1. Introduction to Regression Analysis
Prepared by: Bhakti Joshi
February 03, 2012

2. Assumption of Linearity
Merriam Webster defines it as: “…of relating to, resembling or having a graph that is a line and especially a straight line” “…involving a single dimension”

3. Linearity Examples

4. Examples Savings and interest rates Investment and interest rates
Rise in prices and demand for food
Fall in prices and cost of raw material
Rise in oil prices and costs of transportation
Rise in oil prices and prices of food

5. Regression Analysis
… is a tool to investigate relationships between variables
Dependent Variable
Independent Variable
Error term
Deviation from the actual and explanatory variables
Variable to be studied
Variable that is affected
Variable that is the presumed effect
Known as “actual” variable
Variable that possibly explains the dependent variable
Variable that is the presumed cause
Known as “explanatory” variable

6. Yi = α + β Xi + εi Regression Equation
Dependent Variable
Independent Variable
Error
term
Regression Coefficients
This equation is also described as “Regression of Y on X”

7. Regression Coefficients
‘α’ is the y-intercept or constant of the regression the starting point of the regression line ‘β’ is the slope and represents the rate of change in Yi, with changes in the Xi Also, known as unknown parameters

8. Calculating unknown parameters
Yi = α + β Xi + εi, Assume Σ εi = 0 β = Cov (X,Y) sx2 OR β = AND α = Y - β X Σ
(Xi – X) (Yi – Y) OR Σ
(Xi – X)2 sx r * sY

9. Independent Variable (X) Dependent Variable (Y)
Example 1.
Independent Variable (X)
Dependent Variable (Y) 6 2 5 3 11 9 7 1 8 4

10. Interpretation
The y-intercept or constant ‘α’ is the starting value of the regression line when X equals zero. A negative ‘α’ implies that the line begins from below zero (or the 3rd and 4th quadrants). A positive ‘α’ will imply the line begins from above zero
Interpretation of the slope ‘β’ is two-fold:
It shows the direction of the curve. A negative ‘β’ implies a downward sloping curve and a positive ‘β’ implies an upward sloping curve
It reflects the rate of change in X for Y. If the slope is (positive) then for every 1 unit increase in X variable, the variable Y will increase by If the slope is (negative) then for every 1 unit increase in X variable, the variable Y will decrease by 0.305

11. Estimated Values of Y or Yᶺ ᶺ Y ᶺ X ᶺ Y X Y
= α + β
= α + β X 6 2
3.47= * 1
4.99 = * 6 5 3
3.78= * 2
4.69= * 5 11 9
4.08= * 3
6.52= * 11 7 1
4.39= * 4
5.30= * 7 8
4.69= * 5 4
5.00= * 6
4.38= * 4
5.30= * 7
4.38 = * 4

12. Regression Line ᶺ ᶺ = α + β X Y ᶺ Y = α + β X (11,9) (5,8) (4,7) (4,5)
(5,3)
(6,2)

13. Error Terms ᶺ ε ᶺ Y X Y = Y - ε = Y – (α + βX) 6 2 2 -3.47 = -1.49
-2.99 = 2 – 3.16 – 0.31 * 6 5 3
= -0.48
-1.69 = 3 – 3.16 – 0.31 * 5 11 9
9 – 4.08 = 4.92
2.49 = 9 – 3.16 – 0.31 * 11 7 1
1 – 4.39 = -3.39
-4.30 = 1 – 3.16 – 0.31 * 7 8
8 – 4.69 = 3.31
3.32 = 8 – 3.16 – 0.31 * 5 4
7 – 5 = 2
2.62 = 7 – 3.16 – 0.31 * 4
5 – 5.30 = 1.30
0.62 = 5 – 3.16 – 0.31 * 4

14. Interpretation of Errors
Σ ≠ Σ ᶺ ε ε
Should be close to or equal to Σ ε ᶺ
If = 0 , then the X can explain Y and the data is a good fit Σ ε

15. Example 2 β = 1.64 α = -7.43 Independent Value (X) Dependent Value (Y)12 14 9 8 6 10 11 13 7 3
β = 1.64
α = -7.43

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