet coefficient, as well as two wavelet filters Daubechies and Biorthogonal was used to filter the data.

The application part included the data set which consisted of (132) families from Sulaimani city in Iraqi, Kurdistan Region, then the results obtained between different methods were compared by using {MSE, 𝐹, R2} criteria. The wavelet shrinkage method especially (bior6.8) has been possessed the lowest (MSE) and largest (R2), with more significant (𝐹) and lowest standard error of parameters compared to other.

Finally the most appropriate model was obtained to predict as follow: 𝑦𝑖 = 0.156 + 0.001𝑥2 + 0.001𝑥3 + 0.0000899𝑥5

III

Contents Title Chapter one

Page 1-5

1-1

Introduction

1-2

1-2

Aim

3

1-3

Literature Review

3-5

Chapter two

Theoretical Part

6-17

2-1

Introduction

6

2-2

Simple Liner Regression

7

2-3

Multiple Linear Regression

8-9

2-4

Classical Parameter Estimation Method

10-11

2-5

Properties of the Least Square Estimator

11

2-6

The Assumption of the Model

12

2-7

Heteroscedasticity

13-14

2-8

Reasons of Heteroscedasticity

14-15

2-9

Detection of Heteroscedasticity

15-17

2-10

Treating the Heteroscedasticity of Variance

18

2-10-1

Transformation

18-19

2-10-2

Weighted Least Square

19-21

2-10-3

Wavelets

22-23

2-10-3-1

Wavelet Types

23-24

2-10-3-2

Wavelet Series Expansion

24-25

2-10-3-3

Wavelet Properties

26

2-10-3-4

Multiresolution Analysis

27-29

2-10-3-5

Wavelet Transform

29

2-10-3-6

Discrete Wavelet Transform

29-32 IV

2-10-3-7

Wavelet Filter

32

2-10-3-8

Haar Wavelet

33-34

2-10-3-9

Daubechies Wavelet

35-36

2-10-3-10

Coiflets Wavelet

37-38

2-10-3-11

Biorthogonal Wavelet

39-40

2-10-3-12

Wavelet Shrinkage

41-42

2-10-3-13

Thresholding

43-44

2-10-3-14

Thresholding Rules

45-47

Chapter three

Application Part

48-102

3-1

Introduction

48

3-2

Description of Data

48

3-2-1

Test of Normality

49-50

3-2-2

Test of Homoscedasticity

50-51

3-2-3

Test of Multicollinearity

52

3-2-4

Test of Auto Correlation

53

3-2-5

Linear Regression Estimation Using OLS

54-58

3-3

Treat Heteroscedasticity of Variance

58

3-3-1

Transformation

59-62

3-3-2

Weighted Least Square

63-65

3-3-3

Wavelet Shrinkage

65-88

Chapter Four

Conclusions and Recommendations

89-90

References

92-98

Appendices

99-118

V

List of Tables Table No.

Title

Page

2-1 3-1 3-2 3-3 3-4 3-5 3-6 3-7 3-8 3-9 3-10 3-11 3-12 3-13 3-14 3-15 3-16 3-17 3-18 3-19 3-20 3-21 3-22 3-23 3-24 3-25 3-26 3-27 3-28 3-29 3-30 3-31 3-32

Data Used in Multiple Regression Analysis Test of Normality Test of Multicollinearity Test the Auto Correlation of Residual Regression Parameter and Standard Error for the Model 3-2 Test of Normality Test of Homoscedasticity Test of Multicollinearity Test of Auto Correlation Regression Parameter and Standard Error for the Model 3-3 ANOVA Table for the Model 3-3 Regression Parameter and Standard Error for the Model 3-4 ANOVA Table for the Model 3-4 Regression Parameter and Standard Error for the Model 3-5 ANOVA Table for the Model 3-5 Regression Parameter and Standard Error for the Model 3-6 ANOVA Table for the Model 3-6 Regression Parameter and Standard Error for the Model 3-7 ANOVA Table for the Model 3-7 Regression Parameter and Standard Error for the Model 3-8 ANOVA Table for the Model 3-8 Regression Parameter and Standard Error for the Model 3-9 ANOVA Table for the Model 3-9 Regression Parameter and Standard Error for the Model 3-10 ANOVA Table for the Model 3-10 Regression Parameter and Standard Error for the Model 3-11 ANOVA Table for the Model 3-11 Regression Parameter and Standard Error for the Model 3-12 ANOVA Table for the Model 3-12 Regression Parameter and Standard Error for the Model 3-13 ANOVA Table for the Model 3-13 Comparison between Methods Comparison between the Standard Error of Parameters for all Methods

9 49 52 53 54 55 56 57 57 58 58 60 60 61 62 64 64 68 68 71 71 74 74 77 77 80 80 82 83 85 86 87 88

VI

List of Figures Figure

Title

Page

2-1 2-2 2-3 2-4 2-5 2-6 2-7 2-8 2-9 2-10 2-11 2-12 3-1 3-2 3-3 3-4 3-5 3-6 3-7 3-8 3-9 3-10 3-11 3-12 3-13 3-14 3-15 3-16 3-17 3-18

Heteroscedasticity A Wave and Wavelet Scaling Function and Wavelet Vector Space Discrete Wavelet Transform for Three Levels The Hierarchical Process for DWT Coefficients The Haar Wavelet 𝜓(𝑥) The Haar Scaling Function 𝜙(𝑥) The Daubechies Scaling and Wavelet Functions Wavelet Function and Scaling Function for 𝐶𝑜𝑖𝑓3 and 𝐶𝑜𝑖𝑓5 Biorthogonal Wavelet Wavelet Shrinkage Steps Hard and Soft Thresholding Normal Q-Q plot for Standard Residual Scatter Plot for Standardized Residual vs. Prediction Normal Q-Q plot for Standard Residual Scatter Plot for Standardized Residual vs. Prediction db5/ First Iteration db5/ Second Iteration db5/ Third Iteration bior2.4/ First Iteration bior2.4 / Sixth Iteration bior2.6 / First Iteration bior2.6 / Tenth Iteration bior2.8/ First Iteration bior2.8/ Twelfth Iteration bior4.4 / First Iteration bior4.4 / Eighth Iteration bior5.5 / First Iteration bior6.8 / First Iteration bior6.8 / Sixth Iteration

13 23 28 30 31 34 34 36 38 40 42 47 50 51 55 56 66 66 67 69 70 72 73 75 76 78 79 81 84 84

VII

Abbreviations Abbreviations

Description

ANOVA

Analysis of Variance

OLS

Ordinary Least Square

p-value

Probability Value

SSE

Sum Square of Error

SSR

Sum Square of Regression

MSE

Mean Square Error

R2

Coefficient of Determination

WLS

Weighted Least Square

BLUE

Best Linear Unbiased Estimate

VIF

Variance Inflation Factor

MRA

Multiresolution Analysis

IDWT

Inverse Discrete Wavelet Transform

DWT

Discrete Wavelet Transform

db

Daubechies Wavelets

bior

Biorthogonal Wavelets

𝜂𝑈

Universal Threshold

𝜂𝑠

Sure Threshold

𝑊𝑛(Ht)

Hard Thresholding

𝑊𝑛(St)

Soft Thresholding

VIII

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