WeBWorK Standalone Renderer

You are being given the covariance matrix of a multiple regression with ${\hat\beta}_0,{\hat\beta}_1,\ldots,{\hat\beta}_5,{\hat\beta}_6, \ldots,{\hat\beta}_9$ where $x_1,\ldots,x_5$ are numerical explanatory variables and $x_6,\ldots,x_9$ are 4 binary dummy variables for 4 different categories relative to the baseline category. The categorical explanatory variable has 5 categories.

You are given that the estimated covariance matrix is:

$\left[\begin{array}{cccccccccc} 0.33347 &-0.00123 &0.000379 &-0.00635 &-0.007631 &-0.000744 &-0.036518 &-0.054614 &-0.049256 &-0.043163\cr -0.00123 &1.1\times 10^{-5} &3\times 10^{-6} &0 &3.6\times 10^{-5} &1\times 10^{-6} &1.9\times 10^{-5} &0.000121 &0.0002 &9.9\times 10^{-5}\cr 0.000379 &3\times 10^{-6} &1.9\times 10^{-5} &-3.5\times 10^{-5} &2.6\times 10^{-5} &-1.1\times 10^{-5} &-7.2\times 10^{-5} &1.2\times 10^{-5} &3.2\times 10^{-5} &-1.6\times 10^{-5}\cr -0.00635 &0 &-3.5\times 10^{-5} &0.000281 &7.7\times 10^{-5} &1.9\times 10^{-5} &0.000796 &0.000944 &0.000292 &0.000309\cr -0.007631 &3.6\times 10^{-5} &2.6\times 10^{-5} &7.7\times 10^{-5} &0.00051 &-4\times 10^{-5} &0.001221 &0.002007 &0.001827 &0.001396\cr -0.000744 &1\times 10^{-6} &-1.1\times 10^{-5} &1.9\times 10^{-5} &-4\times 10^{-5} &1.6\times 10^{-5} &-3.2\times 10^{-5} &-6.9\times 10^{-5} &-2.1\times 10^{-5} &6.8\times 10^{-5}\cr -0.036518 &1.9\times 10^{-5} &-7.2\times 10^{-5} &0.000796 &0.001221 &-3.2\times 10^{-5} &0.022795 &0.016794 &0.014729 &0.014576\cr -0.054614 &0.000121 &1.2\times 10^{-5} &0.000944 &0.002007 &-6.9\times 10^{-5} &0.016794 &0.030437 &0.017899 &0.016878\cr -0.049256 &0.0002 &3.2\times 10^{-5} &0.000292 &0.001827 &-2.1\times 10^{-5} &0.014729 &0.017899 &0.025177 &0.017092\cr -0.043163 &9.9\times 10^{-5} &-1.6\times 10^{-5} &0.000309 &0.001396 &6.8\times 10^{-5} &0.014576 &0.016878 &0.017092 &0.029926 \end{array}\right]$

or in R:
acov=matrix(c(0.33347, -0.00123, 0.000379, -0.00635, -0.007631, -0.000744, -0.036518, -0.054614, -0.049256, -0.043163, -0.00123, 1.1e-05, 3e-06, 0, 3.6e-05, 1e-06, 1.9e-05, 0.000121, 0.0002, 9.9e-05, 0.000379, 3e-06, 1.9e-05, -3.5e-05, 2.6e-05, -1.1e-05, -7.2e-05, 1.2e-05, 3.2e-05, -1.6e-05, -0.00635, 0, -3.5e-05, 0.000281, 7.7e-05, 1.9e-05, 0.000796, 0.000944, 0.000292, 0.000309, -0.007631, 3.6e-05, 2.6e-05, 7.7e-05, 0.00051, -4e-05, 0.001221, 0.002007, 0.001827, 0.001396, -0.000744, 1e-06, -1.1e-05, 1.9e-05, -4e-05, 1.6e-05, -3.2e-05, -6.9e-05, -2.1e-05, 6.8e-05, -0.036518, 1.9e-05, -7.2e-05, 0.000796, 0.001221, -3.2e-05, 0.022795, 0.016794, 0.014729, 0.014576, -0.054614, 0.000121, 1.2e-05, 0.000944, 0.002007, -6.9e-05, 0.016794, 0.030437, 0.017899, 0.016878, -0.049256, 0.0002, 3.2e-05, 0.000292, 0.001827, -2.1e-05, 0.014729, 0.017899, 0.025177, 0.017092, -0.043163, 9.9e-05, -1.6e-05, 0.000309, 0.001396, 6.8e-05, 0.014576, 0.016878, 0.017092, 0.029926),10,10)

Please use 3 decimal places for the answers below which are not integer-valued.

Part a)
Based on the (estimated) covariance matrix, what is the SE of ${\hat\beta}_{6} - {\hat\beta}_{8}$ ? Note that ${\hat\beta}_{6} - {\hat\beta}_{8}$ corresponds to the estimated distance of the hyperplanes for two different categories where the binary dummy variables are indexed as $x_{6}$ and $x_{8}$ .

Part b)
Based on the (estimated) covariance matrix, what is the SE of ${\hat\beta}_{6} + {\hat\beta}_{8} -2{\hat\beta}_{7}$ ? Note that ${\hat\beta}_{6} + {\hat\beta}_{8} -2{\hat\beta}_{7}$ corresponds to a contrast to two categories (where the binary dummy variables are indexed as $x_{6}$ ) and $x_{8}$ with a third category (where the binary dummy variable is indexed as $x_{7}$ ).

Hint:

You can earn partial credit on this problem.