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Part a) Ordinary Differential Equation Application ODEs, investments and interest rates: In a period of falling interest rates, the (continuously compounded) short interest rate as a function of t is 1 per annum. (a) What is the value at t = 0 of a contract that makes a payment at time t = 10 of USD 1,000? An investor deposits USD 300 in a bank account at time 0, reinvests all interest payments and also additionally continuously invests USD 300 per annum, until the total value of the deposits reaches USD 3312. At that point the investor stops making additional deposits, but still lets the interest payments accumulate in the account. From our discussion in class, the ODE for the value of deposits, V, over time is then dV dt = =r(t)V(t)+I(t), where I (t) = 300 until V(t) reaches V = 3,312, at which point I(t) instantaneously switches to I (t) = 0. (b) Derive an expression for the value of the asset as a function of time, V T(t), t > 0. Part B) Statistics - Linear Regressions Define the best proportional predictor of Y given X as the ray through the origin, E** (Y|X) = YX, with Y being the value for C that minimizes E(e2), where now e = Y - cX. (a) Show that Y = (XY) / E(X2). (b) Is E** (Y (IX) an unbiased predictor? Explain. (c) Let e = Y - YX. Is Cov(X,e) = 0? = (d) Find the minimized value of E(e2). (e) Compare this E(e2) with those that result when the CEF and the BLP are used. For the joint pdf in the table below: x = 1 xc = 2 x = 3 y = 0 0.15 0.10 0.15 y = 1 0.15 0.30 0.15 (a) Find the conditional expectation function E(Y)X). (b) Find the best linear predictor E*(Y|X). (c) Prepare a table that gives E (Y|x) and E*(Y|x for x = 1,2,3. For the matrix X' = 1 1 1 - 1 4 2 3 5 , compute P = X(X'X) - 1X' and M - I - P. (a) Verify that MP = 0. (b) Let Q 2 1 3 8 = , compute the P and M based on XQ instead of X.

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