ECON1320 Lecture 5

# ECON1320 Lecture 5 - ECON1320 Quantitative Economics and...

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ECON1320 Quantitative Economics and Business Analysis B LECTURE 5 Multiple Regression A Section 14.1 - 14.4

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Today’s Topics Revision: simple linear regression model (chapter 13) Estimation and Interpretation of the multiple regression (MR) model Prediction using the MR model Hypothesis testing: t -test, F -test Regression statistics: r 2 , adjusted r 2 , S e Assumptions of the (MR) model
3 i i 1 0 i ε X β β Y + + = Linear component Some Revision: Simple Linear Regression The population regression model: Population Y intercept Population Slope Coefficient Random Error term Dependent Variable Independent Variable Random Error component

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4 Slope = β 1 Y X Observed Value of Y for X i X i i i 1 0 i ε X β β Y + + = Intercept = β 0 Random Error for this X i value Predicted Value of Y for X i ε i Simple Linear Regression Model
5 i 1 0 i X b b Y ˆ + = The simple linear regression equation provides an estimate of the population regression line Simple Linear Regression Estimate of the regression intercept Estimate of the regression slope Estimated (or predicted) Y value for observation i Value of X for observation i The individual random error terms e i have a mean of zero

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6 Measures of Variation Total variation is made up of two parts: SSE SSR SST + = Total Sum of Squares Regression Sum of Squares Error Sum of Squares - = 2 i ) Y Y ( SST - = 2 i i ) Y ˆ Y ( SSE - = 2 i ) Y Y ˆ ( SSR where: = Average value of the dependent variable Y i = Observed values of the dependent variable i = Predicted value of Y for the given X i value Y ˆ Y
7 SST = total sum of squares Measures the variation of the Y i values around their mean Y SSR = regression sum of squares Explained variation attributable to the relationship between X and Y SSE = error sum of squares Variation attributable to factors other than the relationship between X and Y Measures of Variation

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8 X i Y X Y i SST = ( Y i - Y ) 2 SSE = ( Y i - Y i ) 2 SSR = ( Y i - Y ) 2 _ _ _ Y Y Y _ Y Measures of Variation
9 The coefficient of determination is the portion of the total variation in the dependent variable that is explained by variation in the independent variable The coefficient of determination is also called r-squared and is denoted as r 2 Coefficient of Determination, r 2 1 r 0 2 note: squares of sum total squares of sum regression SST SSR r 2 = =

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10 Standard Error of Estimate The standard deviation of the variation of observations around the regression line is estimated by 2 n ) Y ˆ Y ( 2 n SSE S n 1 i 2 i i YX - - = - = = Where SSE = error sum of squares n = sample size
11 Slope = β 1 Y X Observed Value of Y for X i X i i i 1 0 i ε X β β Y + + = Intercept = β 0 Random Error for this X i value Predicted Value of Y for X i ε i Simple Linear Regression Model

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12 Assumptions of Simple Regression Normality of Error Error values (ε) are normally distributed for any given value of X Homoscedasticity
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ECON1320 Lecture 5 - ECON1320 Quantitative Economics and...

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