PROPERTIES OF THE LEAST SQUARES ESTIMATORS in .NET

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11-3 PROPERTIES OF THE LEAST SQUARES ESTIMATORS
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Strength y (psi) 2158.70 1678.15 2316.00 2061.30 2207.50 1708.30 1784.70 2575.00 2357.90 2277.70
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Age x (weeks) 15.50 23.75 8.00 17.00 5.00 19.00 24.00 2.50 7.50 11.00
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Strength y (psi) 2165.20 2399.55 1779.80 2336.75 1765.30 2053.50 2414.40 2200.50 2654.20 1753.70
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shear strength and age were perfectly deterministic (no error). Does this plot indicate that age is a reasonable choice of regressor variable in this model 11-13. Show that in a simple linear regression model the point ( x, y ) lies exactly on the least squares regression line. 11-14. Consider the simple linear regression model Y 0 . Suppose that the analyst wants to use z x x as the 1x regressor variable. (a) Using the data in Exercise 11-12, construct one scatter plot of the ( xi, yi) points and then another of the ( zi xi x, yi ) points. Use the two plots to intuitively explain how the two models, Y and 0 1x * *z , are related. Y 0 1 (b) Find the least squares estimates of * and * in the model 0 1 * *z . How do they relate to the least Y 0 1 squares estimates 0 and 1
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* 11-15. Suppose we wish to t the model y* i 0 * 1xi x 2 , where y* yi y (i 1, 2, p , n). Find 1 i i the least squares estimates of * and *. How do they relate 0 1 to 0 and 1 11-16. Suppose we wish to t a regression model for which the true regression line passes through the point (0, 0). The appropriate model is Y x . Assume that we have n pairs of data (x1, y1), (x2, y2), p , (xn, yn). Find the least squares estimate of . 11-17. Using the results of Exercise 11-16, t the model Y x to the chloride concentration-roadway area data in Exercise 11-11. Plot the tted model on a scatter diagram of the data and comment on the appropriateness of the model.
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11-3
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PROPERTIES OF THE LEAST SQUARES ESTIMATORS
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The statistical properties of the least squares estimators 0 and 1 may be easily described. Recall that we have assumed that the error term in the model Y is a random 0 1x variable with mean zero and variance 2. Since the values of x are xed, Y is a random vari2 able with mean Y 0 x . Therefore, the values of 0 and 1 depend 0 1x and variance on the observed y s; thus, the least squares estimators of the regression coef cients may be viewed as random variables. We will investigate the bias and variance properties of the least squares estimators 0 and 1. Consider rst 1. Because 1 is a linear combination of the observations Yi, we can use properties of expectation to show that expected value of 1 is E1 1 2
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(11-15)
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Thus, 1 is an unbiased estimator of the true slope
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CHAPTER 11 SIMPLE LINEAR REGRESSION AND CORRELATION
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Now consider the variance of 1. Since we have assumed that V( i) 2 , and it can be shown that V(Yi) V1 1 2 For the intercept, we can show that E1 0 2 and V1 0 2
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(11-16)
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(11-17)
Thus, 0 is an unbiased estimator of the intercept 0. The covariance of the random variables 2 and is not zero. It can be shown (see Exercise 11-69) that cov( , ) x Sxx. 0 1 0 1 2 The estimate of could be used in Equations 11-16 and 11-17 to provide estimates of the variance of the slope and the intercept. We call the square roots of the resulting variance estimators the estimated standard errors of the slope and intercept, respectively. De nition In simple linear regression the estimated standard error of the slope and the estimated standard error of the intercept are se1 1 2 B Sxx