econ 441 - elcbh.com
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جميعنمبالتخلصذلكقبلأوالدراسيالفصلنهايةفيالطالبيتعهد
منعي ُكمامنها،الاستفادةيمكنلابطريقةتمزيقهاطريقعنالملخصات
وفتحصالحصبتسجيلالمعنيالأستاذسيتولىحيثالمحاضراتتسجيل
.اشتراكهمفترةطوالمراتعدةبمشاهدتهاالدورةلطلابالمجال
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الأخرىةالقانونيالإجراءاتباتخاذالمركزحقإلىبالإضافةتسريبه،يتم
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• Simple linear regression: if you have one independent variablesExample: the relationship between income and expenditures
• Multiple linear regression: if you have more than one independent variables
Example: how the expenditures is affected by the gender and amount of income
Simple linear regression is covered in this chapter
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• What is the econometric model of simple linear model
• What is the estimated model of the simple linear model
• Estimate the parameters of estimated model (b0) and (b1)
• Interpret the parameters: what does b0 mean? What does b1 mean
• How good is the model
• Test the parameters and test the model
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Example:
n 1 2 3 4 5 6 7 8
Absent 4 7 15 14 12 7 18 3
Grade 85 81 70 74 71 88 66 89
What is the DV and IV?
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Example:
n 1 2 3 4 5 6 7 8
Absent 4 7 15 14 12 7 18 3
Grade 85 81 70 74 71 88 66 89
Write the econometrics model?
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Example:
n 1 2 3 4 5 6 7 8
Absent 4 7 15 14 12 7 18 3
Grade 85 81 70 74 71 88 66 89
Write the estimated model?
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Example:
n X Y
1 4 85
2 7 81
3 15 70
4 14 74
5 12 71
6 7 88
7 18 66
8 3 89
( )XX i − ( )YYi −( )YYi −( )XX i − ( )2XX i −
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Estimated Model
ii XY 10
+=
What does it mean?
• Intercept interpretation: To interpret the intercept (b0) remember to make all variable equal zero to know what will be the value of dependent variable
• Slope interpretation: b1 it is if X1 (IV) change by one unit, how many units y (DV) will change
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3.93)(int0 =ercept
Given the absent is zero, the student grade will be 93.3
53.1)(1 −==i
i
dX
dYslope
If absent(X) increases by 1 day, the grade (Y) will decrease by 1 mark
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• How well the variations in dependent variable, Y is explained by the variations in
independent variable, X???
(A)total variation in Y
(A)total variations in Y explained by the model (i.e., independent variable, X)
(A)total variations in Y not-explained by the model (i.e., independent variable, X)
( )2 −YYi
( )2 −YYi
( )2 − ii YY
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How good is your estimated regression
model?
Coefficient of Determination:The proportion of variation in dependent variable (Y) explained by the model or independent variable, X
R-Square or R2 = ESS/TSS
0 <= R2 <= 1
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R2 = 0.897. what does it mean ?
89.7% variations in dependent variable, Y is explained by the independent variable, X (or, by the model)
If R2 = 0. what does it mean ?
If R2 = 1. what does it mean ?
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• r is coefficient of the correlation. It shows the relationship between the two variables (positive or negative relationship. Whether there is strong or week relationship.
• -1 <= r <= +1
• r = 1 implies perfect correlation• r = 0 implies no correlation• Closer the value of r to 0, weaker is the relationship between X and Y• Closer is the value of r to 1, stronger is the relationship between X and Y
• R2: Coefficient of Determination: how the dependent variable is explained by the variation of independent variables• It measures the cause and effect relationship between two variables – X and Y, where X is the
cause factor and Y is the effect factor• The value is between 0 and 1• It tests the slope of the model
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Testing the slope (coefficients)
In our example, we will test the slope
Step 1: Set the hypothesis H0: b1 = 0 (absent does not significantly affect the grade of students)
H1: b1 not equal to zero (absent does significantly affect the grade of students)
Step 2: Calculate the test statistics
)( 1
1
SEvaluet =−
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21)(
1
)1()(
XXkn
RSSSE
i −
−−=
Degree of Freedom = n-k-1n = no of observationsk = no of independent variable
Se = 0.211
T- calculated = -1.5/0.211 = -7.2
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Step 3: find t-table (critical t-value)
• Which level of confidence:
Level of Confidence Level of Significance
99% 1% Alpha = 0.0195% 5% Alpha = 0.05
• What is the DoF
Degree of Freedom = n-k-1n = no of observationsk = no of independent variable
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Step 4: Compare the calculated t-value with critical t-value (table t distribution)
Step 5: Conclusion
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• This is the average of Y. You can calculate it by summing the valuesof Y and then dividing it by the number of the sample
Y
X • This is the average of X. You can calculate it by summing the valuesof X and then dividing it by the number of the sample
( )YYi − • The summation always equal 0
( )XX i − • The summation always equal 0
0= ie • This is the summation of the difference between actual Y andestimated Y
( ) 10
2, - unknowns respect to with
− ii YYMinimize