Pennsylvania State University ECON 465-Fall 2016 Homework 3 (due: Thursday, October 27, 2016) 1. Suppose the assumptions of the linear regression model are satisfied for some x and y: \u0002 \u0003 E u\fx = 0. y = 0 + 1 x + u; \b n Suppose we have a random sample (xi , yi ) i=1 . Suppose that 0 6= 0 (the true intercept is nonzero). Suppose however, that we mistakenly assume that the true model is: y = 1 x + u; \u0002 \u0003 E u\fx = 0. That is, suppose that we mistakenly assume that the intercept is zero. The OLS estimator for 1 is therefore the solution to: Minimize n X yi b1 xi \u00012 . Therefore: i=1 | {z Pn i=1 u b2i Pn yi xi b 1 = Pi=1 n 2 = i=1 xi 1 Pn i=1 yi xi n 1 Pn 2 i=1 xi n } (a) Show that we can express b1 as: b1 = 0 1 Pn i=1 xi n 1 Pn 2 i=1 xi n + 1 + 1 Pn i=1 ui xi n 1 Pn 2 i=1 xi n \u0001 \u0001 \u0002 (b) Using part (a), find plim b1 . Show that plim b1 = 1 if and only if E x] = 0. Show \u0001 \u0002 how the sign of plim b1 1 depends on the sign of E x]. (c) Provide an intuitive explanation of why the OLS estimator b1 is inconsistent in this case. How does the formula for b1 in this case differ from the one we studied in class? 2. One of the oldest economic theories of consumption assumes that we can express consumption of individual i as: Consumptioni = C + c Incomei The term C is called \"autonomous consumption\