Borghans, Lex

English

Research Papers in Economics

Economics  Letters  41  (1993)  25-28 
01651765/93/$06.00  0  1993  Elsevier  Science  Publishers  B.V.  All  rights  reserved 
25 
Errors  in  rational  expectations  matter 
Lex  Borghans 
Research  Centre  for  Education  and  the  Labour  Market,  PO  Box  616,  NL-6200  MD  Maastricht,  Netherlands 
Received  23  July  1992 
Accepted  1  September  1992 
Abstract 
Muth  (Econometrica,  1961,  29,  299-306)  and  Sargent  and  Wallace  (Journal  of  Monetary  Economics,  1976,  2, 
169-183)  claimed  that  the  main  results  of  rational  expectations  theory  allow  for  possibly  large  independent  errors.  This 
paper  shows  that  in  such  a  situation  the  pseudo  rational  expectation  is  no  longer  optimal. 
I.  Introduction 
Uncertainty  plays  an  important  role  in  educational  choice.  This  choice  determines  the  future 
occupational  possibilities,  and  since  many  courses  take  several  years,  students  have  to  form 
expectations  about  the  complex  and  non-transparent  labour  market.  The  uncertainty  students  face 
has  been  studied  both  by  cobweb  models  [Freeman  (1971)]  and  by  rational  expectations  models 
[Siow  (1984)  and  Zarkin  (1985)].  The  problem  with  the  cobweb  model  is  that  it  imposes  an 
expectation  behaviour  which  is  a  priori  non-optimal.  On  the  other  hand,  rational  expectations 
theory  assumes  that  all  prediction  errors  are  due  to  uncertainty  about  the  future  state  of  the 
world,  and  therefore  implicitly  assumes  that  students  do  not  make  errors  in  the  forecasts 
themselves.  They  are  assumed  to  have  perfect  insight  in  the  functioning  of  the  labour  market. 
Muth  (1961),  and  Sargent  and  Wallace  (1976)  claim,  however,  that  the  main  results  of  the  rational 
expectations  theory  are  not  influenced  by  the  introduction  of  an  independent  random  error  term. 
In  this  paper  this  proposition  is  challenged  by  showing  that  in  the  case  of  such  random  errors  this 
pseudo  rational  expectation  is  no  longer  optimal. 
2.  Information  versus  interpretation  errors 
Suppose  students  have  to  forecast  the  wage  they  will  earn  after  some  kind  of  schooling,  to 
make  their  enrolment  decision.  In  general  their  prediction,  rvPre, will  not  equal  the  realisation. 
rea. There  are  two  ways  to  explain  the  difference  between  the  realisation  and  the  expectation 
:.e.  the  prediction  error).  Firstly,  following  Arrow  (1951),  it  is  possible  to  assume  that  students’ 
expectations  are  optimal  given  the  information  available  at  the  moment  of  choice,  but  that  there 
will  be  some  future  events,  unpredictable  at  that  moment,  which  influence  the  realised  wage.  In 
this  state  of  the  world  view  on  prediction  errors  the  quality  of  the  forecast  is  determined  by  the 
amount  of  information  available  at  the  moment  of  choice.  It  is  the  world  that  might  deviate  from 26  L.  Borghans  I  Economics  Letters  41  (1993)  25-28 
the  prediction: 
This  information  error,  ere,  is  independent  of  the  prediction.  This  type  of  uncertainty  is  typical  of 
rational  expectations  models. 
A  second  way  to  explain  prediction  errors  is  to  assume  that  students  are  not  fully  capable  of 
interpreting  the  available  information,  The  labour  market  is  very  complex,  and  it  is  not  evident  in 
which  way  available  information  should  be  used  to  deduce  optimal  forecasts  about  future  wages. 
A  student  should  decide  which  past  data  are  relevant  for  the  situation  he  faces,  and  should  decide 
on  the  extent  of  the  similarity  of  these  past  events  and  future  developments.  Furthermore,  there  is 
no  reason  why  theoretical  thought  might  not  provide  ‘data’  relevant  for  the  expectation  formation. 
The  expectation  therefore  depends  critically  upon  some  random  personal  interpretations  about  the 
relevant  evidence: 
W Pre  =  Wrea  +  EeXp  (2) 
This  interpretation  error  is  independent  of  the  realisation.  The  realisation  is  constant,  but  due  to 
errors  in  interpretation  the  prediction  differs  from  the  realisation. 
Of  course,  both  models  are  extreme  variants.  It  might  be  the  case  that  both  types  of  error  occur 
in  a  forecast.  In  that  case  there  exists  an  unobserved  optimal  forecast,  wre,  such  that 
rea 
W  =  w= + l  r= ) 
W 
Pre  =  W=  +  ECXP  . 
(3) 
E ”  is  independent  of  wPrr  and  wre,  while  l  exP is  independent  of  wrea  and  wrC. 
The  possibility  of  errors  in  forecasts  is  also  observed  by  proponents  of  rational  expectations,  in 
which  errors  are  always  due  to  unforecastable  exogenous  shocks.  Muth  (1961,  p.  321)  and  Sargent 
and  Wallace  (1976,  p.  ISO),  however,  state  that  it  is  possible  that  there  are  deviations  from 
rationality,  but  since  these  will  be  independent  errors,  they  do  not  matter  for  the  analysis.  In  the 
next  section  it  is  shown  that  this  proposition  is false.  The  presence  of  errors  in  forecasts  influences 
the  optimality  property  of  the  rational  expectation  and  causes  students  to  deviate  from  this  pseudo 
rational  expectation. 
3.  The  optimal  prediction 
There  are  many  different  approaches  to  making  a forecast  and  there  is  no  reason  to  choose  one 
of  these  methods  in  advance.  One  approach  is  to  base  the  forecast  on  similar  past  events.  The 
student  has  to  select  cases  that  appear  to  him  identical  to  some  extent,  that  can  be  used  to 
construct  a  probability  distribution  about  the  future  wage.  The  problem  is  to  decide  which  past 
events  are  still  seen  as  similar  and  which  are  not.  The  larger  the  class  of  similar  cases  the  less 
informative  the  probability  distribution  will  become,  i.e.  the  information  error  rises. 
On  the  other  hand  it  is  possible  to  restrict  the  class  of  similar  cases.  Such  a  restriction  will  be 
based  on  theoretical  arguments  about  what  is  typical  for  the  event  that  has  to  be  predicted.  In  the 
case  where  the  future  event  is  thought  to  be  very  typical,  it  might  be  the  case  that  no  similar 
events  can  be  found  in  the  empirical  data.  The  only  comparable  events  are  constructions  of  the L.  Borghans  I  Economics  Letters  41  (1993)  25-28  27 
mind.  In  that  case  the  forecast  will  be  purely  theoretical.  The  more  restrictive  the  selection  of 
comparable  events  is,  the  more  informative  the  selection  becomes,  but  also  the  larger  the 
probability  that  a  wrong  selection  is  made,  i.e.  the  interpretation  error  rises. 
The  student  has  to  choose  between  the  information  errors  of  a  very  large  set  of  comparable 
events,  and  the  interpretation  errors  in  the  case  of  a small  set  of  comparable  events.  It  is therefore 
not  sensible  to  distinguish  only  one  (unobservable)  perfect  prediction,  as  in  Eq.  (3),  but  there  will 
be  many  ‘perfect’  predictions,  rankable  from  a  low  degree  of  informativity  to  a  high  degree  of 
informativity.  In  this  paper  the  analysis  is  restricted  to  two  different  unobservable  perfect 
predictions,  but  this  can  easily  be  extended  to  more.  The  ranking  provides  a  trade  off  between 
information  errors  and  interpretation  errors. 
Let  wT’ be  a perfect  expectation  with  a high  degree  of  abstraction,  and  therefore  a  low  degree  of 
informativity,  and  let  WY be  a  more  specific  perfect  expectation.  Equation  (3)  can  be  extended  to 
rea 
W  =  wy  + l  5’ ,  (4) 
re 
W2 =  W;’ +  E’;’ )  (5) 
exP  _ 
w2  -  WY  +  EyP  +  EyP  )  (6) 
WfXP =  wy  +  c;XP  )  (7) 
in  which  the  error-terms  are  always  independent  of  the  other  right-hand-side  variables  and  have  a 
zero  expectation.  WI” is  an  abstract  perfect  prediction  of  the  realised  wage,  but  can  also  be  seen  as 
a  perfect  prediction  of  the  more  specific  perfect  prediction,  WY.  Both  perfect  predictions  are  only 
observed  with  an  additional  measurement  error.  The  specific  prediction,  wtxp,  contains  at  least  the 
error  of  the  abstract  prediction,  wryp. 
In  a  rational  expectations  model  the  measurement  errors,  l  FXP,  equal  0.  In  that  case  the  specific 
prediction  dominates  the  abstract  prediction,  so  it  is  optimal  to  use  the  most  specific  prediction. 
The  propositions  of  Muth,  and  Sargent  and  Wallace,  however,  is  that  this  is  still  the  case  if 
measurement  errors  are  introduced  into  the  model.  On  the  basis  of  an  MSE  loss  function  it  is, 
however,  easy  to  show  that  it  is  possible  to  gain  from  a  combination,  wPre =  Aw~“~ +  (1  -  A)w;~~, 
of  both  expectations.  The  prediction  error  of  this  combined  prediction  equals 
Aw  =  WPre -  WTea  =  A(-Erel)  +  (1  -  A)e;‘p  +  E;XP -  E:e  .  (8) 
The  combination  is  optimal  for  a  value  of  A for  which  the  variance  is  minimal.  Denoting  the 
variances  of  the  errors  by  respectively  v:~,,  afeZ,  (T:,~,  , and  flzXp2, the  variance  of  the  prediction 
error  equals 
c2, =  A'cT;~,  + (1  -  A)2az,p2  +  (T;,~~ +  afe2  .  (9) 
Minimising  Eq.  (9)  with  respect  to  A gives 
This  result  shows  that  only  the  additional  expectation  errors  are  relevant  for  the  choice.  (+zXp2  is 
the  error  of  the  specific  expectation  compared  with  the  abstract  expectation.  By  introducing  the 28  L.  Borghans  I  Economics  Letters  41  (1993)  25-28 
specific  expectation  an  additional  interpretation  error  appears.  v:,,  is  the  additional  information 
error  of  the  abstract  expectation,  compared  with  the  specific  expectation.  The  specific  expectation 
is  optimal  only  if  *,ZXp2  =  0  or  azc,  =  00. The  proposition  of  Muth,  and  Sargent  and  Wallace,  is 
therefore  not  true  in  this  construction.  The  only  exception  is  the  case  in  which  only  one 
expectation  exists,  or  of,,  =  ~0, i.e.  the  abstract  expectation  is  completely  uninformative. 
The  result  (10)  can  be  given  a  Bayesian  interpretation.  wrxp,  with  (of,,,  can  be  regarded  as  a 
prior  distribution  for  the  realised  wage.  The  optimal  prediction  is  the  posterior  distribution  after 
inference  with  w;‘~.  This  Bayesian  interpretation  of  expectation  formation  differs  strongly, 
however,  from  the  usual  models  of  Bayesian  learning  [e.g.  Cyert  and  DeGroot  (1987)].  In  these 
models  the  prior  distribution  is  based  on  some  a  priori  theoretical  insight  into  the  possible 
distribution  of  the  parameters,  while  empirical  events  provide  data  for  inference.  In  the  model  of 
this  paper  the  prior  distribution  is  formed  by  the  frequency  of  past  events,  while  theoretical 
insights  are  viewed  as  data  for  inference.  While  in  the  usual  model  forecasts  improve  from  new 
data,  the  main  improvements  in  this  model  come  from  additional  theoretical  insights. 
4.  Implications 
The  model  in  this  paper  shows  that  if  students  deviate  from  the  rational  expectation  by 
independent  random  errors,  it  becomes  optimal  for  them  to  deviate  from  this  pseudo  rational 
expectation  and  to  make  some  systematic  errors  in  their  forecasts.  This  result  has  two  important 
implications.  First,  such  systematic  deviations  from  the  rational  expectation  make  it  possible  to 
use  policies  which  manipulate  people’s  expectations.  In  Borghans  (1992a)  it  is  shown,  however, 
that  the  possibilities  of  such  expectations  manipulation  policies  are  restricted.  Secondly,  the 
recognition  of  errors  in  forecasts  by  the  students  makes  it  possible  to  model  an  economic  policy 
which  aims  at  the  improvement  of  students’  expectations  by  the  provision  of  public  labour  market 
information  as  described  in  Borghams  (1992b).  Such  a  provision  of  public  labour  market 
information  can  be  interpreted  as  a  decrease  of  errors  in  the  forecasts  made  by  students  (&,,). 
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The market for college-trained manpower: A study in the economics of career choice

Errors in rational expectations matter.

Errors in rational expectations matter.

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