# Theory and Estimation of Regression Models Simple Regression Theory

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Theory and Estimation of Regression Models Simple Regression Theory
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Population Line: E[Y] = B0+B1X Yi = E[Yi]+ui ui E[Yi] = B0+B1Xi Xi
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Population Line: E[Y] = B0+B1X ^ Yi = Yi + ei Estimated Line: ^ ^ ^
ui ^ ^ ^ Yi = B0+B1Xi E[Yi] Xi Sumber:

^ ^ ^ Y = B0+B1X ei ei ei ei ei ei Xi
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Metode Ordinary Least Squares (OLS)
In the Ordinary Least Squares (OLS) method, the criterion for estimating β0 and β1 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:

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

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

Interpretasi Model Regresi
Assume, for example, that the estimated or fitted regression equation is: or Yi = Xi + ei Sumber:

Sumber: www. aaec. ttu. edu/faculty/omurova/aaec_4302/. /Chapter%205
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Interpretasi Model Regresi
Yi = Xi + ei 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). Sumber:

Interpretasi Model Regresi
Yi = Xi + ei 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. Sumber:

Mengukur Goodness of Fit: R2
There are two statistics (formulas) that quantify how well the estimated regression line fits the data: The standard error of the regression (SER) (Sometimes called the standard error of the estimate) R2 - coefficient of determination Sumber:

Mengukur Goodness of Fit: R2
SER agak berbeda dengan simpangan-baku (standard deviasi S) ei (oleh derajat bebasnya): (5.20) Sumber:

Mengukur Goodness of Fit: R2
The term on the left measures the proportion of the total variation in Y not explained by the model (i.e. by X) R2 mengukur proporsi dari total ragam Y yang dapat dijelaskan oleh model (yaitu dijelaskan oleh X) Sumber:

Sifat-sifat Estimator OLS
The Gauss-Markov Theorem states the properties of the OLS estimators; i.e. of the: and They are unbiased E[B0 ]= and E[B1]= Sumber:

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

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