ARE/ECN 240C Time Series Analysis
Fall 2002
Professor Òscar Jordà
Economics, U.C. Davis
PROBLEM SET 5 – SOLUTIONS
Part I – Analytical Questions
Problem 1: Consider the following DGP for the cointegrated random variables z and y
0.8L   1t 
y 
1  0.8L
(1  L) t   (1  0.4 L) 1 
 
1  0.6L   2t 
 0.1L
 zt 
where  ~ N(0, I) with z0 = y0 = 0.
(a) Obtain the autoregressive representation of this DGP.
(b) Obtain the error-correction representation of this DGP.
(c) Deduce the long-run relation between z and y.
(a) Directly inverting the lagged matrices on the right-hand side, we get
1  0.6L  0.8L  yt    1t 

    
  0.1L 1  0.8L  zt    2t 
(b) From the autoregressive representation
 0.8L 
1  0.6 L  0.8L  1  L  0.4 L

  

1  L  0.2 L 
  0.1L 1  0.8L    0.1L
0   0.4 L  0.8L 
1  L
 0.4 

  
  I  
1  2 L
1  L    0.1L 0.2 L 
 0
  0.1
Hence   (0.4  0.1)' ;   (1  2)'
(c) From the ECM, the long run solution is y = 2z.
Problem 2: Consider the following DGP
xt  yt  u1t ; where u1t  u1t 1   1t
xt  yt  u2t ; where u2t  u2t 1   2t
with || < 1, and
 0    12  
  1t 

  ~ D  , 
2 
 0     2 
  2t 
where D denotes a generic distribution.
1
ARE/ECN 240C Time Series Analysis
Fall 2002
Professor Òscar Jordà
Economics, U.C. Davis
(a) Derive the degree of integratedness of the two series, xt and yt. Do your results
depend on any restrictions on the values of , , and  ? Discuss how.
If θ = 1, then xt and yt are I(1) since xt  yt  u1t ~ I (1) . In addition, given || < 1
one needs to impose the condition    . The same linear combination of xt and yt
cannot be simulatenously I(0) and I(1).
(b) Under what coefficient restrictions are xt and yt cointegrated? What are the
cointegrating vectors in such cases?
  1;  1 . Cointegrating vector (1 ).
(c) Choose a particular set of coefficients that ensures xt and yt are cointegrated and
derive the following representations:
I. The moving-average.
II. The autoregressive.
III. The error-correction.
I. MA
 1

 1

1 
 1t  x
 1t 

x
1

1

  

 t 

;  t   


    1 1 L
  1 1 L
y
y
1

1





 t  
 
 2t   t  
 2 t 
1


L
1


L




Note :
1
1  
1

 
 
1  
Hence :


 1


 1
 xt     1t i     i 2 t i
i 0
i 0



i
 y t  i 0  1t i  i 0   2 t i
Aside :
(1  ) is the cointegrat ing vector, while (1  ) is a serial correlatio n feature
II. AR
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ARE/ECN 240C Time Series Analysis
Fall 2002
xt  y t 
Professor Òscar Jordà
Economics, U.C. Davis
 1t
  xt 1    1t 
1  L  1   xt   1
   






 2t
y
y
1

1

(


1
)
t
t

1







   2t 
xt  y t 
1 L
  xt 1 
 xt 
1      1
1       1t 
  
 




 
y
y
1

1
1

(


1
)
1

1








 t 1 

  2 t 
 t
III. ECM
Define the error correction term zt 1  xt 1  yt 1 , then
 (   1)
z u
   t 1 1t
(1   )
y t 
z u
   t 1 2 t
x t 
with
1
( 1t  2 t )
 
1
u2 t 
(   2 t )
   1t
u1t 
(d) Can all the cointegrated systems be represented as an error-correction model?
What are the problem/s of analyzing a VAR in the differences when the system is
cointegrated?
From the Granger representation theorem we know the answer is yes. Analyzing the
VAR in the differences omits the error correction term in the specification. Therefore
we have the classical problems of omitted variable bias.
(e) Suppose that economic theory suggests that xt and yt should be cointegrated with
cointegrating vector [1  + 0.5t]. Describe:
I. How would you test whether this is indeed a cointegrating vector?
Run the OLS regression
xt  (  0.5t ) y t  vt
and test the residuals with an ADF test.
II. What is the likely outcome of the test in short samples? Why?
In short-samples, the cointegrating vector (1 +0.5t) will differ from the
cointegrating vector (1 ). However, as the sample size gets larger, note that
the bias 0.5t disappears very quickly.
3
ARE/ECN 240C Time Series Analysis
Fall 2002
Professor Òscar Jordà
Economics, U.C. Davis
III. What is the likely outcome of the test asymptotically? Why?
Asymptotically the bias disappears sufficiently quickly.
Problem 3: Consider the bivariate VECM
y t  c   ' y t 1   t
iid
 it ~ (0,  2 )
where   (1 ,0)' and   (1,  2 )'. Equation by equation, the system is given by
y1t  c1  1 ( y1t 1   2 y 2t 1 )   1t
y 2t  c2   2t
Answer the following questions:
(a) From the VECM representation above, derive the VECM representation
yt  c  yt 1   t
and the VAR(1) representation
yt  c  Ayt 1   t

   1
0
 1  2 
   1  1  2 
; A   1

0 
1 
 0
(b) Based on the given values of the elements in  and , determine   ,   , such that
 '    0 and  '    0.
 0
 k
    ;     2 ; k  0
k 
 k 
(c) Using the Granger representation theorem determine that
 (1)    (  ' I 2   ' ) 1  , where  (L) is the moving average polynomial
corresponding to the VECM system above and I2 is the identity matrix of order 2.
Hint: you may show this result by showing that  (1) is orthogonal to the
cointegrating space.
Using the hint: (1)’ = 0. It is easy to show that
4
ARE/ECN 240C Time Series Analysis
Fall 2002
Professor Òscar Jordà
Economics, U.C. Davis
 0 2 

1 
  (  ' I 2   ' ) 1   
0
and therefore
1  1  2  0  2 


0
0  0 1 
0
(d) Using the Beveridge-Nelson decomposition and the result in (c), determine the
common trend in the VECM system.
All you need to remember is that from the B-N decomposition, the trends are the
linear combinations captured in (1)yt, which in this case turns out to be 2y1t + y2t.
Notice that this combination is orthogonal to the cointegrating vector.
(e) Show that  ' y t follows an AR(1) process and show that this AR(1) is stable
provided that  2  1  0 . What can you say about the system when 1 = 0?
Let zt  y1t   2 y 2t be the cointegrating vector. From the equations for y1 and y2 we
have
zt  c1  (1  1) y1t 1  1  2 y2t 1   1t   2 c2   2 y2t 1   2 2t
Combining terms
zt  (c1   2 c2 )  (1  1) zt 1  vt ; vt   1t   2 2t
which is an AR(1) whose stationarity requires that |1 + 1| < 1 or the equivalent
condition -2 < 1 < 0. When 1 = 0, zt is no longer stationary, so there is no
cointegration for any value of 2. y1 and y2 are in this case two independent random
walks.
5
ARE/ECN 240C Time Series Analysis
Fall 2002
Professor Òscar Jordà
Economics, U.C. Davis
Problem 4: Consider the following VAR
y t  (1   ) y t 1  xt 1   1t
xt  y t 1  (1   ) xt 1   2 t
(a) Show that this VAR is not-stationary.
Stationarity requires that the values of z satisfying
 1 0  1  

  
 0 1  
  
z  0
(1   ) 
lie outside the unit circle. For z = 1, notice



      0

(b) Find the cointegrating vector and derive the VECM representation.
Notice that
 
 (1)  
 
 
  ( 
 
  )' (1   )
so that
yt   ( yt 1  xt 1 )   1t
xt   ( yt 1  xt 1 )   2 t
(c) Transform the model so that it involves the error correction term (call it z) and a
difference stationary variable (call it wt). w will be a linear combination of x and
y but should not contain z. Hint: the weights in this linear combination will be
related the coefficients of the error correction terms.
Given the ECM in part (b), notice
 yt  xt  zt 1  1t  zt 1  2t
wt  yt  xt  1t  2t
Next
6
ARE/ECN 240C Time Series Analysis
Fall 2002
Professor Òscar Jordà
Economics, U.C. Davis
y t  y t 1   ( y t 1  xt 1 )   1t
xt  xt 1   ( y t 1  xt 1 )   2 t
( y t  xt )  ( y t 1  xt 1 )  zt 1  zt 1   1t   2 t
zt  (1     ) zt 1   1t  
 zt  1    

  
0
 wt  
0  zt 1   1


0  wt 1    
    1t 
 
   2 t 
(d) Verify that y and x can be expressed as a linear combination of w and z. Give an
interpretation as a decomposition of the vector (y x)’ into permanent and
transitory components.
From part (c)
 zt   1
   
 wt    
   xt 
 
  yt 
taking the inverse
1
  
    zt   xt 

    
  1  wt   yt 
and therefore

z 
   t 

xt 
z 
   t 
yt 

wt
 
1
wt
 
wt is I(1) and zt is I(0), which is a version of the Beveridge-Nelson decomposition
proposed by Gonzalo and Granger (1995).
7