Problem 1
Consider the AR (1) model
1 yt     yt 1  t
where  is a non-serial correlated error term with constant variance 2.
a) How would you estimate this model? What do you know about the small
sample and large sample properties of the estimator?
b) Show that if || < 1, then you can write


   i  t i
 2  yt 
1   i 0
c) Use equation (2) to show that the k’th autocorrelation, which we will denote as
 k , is equal to  |k | for the AR (1) model.
Consider the MA (1) model
3 yt    t  t 1
d) Find the autocorrelation coefficient for the MA (1) model. How can the
autocorrelation functions be used to distinguish between the MA(1) and AR(1)
empirically?
Consider the AR (2) model
 4
yt    1 yt 1   2 yt 2  t
Assume that the following conditions hold:
5 1  2  1, 2  1  1,
|  2 | 1
e) Find an expression of the variance of y in terms of 1 , 2 and  2 . Discuss the
conditions under (5) in light of this expression.
f) Find the autocorrelations 1 and 2 for the AR (2) model.
The partial correlation between two variables x and y holding a third variable z
constant is given by
corr ( x, y )  corr  x, z  corr  y, z 
 6  rxy , z 
2
1  corr  x, z  1  corr ( y, z ) 2
g) For both the AR(1) and AR(2) models, use (6) to calculate the partial
autocorrelation between yt and yt  2 holding yt 1 constant, which you can
denote  2|1 . Show that for the AR(2) model,  2|1 equals the coefficient  2 .
Explain how the results can be used to distinguish between different orders for
the autoregressive process.
h) Assuming the parameters of the model where known. How would you predict
yt  k using observations of y up to time t under the AR (1) and MA (1) model?
i) Discuss how the “efficient market hypothesis” relates to the test of   1 in
the AR (1) model.
Problem 2
a) Use GiveWin and PcGive (or any other software of your preference) to
estimate an AR(1) model for each of the following data sets containing
Norwegian interest rate of 3 months maturity:

HF-3MR: High frequency data, 10-minutes frequency for Norwegian 3month rates. This is tick-by-tick data collected by the Central Bank of
Norway. Note that in most days there you will not be able to find
observations between 5.00 p.m. (or later) and 9.00 a.m. (or earlier). This is
due to the fact that the Norwegian interest rate market is not so active
during those hours (as it would be the case for the US rate markets for
example). This implies that one cannot interpret that as missing
observations. Important: The relevant data-column to look at is the last
one.

Daily3mr: This is the daily market 3-month rates for Norway. Important:
The relevant data-column to look at is the first one.

Weekly3mr: This is the weekly market 3-month rates for Norway.
Important: The relevant data-column to look at is the first one.

Month3mr: This is the monthly market 3-month rates for Norway.
Important: The relevant data-column to look at is the second one.
b) How would you interpret the estimates? On the basis of the estimates that you
have obtained, how would you judge the proposition 2.4 in Gourieroux and
Jassiak’s book?
c) Use the datasets HF-3MR and daily3mr to construct, for each of the data sets,
one subset of only 10 days. Then draw a figure showing how these interest
rate moves within 10 days. Construct another subset of 10-days (choose any
period you wish) for each data set, draw a figure of them and just compare
them with the first subset you constructed. What would you conclude?