1. Let gMt be the annual growth in the money supply and let unemt be the unemployment rate. Assuming that unemt follows a stable AR(1) process, explain in detail how you would test whether gM Granger...

1. Let
gMt
be the annual growth in the money supply and let
unemt
be the unemployment rate. Assuming that
unemt
follows a stable AR(1) process, explain in detail how you would test whether
gM
Granger causes
unem.

2 Suppose that
yt
follows the model
yt
5 a 1 d1zt21 1
ut ut
5 rut21 1
et
E1et
0It21 2 5 0, where
It21 contains
y
and
z
dated at
t
2 1 and earlier.

(i) Show that E1yt11 0It
2 5 11 2 r2a 1 ryt
1 d1zt
2 rd1zt21. (Hint: Write
ut21 5
yt21 2 a 2 d1zt22 and plug this into the second equation; then, plug the result into the first equation

and take the conditional expectation.)

(ii) Suppose that you use
n
observations to estimate a, d1, and r. Write the equation for forecasting
yn11.

(iii) Explain why the model with one lag of
z
and AR(1) serial correlation is a special case of the Model
yt
5 a0 1 ryt21 1 g1zt21 1 g2zt22 1
et.

(iv) What does part (iii) suggest about using models with AR(1) serial correlation for forecasting?