Professor Zellner has greatly contributed to econometrics in many aspects. This paper compiles a line of research developed by him and associates tha t goes back to the early 1970s, in which time-series techniques and Bayesian analysis are used in the construction of econometric models.Modelo econométrico; Modelo matemático; Análisis de series temporales;

Espasa, Antoni

English

Research Papers in Economics

Joumal 01 Forecasting,  Vol.  13,  235-239 (1994) 
Discussion 
Comments on  'Time-series Analysis, 
Forecasting and  Econometric Modelling: 
The Structural Econometric Modelling, 
Time-series Analysis  (SEMTSA) Approach', 
by  A.  Zellner. 
ANTONI ESPASA 
Universidad Carlos  111  de  Madrid,  Spain 
Professor  Zellner  has  greatly  contributed  to  econometrics  in  many  aspects.  This  paper 
compiles a line of research developed by him and associates that goes back to the early 1970s, 
in  which  time-series  techniques  and  Bayesian  analysis  are  used  in  the  construction  of 
econometric models. 
The starting point in this strategy is that the economic analysis must be based on statistical 
models which have been tested against data. Therefore the aim is to produce good models in 
order to forecast and explain reality and to serve policy makers. 
Trying to obtain all  these  objectives at once is  risky.  On the other hand, before using  a 
model the analyst must have evidence that the model is worth while. This can be obtained by 
submitting the model to the proof of forecasting. Therefore a first aim in modelling could be 
to obtain models that forecast well. 
In this respect another principIe of this strategy is to start from simple models, which means 
modelling  specific  variables  and  then  trying  to  put  them  together.  From  a  general  multi-
equational econometric model (SEM) the single transfer function corresponding to a particular 
variable can be derived. This transfer function constitutes the starting point of the SEMTSA 
methodology. 
The variable chosen in Zellner's paper is the growth on real GDP (y) and a dynamic leading 
indicator model is  constructed. The final  model is 
CX3 (L )Yt =  e + WI (L )SRt- 1 + (3GMt-1 + 'Y WSRt- 1 + Ut,  (1) 
where  cx3(L)  and  wI(L) are  polynomials  on  the  lag  operator  L  of order  three  and  one, 
respectively.  This is  an extension of the univariate model: 
(2) 
In passing from model (2) to model (1) sorne care must be taken in computing the dynamics 
of the endogenous variable.  In order to clarify the discussion it is  convenient to reformulate 
model (1) as 
_  *  WI (L)  (3  'Y  Ut 
Yt - e  + cx3(L) SRt- 1 + cx3(L) GMt- 1 + cx3(L) SRt- 1 + cx3(L)'  (3) 
eee 0277-6693/94/020235-05 
©  1994  by  John Wiley &  Sons, Ud. 
1236  Journal 01 Forecasting  Vol.  13,  ¡ss.  No.  2 
8---------·8---------· ® 
While  - Smoolhco local trenos 
nOlse  - Oscillations 
Figure I 
Model (2) can be explained by the use of Figure 1.  In this figure,  by passing a white-noise 
process through a filter-the polynomial CP3 (L) -I-we obtain a process that generates the YI 
series,  which can contain smoothed local  trends and periodic oscillations.  In particular, in 
model (2)  the couple for conjugate complex roots in CP3 (L) generates periodic oscillations in 
YI.  In univariate models the input (al) which generates the data (YI) has no structure at all and 
is  white noise.  Therefore the structure in the output is  due completely to the filter. 
In single transfer function (TF) equations data generation can be summarized by Figure 2. 
The lower part of the figure shows that from a white noise UI, through a noise filter we obtain 
the residual component, NI, of YI.  In models (1)  or (3) this component is 
NI =  ulfcp3(L). 
As in the univariate model, all the structure in NI is  due to the noise filter. 
In the upper part of Figure 2 the inputs Xj¡' j  =  1, ... , k, have a structure by themselves and 
contribute to the generation of YI  by passing through a corresponding input filter j. Thus the 
contribution of the rate of change of real stock prices, SR, in YI,  in models (1) or (3)  is 
SR*=WI(L) SR 
I  cx3(L)  1-1·  (4) 
However, since the Xjl inputs have a structure one could think of the filter being as simple as 
possible. In particular, periodic oscillations in the input filters could be an indication that sorne 
relevant input is  missing in the model. Therefore, before accepting a model with input filters 
which generate oscillations it seems worth exploring if there are relevant missing inputs. 
The question is that in extending a univariate modellike (2) to a single TF modellike model 
(1)  it does not seem to be always a  good strategy to keep the autoregressive structure of the 
(9-----. 
Input 
filler  - - ---. 
------------------------------------------- ...... 
@-----. 
8-----> 
InpUI • 
filler 
Noise 
filler 
k 
- - - -->  ~------------~~ 
- - - --> 
Figure 2 
YI 
2Antoni Espasa  Time-series Analysis, Forecasting and Econometric Modelling  237 
endogenous variable. It appears preferable to keep it only to define the noise filter and design 
each  input  filter  by  itself,  avoiding  rational  filters  with  complex  or negative  autoregressive 
roots;  for  a  further  discussion  of this  point  see  Espasa  (1982).  Keeping  the  cx3(L)  in  the 
endogenous variable in model (1) implies,  for instance, that a positive innovation in  the rate 
of change of real stock prices in year t will produce positive and negative contributions in the 
rate of growth of real GDP in  the following  years.  A justification for  these  future negative 
contributions does not seem clear to me. 
In macroeconomic variables a missing input could be the relative price of energy. This can 
be  illustrated with  an example  for  the Spanish economy.  Andrés  el al.  (1990)  estimate the 
following model for real private investment Ut):  • 
I  = - 2.39  (2.44 - 1.66L  2)  Y _  +  2.24  (1  _ L )CU 
t  c/>2(L)  +  c/>2(L)  t  1  c/>2(L)  t 
(-1.52+0.57L) (C/P)  -0.85(1  L)2  at  +  t-l +-- - 7rt+---
c/>2 (L)  c/>2 (L)  c/>2 (L), 
(5) 
where  Y, CU, Cf P, and 7r  are real GDP, the degree of utilization of the productive capacity, 
the cost of use of capital,  and the inflation rate,  respectively.  The variables are in logs  and 
c/>2(L)=(1-0.56L+0.27L2). 
Since  the  polynomial  c/>2 (L)  has  a  couple  of conjugate  complex  roots  all  the  dynamic 
functions  relating the explanatory variables  to investment generate oscillations.  Considering 
the contributions of the explanatory variables, and, in particular, the contribution of GDP 
y* _  (2.44 - 1.66L 2) 
t  - 1 _ 0.56L + 0.27 L 2  Yt- l  (6) 
and comparing Yt- l with Yt it can be seen that the fi!ter generates oscillations in  Yt* that are 
important in the years  around the oil  crises.  This suggest  that the relative  prices  of ener[ 
(RPE) could be an input that should be considered. Espasa and Senra (1992) introduce that 
variable and found that a  dynamic  relationship of investment with  GDP depends  on these 
relative prices and they estimate this relationship as: 
Y t** = [1.25 - 0.004(1 + L + L2)RPEt_ ¡] Yt-l + 1.10(1 + L) ~  Yt-l, 
where  ~ = (1  - L). 
With this  factor the resulting model for investment is 
(7) 
It = - 5.83 + Yt* + 1.04(1 - L) Ut-l - 0.83(CfP)t-1 - 0.27(1 - L )27rt + et/(1 + O. 79L 2).  (8) 
Model (8)  says  that under stable relative prices of energy (RPE equals zero) the dynamic 
function  relating  investment  with  GDP and its  rate of growth is  very  simple  and does  not 
generate  oscillations.  On  the  other  hand,  the  dynamic  functions  for  all  the  inputs  do 
not contain an autoregressive part. The autoregressive factor only appears in the residuals. 
Model  (8)  improves  considerably  the  fit  by  reducing  by  one  third  the  error  variance, 
var(et) = 0.66 var (at). 
With respect to the question of forecasting an aggregate variable directly from a model that 
explains it or by aggregating the forecasts from models which explain the components of this 
aggregate,  I would like to comment from my experience in  forecasting different variables of 
the Spanish economy for  several  years.  The experience shows  that if the components have 
different  seasonal  behaviour  or trends  then  it  is  worth  forecasting  the  aggregate  from  the 
component models. In other words, if the components are not co-integrated then the forecast 
3238  Journal of  Forecasting  Vol.  13,  [ss.  No. 2 
of the aggregate can be improved by modelling the components. In Espasa and Pérez (1979), 
Espasa el al.  (1984),  and Espasa y Matea (1991)  there is  evidence of improving the forecast 
of money aggregates  and inflation  in  the CPI by  decomposing these  aggregates into a  few 
components which have quite different trends and/or seasonal behaviour. 
REFERENCES 
Andrés,  J., Escribano,  A.,  Molinas,  C.  and Taguas,  D.,  La inversión  en  España:  Econometr{a  con 
restricciones de equilibrio, Barcelona and Madrid: Antoni Bosch e Instituto de Estudios Fiscales, 1990. 
Espasa, A., 'Relationships between variables: The short and long run effects', in Charatsis, E. G. (ed.), 
Selected  Papers  on  Contemporary  Econometric  Problems,  The  Athens  School  of Economics  and 
Business Science,  1982. 
Espasa,  A.  and  Matea,  M.  Ll.,  'Underlying  inflation  in  the  Spanish  economy:  estimation  and 
methodology', Working paper No. 91-30,  Universidad Carlos III de Madrid,  1991. 
Espasa, A., Molina, A. and Ortega, E., 'Forecasting the rate of inflation by means of the consumer price 
index', Fourth International Symposium on Forecasting, London, July 1984. 
Espasa, A.  and Pérez,  J., 'Within month predictions for  monetary aggregates and Spanish monetary 
policy implementation', Fourth Meeting of the Association d'Econometrie Appliquée, Rome, February 
1979. 
Espasa,  A.  and  Senra,  E.  'Consideraciones sobre  la  función  de  inversión  en  el  modelo MOISEES', 
Documento de trabajo No. 92-14, Universidad Carlos III de Madrid, 1992. 
Author's biography: 
Antoni Espasa is Professor of Econometrics at the Universidad Carlos IlI, Madrid. He obtained his PhD 
from the London School of Economics in  1975.  He has been Chief Economist at the Research Unit of 
the Banco de España. His published research includes studies in spectral econometrics, dynamic mode1s, 
applied econometrics, forecasting,  and signal extraction in  time series. 
Author's address: 
Antoni Espasa, Universidad Carlos III de Madrid, Calle Madrid, 126, 28903  Getafe, Madrid, Spain. 
Response to Cornrnents of A.  Espasa 
A.  ZELLN~R 
I  thank Professor Espasa for  his  thoughtful  comments  on my  papero  He provides a  brief 
summary of our structural econometric modelling,  time-series analysis (SEMTSA) approach 
to econometric  model  construction.  Then  he  goes  on to discuss  one of our autoregressive 
leading indicator models in the context of transfer function modelling, the possibility of adding 
an additional input variable and aggregation issues. 
4A. Zel/ner  Response lo  Commenls 01 A. Espasa  239 
General transfer function modelling techniques were employed in Zellner and Palm (1974, 
1975). In the latter work we found that even with a large sample of monthly data for the USA, 
' ... the parameter estimates are in general rather imprecise.  This is  somewhat disquieting as 
the sample size is  relatively large'  (p.  46).  Looking back on this fmding  in  the light of later 
work, it may be that the transfer functions' parameters are not constant but time varying. As 
noted in my paper, there are many reasons for parameters to be time varying and time-varying 
parameter models tended to perform better in  forecasting than did fixed-parameter models. 
Also, Palm and I noted difficulties in selecting a particular transfer function specification from 
a multitude of possibilities even with a large sample of data. In our present work with annual 
macroeconomic data, for each country, we  have a small sample of just 20  observations with 
which to begin our forecasting experiments and thus cannot entertain extremely complicated 
models with many parameters. 
As  regards Espasa's query regarding an innovation in the growth rate of real stock prices 
producing positive and negative contributions to the rate of growth of GDP in following years, 
it is  well known in economics and other areas that when a shock hits a system it can produce 
oscillations,  a  characteristic of many physical  systems and of many models  of the business 
cycle.  Further, innovations in the stock price variable might be produced from a variety of 
sources,  as  explained  in  my  paper,  including  changes  in  the  relative  price  of energy.  This 
'catch-all' property of the stock price leading indicator variable is a major reason for including 
it in our models. 
While it is possible to view the transfer function model-building process as shown in Figure 
2 in Professor Espasa's comments, it should be appreciated that this approach does not take 
account of the relation of the transfer functions to structural econometric models, as explained 
in my paper and illustrated with examples in previous work. Neglecting the link with structural 
equations can result in loss of important information about the system being modelled. Also, 
as  indicated  in  my  paper,  the  input  x-variables  need  not  be  generated  by  independent 
processes.  Intercorrelations among input variables  can cause  well-known  problems in  usual 
transfer  function  model  identification  and  estimation  procedures.  While  there  are  index, 
component,  etc.  techniques  for  attempting to cope with this  range of problems,  with  just 
20-30 observations, they did not appear to be feasible, and instead we opted for use of several 
'catch-all' leading indicator variables that Burns and Mitchell discovered in their research with 
pre-Second World War data for several countries. Note, however, that addition of a measure 
of world output growth led to an elaboration of our model that produced improvement in 
forecasting performance for many countries. 
Clearly, there are many additional variables that may lead to improved forecasts. However, 
including them all  in a single relation with limited data is  impractical. As  I indicated in my 
paper,  what  is  practical  is  to  disaggregate  by  industrial  sector  and  to  develop  sectoral 
econometric models with their associated transfer functions incorporating specific and general 
input variables. Then forecasts of sectors' outputs as well as forecasts of aggregate output can 
be obtained. It will be of great interest to determine whether disaggregation will lead to good 
sector  forecasts  and  better  aggregate  forecasts  as  indicated  in  sorne  of Espasa's  and  our 
theoretical  work.  Finally,  I  believe  that the  sectoral  models  so  developed  will  be  of great 
interest and value.  Our preliminary work on this problem, involving use of demand, supply, 
and entry equations for each industrial sector, has yielded interesting non-linear models that 
are currentIy being analysed. 
Thanks again to Professor Espasa for his stimulating comments and to the organizers of this 
productive meeting for inviting me  to participate in it. 
5

Consideraciones sobre la función de inversión en el modelo MOISEES', Documento de trabajo No. 92-14, Universidad Carlos III de Madrid,

Forecasting the rate of inflation by means of the consumer price index',

La inversión en España: Econometr{a con restricciones de equilibrio, Barcelona and Madrid: Antoni Bosch e Instituto de Estudios Fiscales,

Relationships between variables: The short and long run effects',

Underlying inflation in the Spanish economy: estimation and methodology', Working paper No. 91-30, Universidad Carlos III de

Within month predictions for monetary aggregates and Spanish monetary policy implementation', Fourth Meeting of the Association d'Econometrie Appliquée,

Zel/ner Response lo Commenls 01 A. Espasa 239 General transfer function modelling techniques were employed in Zellner and Palm

Comments on time-series analysis, forecasting and econometric modelling: The structural econometric modelling, time-series analysis (SEMTSA) approach, by A. Zellner.

http://e-archivo.uc3m.es/bitstream/10016/3067/1/Comments_espasa_JOF_1994_ps.pdf

Comments on time-series analysis, forecasting and econometric modelling: The structural econometric modelling, time-series analysis (SEMTSA) approach, by A. Zellner.

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