(Wt)t20 be a standard Brownian motion. In each of the following identify whether the item is generally a stochastic process, Let W a random variable, a deterministic function of t, or a real number. (a) W2 (b) So W?dt (e) E [[" W;dt]

Consider random variables Y and X = 7Y + 9 with moment-generating functions 1 e2t – e4t 1- et ebt – emt m, (t) 2 m – b 1– ect Identify the values of the parameters (b, m, c) for X.

3. The expectation operator E can be applied to a random vector. Specif- ically, if X = [X1 X2 …… Xn]", then EX = [EX1 EX2 .… EXn]". Show that E[(X – EX)(X – EX)"]= [cov(X¡, X;)]"j=1 •

Q.5 Suppose we have an economy where rational agents form the rational expectation about the economy. Let Yt, mi, Pt be the logs of real output, money stock and price level in period t, respectively. We assume the following relations hold: (1)у, — т, — P. (2)y —D у + Ф(р, — Е-1 (р.)) + и (3)m, 3 т where y is the level of output and u; is a random variable with zero mean and variance. Calculate the reduced forms for y, and p,