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1 Theory and Estimation of Regression Models Simple Regression Theory Sumber:

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2 Population Line: uiui Y i = E[Y i ]+u i Xi E[Y] = B 0 +B 1 X E[Y i ] = B 0 +B 1 X i Sumber:

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3 Population Line: eiei Y i = Y i + e i Xi E[Y] = B 0 +B 1 X Y i = B 0 +B 1 X i ^ ^^^ Estimated Line: Y = B 0 +B 1 X ^^^ uiui E[Y i ] Sumber:

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4 eiei Xi Y = B 0 +B 1 X ^^^ eiei eiei eiei eiei eiei Sumber:

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Metode Ordinary Least Squares (OLS) In the Ordinary Least Squares (OLS) method, the criterion for estimating β 0 and β 1 is to make the sum of the squared residuals (SSR) of the fitted regression line as small as possible i.e.: Minimize SSR = minimize = minimize Sumber:

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6 Rumus estimator OLS : (5.12) (5.13) Sumber: Metode Ordinary Least Squares (OLS)

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7 Garis regresi yang diestimasi dengan menggunakan metode OLS mempunyhai ciri- ciri : 1. (i.e. the sum of its residuals is zero) 2. It always passes through the point The residual values (e i ’s) are not correlated with the values of the independent variable (X i ’s) Sumber: Metode Ordinary Least Squares (OLS)

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8 Interpretasi Model Regresi Assume, for example, that the estimated or fitted regression equation is: or Y i = X i + e i Sumber:

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10 The value of = 0.15 indicates that if the average cotton price received by farmers in the previous year increases by 1 cent/pound (i.e. X=1), then this year’s cotton acreage is predicted to increase by 0.15 million acres (150,000 acres). Y i = X i + e i Sumber: Interpretasi Model Regresi

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11 The value of = 3.7 indicates that if the average cotton price received by farmers in the previous year was zero (i.e. =0), the cotton acreage planted this year will be 3.7 million (3,700,000) acres; sometimes the intercept makes no practical sense. Y i = X i + e i Sumber: Interpretasi Model Regresi

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12 There are two statistics (formulas) that quantify how well the estimated regression line fits the data: 1. The standard error of the regression (SER) (Sometimes called the standard error of the estimate) 2. R 2 - coefficient of determination Sumber: Mengukur Goodness of Fit: R 2

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13 SER agak berbeda dengan simpangan-baku (standard deviasi S) e i (oleh derajat bebasnya): (5.20) Sumber: Mengukur Goodness of Fit: R 2

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The term on the left measures the proportion of the total variation in Y not explained by the model (i.e. by X) R 2 mengukur proporsi dari total ragam Y yang dapat dijelaskan oleh model (yaitu dijelaskan oleh X) Sumber:

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15 The Gauss-Markov Theorem states the properties of the OLS estimators; i.e. of the: and They are unbiased E[B 0 ]= and E[B 1 ]= Sumber: Sifat-sifat Estimator OLS

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16 Sifat-sifat Estimator OLS If the dependent variable Y (and thus the error term of the population regression model, u i ) has a normal distribution, the OLS estimators have the minimum variance Sumber:

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17 BLUE – Best Linear Unbiased Estimator Unbiased => bias of β j = E(β j ) - β j = 0 Best Unbiased => minimum variance & unbiased Linear => the estimator is linear ^ ^ Sumber: Sifat-sifat Estimator OLS

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