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Lecture 9 Regression Analysis

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**Relationship between “Cause” and “Effect”t **

Regression analysis establishes relationship between a dependent variable and independent variables Relationship between “Cause” and “Effect”t Relationship between variables

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**Usefulness of regression analysis**

Regression analysis is a vary widely used tool for research. It shows type and magnitude of relationship between two variables.

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**Example of Usefulness of Regression Analysis**

: Shows for example whether there is any relationship between an increase in household income (Y) land an increase in consumption (C ). Whether there is positive or negative relationship between Y and C. Whether if : Y C or reverse How much of an increase in income (Y) is spent on consumption ( C ).

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**Example of Usefulness of Regression Analysis**

Regression is also used for prediction and forecasting, Regression analysis allows to measure confidence or significance level of the findings.

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**Example of Usefulness of Regression Analysis**

Increase in traffic jam (hours of non-movement) depends on Increase in number of cars in Dhaka City. (+ dependency) A decrease in number of School drop-out depends on an increase I income of parents.(-ve dependency) An increase in household income leads to an increase in household consumption.

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**Other Logical Examples of Positive and Negative Dependency**

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**Forms of regression models**

A regression model relates dependent variable Y to be a function/relation of independent variable X. Symbolically, Y = f (Xi) Where i = 1,2,3,4,…

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**Diagrammatic Representation of Regression Model**

Consumption Expenditure(,000Tk) Each dot represent sample data for Income and Expenditure for each sample household 100 90 130 Income of the Household (,000 Tk) 120

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**Consumption Expenditure ( C )**

C = a + by Regression analysis draw a mean /average line with equation C = a + b Y so that difference between sample data and estimated data is minimized. Income of the Household (Y) Does dotted line minimize deviations?

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**Deviations between sample value and the mean value**

Mean value line

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**Diagrammatic Representation of Regression Equation**

In mean or average line, square of the deviation ( C i) for each of the sample from mean ( C )is minimized. Why ? Because simple sum of difference from mean is always zero.

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**Example C - C Sum is zero 1 -1 0 (C – C)**2 1 0**

(C – C)**2 Sum of square is + number Y 10 8 9 Av Y is 9 C 6 7 Av C is 7 Dependent variable

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Formula for Regression coefficient b when sum of square is minimized , b = (Ci – C) (Yi –Y) (Yi – Y) 2 i = 1,2, ….n

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General Formula If Y is dependent variable and X is independent variable e.g. Y = f (x) then Regression coefficient = Sum of (Xi –X) (Yi – Y) Sum of (Xi –X)**2

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Example : Given the following data C = f (Y), predict Consumption level for a household with annual income of 500 thousand Taka Annual Income (Y) (,000Tk) 100 150 200 250 300 Annual Expenditure (C ) 80 90 110 120

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Example : Given the following data, predict Consumption level for a household with annual income of 500 thousand Taka. (Fig in,000Tk) Annual Income (Y) Av Y = 200 Yi - Y 100 -100 150 -50 200 250 50 300 Annual Expenditure (C ) Av C = 100 Ci - C 80 -20 90 -10 110 10 120 20

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**Example (Ci – C) (Yi –Y) = 2000 +500 + 0 + 500 + 2000 = 5000**

= = 5000 (Yi – Y) 2 = = 25000 Therefore b = (Ci – C) (Yi –Y) / (Yi – Y) 2 = 0.2

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**Calculated Regression Equation Example**

C = a + b Y Or C = a Y or C = a Y Or a = C -0.2 Y Or a = x 200 = 100 – 40 = 60 Therefore C = Y

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**Calculated Regression Equation Example**

C = Y What kind of relationship between Y and C ? How much consumption increases for Tk 1000 increase in income ?

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**C = 60 +0.2 Y What is consumption, when income is zero?**

What is predicted consumption, when income is Tk 500,000?

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**Correlation : A measure of simple relationship**

Correlation shows only associanship or relationship between two variables. Whereas Regression analysis shows dependency relationship Correlation between two variables ( for example Income and Expenditure) is measured by a formula shown as ;

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**Formula of Correlation coefficient r is (Ci – C) (Yi –Y) (Yi – Y) 2 (Ci – C)2**

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**Formula of Correlation coefficient r in terms of regression coefficient r**

(Yi – Y)**2 r = b (Ci – C )**2

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The End

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Class Assignment Given the following data, calculate correlation coefficient between Income and Expenditure. Also predict how much Consumption will increase for a 1000 Tk increase in household income? Annual Income (Y) (,000Tk) 110 160 210 260 310 Annual Expenditure (C ) 75 85 95 105 115

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The End

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