2 the larger the sum of squares the smaller the

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2. The larger the sum of squares, , the smaller the variances of the least squares estimators and the more precisely we can estimate the unknown parameters. 3. The larger the sample size N , the smaller the variances and covariance of the least squares estimators. MAJOR POINTS ABOUT THE VARIANCES AND COVARIANCES OF b 1 AND b 2 2.4 Assessing the Least Squares Fit 2.4.4 The Variances and Covariances of b 1 and b 2 ² ³ 2 ¦ ´ x x i
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Principles of Econometrics, 4t h Edition Page 30 Chapter 2: The Simple Linear Regression Model Figure 2.11 The influence of variation in the explanatory variable x on precision of estimation (a) Low x variation, low precision (b) High x variation, high precision The variance of b 2 is defined as > @ 2 2 2 2 ) ( ) var( b E b E b ´ 2.4 Assessing the Least Squares Fit 2.4.4 The Variances and Covariances of b 1 and b 2 Principles of Econometrics, 4t h Edition Page 31 Chapter 2: The Simple Linear Regression Model 2.5 The Gauss-Markov Theorem Under the assumptions SR1-SR5 of the linear regression model, the estimators b 1 and b 2 have the smallest variance of all linear and unbiased estimators of b 1 and b 2 . They are the Best Linear Unbiased Estimators (BLUE) of b 1 and b 2 GAUSS-MARKOV THEOREM
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Principles of Econometrics, 4t h Edition Page 32 Chapter 2: The Simple Linear Regression Model If we make the normality assumption (assumption SR6 about the error term) then the least squares estimators are normally distributed: (2.17) (2.18) Probability Distributions of the Least Squares Estimators ² ³ 2 2 1 1 2 σ ~ β , i i x b N N x x § · ¨ ¸ ¨ ¸ ´ © ¹ ¦ ¦ ² ³ 2 2 2 2 σ ~ β , i b N x x § · ¨ ¸ ¨ ¸ ´ © ¹ ¦ Principles of Econometrics, 4t h Edition Page 33 Chapter 2: The Simple Linear Regression Model If assumptions SR1-SR5 hold, and if the sample size N is sufficiently large , then the least squares estimators have a distribution that approximates the normal distributions shown in Eq. 2.17 and Eq. 2.18 2.6 The Probability Distributions of the Least Squares Estimators A CENTRAL LIMIT THEOREM
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Principles of Econometrics, 4t h Edition Page 34 Chapter 2: The Simple Linear Regression Model Since the expectation is an average value we might consider estimating e 2 as the average of the squared error, i.e. The random errors e i are unobservable. The least squares residuals are obtained by replacing the unknown parameters by their least squares estimates, : Subtractiing with the number of regression parameters in the denominator of the model produces an unbiased estimator: so that 2 ˆ ˆ 2 2 ´ ¦ N e i V ² ³ 2 2 ˆ σ σ E 2.7 Estimating the Variance of the Error Term Variance of Error Term N e i ¦ 2 2 ˆ V i i i i i x b b y y y e 2 1 ˆ ˆ ´ ´ ´ Principles of Econometrics, 4t h Edition Page 35 Chapter 2: The Simple Linear Regression Model 2.7.2 Calculations for the Food Expenditure Data 2 2 ˆ 304505.2 ˆ σ 8013.29 2 38 i e N ´ ¦ Table 2.3 Least Squares Residuals 2.7 Estimating the Variance of the Error Term
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Principles of Econometrics, 4t h Edition Page 36 Chapter 2: The Simple Linear Regression Model
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