Suppose that Y, = m(a:,;) + 5,, where as, are xed constants and e, 1131 'P such that lE(e) = 0 and V(e) = 02 R, where X is just the space containing the Lag's. Suppose you construct an estimator {a of the function m (e.g., this could be a linear function, but it does not need to be). At some xed 330, the mean squared error of the estimator in is dened as lawman} = E [{YO mar] where the expectation is over Yo. Note: we are used to having an estimator for a parameter, but now we have an estimator for a whole function m. In general, a lot of what we've already learned applies for estimating a function, but everything does start to become more complicated. In this question, I only want you to show the following decomposition for the mean squared error: IE [{%($0)}2] =02 +Bias{a(wo)}2+V{a(wo)} (1) where Yo is a random variable with Yo | x0 ~ So so that E(Yo) = x0 and V(Yo) = V(EO) You can think of Yo as a new outcome calculated for someone with covariate co. So, Yo is independent from Y1, . .., Yn and from m, i.e. an observation not used in constructing the estimate m(x0). The second term on the RHS of eq. (1) is Bias{m(xo)} = E{m(xo) -m(xo)} = E{m(xo)} - m(x0). This proof is not that easy, so we'll work through it in a few parts: (a) Show that the MSE can be rewritten as E { Yo - m(ro)}? = E{(Y -m) ? } +E (m-m) +E (m-m)?] + 2E {(Y - m) (m - m)} + 2E{(Y - m) (m -m) } + 2E [(m - m) (m - m)] where Y = Yo m =m(xo) m = Elm(xo)} m = m(xo). (b) Prove that E{ (Y - m)?} = 02. (c) Prove that E {(m - m)?} = Bias{m(x0)}2 (d) Prove that E {(m -m)2} = Vim(x0)} (e) Explain why Yo Im(x0). (f) Prove that E(Y - m) = 0. (g) Use the previous two parts to argue that E{(Y - m)(m - m)} = 0, E{(Y - m)(m - m)} = 0. (h) Explain why E[(m -m) (m -m)] = 0. Hint: m and m are fixed and thus can be factored out