ECON 710 Midterm: ECON 710 UW Madison Midterm 2016a

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31 Jan 2019
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@x m(x) = c1 + 2c2x, then (cid:18) = e(cid:2) @ = c1 + c2 cov(cid:0)x; x2(cid:1) (cid:27)2 x. Thus cov(cid:0)x; x2(cid:1) = sx + 3(cid:22)x(cid:27)2 x + (cid:22)3 x (cid:0) (cid:22)x(cid:27)2 x (cid:0) (cid:22)3 x = sx + 2(cid:22)x(cid:27)2 x: (cid:12) 1 = c1 + c2 sx + 2(cid:22)x(cid:27)2 x (cid:27)2 x. First, c2 = 0 occurs when the true regression is linear. Thus, as seems natural, the linear approximation will equal the average derivative when the true regression is linear. Second, sx = 0 occurs when the third centered moment of xi is zero. This occurs when the distribution of xi is symmetric about its mean. Thus the linear approximation will equal the average derivative when the true regression is quadratic and xi is symmetric about its mean. Roughly, the bias on the two sides of (cid:22)x cancel out. Since expectation is a linear operator b(cid:18) is therefore unbiased.

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