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I have problem with summation when it is in matrices. I want to make mathematica compute this:

enter image description here

Large X I have added dataset for as well as e. Would anyone be able to show me how to compute this in mathematica.

Due to comments Ill explain a bit more, X' is Transpose[X] so yes it would be a 2 by 2 matrix.

X= {{20.08, 1}, {19.78, 1}, {19.86, 1}, {19.4, 1}, {19.66, 1}, {19.32, 
  1}, {18.99, 1}, {19.29, 1}, {19.43, 1}, {19.52, 1}, {20.07, 
  1}, {20.13, 1}, {20.02, 1}, {19.9, 1}, {20.11, 1}, {20.44, 
  1}, {20.37, 1}, {20.31, 1}, {19.7, 1}, {20.12, 1}, {19.63, 
  1}, {19.76, 1}, {19.34, 1}, {19.66, 1}, {19.85, 1}, {20.32, 
  1}, {20.5, 1}, {20.9, 1}, {21.21, 1}, {21.15, 1}, {21.18, 
  1}, {21.02, 1}, {21.29, 1}, {20.92, 1}, {20.83, 1}, {21.22, 
  1}, {20.73, 1}, {20.76, 1}, {21.16, 1}, {21.31, 1}, {21.94, 
  1}, {22.7, 1}, {22.95, 1}, {22.8, 1}, {22.7, 1}, {22.9, 1}, {22.6, 
  1}, {22.4, 1}, {22.34, 1}, {22.43, 1}, {22.07, 1}, {22.01, 
  1}, {22.17, 1}, {22.1, 1}, {22.86, 1}, {23.26, 1}, {23.38, 
  1}, {23.59, 1}, {23.59, 1}, {23.53, 1}, {23.5, 1}, {23.78, 
  1}, {23.38, 1}, {23.52, 1}, {23.3, 1}, {23.48, 1}, {23.83, 
  1}, {24.2, 1}, {24.49, 1}, {24.69, 1}, {23.98, 1}, {23.79, 
  1}, {23.7, 1}, {23.26, 1}, {23.47, 1}, {23.95, 1}, {24.38, 
  1}, {24.75, 1}, {25.2, 1}, {24.68, 1}, {24.43, 1}, {24.73, 
  1}, {24.3, 1}, {24.97, 1}, {25.07, 1}, {25.17, 1}, {26.07, 
  1}, {26.41, 1}, {26.24, 1}, {26.83, 1}, {26.39, 1}, {26.64, 
  1}, {27.34, 1}, {26.58, 1}, {26.52, 1}, {26.58, 1}, {27., 
  1}, {26.54, 1}, {26.02, 1}, {26.33, 1}, {26.07, 1}, {25.81, 
  1}, {26.33, 1}, {26.3, 1}, {26.39, 1}, {26.26, 1}, {25.75, 
  1}, {26.21, 1}, {27.04, 1}, {26.74, 1}, {26.4, 1}, {25.89, 
  1}, {26.37, 1}, {26.28, 1}, {26.54, 1}, {26.66, 1}, {26.24, 
  1}, {25.35, 1}, {25.19, 1}, {24.07, 1}, {24.96, 1}, {25.29, 
  1}, {25.47, 1}, {25.13, 1}, {25.24, 1}, {24.99, 1}, {25.59, 
  1}, {25.68, 1}, {25.53, 1}, {26.68, 1}, {26.56, 1}, {27.04, 
  1}, {27.23, 1}, {27.34, 1}, {26.88, 1}, {29.66, 1}, {29.66, 
  1}, {29.11, 1}, {27.86, 1}, {27.36, 1}, {27.84, 1}, {28.27, 
  1}, {28.11, 1}, {27.93, 1}, {28.05, 1}, {28.09, 1}, {27.96, 
  1}, {27.65, 1}, {27.67, 1}, {27.32, 1}, {27.39, 1}, {27.48, 
  1}, {27.68, 1}, {28.35, 1}, {28.34, 1}, {28.05, 1}, {27.14, 
  1}, {27.32, 1}, {26.91, 1}, {27.12, 1}, {27.06, 1}, {27.9, 
  1}, {27.99, 1}, {27.7, 1}, {27., 1}, {27.11, 1}, {27.3, 1}, {27.12, 
  1}, {27.78, 1}, {28.07, 1}, {28.23, 1}, {28.17, 1}, {29.24, 
  1}, {29.48, 1}, {29.19, 1}, {29.65, 1}, {29.26, 1}, {29.62, 
  1}, {30., 1}, {30.44, 1}, {31.03, 1}, {30.93, 1}, {30.26, 
  1}, {31.27, 1}, {31.34, 1}, {32.75, 1}, {33.55, 1}, {33.17, 
  1}, {34.31, 1}, {34.14, 1}, {33.88, 1}, {34.89, 1}, {34.14, 
  1}, {32.93, 1}, {33.01, 1}, {33.87, 1}, {34.15, 1}, {34., 
  1}, {33.38, 1}, {33.09, 1}, {32.74, 1}, {33.43, 1}, {34.06, 
  1}, {33.94, 1}, {33.1, 1}, {33.08, 1}, {32.25, 1}, {32.35, 
  1}, {33.17, 1}, {32.57, 1}, {32.94, 1}, {33.18, 1}, {33.19, 
  1}, {32.97, 1}, {32.88, 1}, {32.11, 1}, {33.12, 1}, {33.82, 
  1}, {34.07, 1}, {35.1, 1}, {35.69, 1}, {35.47, 1}, {34.98, 
  1}, {34.63, 1}, {35.62, 1}, {36.3, 1}, {36.49, 1}, {36.29, 
  1}, {36.64, 1}, {36.96, 1}, {36.98, 1}, {37.01, 1}, {36.56, 
  1}, {38.13, 1}, {38.87, 1}, {38.86, 1}, {38.87, 1}, {40.22, 
  1}, {39.16, 1}, {39.35, 1}, {39.73, 1}, {39.73, 1}, {39.51, 
  1}, {40.04, 1}, {40.2, 1}, {40.12, 1}, {40.77, 1}, {40.85, 
  1}, {40.65, 1}}



    e={-39.5086, -34.3476, -21.3119, -17.2624, -22.8212, -12.6281, \
-4.05305, -10.284, -29.2365, -28.9788, -47.4106, -52.2288, -57.3004, \
-57.044, -21.7327, -14.2378, -13.6716, -15.6634, -3.07337, -9.56075, \
0.0428573, 17.8984, 7.95581, 9.61876, 11.0461, 7.3986, 15.974, \
6.7027, 0.0836829, 3.19188, 7.67778, 15.2763, 24.9694, 16.4266, \
20.9189, 18.9357, 17.7393, 16.6252, 19.9538, 19.0034, 14.0373, \
17.3368, 30.486, 25.6365, 28.4368, 24.1961, 32.2371, 28.1878, 26.836, \
21.6437, 20.0929, 14.5311, 15.5325, 20.0288, 4.56828, -2.70303, \
-5.26942, -5.80811, -15.5681, -23.0899, -15.6158, -8.19073, -8.70942, \
-22.0519, -30.5952, -41.6398, -44.3709, -37.6881, -42.0611, -45.5617, \
-42.4614, -28.0188, -37.4065, -47.103, -16.4617, -23.8073, -22.8127, \
-23.1099, -34.5714, -34.0337, -12.4429, -11.6738, -9.01844, -10.4266, \
4.10304, 18.3527, -0.190233, 10.8367, 11.3732, 10.7993, 15.0527, \
20.6719, 38.3996, 38.3901, 44.6483, 43.0501, 34.8527, 24.5722, \
26.1599, 11.7909, 23.8498, 14.9986, 9.23092, 10.165, 5.15272, \
-1.32286, 7.33681, 5.00731, 6.98059, 22.2116, 17.1747, 17.4044, \
14.8388, 14.3011, 21.4022, 33.8758, 40.5732, 38.6981, 37.4166, \
43.9763, 26.3214, 28.5563, 29.1317, 37.8048, 43.5765, 41.9573, \
36.5953, 42.233, 56.2035, 38.1798, 40.8262, 47.4906, 47.248, 47.1996, \
49.3891, 4.39601, -3.47399, -8.86219, 24.9919, 25.9935, 17.508, \
-4.48744, -4.30891, -4.54432, 2.20928, -1.59285, 16.9316, 24.1806, \
22.2945, 19.3957, 12.3694, 12.9571, 7.54649, -7.9417, -12.0437, \
-18.9007, -14.6697, -20.2643, -5.04498, -8.59367, -3.73548, -12.6302, \
-17.5525, -16.7896, -21.9673, -25.3056, -21.4283, -27.1137, -24.0638, \
-17.3968, -11.9953, -11.0271, -19.4666, -22.6394, -10.5365, -20.936, \
-18.7627, -25.7619, -33.4171, -23.1605, -37.4145, -31.1141, -37.136, \
-52.7573, -63.4735, -84.8161, -99.2387, -93.7135, -100.639, -96.9627, \
-104.754, -124.455, -119.213, -112.161, -111.235, -112.446, -113.121, \
-106.63, -90.9322, -70.3292, -74.0381, 37.6777, 19.6116, 25.208, \
62.8927, 57.3488, 60.1155, 58.3352, 66.6165, 70.2785, 64.5913, \
57.2585, 56.1704, 55.0572, 57.7095, 55.078, 47.1967, 30.6944, \
27.9336, 32.3462, 25.7823, 27.589, 33.3226, 32.5037, 13.9685, \
14.9823, 9.79964, 27.0003, 33.9492, 33.5921, 29.756, 24.1719, \
30.0634, 10.1723, -2.36215, 10.3259, 18.4378, 3.62343, 13.0049, \
2.67227, -12.503, -0.312969, 0.043751, 6.55802, 5.49949, 21.1638, \
25.3716, 20.8474, 29.628}

I have added the explanation from the book:

enter image description here

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  • $\begingroup$ is $X'$ supposed to be the transpose of $X$? Then $X'X$ is just a 2 by 2 matrix. What is $N$ in there? You can say few words to explain the symbols used. $\endgroup$ – Nasser Dec 27 '13 at 7:28
  • $\begingroup$ That is correct it is just the transpose. $\endgroup$ – ALEXANDER Dec 27 '13 at 7:31
  • $\begingroup$ But then, the result is a matrix. Yet, you say there is a linear sum there. This does not add up. How can sum[x x',{1,N}] be a sum over a matrix? a matrix is 2D not 1D. May be someone better than me in math can figure it out. $\endgroup$ – Nasser Dec 27 '13 at 7:35
  • $\begingroup$ Nope, everything in there is taken straight out of textbook. Should be correct! It is also given proof in the textbook, however did not think it was any reason to post it because I am only interested in how to make mathematica due the calculation. $\endgroup$ – ALEXANDER Dec 27 '13 at 7:37
  • $\begingroup$ I could add the "proof" if that makes a difference? Could add the explanation, could be that I have misunderstood! $\endgroup$ – ALEXANDER Dec 27 '13 at 7:39
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Please beware that this is NOT optimized. I just wrote down the formulas for easy understanding:

m =  SparseArray[{p_, q_} :> Sum[x[[i, p]] x[[i, q]], {i, Length@x}], {2, 2 }];
me = SparseArray[{p_, q_} :> Sum[e[[i]]^2 x[[i, p]] x[[i, q]], {i, Length@x}], {2, 2}];
mi = Inverse[m];
(v = mi.me.mi) // MatrixForm

$\left( \begin{array}{cc} 0.187057 & -4.58584 \\ -4.58584 & 116.389 \\ \end{array} \right)$

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  • $\begingroup$ Would you be able to optimize it? $\endgroup$ – ALEXANDER Dec 28 '13 at 1:03
  • $\begingroup$ @ALEXANDER I don't think it will serve any purpose, since Mathematica already has its own least squares algorithms implemented $\endgroup$ – Dr. belisarius Dec 28 '13 at 1:08
  • $\begingroup$ But will the least square in mathematica be able to create newey west standard errors and white standard errors? $\endgroup$ – ALEXANDER Dec 28 '13 at 1:13
  • $\begingroup$ @ALEXANDER If you have heteroskedasticity, take a look at ARCH and GARCH in the docs $\endgroup$ – Dr. belisarius Dec 28 '13 at 1:25
  • $\begingroup$ The problem is that I need to be able to handle it with Newey west standard errors or white standard errors, I am aware of ARCH and GARCH. $\endgroup$ – ALEXANDER Dec 28 '13 at 1:33

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