PROBLEM SET 1
Exercise 1
A demand/supply model is specified by.
Pt   0  1Qt   2Yt  ut
(demand )
Qt   0  1 Pt 1   2 ( Pt 1  Pt 2 )  vt (sup ply )
where P indicates price, Q quantity, and Y income.
(1)
(2)
Discuss the identifiability of the structural parameters ( the  ' s and the  ' s ) of
this model.
Derive the AR equations for Pt and Qt , and show that the AR coefficients are
identical.
Exercise 2
We consider the following AR(2) process:
Yt  c  1Yt 1  2Yt 2   t
t  0, 1, 2, 3,....
where  t is WN (0,  )
2
(1) Give the characteristic equation for this process.
(2) What is the requirements for this process to be stationary?
(3) Show that the stationary conditions on the roots imply that
(a) 1  2  1, (b) 2  1  1 and (c) 2  1
Exercise 3
A macroeconomic model for the log of U.S. real GNP postulates that
A( L)(Yt   0  1t )   t
where the shift polynomial A( L) is given by A( L)  1  1L   2 L2
An OLS regression yields:
Yt  0.321  0.003t  1.335Yt 1  0.401Yt 2  ut
(1) Determine the estimates of 1 ,  2 ,  0 , 1
(2) Does the estimates of 1 and  2 satisfy the conditions (a), (b) and (c) listed in
Exercise 2?
(3) Compute the roots of the corresponding characteristic equation.
(4) Does the empirical regression above describe a stationary process?
An alternative specification fitted to the same data yields:
Yt  0.003  0.369Yt 1  vt
What are the roots of the characteristic equation corresponding to this AR process?
Exercise 4
Consider the difference equation
Yt  1.6Yt 1  0.89Yt 2  0 t  2,3, 4,.....
(1) Give the characteristic equation, and find its roots.
(2) Give the expression for Yt if Y0  0.8 and Y1  0.9
(3) If  t is WN (0,  2 ) and t  0, 1, 2, 3,..... , is there a stationary time series
satisfying the AR process:
Yt  1.6Yt  0.89Yt 2   t ?
Let the time series Yt , t  2,3, 4,5,..... be defined by
Yt  1.6Yt 1  0.89Yt 2   t
where Y0  Y1  0 and  t is WN (0,  2 )
(4) Give the covariance matrix for (Y2 , Y3 , Y4 )
Exercise 5
Let Z n  be sequence of uncorrelated real-valued variables with zero means and unit
variances, and define the MA-process
r
Yn    i Z n i
i 0
for constants  0 , 1 ,  2 ,....,  r . Show that Y is stationary and find its auto-covariance
function.
Suppose now that Yn  is an auto-regressive stationary process in that it satisfies
Yn  Yn1  Z n
  n  
For some   1 and Z n  is as above. Show that Y has auto-covariance function
m
c ( m) 
(1   2 )
Exercise 6
Let A and B be uncorrelated random variables, each of which has mean
0 and var iance 1. Fix a number   0,   and define the process
X n  A cos( n)  B sin( n)
Show that E ( X n )  0 for all n. Calculate the auto-covariance function for
Explain if this process  X n  is stationary.
X n .