Palm, Franz C.

English

Research Papers in Economics

JOURNAL OF 
Econometrics 
ELSEVEX  Journal  of  Econometrics  69  (1995)  333-335 
Bayesian  model  selection  and  prediction  with 
empirical  applications 
Comments 
Franz  C. Palm 
Department of Quantitative Economics, Universi~ of Limburg, 6200 MD Maastricht, The Netherlands 
Key  words:  Model  selection;  PIC;  Forecasting 
JEL  classijcation:  211 
In  the  preceding  article  Peter  Phillips  has  written  a  very  stimulating  paper  in 
which  he  uses  the  posterior  information  criterium  (PIC)  put  forward  in joint 
work  with  Werner  Ploberger  to  select  forecast  models.  PIC  is applied  to  the 
fourteen  Nelson-Plosser  and  Schotman-van  Dijk  macro-economic  time  series 
for the U.S. economy.  The  selected  models  are found  to be parsimonious  and  to 
perform  well in forecasting  over  the  period  197&1988.  They  generally  outper- 
form  a  fixed-format  AR(3)  model  with  linear  trend.  The  question  of  whether 
a  fixed-format  model  or  an  evolving-format  model  is  more  appropriate  for 
forecasting  the series at hand  is of great  practical  importance.  The  issue whether 
the  use of PIC  leads  to the selection  of models  which  in forecasting  outperform 
models  selected  using other  criteria,  such  as full-fledged  posterior  odds,  BIC, or 
AIC, is of even greater  practical  significance.  It is not  addressed  in the paper  but 
it should  be considered  before  PIC  can  be advocated  as a panacea  when there  is 
model  uncertainty.  For  instance,  Zellner  (1978)  has  made  the  relationship 
between  AIC  and  the  posterior  odds  criterium  (POC)  explicit  by showing  that 
AIC  is a truncated  version  of the  POC. 
From  a Bayesian  point  of view, the posterior  odds  ratio  is the criterion  to be 
used when the loss structure  is proportional  to the identity  matrix.  Now  PIC can 
be  interpreted  as  a  posterior  odds  ratio  under  a  specific  prior  distribution. 
Certainly,  an  attractive  feature  of  PIC,  in  particular  for  automated  decision- 
making,  is that  it  can  be  easily  computed.  The  prior  probability  distribution 
function  (pdf)  on the regression  coefficients  fl is uniform  and centred  at zero. The 
regression  coefficients  are  assumed  to  be  a  priori  independent.  Furthermore, 
0304-4076/95/%09.50  0  1995 Elsevier  Science  S.A.  All  rights  reserved 
SSDI  030440769401673  N 334  F. C.  Palm  / Journal  of  Econometrics  69  (1995)  333-335 
PIC  is conditioned  on  the  maximum  likelihood  estimate  of  the  disturbance 
variance  ~~ under  the  more  complex  model. 
In  my  view,  the  prior  distribution  underlying  PIC  has  some  undesirable 
features.  First,  to the extent  that  difference-stationarity  becomes  more  probable, 
the  need  for  allowing  for  trend-stationarity  decreases  and  vice  versa.  The 
coefficients  h and  6, should  not  be assumed  a priori  independent,  but  negatively 
correlated.  Second,  by centering  the prior  pdf of h and bk, k  =  1,2, . . . , r, on zero, 
one clearly  centres  the model  (2) on the unit root  hypothesis,  a feature  that  might 
be appropriate  in some instances  but not  in general.  Third,  the error  variance  g2 
is estimated  instead  of being integrated  out.  It would  however  not  be difficult  to 
integrate  with  respect  to  rr2 yielding  under,  e.g., Jeffreys’  prior  pdf  a  Student-r 
predictive  density  function.  For  its use in forecasting,  I refer,  among  others,  to 
Min and Zellner  (1993). In fact, as shown  by Zellner  (1977), Jeffreys’ prior  can be 
interpreted  as a maximal  data  information  prior  pdf. In the presence  of evolving- 
format  models,  conditioning  on  the  likelihood  estimate  of 0’  instead  is hardly 
justifiable  in terms  of large-sample  properties.  In fact, more  generally,  the use of 
the  likelihood  ratio  as an approximate  POC  may  not  be justified  either  under 
evolving-format  models. 
There  are several  other  features  of Phillips’ approach  on which I would  like to 
briefly comment.  First, it seems that  the approach  is especially  suited when there 
is uncertainty  about  the  structure  of the  data-generating  process.  Under  model 
uncertainty,  for  a given  time  series a model  is assumed  to  be appropriate  with 
a given probability  and to belong  to a set of models  considered.  Sample evidence 
is then  used to select a model.  Under  the changing  format  models,  it is probably 
more  appropriate  to  explicitly  model  the  structural  changes  than  to  let  the 
sample  evidence  determine  which  model  from  a  given  set  is  selected.  Time- 
varying  parameter  and  regime  shift  models  could  be  considered  in  that  case 
rather  than  selecting  a sequence  of evolving-format  models. 
Second,  the notion  of model  complexity  used by Phillips  is not  unequivocally 
defined.  Is it synonymous  of parameter  parsimony?  Then,  if two models  with the 
same number  of parameters  but  with different  sets of explanatory  variables  are 
equally  complex,  which  model  is used  to  estimate  the  disturbance  variance? 
Third,  what  is  the  meaning  of  model-encompassing  when  a  fixed-format 
model  is compared  to  an evolving  sequence  of alternative  models  and  is found 
not  to  encompass  the  evolving-format  models?  Should  this  finding  be  inter- 
preted  as evidence  of a misspecification  in a fixed-format  model  or as evidence  in 
favour  of a time-varying  model  structure. 
To  conclude,  Phillips’  very  stimulating  paper  contains  an  interesting  ap- 
proach  to  model  selection,  which  is  of  great  potential  practical  importance. 
Further  investigation  of  its  properties  and  comparison  with  approaches 
using other  model  selection  criteria  such as POC  or  using time-varying  param- 
eter  (see, e.g.,  Min  and  Zellner,  1993) or  switching  regime  models  would  be 
desirable. F.C.  Palm  J Journal  of Econometrics  69  (I 995)  333-335  335 
References 
Mm,  Chung-Ki  and  Arnold  Zellner,  1993,  Bayesian  and  non-Bayesian  methods  for  combining 
models  and  forecasts  with  applications  to  forecasting  international  growth  rates,  Journal  of 
Econometrics  56,89-l  18. 
Zellner,  A.,  1977,  Maximal  data  information  prior  distributions,  in:  A.  Aykac  and  C.  Brumat,  eds., 
New  developments  in  the  applications  of  Bayesian  methods  (North-Holland,  Amsterdam) 
21 l-232. 
Zellner,  A.,  1978,  Jeffreys-Bayes  posterior  odds  ratio  and  the  Akaike  information  criterion  for 
discriminating  between  models,  Economics  Letters  1, 337-342. 

Bayesian model selection and prediction with empirical applications: comment.

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