Consider linear regression model y = XB + e with transposed design matrix XT given by...

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Consider linear regression model y = XB + e with transposed design matrix XT given by > t(data.frame (X)) [, 1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] X1 -1 -1 -1 1 -1 1 X2 -1 1 -1 -1 X3 -1 1 -1 1 1 1 1 X4 -1 -1 -1 -1 The model does not include an intercept. The transpose of the response vector y" is given by > t(data.frame (y=y)) [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] y -0.71 -1.1 -0.12 1.4 -1.1 -4 -0.83 -0.25 -2.5 1.5 Assuming the standard assumption that e- N(0, o In), answer the following: (a) Which of the 10 data points have the highest influence on the regression, based on the PRESS residuals? (b) Using the PRESS statistic, among the the following two models, which one should be selected: (1) the model that drops a1 or (2) the model that drops x2? Now, suppose that we realize that the standard assumption on errors was not true. Instead, the following holds E; - N(0,0??), i = 1,...,n with {€1, €2, ...,En} assumed independent and o? unknown. Answer the following under this new assumption: (c) Is n-4 an unbiased estimator of o?, where e is the usual residual vector based on the least- squares estimate? Justify your answer. (d) Is the least-squares estiamte B = (XTX)-XTy unbiased for 3? Justify your answer. (e) Find a 95% confidence interval for ß1 + B2 + B3 + B4. Consider linear regression model y = XB + e with transposed design matrix XT given by > t(data.frame (X)) [, 1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] X1 -1 -1 -1 1 -1 1 X2 -1 1 -1 -1 X3 -1 1 -1 1 1 1 1 X4 -1 -1 -1 -1 The model does not include an intercept. The transpose of the response vector y" is given by > t(data.frame (y=y)) [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] y -0.71 -1.1 -0.12 1.4 -1.1 -4 -0.83 -0.25 -2.5 1.5 Assuming the standard assumption that e- N(0, o In), answer the following: (a) Which of the 10 data points have the highest influence on the regression, based on the PRESS residuals? (b) Using the PRESS statistic, among the the following two models, which one should be selected: (1) the model that drops a1 or (2) the model that drops x2? Now, suppose that we realize that the standard assumption on errors was not true. Instead, the following holds E; - N(0,0??), i = 1,...,n with {€1, €2, ...,En} assumed independent and o? unknown. Answer the following under this new assumption: (c) Is n-4 an unbiased estimator of o?, where e is the usual residual vector based on the least- squares estimate? Justify your answer. (d) Is the least-squares estiamte B = (XTX)-XTy unbiased for 3? Justify your answer. (e) Find a 95% confidence interval for ß1 + B2 + B3 + B4.

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Model1 Call lmformula y x1 1 data data Residuals Min 1Q Median 3Q Max 40000 14800 06700 02025 15000 Coefficients Estimate Std Error t value Prt x1 038... View the full answer