Wolff, Christian C.P.

English

Research Papers in Economics

Economics  Letters  20  (1986)  373-376 
North-Holland 
373 
EXCHANGE  RATE  MODELS  AND  INNOVATIONS 
A  Derivation 
Christian  C.P.  WOLFF  * 
London  Business  School,  London  NWI  ISA,  UK 
Received  6  November  1985 
An  explicit  relationship  between  innovations  in  the  spot  exchange  rate  and  innovations  in  its  driving  variables  is derived  in  the 
context  of  the  ‘asset  market  theory’  of  the  exchange  rate.  The  relationship  forms  a  theoretical  basis  for  regression  results  in 
‘news’  form  that  have  been  reported  in  the  literature. 
1. Introduction 
In  recent  years  the  distinction  between  anticipated  and  unanticipated  movements  in  the  exchange 
rate  and  its  driving  variables  has  been  emphasised  in  the  literature  [see,  e.g.,  Frenkel  and  Mussa 
(1980)  Frenkel  (1981)  and  Mussa  (1984)].  The  essence  of  this  line  of  thinking  is  embodied  in  the 
‘asset  market  theory’  of  the  exchange  rate  that  is  presented  in  Frenkel  and  Mussa  (1980).  The 
framework  that  was  developed  by  Frenkel  and  Mussa  views  the  exchange  rate  as  a  highly  sensitive 
asset  price  which  is  immediately  affected  by  an  influx  of  new  information.  This  approach  is  generally 
taken  to  imply  that  empirical  research  on  the  determinants  of  exchange  rates  should  relate 
innovations  in  exchange  rates  to  innovations  in  a  relevant  vector  of  explanatory  variables.  The  idea 
was  first  implemented  empirically  by  Frenkel  (1981). 
While  the  innovations  or  ‘news’  approach  is  intuitively  appealing,  no  theoretical  derivation  has 
been  presented  in  the  literature  that  links  the  general  class  of  ‘news’  models  that  is  estimated  in 
Frenkel  (1981)  with  the  theoretical  model  of  Frenkel  and  Mussa  (1980).  Such  a  derivation  is 
presented  in  the  next  section  of  this  paper. 
2.  A  derivation 
Consider  the  following  simple  model  of  exchange  rate  determination: 
s(t)=jYz(t)+aE[s(t+l)-s(t)It],  H[L]z(t)  =F[L]u(t).  (l),(2) 
Eq.  (1)  is  due  to  Frenkel  and  Mussa  (1980)  and  states  that  the  logarithm  of  the  equilibrium  spot 
*  This  paper  is  drawn  from  Section  V.2  of  the  author’s  doctoral  dissertation  at  the  University  of  Chicago.  Helpful  comments 
from  Michael  L.  Mussa  are  gratefully  acknowledged. 
0165-1765/86/$3.50  8  1986,  Elsevier  Science  Publishers  B.V.  (North-Holland) 374  C.  C.  P.  Wolff /  E,~chu,~gc  rute nwdels  rrr~d  I~~~O~YI~IO~J 
exchange  rate,  s(t),  is  determined  not  only  by  a  set  of  current  market  fundamentals,  but  also  by  the 
expected  rate  of  change  of  the  exchange  rate,  E[s(t  +  1) -s(t)  1  t],  which  motivates  domestic  and 
foreign  residents  to  move  into  or  out  of  foreign  exchange  depending  on  whether  the  relative  price  of 
foreign  exchange  is  expected  to  rise  or  fall.  The  vector  /3  is  a  vector  of  parameters.  u  (u  >  0)  is  a 
scalar  parameter  and  E[.  1  t] denotes  an  expected  value  conditional  on  information  available  at  time  1. 
Eq.  (1)  represents  a  general  relationship  which  can  be  derived  from  a  variety  of  models  of  exchange 
rate  determination  that  generally  differ  in  their  interpretations  of  the  elements  of  the  vector  z(t).  In 
eq.  (2)  it  is  assumed  that  z(t)  is  generated  by  a  general  vector  ARMA  time  series  process.  H[  L]  and 
F[L]  are  square  matrices.  assumed  of  full  rank,  whose  elements  are  finite  polynomials  in  the  lag 
operator  L.  The  vector  -_(t)  is  assumed  to  be  m-dimensional  and  thus  F[  L]  and  H[ L]  are  m  x  m 
matrices.  Further.  we  assume  that  the  VI X  I vector  of  innovations  P(Z)  has  a  zero  mean,  an  identity 
covariance  matrix  and  no  serial  correlation,  that  is. 
E[L+)]  =O,  E[~(r)u’(r)]  =&I,  (3).(4) 
where  I  is  an  1?1  X  m  unit  matrix  and  a,,  is  the  Kronecker  delta.  Since  we  assume  initially  that  the 
process  is  stationary  and  invertible  (stationarity  will  be  relaxed  below),  all  roots  of  11  H [  L]  I/  =  0  and, 
/I F[  L]  II  =  0  lie  outside  the  unit  circle  (II.11 indicates  the  determinant  of  a  matrix).  Since  H[  L]  is 
assumed  to  have  full  rank,  (2)  can  be  solved  for  z(t). 
=(t)  =  (H*[L]/IIH[L]  II)F[Llc(Q,  (5) 
where  H*[  L]  is  the  adjoint  matrix  (the  transpose  of  the  matrix  of  cofactors)  associated  with  H[L]. 
Eq.  (5)  expresses  z(t)  as  an  infinite,  invertible  vector  moving  average  process.  For  simplicity  of 
notation,  we  define 
B[L]  =  (H*[~]/I~H[L]  II)F[L].  (6) 
The  matrix  B[L]  can  be  written  as 
B[L]=B,,+B,L+B,L2+BjL3+... 
and  eq.  (5)  now  reads 
(7) 
z(t)=  B[L]c(r).  (8) 
Since  the  vector  z(t)  is shown  to  have  an  invertible  moving  average  representation  in  eqs.  (5)  and  (8) 
we  can  conclude  that  the  sequence  of  matrices  [B,;  j  =  0,  1,  2,  .] is  absolutely  summable  [see,  e.g., 
Fuller  (1976.  ch.  II)].  That  is.  if  we  define  I$’  to  be  the  k,lth  element  of  the  matrix  B,, 
lim  5  Ih:‘l  <  w  for  all  k,l,  (9) 
)I+  wz 
J=o 
where  1  . 1 denotes  an  absolute  value.  This  result  will  be  used  below. 
If  we  assume  that  expectations  in  eq.  (1)  are 
with  the  validity  of  (1)  in  all  future  periods, 
appropriate  boundary  condition  gives 
formed  rationally  in  the  sense  that  they  are  consistent 
then  forward  iteration  of  (1)  and  application  of  an 
tiij)  It]  (10) C. C. P.  Wolff  /  Exchange  rate  models  and  innovations  315 
[see,  e.g.,  Mussa  (1984)].  The  exchange  rate  that  is  currently  expected  to  prevail  at  time  t + i  (i  2  0) 
depends  on  a  weighted  average  of  expected  future  z’s.  Setting  i =  0  in  eq.  (10)  and  applying  the 
unexpected  change  operator  D”[s(t)]  =  s(t  +  1) -  E[s(t  +  1) 1  t]  gives 
W4t>l  = [&];zJ&]  ‘jY(E[z(t+j+l)It+l]  -E[z(t+j+l)]t]).  (11) 
The  unexpected  change  in  the  exchange  rate  is  a  weighted  average  of  changes  in  expectations 
concerning  the  future  z-vectors  resulting  from  new  information  that  is  received  between  times  t  and 
t +  1.  Substituting  (8)  into  (11)  gives 
N4~)l  = [j+#[&]  ‘V(E[B[L]c(t+j+l)It+l]  -E[B[L]u(t+j+l)It]).  (12) 
Using  the  definition  of  B[L]  that  is  given  in  (7)  we  can  thus  derive 
D”[.4t,l  = [&JB’(B,+  [+4+  [&]*B*+  .  ..ju(t+l). 
Making  use  of  the  result  in  (9),  we  can  show  that  the  infinite  series, 
4-t  [&I”’  +  [&]iB*+  . .  =  ;J&]‘b 
(13) 
(14) 
converges  to  a  finite  matrix.  Again,  let  b:’  be  the  k,lth  element  of  the  matrix  B,.  Then  we  have 
‘b;‘s  lim  5  a 
[  1  n+m,=a  1+a 
‘lb;‘1  <  00,  (15) 
where  the  first  inequality  follows  from  the  definition  of  an  absolute  value  and  the  second  from  (9). 
Thus,  the  infinite  series  in  (14)  converges  to  a  finite  matrix,  say  C, 
lim  i  [&]‘B,=C. 
n-p  ,=rJ  (16) 
Note  that  convergence  does  not  critically  depend  on  the  stationarity  assumption  that  was  made  with 
regard  to  the  process  in  eq.  (2).  Convergence  is  also  obtianed  for  all  non-stationary  processes  that 
satisfy  the  right-hand  side  inequality  in  (15). 
Combining,  finally,  eq.  (13)  and  (16),  we  obtain  the  relationship  we  set  out  to  derive, 
D”[s(t)]  = [#3'c:u(t+l).  (17) 
Eq.  (17)  is  a linear  relationship  between  the  innovation  in  the  spot  exchange  rate  between  t and  t +  1 
and  innovations  in  the  elements  of  the  z-vector  in  the  corresponding  period.  Given  appropriate 
measures  of  innovations  in  the  exchange  rate  and  its  driving  variables,  the  validity  of  (17)  can  be 
examined  empirically.  Since  innovations  are  inherently  unobservable,  any  empirical  study  on  the 316  C.C.P.  Wolff  /  Exchange  rate  models  und  innm~atrom 
basis  of  (17)  involves  a joint  examination  of  the  model  and  the  method  that  is  used  to  construct 
innovations.  Eq.  (17)  forms  a theoretical  basis  for  the  ‘news’  regressions  that  are  presented  in  Frenkel 
(1981). 
It  should  be  noted  that  estimated  coefficients  in  regression  equations  that  are  in  innovations  form 
do  not  have  the  same  interpretations  as  those  estimated  from  standard  structural  models.  That  is, 
when  estimating  an  equation  based  on  (17)  one  does  not  recover  the  parameter  vector  ,O, but  a 
complicated  vector  of  coefficients  that  involves  the  elements  of  the  vector  ,B, the  elements  of  the 
matrix  C  and  the  scalar  u. 
References 
Frenkel.  J.A.,  1981,  Flexible  exchange  rates,  prices  and  the  role  of  ‘news’:  Lessons  from  the  197Os,  Journal  of  Political 
Economy  89,  Aug..  665-705. 
Frenkel,  J.A.  and  M.L.  Mussa.  1980,  The  efficiency  of  foreign  exchange  markets  and  measures  of  turbulence,  American 
Economic  Review  70.  May,  374-381. 
Fuller,  W.A.,  1976,  Introduction  to  statistical  time  series  (Wiley,  New  York). 
Mussa.  M.L.,  1984.  The  theory  of  exchange  rate  determination,  in:  J.F.O.  Bilson  and  R.C.  Marston.  eds.,  Exchange  rate  theory 
and  practice  (University  of  Chicago  Press,  Chicago,  IL)  13-58. 
Wolff,  C.C.P.,  1985,  Exchange  rate  models.  parameter  variation  and  innovations:  A  study  on  the  forecasting  performance  of 
empirical  models  of  exchange  rate  determination,  Unpublished  Ph.D.  dissertation  (University  of  Chicago,  Chicago.  IL). 

Exchange rate models. parameter variation and innovations: A study on the forecasting performance of empirical models of exchange rate determination, Unpublished Ph.D. dissertation

Exchange rate models and innovations..

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