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Regression Analysis Lecture 9

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Regression analysis establishes relationship between a dependent variable and independent variables Relationship between Cause and Effecttt 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 : 1.Shows for example whether there is any relationship between an increase in household income (Y) land an increase in consumption (C ). 2.Whether there is positive or negative relationship between Y and C. Whether if : Y C or reverse 1.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) Income of the Household (,000 Tk) 0 Each dot represent sample data for Income and Expenditure for each sample household

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Consumption Expenditure ( C ) Income of the Household (Y) 0 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. 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 Y1089 Av Y is 9 C867 Av C is 7 Dependent variable C - C Sum is zero (C – C)**211 0 Sum of square is + number

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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) Annual Expenditure (,000Tk) (C )

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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 Annual Expenditure (C ) Av C = 100 Ci - C

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Example (Ci – C) (Yi –Y) = = 5000 (Yi – Y) 2 = = 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 = 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 (Ci – C )**2 r = b

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The End

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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) Annual Expenditure (,000Tk) (C ) Class Assignment

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The End

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