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ECONOMETRICS I CHAPTER 3: TWO VARIABLE REGRESSION MODEL: THE PROBLEM OF ESTIMATION Textbook: Damodar N. Gujarati (2004) Basic Econometrics, 4th edition, The McGraw-Hill Companies

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3.1 THE METHOD OF ORDINARY LEAST SQUARES PRF: SRF: How is SRF determined? We do not minimize the sum of the residuals! Why not?

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Least squares criterion

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3.1 THE METHOD OF ORDINARY LEAST SQUARES We adopt the least-squares criterion We want to minimize the sum of the squared residuals. This sum is a function of estimated parameters: Normal equations:

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3.1 THE METHOD OF ORDINARY LEAST SQUARES Solving the normal equations simultaneously, we obtain the following: Beta2-hat can be alternatively expressed as the following:

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Three Statistical Properties of OLS Estimators I. The OLS estimators are expressed solely in terms of the observable quantities (i.e. X and Y). Therefore they can easily be computed. II. They are point estimators (not interval estimators). Given the sample, each estimator provide only a single (point) value of the relevant population parameter. III.Once the OLS estimates are obtained from the sample data, the sample regression line can be easily obtained.

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The properties of the regression line 1.It passes through the sample means of Y and X.

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The properties of the regression line 2.

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The properties of the regression line 3. The mean value of the residuals is zero.

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The properties of the regression line 4. 5.

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3.2 The Classical Linear Regression Model: The Assumptions Underlying the Method of Least Squares

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Example of perfect multicollinearity: X 1 = 2X 2 +X 3 YX1X1 X2X2 X3X

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PRECISION OR STANDARD ERRORS OF LEAST SQUARES ESTIMATES var: variance se: standard error : the constant homoscedastic variance of ui : the standard error of the estimate : OLS estimator of

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Gauss – Markov Theorem An estimator, say the OLS estimator, is said to be a best linear unbiased estimator (BLUE) of β 2 if the following hold:

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The coefficient of determination r 2 TSS: total sum of squares ESS: explained sum of squares RSS: residual sum of squares

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The coefficient of determination r 2 The quantity r 2 thus defined is known as the (sample) coefficient of determination and is the most commonly used measure of the goodness of fit of a regression line. Verbally, r 2 measures the proportion or percentage of the total variation in Y explained by the regression model.

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The coefficient of determination r 2

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The coefficient of determination r 2

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The coefficient of correlation r r is the sample correlation coeffient

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Some of the properties of r

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Homework Study the numerical example on pages There will be questions on the midterm exam similar to the ones in this example. Data on page 88:

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Homework

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