ECON 710 Midterm: ECON 710 UW Madison Midterm 2016
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The exam questions all concern the following setting. The random variables are (y; x) with y 2 r and x 2 r and there is a random sample fyi; xi : i = 1; :::; ng from (y; x). De(cid:133)ne the conditional mean m(x) = e (yijxi = x). A researcher is interested in estimating the average derivative (cid:18) = e(cid:20) @ Assume that the true conditional mean takes the form m(x) = c0 + c1x + c2x2 (1) but this is not necessarily known by the researcher. Speci(cid:133)cally, they estimate by least-squares yi =b(cid:12) 0 +b(cid:12) 1xi +bei and then set b(cid:18) = b(cid:12) 1. Let ((cid:12) 0; (cid:12) 1) denote the population version of this regression (the best linear prediction coe cients). Find an expression for the bias in (cid:12) 1 for (cid:18), e. g. the di erence (cid:12) 1 (cid:0) (cid:18), in terms of c0, c1, c2 and the moments of xi:
