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
(a) What is the value at t = 0 of a contract that makes a payment at time t = 10 of USD
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
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
x = 3
y = 0
y = 1
(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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